Integral Equations and Operator Theory - Volume 66

Integr. Equ. Oper. Theory 66 (2010), 1–20 DOI 10.1007/s00020-009-1738-2 Published online January 13, 2010 © Birkhäuser V...

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Integr. Equ. Oper. Theory 66 (2010), 1–20 DOI 10.1007/s00020-009-1738-2 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

Weyl Type Theorems for Left and Right Polaroid Operators Pietro Aiena, Elvis Aponte and Edixon Balzan Abstract. A bounded operator deﬁned on a Banach space is said to be polaroid if every isolated point of the spectrum is a pole of the resolvent. In this paper we consider the two related notions of left and right polaroid, and explore them together with the condition of being a-polaroid. Moreover, the equivalences of Weyl type theorems and generalized Weyl type theorems are investigated for left and a-polaroid operators. As a consequence, we obtain a general framework which allows us to derive in a uniﬁed way many recent results, concerning Weyl type theorems (generalized or not) for important classes of operators. Mathematics Subject Classiﬁcation (2010). Primary 47A10, 47A11. Secondary 47A53, 47A55. Keywords. Key words and phrases: Localized SVEP, semi B-Brower operators, left and right Drazin invertibility, Weyl’s theorem, property (w).

1. Introduction and preliminaries Let L(X) be the algebra of all bounded linear operators acting on an inﬁnite dimensional complex Banach space X and if T ∈ L(X) let us denote by α(T ) the dimension of the kernel ker T and by β(T ) the codimension of the range T (X). Recall that the operator T ∈ L(X) is said to be upper semi-Fredholm, T ∈ Φ+ (X), if α(T ) < ∞ and the range T (X) is closed, while T ∈ L(X) is said to be lower semi-Fredholm, T ∈ Φ− (X), if β(T ) < ∞. If either T is upper or lower semi-Fredholm then T is said to be a semi-Fredholm operator, while if T is both upper and lower semi-Fredholm then T is said to be a Fredholm operator. If T is semi-Fredholm then the index of T is deﬁned by ind (T ) := α(T ) − β(T ). A bounded operator T ∈ L(X) is said to be a Weyl operator, T ∈ W (X), if T is a Fredholm operator having index 0. The The ﬁrst author was supported by ex-60 2008, Fondi Universit´ a di Palermo.

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classes of upper semi-Weyl’s and lower semi-Weyl’s operators are deﬁned, respectively: W+ (X) := {T ∈ Φ+ (X) : ind T ≤ 0}, W− (X) := {T ∈ Φ− (X) : ind T ≥ 0}. Clearly, W (X) = W+ (X) ∩ W− (X). The Weyl spectrum and the upper semiWeyl spectrum are deﬁned, respectively, by / W (X)}. σw (T ) := {λ ∈ C : λI − T ∈ and σuw (T ) := {λ ∈ C : λI − T ∈ / W+ (X)}. The ascent of an operator T ∈ L(X) is deﬁned as the smallest nonnegative integer p := p(T ) such that ker T p = ker T p+1 . If such integer does not exist we put p(T ) = ∞. Analogously, the descent of T is deﬁned as the smallest non-negative integer q := q(T ) such that T q (X) = T q+1 (X), and if such integer does not exist we put q(T ) = ∞. It is well-known that if p(T ) and q(T ) are both ﬁnite then p(T ) = q(T ), see [1, Theorem 3.3]. Moreover, if λ ∈ C then 0 < p(λI − T ) = q(λI − T ) < ∞ if and only if λ is a pole of the resolvent of T . In this case λ is an eigenvalue of T and an isolated point of the spectrum σ(T ), see [28, Prop. 50.2]. A bounded operator T ∈ L(X) is said to be Browder (resp. upper semi-Browder, lower semi-Browder) if T ∈ Φ(X) and p(T ) = q(T ) < ∞ (resp. T ∈ Φ+ (X) and p(T ) < ∞, T ∈ Φ− (X) and q(T ) < ∞). Denote by B(X), B+ (X) and B− (X) the classes of Browder operators, upper semi-Browder operators and lower semiBrowder operators, respectively. Clearly, B(X) ⊆ W (X), B+ (X) ⊆ W+ (X) and B− (X) ⊆ W− (X). Let σb (T ) := {λ ∈ C : λI − T is not Browder} denote the Browder spectrum and let σub (T ) denote the upper semi-Browder spectrum of T , deﬁned as σub (T ) := {λ ∈ C : λI − T is not upper semi-Browder}. Clearly, σw (T ) ⊆ σb (T ) and σuw (T ) ⊆ σub (T ). Lemma 1.1. If T ∈ L(X) and p = p(T ) < ∞ then the following statements are equivalent: (i) There exists n ≥ p + 1 such that T n (X) is closed; (ii) T n (X) is closed for all n ≥ p. Proof. Deﬁne ci (T ) := dim(ker T i / ker T i+1 ). Clearly, p = p(T ) < ∞ entails that ci (T ) = 0 for all i ≥ p, so ki (T ) := ci (T ) − ci+1 (T ) = 0 for all i ≥ p. The equivalence then easily follows from [33, Lemma 12]. The concept of semi-Fredholm operators has been generalized by Berkani ([16], [17]) in the following way: for every T ∈ L(X) and a nonnegative integer n let us denote by T[n] the restriction of T to T n (X) viewed as a map from the space T n (X) into itself (we set T[0] = T ). T ∈ L(X) is said to be

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semi B-Fredholm (resp. B-Fredholm, upper semi B-Fredholm, lower semi BFredholm,) if for some integer n ≥ 0 the range T n (X) is closed and T[n] is a semi-Fredholm operator (resp. Fredholm, upper semi-Fredholm, lower semiFredholm). In this case T[m] is a semi-Fredholm operator for all m ≥ n ([17]). This enables one to deﬁne the index of a semi B-Fredholm as ind T = ind T[n] . A bounded operator T ∈ L(X) is said to be B-Weyl (respectively, upper semi B-Weyl, lower semi B-Weyl) if for some integer n ≥ 0 T n (X) is closed and T[n] is Weyl (respectively, upper semi-Weyl, lower semi-Weyl). The classes of operators previously deﬁned generate the B-Weyl spectrum σbw (T ), the upper B-Weyl spectrum σusbw (T ), and the lower B-Weyl spectrum σlsbw (T ). Analogously, a bounded operator T ∈ L(X) is said to be B-Browder (respectively, (respectively, upper semi B-Browder, lower semi B-Browder) if for some integer n ≥ 0 T n (X) is closed and T[n] is Browder (respectively, upper semi-Browder, lower semi-Browder). The B-Browder spectrum is denoted by σbb (T ), the upper semi B-Browder spectrum by σusbb (T ) and the lower semi B-Browder spectrum by σlsbb (T ). This note also deals with the concept of Drazin invertibility which has been introduced in a more abstract setting than operator theory, see [25]. In the case of the Banach algebra L(X), T ∈ L(X) is said to be Drazin invertible (with a ﬁnite index) if and only if p(T ) = q(T ) < ∞ and this is equivalent to saying that T = T0 ⊕ T1 , where T0 is nilpotent and T1 is invertible, see [31, Corollary 2.2] and [29, Prop. A]. Every B-Fredholm operator T admits on a Banach space the representation T = T0 ⊕ T1 , where T0 is nilpotent and T1 is Fredholm [18], so every Drazin invertible operator is B-Fredholm. Drazin invertibility for bounded operators suggests the following deﬁnitions. Deﬁnition 1.2. T ∈ L(X) is said to be left Drazin invertible if p := p(T ) < ∞ and T p+1 (X) is closed, while T ∈ L(X) is said to be right Drazin invertible if q := q(T ) < ∞ and T q (X) is closed. Clearly, T ∈ L(X) is both right and left Drazin invertible if and only if T is Drazin invertible. In fact, if 0 < p := p(T ) = q(T ) then T p (X) = T p+1 (X) is the kernel of the spectral projection associated with the spectral set {0}, see [28, Prop. 50.2]. The left Drazin spectrum is then deﬁned as σld (T ) := {λ ∈ C : λI − T is not left Drazin invertible}, the right Drazin spectrum is deﬁned as σrd (T ) := {λ ∈ C : λI − T is not right Drazin invertible}, and the Drazin spectrum is deﬁned as σd (T ) := {λ ∈ C : λI − T is not Drazin invertible}. Obviously, σd (T ) = σld (T ) ∪ σrd (T ). Theorem 1.3. [5] For every T ∈ L(X) we have σusbb (T ) = σld (T ),

σlsbb (T ) = σrd (T ),

σbb (T ) = σd (T ).

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Deﬁne ∆(T ) := {n ∈ N : m ≥ n, m ∈ N ⇒ T n (X) ∩ ker T ⊆ T m (X) ∩ ker T }. The degree of stable iteration is deﬁned as dis(T ) := inf ∆(T ) if ∆(T ) = ∅, while dis(T ) = ∞ if ∆(T ) = ∅. The following class of operators has been introduced by Labrousse [30] Deﬁnition 1.4. T ∈ L(X) is said to be quasi-Fredholm of degree d, if there exists d ∈ N such that: (a) dis(T ) = d, (b) T n (X) is a closed subspace of X for each n ≥ d, (c) T (X) + ker T d is a closed subspace of X. It should be noted that by Proposition 2.5 of [17] every semi B-Fredholm operator is quasi-Fredholm, in particular every left o right Drazin invertible operator is quasi-Fredholm. The analytic core of T ∈ L(X) is deﬁned K(T ) := {x ∈ X : there exist c > 0 and a sequence (xn )n≥1 ⊆ X such that T x1 = x, T xn+1 = xn for all K(T ), and n ∈ N, and ||xn || ≤ cn ||x||for all n ∈ N}. Note that T (K(T )) = ∞ K(T ) is contained in the hyper-range of T deﬁned by T ∞ (X) := n=0 T n (X), see [1, Chapter 1] for details. A bounded operator T ∈ L(X) is said to besemi-regular if T (X) is ∞ closed and N ∞ (T ) ⊆ T ∞ (X), where N ∞ (T ) := n=1 ker T n denotes the hyper-kernel of T . In the sequel we need the following result: Theorem 1.5. If T ∈ L(X) is quasi-Fredholm then there exists ε > 0 such that N ∞ (λI − T ) ⊆ (λI − T )∞ (X) for all 0 < |λ| < ε. If T is semi B-Fredholm then λI − T is semi-regular in a suitable punctured open disc centered at 0. Proof. Observe ﬁrst that if T is quasi-Fredholm of degree d then T n (X) is closed for all n ≥ d, so T ∞ (X) is closed. Furthermore, by Theorem 3.4 of [27] the restriction T |T ∞(X) is onto, so T (T ∞(X)) = T ∞ (X). Let T0 := T |T ∞(X). Clearly, T0 is onto and hence λI − T is onto for all |λ| < ε, where ε := γ(T0 ) is the minimal modulus of T0 , see [1, Lemma 1.30]. Therefore, (λI − T )(T ∞ (X)) = T ∞ (X) for all |λ| < ε. Since T ∞ (X) is closed, by [1, Theorem 1.22] it then follows that T ∞ (X) ⊆ K(λI − T ) ⊆ (λI − T )∞ (X) for all |λ| < ε. By part (ii) of Theorem 1.3 of [1] we have N ∞ (λI − T ) ⊆ T ∞ (X) for all λ = 0, so we conclude that N ∞ (λI − T ) ⊆ (λI − T )∞ (X) for all 0 < |λ| < ε, and the ﬁrst assertion is proved. To show the second assertion, suppose that T is semi B-Fredholm. Then there exists an open disc D centered at 0 such that λI − T is semi-Fredholm for all λ ∈ D \ {0} (this follows as a particular case of a result proved in [17, Corollary 3.2] for operators having topological uniform descent for n ≥ d). Since semi-Fredholm operators have closed range then the last assertion easily follows.

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2. Left and right polaroid operators The ﬁrst part concerns the basic properties of left and right polaroid operators. In the sequel by T we shall denote the dual of T ∈ L(X). Denote by M ⊥ the annihilator of M ⊆ X, while by ⊥ N we denote the pre-annihilator of N ⊆ X . Theorem 2.1. For every T ∈ L(X) the following equivalences hold: (i) T is left Drazin invertible ⇔ T is right Drazin invertible. (ii) T is right Drazin invertible ⇔ T is left Drazin invertible. (iii) T is Drazin invertible if and only if T is Drazin invertible. Proof. (i) Suppose that T is left Drazin invertible. Then p := p(T ) < ∞ and T p+1 (X) is closed, and hence also T p+1 (X ) is closed. By Lemma 1.1 p T p (X) is closed, and consequently also T (X ) is closed. From the equality p p+1 ker T = ker T and from the classical closed range theorem we then deduce that T (X ) = [ker T p ]⊥ = [ker T p+1 ]⊥ = T p

p+1

(X ).

This shows that T has ﬁnite descent q := q(T ) ≤ p and since T q (X ) = p T (X ) is closed it then follows that T is right Drazin invertible. Conversely, suppose that T is right Drazin invertible. Then q := q(T ) q q q+1 < ∞ and T (X ) is closed. From the equality T (X ) = T (X ) and from the closed range theorem we then obtain: ker T q =⊥ [T (X )] =⊥ [T q

q+1

(X )] = ker T q+1 ,

and hence p := p(T ) ≤ q. Since T (X ) is closed then also T q+1 (X) is closed and by Lemma 1.1 it then follows that T p+1 (X) is closed, so T is a left Drazin invertible. (ii) This may be proved in a similar way of part (i). (iii) Obviously, since both left and right Drazin invertibility entails Drazin invertibility. q+1

Recall that T ∈ L(X) is said to be bounded below if T is injective with closed range. The classical approximate point spectrum is deﬁned by σa (T ) := {λ ∈ C : λI − T is not bounded below}. If σs (T ) denotes the surjectivity spectrum it is well known that σa (T ) = σs (T ) and σs (T ) = σa (T ) for all T ∈ L(X). Deﬁnition 2.2. Let T ∈ L(X), X a Banach space. If λI − T is left Drazin invertible and λ ∈ σa (T ) then λ is said to be a left pole of the resolvent of T . If λI − T is right Drazin invertible and λ ∈ σs (T ) then λ is said to be a right pole of the resolvent of T . It is easily seen that λ is a pole of T if and only if λ is both a left and a right pole of T . In fact, if λ is a pole of T then λI − T is Drazin invertible, so λI − T is both left and right Drazin invertible. Moreover, λ is both a left

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and a right pole of T , since the condition 0 < p(λI − T ) = q(λI − T ) < ∞ entails that λ ∈ σa (T ) as well as λ ∈ σs (T ). The converse is clear. Remark 2.3. Suppose that for a linear operator T we have α(T ) < ∞. Then α(T n ) < ∞ for all n ∈ N. This may be easily seen by an inductive argument. Suppose that dim ker T n < ∞. Since T (ker T n+1 ) ⊆ ker T n then the restriction T0 := T | ker T n+1 : ker T n+1 → ker T n has kernel equal to ker T so the canonical mapping Tˆ : ker T n+1 / ker T → ker T n is injective. Therefore we have dim ker T n+1 / ker T ≤ dim ker T n < ∞, and since dim ker T < ∞ we then conclude that dim ker T n+1 < ∞. Analogously, if β(T ) < ∞ then β(T n ) < ∞ for all n ∈ N. In fact, suppose β(T n ) < ∞. Since the map T˜ : T n (X)/T n+1(X) → T n+1 (X)/T n+2(X), deﬁned by T˜ (z + T n+1 (X)) = T z + T n+2 (X),

z ∈ T n (X),

is onto, then dim T n+1 (X)/T n+2 (X) ≤ dim T n (X)/T n+1 (X). This easily implies that β(T n+1 ) < ∞. Lemma 2.4. Let T ∈ L(X). Then we have: (i) T is upper semi B-Fredholm and α(T ) < ∞ if and only if T ∈ Φ+ (X). (ii) T is lower semi B-Fredholm and β(T ) < ∞ if and only T ∈ Φ− (X). Proof. (i) If T is upper semi B-Fredholm then there exists n ∈ N such that T n (X) is closed and T[n] is upper semi-Fredholm. Since α(T ) < ∞ then α(T n ) < ∞, hence T n is upper semi Fredholm. From the classical Fredholm theory then also T is upper semi-Fredholm. The converse is obvious. Part (ii) may be proved in a similar way. Deﬁnition 2.5. A bounded operator T ∈ L(X) is said to be left polaroid if every isolated point of σa (T ) is a left pole of the resolvent of T . T ∈ L(X) is said to be right polaroid if every isolated point of σs (T ) is a right pole of the resolvent of T . T ∈ L(X) is said to be polaroid if every isolated point of σ(T ) is a pole of the resolvent of T . Theorem 2.6. If T ∈ L(X) is both left and right polaroid then T is polaroid. Proof. If iso σ(T ) = ∅ there is nothing to prove. Suppose then λ ∈ iso σ(T ) = ∅. Since the boundary the spectrum is contained in σa (T ), see [1, Theorem 2.42], then λ ∈ iso σa (T ), so λ is a left pole and hence p(λI − T ) < ∞. On the other hand, λ ∈ σs (T ), (otherwise we have 0 = q(λI − T ) = p(λI − T ) and hence λ ∈ / σ(T )). Therefore, λ ∈ iso σs (T ) and since T is right polaroid then q(λI − T ) < ∞, from which conclude that λ is a pole of the resolvent of T . The following example shows that the converse of Theorem 2.6, in general, does not hold.

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Example 2.7. Let R denote the right shift on 2 (N) deﬁned by R(x1 , x2 , . . . ) := (0, x1 , x2 , . . . )

(xn ) ∈ 2 (N),

and let Q be the weighted left shift deﬁned by Q(x1 , x2 , . . . ) := (x2 /2, x3 /3, . . . ) (xn ) ∈ 2 (N). Q is a quasi-nilpotent operator, σ(R) = D(0, 1), where D(0, 1) denotes the closed unit disc of C, and σa (R) = Γ, where Γ is the unit circle of C. Moreover, if en := (0, ..., 0, 1, 0...), where 1 is the n-th term, then en+1 ∈ ker Qn+1 while / ker Qn for every n ∈ N, so p(Q) = ∞. en+1 ∈ Deﬁne T := R ⊕ Q on X := 2 (N) ⊕ 2 (N). Clearly, σ(T ) = D(0, 1), and σa (T ) = Γ ∪ {0}. We have p(T ) = p(R) + p(Q) = ∞, so 0 is not a left pole. Therefore, T is polaroid, since iso σ(T ) = ∅, but not left polaroid. It is easily seen that the dual T is polaroid but not right polaroid, since q(T ) = ∞. The concept of left and right polaroid are dual each other: Theorem 2.8. If T ∈ L(X) then the following equivalences hold: (i) T is left polaroid if and only if T is right polaroid. (ii) T is right polaroid if and only if T is left polaroid. (iii) T is polaroid if and only if T is polaroid. Proof. (i) Suppose that T is left polaroid. If iso σs (T ) = ∅ there is nothing to prove. Suppose that λ ∈ iso σs (T ). Then λ is an isolated point of σa (T ), hence, by assumption, λI − T is left Drazin invertible. By Theorem 2.1 then λI − T is right Drazin invertible, so λ is a right pole of the resolvent of T . Conversely, suppose T right polaroid and let λ be an isolated point of σa (T ). Then λ ∈ iso σs (T ) and hence is a right pole of the resolvent of T . Therefore, λI − T is right Drazin invertible so, by Theorem 2.1, λI − T is left Drazin invertible. Thus, λ is a left pole of the resolvent of T . (ii) The proof is similar to that of part (i). (iii) It is well-known that σ(T ) = σ(T ) and that λ is a pole of the resolvent of T if and only if λ is a pole of the resolvent of T . Deﬁnition 2.9. A bounded operator T ∈ L(X) is said to be a-polaroid if every λ ∈ iso σa (T ) is a pole of the resolvent of T . Trivially, T a-polaroid ⇒ T left polaroid.

(1)

Moreover, iso (T ) ⊆ σa (T ) for every T ∈ L(X), since the boundary of σ(T ) is contained in σa (T ), from which we easily obtain: T a-polaroid ⇒ T polaroid.

(2)

The following example provides an operator that is left polaroid but not a-polaroid.

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Example 2.10. Let R ∈ 2 (N) be the unilateral right shift deﬁned as R(x1 , x2 , . . . ) := (0, x1 , x2 , · · · ) for all (xn ) ∈ 2 (N), and U (x1 , x2 , . . . ) := (0, x2 , x3 , · · · ) for all (xn ) ∈ 2 (N). If T := R ⊕ U then σ(T ) = D(0, 1), so iso σ(T ) = ∅. Moreover, σa (T ) = Γ∪{0}, Γ the unit circle, so iso σa (T ) = {0}. Since R is injective and p(U ) = 1 it then follows that p(T ) = p(R) + p(U ) = 1. Furthermore, T ∈ Φ+ (X) and hence T 2 ∈ Φ+ (X), so that T 2 (X) is closed. Therefore 0 is a left pole and hence T is left polaroid. On the other hand q(R) = ∞, so that q(T ) = q(R) + q(U ) = ∞, so T is not a-polaroid. Note that T is also polaroid. In the case of Hilbert space operators T ∈ L(H) instead of the dual T it is more appropriate to consider the Hilbert adjoint T ∗ . By means of the classical Fr´echet- Riesz representation theorem we know that if U is the conjugate-linear isometry that associates to each y ∈ H the linear form x → x, y then λI − T ∗ = (λI − T )∗ = U −1 (λI − T ) U.

(3)

This obviously implies that σa (T ∗ ) = σa (T ) and σs (T ∗ ) = σs (T ). Theorem 2.11. If T ∈ L(H), H a Hilbert space, then the following equivalences hold: (i) T is left polaroid if and only if T ∗ is right polaroid. (ii) T is right polaroid if and only if T ∗ is left polaroid. (iii) T is polaroid if and only if T ∗ is polaroid. Proof. (i) Suppose that T is left polaroid. If λ ∈ iso σs (T ∗ ) then λ ∈ iso σs (T ) = iso σa (T ), so λ is a left pole of T , hence p := p(λI − T ) < ∞ and (λI − T )p+1 (H) is closed. From ker (λI − T )p = ker (λI − T )p+1 we see that (λI − T ∗ )p (H) = [ker (λI − T )p ]⊥ = [ker (λI − T )p+1 ]⊥ = (λI − T ∗ )p+1 (H), where by N ⊥ we denote the orthogonal of N ⊆ H. Therefore q := q(λI − T ∗) ≤ p < ∞ and consequently (λI − T ∗ )q (H) = (λI − T ∗ )p (H) is closed, by Lemma 1.1. Thus λ is a right pole of T ∗ and T ∗ is right polaroid. Conversely, suppose that T ∗ is right polaroid and let λ ∈ iso σa (T ). Then λ ∈ iso σs (T ) = iso σs (T ∗ ), thus λ is a right pole of T ∗ . Consequently, q := q(λI − T ∗ ) < ∞ and (λI − T ∗ )q (H) = (λI − T ∗ )q+1 (H) is closed. Since ker(λI − T )q+1 = [(λI − T ∗ )q+1 (H)]⊥ = [(λI − T ∗ )q (H)]⊥ = ker(λI − T )q it then follows that p := p(λI − T ) ≤ q < ∞. By Lemma 1.1 (λI − T )p+1 (H) is closed, so λ is a left pole of T . (ii) This may be proved in a similar way of part (i). (iii) See Theorem 2.5 of [8].

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The following property has relevant role in local spectral theory, see the recent monographs by Laursen and Neumann [32] and [1]. Deﬁnition 2.12. Let X be a complex Banach space and T ∈ L(X). The operator T is said to have the single valued extension property at λ0 ∈ C (abbreviated SVEP at λ0 ), if for every open disc D of λ0 , the only analytic function f : D → X which satisﬁes the equation (λI − T )f (λ) = 0 for all λ ∈ D is the function f ≡ 0. An operator T ∈ L(X) is said to have SVEP if T has SVEP at every point λ ∈ C. Evidently, T ∈ L(X) has SVEP at every isolated point of the spectrum. We also have p(λI − T ) < ∞ ⇒ T has SVEP at λ,

(4)

and dually

q(λI − T ) < ∞ ⇒ T has SVEP at λ, (5) see [1, Theorem 3.8]. Furthermore, from deﬁnition of localized SVEP it easily seen that σa (T ) does not cluster at λ ⇒ T has SVEP at λ,

(6)

σs (T ) does not cluster at λ ⇒ T has SVEP at λ.

(7)

and dually, The quasi-nilpotent part of T ∈ L(X) is deﬁned as the set 1

H0 (T ) := {x ∈ X : lim T nx n = 0}. n→∞

Clearly, ker T n ⊆ H0 (T ) for every n ∈ N. Moreover, T is quasi-nilpotent if and only if H0 (λI − T ) = X, see Theorem 1.68 of [1]. Note that H0 (T ) generally is not closed and ([1, Theorem 2.31 ] H0 (λI − T ) closed ⇒ T has SVEP at λ.

(8)

Remark 2.13. All the implications (4), (5), (6), (7) and (8) are equivalences whenever λI − T is a quasi-Fredholm operator, in particular whenever λI − T is a semi B-Fredholm operator, see [3]. Theorem 2.14. Suppose that T ∈ L(X). Then the following assertions hold: (i) If T has SVEP then the properties of being polaroid, a-polaroid and left polaroid for T are all equivalent. (ii) If T has SVEP then the properties of being polaroid, a-polaroid and left polaroid for T are all equivalent. Proof. (i) Suppose that T has SVEP. Then σ(T ) = σa (T ), see Corollary 2.44 of [1], hence T is polaroid precisely when T is a-polaroid. By (1) the apolaroid condition for T entails that T is left polaroid. Conversely, if T is left polaroid and λ ∈ iso σ(T ) then λ ∈ iso σa (T ), again because the boundary of the spectrum is contained in σa (T ), hence λ is a left pole, so that then p(λI − T ) < ∞. On the other hand, since λI − T is left Drazin invertible, in

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particular quasi-Fredholm the SVEP of T at λ entails that q(λI − T ) < ∞, see [3, Theorem 2.11], and consequently λ is a pole of the resolvent of T . Therefore T is polaroid. (ii) If T has SVEP, by Corollary 2.44 of [1] then σ(T ) = σs (T ) and hence σ(T ) = σa (T ), so T is polaroid if and only if T is a-polaroid. Again from (1) the a-polaroid condition for T entails that T is left polaroid. Suppose now that T is left polaroid. If λ ∈ iso σ(T ) = iso σ(T ) = iso σa (T ) then λI − T is left Drazin invertible, hence by Theorem 2.1 λI − T is right Drazin invertible, so that q(λI−T ) < ∞. On the other hand, λI−T is quasi-Fredholm so,by Theorem [3, 2.7], the SVEP for T at λ entails, that p(λI − T ) < ∞. Hence λ is a pole of T so that T is polaroid, or equivalently T is polaroid.

3. Weyl’s type theorem In this section we show the equivalence of some Weyl type theorems in the case that T is left polaroid or a-polaroid. If T ∈ L(X) deﬁne E(T ) := {λ ∈ iso σ(T ) : 0 < α(λI − T )}, and E a (T ) := {λ ∈ iso σa (T ) : 0 < α(λI − T )}. Evidently, E(T ) ⊆ E a (T ) for every T ∈ L(X). Deﬁne π00 (T ) : {λ ∈ iso σ(T ) : 0 < α(λI − T ) < ∞}, and a π00 (T ) : {λ ∈ iso σa (T ) : 0 < α(λI − T ) < ∞}.

Let p00 (T ) := σ(T ) \ σb (T ), i.e. p00 (T ) is the set of all poles of the resolvent of T having ﬁnite rank. Clearly, for every T ∈ L(X) we have a (T ). p00 (T ) ⊆ π00 (T ) ⊆ π00

(9)

It should be noted that the condition p00 (T ) = π00 (T ) is equivalent to saying that there exists p := p(λI − T ) ∈ N such that H0 (λI − T ) = ker (λI − T )p

for all λ ∈ π00 (T ),

(10)

see [7, Theorem 2.2]). Also the condition of being polaroid may be characterized by means of the quasi-nilpotent part: T is polaroid if and only if there exists p := p(λI − T ) ∈ N such that H0 (λI − T ) = ker (λI − T )p

for all λ ∈ iso σ(T ),

(11)

[4, Theorem 2.9]. Trivially, for every polaroid operator p00 (T ) = π00 (T ). a Lemma 3.1. If T is a-polaroid then p00 (T ) = π00 (T ) = π00 (T ) and E(T ) = a E (T ).

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a Proof. To show the ﬁrst equalities it suﬃces, by (9), to prove that π00 (T ) ⊆ a p00 (T ). Let λ ∈ π00 (T ). Then λ is isolated in σa (T ) and hence the a-polaroid condition entails that p(λI − T ) = q(λI − T ) < ∞. Moreover, α(λI − T ) < ∞, so by Theorem 3.4 of [1] we have β(λI − T ) < ∞, hence λ ∈ p00 (T ). To show the equality E(T ) = E a (T ) it suﬃces to prove the inclusion E a (T ) ⊆ E(T ). If λ ∈ E a (T ) then λ ∈ iso σa (T ) and hence is a pole of T , so λ is an isolated point of σ(T ).

We give now the deﬁnition of Weyl’s theorem and of some of its variants. The symbols here used could generate a certain confusion, but these are the most used in literature. Deﬁnition 3.2. A bounded operator T ∈ L(X) is said to satisfy Weyl’s theorem, in symbol (W ), if σ(T ) \ σw (T ) = π00 (T ). T ∈ L(X) is said to satisfy a a-Weyl’s theorem, in symbol (aW ), if σa (T ) \ σuw (T ) = π00 (T ). T ∈ L(X) is said to satisfy property (w), if σa (T ) \ σuw (T ) = π00 (T ). Weyl’s theorem for T entails Browder’s theorem for T , i.e. σw (T ) = σb (T ). Note that Browder’s theorem for T and Browder’s theorem for T ∗ are equivalent, since σw (T ) = σw (T ∗ ) and σb (T ) = σb (T ∗ ). Furthermore, by [2, Theorem 3.1], (W ) holds for T ⇔ Browder’s theorem holds for T and p00 (T ) = π00 (T ). Either a-Weyl’s theorem or property (w) entails Weyl’s theorem. Property (w) and a-Weyl’s theorem are independent, see [10]. The following result gives a very simple and useful framework for establishing Weyl’s theorem for several classes of operators: Theorem 3.3. It T ∈ L(X) is polaroid and either T or T has SVEP then both T and T satisfy Weyl’s theorem. Proof. The SVEP of either T or T entails Browder’s theorem for T , or equivalently Browder’s theorem for T . The polaroid condition for T entails that p00 (T ) = π00 (T ), so Weyl’s theorem holds for T . If T is polaroid then T is polaroid and hence p00 (T ) = π00 (T ), so Weyl’s theorem also holds for T . Deﬁnition 3.4. A bounded operator T ∈ L(X) is said to satisfy generalized Weyl’s theorem, in symbol (gW ), if σ(T )\σbw (T ) = E(T ). T ∈ L(X) is said to satisfy generalized a-Weyl’s theorem, in symbol (gaW ), if σa (T ) \ σubw (T ) = E a (T ). T ∈ L(X) is said to satisfy generalized property (w), in symbol (gw), if σa (T ) \ σubw (T ) = E(T ). For a bounded operator T ∈ L(X), deﬁne Πa00 (T )

:= σa (T )\ σubb (T ) = {λ ∈ σa (T ) : λI − T is upper semi B-Browder}.

Since σubb (T ) = σld (T ), by Theorem 1.3, it is clear that Πa00 (T ) is the set of all left poles of the resolvent. Lemma 3.5. If T ∈ L(X) then Πa00 (T ) ⊆ E a (T ).

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Proof. If λ ∈ Πa00 (T ) then λ ∈ σa (T ), p(λI −T ) < ∞ and λI −T is left Drazin invertible. The condition p(λI − T ) < ∞ entails, by Remark 2.13, that σa (T ) does not cluster at λ. On the other hand we have 0 < α(λI − T ), otherwise if were α(λI − T ) = 0 then p(λI − T ) = 0 and from deﬁnition of left Drazin invertibility (λI − T )(X) would be closed, hence λ ∈ / σa (T ), a contradiction. Recall that T ∈ L(X) is said to satisfy generalized Browder’s theorem if σbb (T ) = σbw (T ), while T ∈ L(X) is said to satisfy generalized a-Browder’s theorem if σubb (T ) = σubw (T ). Browder’s Theorem and generalized Browder’s theorem are equivalent, and analogously a-Browder’s theorem and generalized a-Browder’s theorem are equivalent, see [5] or [14]. Generalized a-Weyl’s theorem, as well as generalized property (w), entails generalized a-Browder’s theorem. Theorem 3.6. T ∈ L(X) satisﬁes (gaW ) if and only if T satisﬁes generalized a-Browder’s theorem and E a (T ) = Πa00 (T ). In particular, every left polaroid operator which has SVEP satisﬁes (gaW ). Proof. The ﬁrst assertion has been proved in [9]. The second assertion is clear: for every left polaroid operator we have E a (T ) = Πa00 (T ). The SVEP for T entails Browder’s theorem, or equivalently generalized Browder’s theorem. It should be noted that if T or T (or T ∗ , for Hilbert space operators) has SVEP then a-Browder’s theorem holds for T , or, equivalently, generalized a-Browder’s theorem, holds for T , see [6]. In the following diagram we resume the relationships between all Weyl type theorems: (gw) ⇒ (w) ⇒ (W ) (gaW ) ⇒ (aW ) ⇒ (W ), see [15, Theorem 2.3], [10] and [19]. Generalized property (w) and generalized a-Weyl’s theorem are also independent, see [15]. Furthermore, (gw) ⇒ (gW ) ⇒ (W ) (gaW ) ⇒ (gW ) ⇒ (W ), see [15] and [19]. The converse of all these implications in general does not hold. However, we have: Theorem 3.7. Let T ∈ L(X). Then we have: (i) If T is left-polaroid, then (aW ) and (gaW ) for T are equivalent. If T is right-polaroid then (aW ) and (gaW ) for T are equivalent. (ii) If T is polaroid, then (W ), and (gW ) for T are equivalent. Analogously, (W ), and (gW ) for T are equivalent.

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Proof. (i) As already observed, generalized property (gaW ) entails property (aW ) without any assumption on T . Suppose now that property (aW ) holds for T , i.e. σa (T ) \ σuw (T ) = a (T ). We have to prove that σa (T ) \ σubw (T ) = E a (T ). We show ﬁrst that π00 the inclusion σa (T ) \ σubw (T ) ⊆ E a (T ) holds without any assumption on T . Let λ ∈ σa (T ) \ σubw (T ). We can suppose that λ = 0. Therefore, 0 ∈ σa (T ) and T is upper semi B-Fredholm with index less or equal than 0. By [17, Corollary 3.2] then there exists ε > 0 such that µI − T ∈ W+ (X) for all 0 < |µ| < ε. We claim that T has SVEP at every µ. If µ ∈ / σa (T ) this is obvious. Suppose a (T ), so µ is an isolated point that µ ∈ σa (T ). Then µ ∈ σa (T ) \ σuw (T ) = π00 of σa (T ) and hence T has SVEP at µ. The following argument shows that T has SVEP at 0. Let f : D0 → X be an analytic function deﬁned on an open disc D0 centered at 0 for which the equation (λI − T )f (λ) = 0 for all λ ∈ D0 . Take 0 = µ ∈ D0 and let D1 be an open disc centered at µ contained in D0 . The SVEP of T at µ implies that f ≡ 0 on D1 and hence, from the the identity theorem for analytic functions, it then follows that f ≡ 0 on D0 , so T has SVEP at 0. But T is upper semi B-Fredholm, so, by Theorem 2.7 of [3], 0 ∈ iso σa (T ). Suppose now that α(T ) = 0. By Lemma 3.5 then T ∈ Φ+ (X), so T (X) is closed and, consequently, 0 ∈ / σa (T ), a contradiction. Therefore α(T ) > 0, from which we conclude 0 ∈ E a (T ) and hence σa (T ) \ σubw (T ) ⊆ E a (T ). Suppose now that T is left polaroid and let λ ∈ E a (T ). Then λ is an isolated point of σa (T ), and hence by the left polaroid condition λ is a left pole of T . In particular, λI − T is left Drazin invertible or equivalently, by Theorem 1.3, an upper semi B-Browder operator. Since σubw (T ) ⊆ σubb (T ) we then have λ ∈ σa (T ) \ σubb (T ) ⊆ σa (T ) \ σubw (T ). Therefore, generalized a-Weyl’s theorem holds for T . The assertion concerning right-polaroid operators is obvious by Theorem 2.8. (ii) We have only to show that Weyl’s theorem entails generalized Weyl’s theorem. Suppose ﬁrst that λ0 ∈ E(T ). Since T is polaroid then λ0 is a pole of T , hence 0 < p(λ0 I − T ) = q(λ0 I − T ) < ∞. Therefore, λ0 I − T is Drazin invertible or equivalently, by Theorem 1.3, λ0 I − T is B-Browder and hence B-Weyl. Consequently, λ0 ∈ σ(T ) \ σbw (T ) and hence E(T ) ⊆ σ(T ) \ σbw (T ). Conversely, assume that λ0 ∈ σ(T ) \ σbw (T ). Then λ0 I − T is B-Weyl and hence, again by [17, Corollary 3.2], there exists ε > 0 such that λI − T is Weyl for all 0 < |λ − λ0 | < ε. By Theorem 1.5 we know that λI − T is semi-regular in a punctured open disc centered at λ0 , so we can assume that ker (λI − T ) ⊆ N ∞ (λI − T ) ⊆ (λI − T )∞ (X) for all 0 < |λ − λ0 | < ε. As observed above Weyl’s theorem for T entails Browder’s theorem for T , i.e. σw (T ) = σb (T ). Therefore, λI − T is Browder for all 0 < |λ − λ0 | < ε and, consequently, p(λI − T ) = q(λI − T ) < ∞. By Lemma 3.2 of [1] we then have ker (λI − T ) = ker (λI − T ) ∩ (λI − T )∞ (X) = {0},

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thus α(λI − T ) = 0 and since λI − T is Weyl we then conclude that also β(λI − T ) = 0, so λI − T is invertible for all 0 < |λ − λ0 | and hence λ0 ∈ iso σ(T ). To show that λ0 ∈ E(T ) it remains to prove that α(λ0 I − T ) > 0. Suppose that α(λ0 I − T ) = 0. Since λ0 I − T is B-Weyl then, by Lemma 2.4, λ0 I − T is Weyl and since α(λ0 I − T ) = 0 it then follows that λ0 I − T is invertible, a contradiction since λ0 ∈ σ(T ). Therefore, λ0 ∈ E(T ), so generalized Weyl’s theorem holds for T . The last assertion is clear: T is also polaroid. Corollary 3.8. If T ∈ L(X) is a-polaroid then (aW ), (gaW ), (w), (gw) for T are equivalent. Proof. Every a-polaroid operator is left polaroid so, by part (i) of Theorem 3.7, (aW ) and (gaW ) are equivalent. Property (w) and (aW ) are equiva alent, since by Lemma 3.1 we have π00 (T ) = π00 (T ). By Lemma 3.1 we also a have E(T ) = E (T ), from which it easily follows that (gaW ) and (gw) are equivalent. In the following example we show that the result of part (ii) of Theorem 3.7 does not hold if we replace the condition of being a-polaroid by the weaker conditions of being left polaroid or polaroid. Example 3.9. Let R and U be deﬁned as in Example 2.10. As observed before T := R ⊕ U is both left polaroid and polaroid. Moreover, σa (T ) = Γ ∪ {0} and iso σ(T ) = π00 (T ) = ∅, so σa (T ) \ σuw (T ) = {0} = π00 (T ), i.e. T does a not satisfy property (w). On the other hand, we have π00 (T ) = {0}, hence T satisﬁes a-Weyl’s theorem. In the following result we show that if T is polaroid and T has SVEP, (respectively, T has SVEP), we can say much more: all Weyl type theorems, generalized or not, are equivalent and hold for T , (respectively, for T ). Theorem 3.10. Let T ∈ L(X) be polaroid. Then we have (i) If T has SVEP then (W ), (aW ), (w), (gW ), (gaW ) and (gw) hold for T . Moreover, T satisﬁes (gW ). (ii) If T has SVEP then (W ), (aW ), (w), (gW ), (gaW ) and (gw) hold for T . Moreover, T satisﬁes (gW ). Proof. (i) T satisﬁes (W ) by Theorem 3.3. The ﬁrst statement is then proved if we show that (W ) is equivalent to each one of the other Weyl type theorems for T , generalized or not. Since T has SVEP (W ) and (aW ) for T are equivalent, by part (i) of [10, Theorem 2.16]. By Theorem 2.14 T is a-polaroid hence, by Theorem 3.8, (aW ), (gaW ), (w), (gw) for T are equivalent. Finally, by part (ii) of Theorem 3.7, (W ) and (gW ) for T are equivalent. By Theorem 3.3 T satisﬁes (W ) and since T is polaroid then, by part (ii) of Theorem 3.7, (gW ) holds for T . (ii) T satisﬁes (W ) by Theorem 3.3, so it suﬃces to prove that (W ) is equivalent to each one of the other Weyl type theorems, generalized or not, for T . Since, by Theorem 2.14, T is a-polaroid from Theorem 3.8 it then

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follows that (aW ), (gaW ), (w), (gw) are equivalent for T . The SVEP for T entails by part (ii) of [10, Theorem 2.16] that (W ) and (aW ) are equivalent for T , while (W ) and (gW ) for T are equivalent by part (ii) of Theorem 3.7. By Theorem 3.3 T satisﬁes (W ) and since T is polaroid this is equivalent to (gW ) for T , by part (ii) of Theorem 3.7. Let H(σ(T )) denote the set of all analytic functions deﬁned on an open neighborhood of σ(T ) deﬁne, by the classical functional calculus, f (T ) for every f ∈ H(σ(T )). Lemma 3.11. Suppose that f ∈ H(σ(T )) is non constant on each of the components of its domain. If T is left polaroid (respectively, right polaroid, polaroid), then f (T ) is left polaroid (respectively, right polaroid, polaroid). Proof. Let λ0 ∈ iso σa (f (T )). We have to show that λ0 is a left pole of f (T ). Since σa (T ) satisﬁes the spectral mapping theorem we have λ0 ∈ iso f (σa (T )). We show that λ0 ∈ f (iso σa (T )). Let µ0 ∈ σa (T ) be such that f (µ0 ) = λ0 . Denote by Ω the open and connected component of the domain of f which contains µ0 . Suppose that µ0 is not isolated in σa (T ). Then there exists a sequence (µn ) ⊂ σa (T ) of distinct scalars such that µn → µ0 . Clearly, for n suﬃciently large µn ∈ Ω and since K := {µ0 , µ1 , µ2 , . . . } is compact subset of Ω, the classical principle of isolated zeros of analytic functions says to us that f may assume the value λ0 = f (µ0 ) only a ﬁnite number of points of K, so for n suﬃciently large f (µn ) = f (µ0 ) = λ0 , and since f (µn ) → f (µ0 ) = λ0 it then follows that λ0 is not an isolated point of f (σa (T )), a contradiction. Hence λ0 = f (µ0 ), with µ0 ∈ iso σa (T ). Since T is left polaroid then µ0 is a left pole of T and by [11, Theorem 2.9] it then follows that λ0 is a left pole of f (T ), which proves that f (T ) is left polaroid. The proofs for right polaroid and polaroid operators are analogous, just use the spectral mapping theorems for σs (T ) and σ(T ), respectively, and [11, Theorem 2.9]. The result of Theorem 3.10 may be considerably extended as follows Theorem 3.12. Let T ∈ L(X) be polaroid an suppose that f ∈ H(σ(T )) is not constant on each of the components of its domain. Then we have (i) If T has SVEP then (W ), (aW ), (w), (gW ), (gaW ) and (gw) hold for f (T ). (ii) If T has SVEP then (W ), (aW ), (w), (gW ), (gaW ) and (gw) hold for f (T ). Proof. (i) If T ∗ has SVEP then f (T ) = f (T ) has SVEP, see [1, Theorem 2.40]. Moreover, T is left polaroid by Theorem 2.14, so f (T ) is left polaroid by Lemma 3.11. Again by Theorem 2.14, the SVEP of f (T ) entails that f (T ) is polaroid, hence Theorem 3.10 applies to f (T ). (ii) Argue as in the proof of part (i), just replace T with T .

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Remark 3.13. Obviously, in the case of Hilbert space operators, the condition T has SVEP in Theorem 3.12 may be replaced by the SVEP of the Hilbert adjoint T ∗ . Theorem 3.14. Let T ∈ L(X) . Then we have (i) If T ∈ L(X) is left-polaroid and has SVEP, f ∈ H(σ(T )) is not constant on each of the components of its domain, then (aW ) holds for f (T ), or equivalently (gaW ) holds for f (T ). (ii) If T ∈ L(X) is polaroid and has SVEP, f ∈ H(σ(T )) is not constant on each of the components of its domain, then (W ) holds for f (T ), or equivalently (gW ) holds for f (T ). Proof. (i) If T is left polaroid then f (T ) is left polaroid, by Lemma 3.11. By [1, Theorem 2.40] f (T ) has SVEP, hence Theorem 3.6 applies to f (T ). The equivalence of (aW ) and (gaW ) follows from Theorem 3.7. (ii) If T is polaroid then f (T ) is polaroid and has SVEP, so Theorem 3.3 applies to f (T ). The equivalence of (W ) and (gW ) follows from Theorem 3.7.

4. Some applications Weyl type theorems, in their classical and more recently in their generalized form, have been studied by a large number of authors. The results of the previous sections give us an unifying theoretical framework for establishing all Weyl type theorems for a large number of the commonly considered classes of operators. It should be noted that for these classes of operators, Weyl type theorems, or their generalized versions, have been proved in several papers. The classes of polaroid operators introduced in the previous sections are rather large. In the sequel we list some, by no means all, of these classes of operators. In the following H always denotes a Hilbert space. (a) A bounded operator T ∈ L(X) is said to belong to the class H(p) if there exists a natural p := p(λ) such that: H0 (λI − T ) = ker (λI − T )p

for all λ ∈ C.

(12)

From the implication (8) we see that every operator T which belongs to the class H(p) has SVEP. Moreover, from (11) it follows that every H(p) operator T is polaroid. Consequently, by Theorem 3.14, f (T ) satisﬁes Weyl’s theorem, or equivalently, (gW ) for every f ∈ H(σ(T )) which is not constant on each of the components of its domain. Note that, by [34, Theorem 3.4], T ∈ H(p) if and only if there exists a f ∈ H(σ(T )), non constant on each of the components of its domain, such that f (T ) is H(p) and this is equivalent to saying that f (T ) is H(p) for all f ∈ H(σ(T )). Consequently, if T is algebraically H(p) (i.e. there exists a non trivial polynomial h such that h(T ) is H(p)) then T is H(p), so (W ) and (gW ) hold for f (T ). The class H(p) has been introduced by Oudghiri in [34] and in [12] this

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class of operators has been studied for p := p(λ) = 1 for all λ ∈ C. Property H(p) is satisﬁed by every generalized scalar operator, and in particular for p-hyponormal, log-hyponormal or M-hyponormal operators on Hilbert spaces, see [34]. Therefore, algebraically p-hyponormal or algebraically M hyponormal operators are H(p). (W ) or (gW ) for f (T ), where T is algebraically M -hyponormal, have been proved in diﬀerent papers [23] and [24], respectively. In [20, Theorem 3.3] it is shown that if T ∈ L(H) is such that T ∗ is p-hyponormal or M - hyponormal then (gaW ) holds for f (T ) for all f ∈ H(σ(T )). Since T ∗ is H(p), hence polaroid (or, equivalently, T is polaroid), then, by Theorem 3.12 and Remark 3.13, all Weyl’s theorems (generalized or not) hold (and are equivalent!) for f (T ). Finally, Theorem 3.14 subsumes also [21, Theorem 3.4]: if T is analytically hyponormal (i.e. there exists h ∈ H(σ(T )) for which h(T ) is hyponormal) then Weyl’s theorem holds for f (T ). Clearly, since analytically hyponormal operators are H(p), then (W ) and (gW ) are equivalent for f (T ), (b) A bounded operator T ∈ L(X) on a Banach space X is said to be paranormal if T x2 ≤ T 2 xx holds for all x ∈ X. Every paranormal operator on a Hilbert space has SVEP [7]. An operator T ∈ L(H) for which there exists a complex nonconstant polynomial h such that h(T ) is paranormal is said to be algebraically paranormal. Every algebraic paranormal operator T deﬁned on a Hilbert space is polaroid, see [7]. Moreover, the SVEP for h(T ) entails the SVEP for T , see [1, Theorem 2.40]. (gW ) for f (T ), T algebraically paranormal, has been proved in [35, Theorem 3.1] and [23, Theorem 4.14], but this is immediate from Theorem 3.14 and (gW ) for f (T ) is equivalent to (W ). In [35, Theorem 3.2] it has been also proved that (gaW ) holds for T if T ∗ is algebraically paranormal. This result is clear from Theorem 3.10, and in particular (gaW ) for T is equivalent to any Weyl type theorem, generalized or not. Furthermore, Theorem 3.12 and Remark 3.13 extend to f (T ) all Weyl type theorems. Similar considerations may be done for analytically paranormal operators on Hilbert spaces studied in [22]. Analytically paranormal operators T are polaroid and have SVEP, again by [1, Theorem 2.40], so that (W ) holds for f (T ), for f ∈ H(σ(T )) non constant on each of the components of its domain, and this is equivalent to saying that (gW ) holds for f (T ). (gW ) for f (T ), T analytically paranormal, has been proved [22, Theorem 3.1, part (a)]. (gaW ) for f (T ∗ ), T analytically paranormal, was proved in [22, Theorem 3.1, part (b)]. This is clear by Theorem 3.12 and Remark 3.13 and, again, all Weyl type theorems, generalized or not, are equivalent for f (T ). Also the result of [22, Theorem 3.2, part] easily follows from Theorem 3.12 and Remark 3.13: If T ∗ ∈ L(H) is analytically paranormal then all Weyl type theorems, generalized or not, are satisﬁed by f (T ) and all of them are equivalent. (c) A bounded operator T ∈ L(H) is a class A operator if |T |2 ≤ |T 2 |, and T is said to be a quasi-class A operator if T ∗ |T |2 T ≤ T ∗ |T 2 |T. If

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T ∈ L(H) is algebraically quasi-class operator (i.e. there exists a nonconstant polynomial h such that h(T ) is a quasi-class A operator), then T is polaroid ([13, Lemma 2.3]) and since h(T ) has SVEP ([26, Lemma 1.5], also T has SVEP. Consequently, by Theorem 3.14 f (T ) satisﬁes (W ), or equivalently (gW ). If T ∗ is an algebraically quasi-class operator then T ∗ , or equivalently T , is polaroid and the SVEP for T ∗ entails, by Theorem 3.12 and Remark 3.13, that all Weyl’s theorems (generalized or not) hold (and are equivalent!) for f (T ). Therefore Theorem 3.12 subsumes and extends [13, Theorem 2.4 and Theorem 3.3]. (d) Every multiplier T of a semi-simple commutative Banach algebra A, (see [1, Chapter 4] for deﬁnitions and details) is H(1), see [12], in particular every convolution Tµ operator of L1 (G), L1 (G) the group algebra of a locally compact Abelian group G is H(1). Therefore, T is polaroid and has SVEP so that (W ), or equivalently (gW ) holds for f (T ). If A is regular and Tauberian, this is the case for instance of the group algebra L1 (G), G a compact Abelian group, then σ(T ) = σa (T ) for every multiplier T , see Corollary 5.88 of [1]. Therefore, if A is regular and Tauberian every multiplier T is a-polaroid. In particular, every convolution operator Tµ on L1 (G) whenever G is compact, is a-polaroid. Consequently, by Theorem 3.10, (W ), (aW ), (w), (gW ), (gaW ) and (gw) for f (T ) are equivalent and f (T ) satisﬁes all of them. Another example of a-polaroid operator is given by a multiplier T of a Banach algebra A with an orthogonal basis. In fact also in this case σ(T ) = σa (T ), see Theorem 4.46 of [1], so that Theorem 3.10 applies to T .

References [1] P. Aiena Fredholm and local spectral theory, with application to multipliers. Kluwer Acad. Publishers (2004). [2] P. Aiena. Classes of Operators Satisfying a-Weyl’s theorem. Studia Math. 169 (2005), 105-122. [3] P. Aiena Quasi Fredholm operators and localized SVEP, Acta Sci. Math. (Szeged), 73 (2007), 251-263. [4] P. Aiena, M. T. Biondi, F. Villaf˜ ane Property (w) and perturbations III J. Math. Anal. Appl. 353 (2009), 205-214. [5] P. Aiena, M. T. Biondi, C. Carpintero On Drazin invertibility, Proc. Amer. Math. Soc. 136, (2008), 2839-2848. [6] P. Aiena, C. Carpintero, E. Rosas. Some characterization of operators satisfying a-Browder theorem. J. Math. Anal. Appl. 311, (2005), 530-544. [7] P. Aiena, J. R. Guillen Weyl’s theorem for perturbations of paranormal operators. Proc. Amer. Math. Soc.35, (2007), 2433-2442. [8] P. Aiena, J. Guillen, P. Pe˜ na Property (w) for perturbation of polaroid operators. Linear Alg. and Appl. 424 (2008), 1791-1802. [9] P. Aiena, T. L. Miller On generalized a-Browder’s theorem. Studia Math. 180, 3, (2007), 285300.

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[10] P. Aiena, P. Pe˜ na A variation on Weyl’s theorem. J. Math. Anal. Appl. 324 (2006), 566-579. [11] P. Aiena, J. E. Sanabria On left and right poles of the resolvent. Acta Sci. Math. 74 (2008),669-687. [12] P. Aiena, F. Villaf˜ ane Weyl’s theorem for some classes of operators. Int. Equa. Oper. Theory 53, (2005), 453-466. [13] J. An, Y. M. Han Weyl’s theorem for algebraically Quasi-class A operators. Int. Equa. Oper. Theory 62, (2008), 1-10. [14] M. Amouch, H. Zguitti On the equivalence of Browder’s and generalized Browder’s theorem. Glasgow Math. Jour. 48, (2006), 179-185. [15] M. Berkani, M. Amouch On the property (gw). Mediterr. J. Math. 5 (2008), no. 3, 371–378. [16] M. Berkani On a class of quasi-Fredholm operators. Int. Equa. Oper. Theory 34 (1), (1999), 244-249. [17] M. Berkani, M. Sarih On semi B-Fredholm operators, Glasgow Math. J. 43 (2001), 457-465. [18] M. Berkani Index of B-Fredholm operators and generalization of a Weyl’s theorem, Proc. Amer. Math. Soc., vol. 130, 6, (2001), 1717-1723. [19] M. Berkani, J. J. Koliha Weyl type theorems for bounded linear operators, Acta Sci. Math. (Szeged) 69 (2003), no. 1-2, 359–376. [20] X. Cao, M. Guo, B. Meng Weyl type theorems for p-hyponormal and M hyponormal operators, Studia Math. 163 (2) (2004), 177-187. [21] X. Cao Weyl’s type theorem for analytically hyponormal operators, Linear Alg. and Appl. 405 (2005), 229-238. [22] X. Cao Topological uniform descent and Weyl’s type theorem, Linear Alg. and Appl. 420 (2007), 175-182. [23] R. E. Curto, Y. M. Han Generalized Browder’s and Weyl’s theorems for Banach spaces operators, J. Math. Anal. Appl. 2 (2007), 1424-1442. [24] R. E. Curto, Y. M. Han Weyl’s theorem for algebraically paranormal operators, Integ. Equa. Oper. Theory 50, (2004), No.2, 169-196. [25] M. P. Drazin Pseudoinverse in associative rings and semigroups. Amer. Math. Monthly 65, (1958), 506-514. [26] B.P. Duggal, I. H. Jeon, I. H. Kim On Weyl’s theorem for quasi-class A operators J. Korean Math. Soc. 43, (2006), 899-909. [27] S. Grabiner Uniform ascent and descent of bounded operators J. Math. Soc. Japan 34 (1982), 317-337. [28] H. Heuser Functional Analysis, Marcel Dekker, New York 1982. [29] J. J. Koliha Isolated spectral points Proc. Amer. Math. Soc. 124 (1996), 34173424. [30] J.P. Labrousse Les op´erateurs quasi-Fredholm., Rend. Circ. Mat. Palermo, XXIX 2, (1980) [31] D. C. Lay Spectral analysis using ascent, descent, nullity and defect. Math. Ann. 184 (1970), 197-214. [32] K. B. Laursen, M. M. Neumann Introduction to local spectral theory., Clarendon Press, Oxford 2000.

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[33] M. Mbekhta, V. M¨ uller On the axiomatic theory of the spectrum II. Studia Math. 119 (1996), 129-147. [34] M. Oudghiri Weyl’s and Browder’s theorem for operators satisfying the SVEP Studia Math. 163, 1, (2004), 85-101. [35] H. Zguitti A note on generalized Weyl’s theorem, J. Math. Anal. Appl. 316 (1) (2006), 373-381. Pietro Aiena Dipartimento di Metodi e Modelli Matematici Facolt` a di Ingegneria Universit` a di Palermo 90128 Palermo Italy e-mail: [email protected] Elvis Aponte Departamento de Matem´ aticas Facult´ ad de Ciencias UCLA Barquisimeto Venezuela e-mail: [email protected] Edixon Balzan Departamento de Matem´ aticas Facult´ ad de Ciencias Universidad del Zulia Maracaibo Venezuela e-mail: [email protected] Submitted: May 20, 2009.

Integr. Equ. Oper. Theory 66 (2010), 21–40 DOI 10.1007/s00020-009-1736-4 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

Weighted Composition Operators from H ∞ to the Bloch Space of a Bounded Homogeneous Domain Robert F. Allen and Flavia Colonna Abstract. Let D be a bounded homogeneous domain in Cn . In this paper, we study the bounded and the compact weighted composition operators mapping the Hardy space H ∞ (D) into the Bloch space of D. We characterize the bounded weighted composition operators, provide operator norm estimates, and give suﬃcient conditions for compactness. We prove that these conditions are necessary in the case of the unit ball and the polydisk. We then show that if D is a bounded symmetric domain, the bounded multiplication operators from H ∞ (D) to the Bloch space of D are the operators whose symbol is bounded. Mathematics Subject Classiﬁcation (2010). Primary 47B38; Secondary 32A18, 30D45. Keywords. Weighted composition operators, Bloch space, homogeneous domains.

1. Introduction Let X be a Banach space of holomorphic functions on a bounded domain D in Cn . For ψ a complex-valued holomorphic function on D and ϕ a holomorphic self-map of D, we deﬁne a linear operator Wψ,ϕ on X, called the weighted composition operator with multiplicative symbol ψ and composition symbol ϕ, by Wψ,ϕ f = ψ(f ◦ ϕ), f ∈ X. Setting Mψ f = ψf and Cϕ f = f ◦ ϕ, we may write Wψ,ϕ = Mψ Cϕ . Then Mψ is called multiplication operator with symbol ψ and Cϕ is called composition operator with symbol ϕ. The study of the weighted composition operators on the Bloch space began with the work of Ohno and Zhao in [19] where the operators from the Bloch space of the open unit disk D into itself were considered. In higher dimensions, these operators on the Bloch space have been studied by Chen,

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Stevi´c and Zhou in [6] (which was motivated by papers and [21] and [7]), and by the authors in [4]. For related work, see [27]. In [18], Ohno investigated the weighted composition operators between the Bloch space of D and the Hardy space H ∞ (D) of bounded analytic functions on D. Characterizations of the boundedness and the compactness of the weighted composition operators from the Bloch space to H ∞ were given by Hosokawa, Izuchi and Ohno in [14] in the one-dimensional case, and by Li and Stevi´c in the case of the unit ball [17]. In [22], Stevi´c determined the norm of the bounded weighted composition operators from the Bloch space and the little Bloch space to the weighted Hardy space Hµ∞ of the unit ball (where µ is a weight). The weighted composition operators from a larger class of spaces known as the α-Bloch spaces to H ∞ on the polydisk where studied by Li and Stevi´c in [16]. In [4], we characterized the boundedness, determined the operator norm, and gave a suﬃcient condition for compactness in the case of a general bounded homogeneous domain in Cn . The study of the weighted composition operators from H ∞ to the Bloch space in several variables was carried out by Li and Stevi´c when the ambient space is the unit polydisk [15]. For the case of the unit ball, the study of these operators from H ∞ into the α-Bloch spaces was carried out by Li and Stevi´c in [17] and Zhang and Chen in [26]. In this paper, we analyze the weighted composition operators from H ∞ into the Bloch space on a bounded homogeneous domain in Cn . In Section 2, we give the background on the bounded homogeneous domains in Cn and special class of such domains that have a canonical representation (up to biholomorphic transformation) due to Cartan [5], the bounded symmetric domains. We then review the notion of the Bloch space of a bounded homogeneous domain [12], [23], and of a subspace we refer to as the ∗-little Bloch space. We also recall the deﬁnition of little Bloch space on a bounded symmetric domain [24]. In Section 3, in the environment of a general bounded homogeneous domain, we characterize the bounded weighted composition operators from H ∞ into the Bloch space and into the ∗-little Bloch space, thereby extending the results of Ohno [18] in the one-dimensional case, of Li and Stevi´c for the polydisk [15] and for the unit ball [17], and of Zhang and Chen for the unit ball [26]. We also give estimates on the operator norm. In Section 4, we describe suﬃcient conditions for the compactness of a weighted composition operator from H ∞ to either the Bloch space or the ∗-little Bloch space of a bounded homogeneous domain. We conjecture these conditions to be necessary, and prove the necessity when the domain is the unit polydisk and the unit ball. In the latter setting, we obtain a result equivalent to a special case of Theorem 2 in [26]. Furthermore, we show that compactness and boundedness are equivalent when the operator maps H ∞ (Bn ) into B0 (Bn ), a result not observed in [26].

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In Section 5, we show that the bounded multiplication operators from H ∞ into the Bloch space (respectively, the ∗-little Bloch space) of a bounded symmetric domain are precisely those whose symbol is bounded (respectively, in the ∗-little Bloch space and bounded). Furthermore, we obtain operator norm estimates in terms of the Bergman constant of the domain. We then discuss the boundedness of the composition operators on a bounded homogeneous domain and establish operator norm estimates. While composition operators from H ∞ to the α-Bloch spaces of the polydisk have been studied by Stevi´c in [21], to our knowledge, there are no results in the literature for the multiplication or the composition operators between H ∞ and the Bloch spaces on general bounded homogeneous or symmetric domains. Finally, in Section 6, we show there are no isometries amongst the multiplication or composition operators from the Hardy space H ∞ into the Bloch space when the domain is the unit disk. We also show that there are no isometric weighted composition operators from the Bloch space into H ∞ if the ambient space is the unit polydisk. We conjecture that, likewise, there are no isometric weighted composition operators from H ∞ to the Bloch space.

2. Preliminaries Let D be a domain in Cn . We denote by H(D) the set of holomorphic functions from D into C, and by Aut(D) the set of biholomorphic maps of D. The space H ∞ (D) of bounded holomorphic functions on D is a Banach algebra equipped with norm ||f ||∞ = supz∈D |f (z)|. A domain D is homogeneous if Aut(D) acts transitively on D. Every homogeneous domain is equipped with a canonical metric invariant under the action of Aut(D), called the Bergman metric [13]. A domain D is symmetric at z0 ∈ D if there exists an involution ϕ ∈ Aut(D) for which z0 is an isolated ﬁxed point. A domain that is symmetric at every point is called symmetric. A bounded symmetric domain is homogeneous and a bounded homogeneous domain that is symmetric at one point is symmetric [13]. The unit ball Bn = {z = (z1 , . . . , zn ) ∈ Cn : ||z|| = 2

n

|zk |2 < 1}

k=1

and the unit polydisk Dn = {(z1 , . . . , zn ) ∈ Cn : |zk | < 1, k = 1, . . . , n} are bounded symmetric domains since they are homogeneous and symmetric at the origin via the map z → −z. While bounded homogeneous domains in dimensions 2 and 3 are symmetric, there are examples of bounded homogeneous domains in dimensions greater than 3 which are not symmetric [20]. In [5], Cartan showed that every bounded symmetric domain in Cn is biholomorphically equivalent to a ﬁnite product of irreducible bounded symmetric domains, unique up to order. He classiﬁed the irreducible bounded

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symmetric domains into six classes, four of which are known as the Cartan classical domains, and the other two, each consisting of a single domain, are known as the exceptional domains. A bounded symmetric domain written as such a product is said to be in standard form. The Cartan classical domains are deﬁned as: RI = {Z ∈ Mm,n (C) : Im − ZZ ∗ > 0}, for m ≥ n ≥ 1, RII = {Z ∈ Mn (C) : Z = Z T , In − ZZ ∗ > 0}, for n ≥ 2, RIII = {Z ∈ Mn (C) : Z = −Z T , In − ZZ ∗ > 0}, for n ≥ 5, RIV = {z = (z1 , . . . , zn ) ∈ Cn : A > 0, ||z|| < 1} , for n ≥ 5, where Mm,n (C) denotes the set of m × n matrices with complex entries, Mn (C) = Mn,n (C), Z T and Z ∗ are the transpose and the adjoint of Z, 2 respectively, and A = | zk2 |2 + 1 − 2 ||z|| . See [11] for a description of the exceptional domains. The notion of Bloch function in higher dimensions was ﬁrst introduced by Hahn in [12]. In [23] and [24], Timoney studied in depth the Bloch functions on a bounded homogeneous domain. In this paper, we conform to his deﬁnition and notation, as follows. Let D be a bounded homogeneous domain. For z ∈ D and f ∈ H(D), deﬁne |∇(f )(z)u| , Qf (z) = sup 1/2 n H z (u, u) u∈C \{0} where ∇(f )(z) is the gradient of f at z, for u = (u1 , . . . , un ), ∇(f )(z)u =

n ∂f (z)uk , ∂zk

k=1

and Hz is the Bergman metric on D at z. The Bergman metric for the unit ball Bn is deﬁned as n + 1 (1 − ||z||2 ) u, v + u, z z, v

, · 2 2 (1 − ||z|| )2 where u, v ∈ Cn , z ∈ Bn , and u, v = uj vj . For the unit polydisk Dn , the Bergman metric is deﬁned as Hz (u, v) =

Hz (u, v) =

n

uj vj

j=1

(1 − |zj | )2

2

,

where u, v ∈ Cn and z ∈ Dn . The Bloch space B(D) on a bounded homogeneous domain D is the set of all functions f ∈ H(D) for which βf = sup Qf (z) < ∞. z∈D

Timoney proved that B(D) is a Banach space under the norm ||f ||B = |f (z0 )| + βf ,

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where z0 is some ﬁxed point in D [23]. For convenience, we shall assume throughout that 0 ∈ D and choose z0 = 0. The ∗-little Bloch space of D is the subspace of B(D) deﬁned as B0∗ (D) = f ∈ B(D) : lim∗ Qf (z) = 0 , z→∂ D

∗

where ∂ D is the distinguished boundary of D. Timoney deﬁned the little Bloch space B0 (D) of a bounded symmetric domain D to be the closure of the polynomials in B(D). Its elements f satisfy the condition lim Qf (z) = 0.

z→∂ ∗ D

If D is the unit ball, then ∂D = ∂ ∗ D and thus B0 (D) = B0∗ (D). When D = Bn , B0 (D) is a proper subspace of B0∗ (D), and B0∗ (D) is a non-separable subspace of B(D) [24]. In [23], Timoney proved that the space H ∞ (D) of bounded holomorphic functions on a bounded homogeneous domain D is a subspace of B(D) and for each f ∈ H ∞ (D), ||f ||B ≤ |f (0)|+c ||f ||∞ where c is a constant depending only on the domain D. In [8], Cohen and the second author deﬁned the Bloch constant of a bounded homogeneous domain D in Cn as cD = sup{Qf (z) : f ∈ H ∞ (D), ||f ||∞ ≤ 1, z ∈ D}. The precise value of the Bloch constant was calculated for each Cartan classical domain to be the reciprocal of the inner radius of the domain with respect to the Bergman metric, that is, 1 . cD = inf ξ∈∂D H0 (ξ, ξ)1/2 In [25], Zhang determined the Bloch constant for the two exceptional domains RV of dimension 16 and RV I of dimension 27. The value of cD for each irreducible bounded symmetric domain D is shown in Theorem 5.2. By Theorem 3 of [8], extended to include the exceptional domains, if D = D1 × · · · × Dk is a bounded symmetric domain in standard form, then cD = max cDj .

(1)

1≤j≤k

Furthermore, it was shown that there exist polynomial functions f (hence in the little Bloch space of D) such that ||f ||∞ ≤ 1 and cD = Qf (0). Let D be a bounded homogeneous domain. In [2], we showed that a function f ∈ H(D) is Bloch if and only if there exists c > 0 such that |f (z) − f (w)| ≤ cρ(z, w), for each z, w ∈ D, where ρ is the Bergman distance on D. Furthermore, the Bloch seminorm of f is precisely the inﬁmum of all such constants c. As a consequence, we obtain that for a Bloch function f on D, |f (z) − f (w)| ≤ ||f ||B ρ(z, w), for each z, w ∈ D.

(2)

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3. Boundedness From now on, unless speciﬁed otherwise, we shall assume D is a bounded homogeneous domain, ψ ∈ H(D), and ϕ = (ϕ1 , . . . , ϕn ) is a holomorphic self-map of D. Deﬁne θψ,ϕ = supz∈D |ψ(z)|θϕ (z), where θϕ (z) = sup{Qf ◦ϕ (z) : f ∈ H ∞ (D), ||f ||∞ ≤ 1}. For z ∈ D, denote by Jϕ(z) the Jacobian matrix of ϕ at z, that is, the ∂ϕ matrix whose (j, k)-entry is ∂zkj (z). Furthermore, deﬁne Bϕ (z) =

sup u∈Cn \{0}

Hϕ(z) (Jϕ(z)u, Jϕ(z)u)1/2 . Hz (u, u)1/2

The quantity Bϕ = supz∈D Bϕ (z) is bounded above by a constant dependent only on D [23]. By the invariance of the Bergman metric under the action of Aut(D), if ϕ ∈ Aut(D), then for each z ∈ D, u ∈ Cn , Hϕ(z) (Jϕ(z)u, Jϕ(z)u) = Hz (u, u). For f ∈ B(D) and z ∈ D, Qf ◦ϕ (z) ≤ Bϕ (z)Qf (ϕ(z)). Taking the supremum over all functions f ∈ H ∞ (D) with ||f ||∞ ≤ 1, we obtain (3) θϕ (z) ≤ cD Bϕ (z) for each z ∈ D. Theorem 3.1. (a) The weighted composition operator Wψ,ϕ : H ∞ (D) → B(D) is bounded if and only if ψ ∈ B(D) and θψ,ϕ is ﬁnite. (b) Wψ,ϕ : H ∞ (D) → B0∗ (D) is bounded if and only if ψ ∈ B0∗ (D), θψ,ϕ is ﬁnite, and lim |ψ(z)|θϕ (z) = 0.

z→∂ ∗ D

(4)

Furthermore, if Wψ,ϕ is bounded as an operator into B(D) or B0∗ (D), then the norm of Wψ,ϕ satisﬁes the estimates ||ψ||B ≤ ||Wψ,ϕ || ≤ ||ψ||B + θψ,ϕ .

(5)

Proof. (a) Assume ﬁrst Wψ,ϕ is bounded. Then ||ψ||B = ||Wψ,ϕ 1||B , thus ψ ∈ B(D) and the lower estimate of (5) holds. For each f ∈ H ∞ (D) with ||f ||∞ ≤ 1, and each z ∈ D, we have ∇(ψ(f ◦ ϕ))(z) = ψ(z)∇(f ◦ ϕ)(z) + f (ϕ(z))∇(ψ)(z)

(6)

so that |ψ(z)| Qf ◦ϕ (z) ≤ ||Wψ,ϕ f ||B + Qψ (z) ≤ ||Wψ,ϕ || + ||ψ||B . Thus, taking the supremum over all such functions f and over all z ∈ D, we see that θψ,ϕ is ﬁnite.

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Conversely, assume ψ ∈ B(D) and θψ,ϕ is ﬁnite. Then for f ∈ H ∞ (D), with f ∞ ≤ 1, we have Qψ(f ◦ϕ) (z) ≤ |ψ(z)|Qf ◦ϕ (z) + |f (ϕ(z))|Qψ (z) ≤ θψ,ϕ + βψ . Thus, Wψ,ϕ f ∈ B(D) and ||ψ(f ◦ ϕ)||B ≤ |ψ(0)f (ϕ(0))| + θψ,ϕ + βψ ≤ ||ψ||B + θψ,ϕ , proving the boundedness of Wψ,ϕ and the upper estimate of (5). (b) From part (a), it suﬃces to show that if Wψ,ϕ is bounded, then (4) holds and, conversely, if ψ ∈ B0∗ (D), θψ,ϕ is ﬁnite, and (4) holds, then for each f ∈ B0∗ (D), ψ(f ◦ ϕ) ∈ B0∗ (D). Assume Wψ,ϕ is bounded. Then ψ = Wψ,ϕ 1 ∈ B0∗ (D) and, for each f ∈ H ∞ (D) and each z ∈ D, from (6) we have |ψ(z)|Qf ◦ϕ (z) ≤ Qψ(f ◦ϕ) (z) + Qψ (z) ||f ||∞ . Taking the limit as z → ∂ ∗ D, we obtain |ψ(z)|Qf ◦ϕ (z) → 0. Since this holds for each f ∈ H ∞ (D), we deduce (4). Next assume ψ ∈ B0∗ (D), θψ,ϕ is ﬁnite, and (4) holds. Then for each nonzero function f ∈ H ∞ (D) and each z ∈ D, using the function f˜ = f / ||f ||∞ , we see that Qf ◦ϕ (z) ≤ θϕ (z) ||f ||∞ . Thus, for each f ∈ H ∞ (D) and each z ∈ D, we obtain Qψ(f ◦ϕ) (z) ≤

Qψ (z) f ||∞ + |ψ(z)|Qf ◦ϕ (z)

≤

(Qψ (z) + |ψ(z)|θϕ (z)) ||f ||∞ .

Taking the limit as z → ∂ ∗ D, we conclude that ψ(f ◦ ϕ) ∈ B0∗ (D).

In the special case of D = Bn , the ﬁniteness of θψ,ϕ follows from (4) since ∂ ∗ Bn = ∂Bn . From this, we obtain the following characterization of the bounded weighted composition operators from H ∞ (Bn ) into B0 (Bn ). Corollary 3.2. Wψ,ϕ : H ∞ (Bn ) → B0 (Bn ) is bounded if and only if ψ ∈ B0 (Bn ) and lim |ψ(z)| θϕ (z) = 0.

||z||→1

In the case of D = D, we can improve the lower bound on the operator norm of Wψ,ϕ . Lemma 3.3. Let ψ ∈ H(D) and ϕ a holomorphic self-map of D. Then θψ,ϕ = sup |ψ(z)| z∈D

(1 − |z| ) |ϕ (z)| 2

1 − |ϕ(z)|2

.

(7)

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Proof. Let f ∈ H ∞ (D) such that ||f ||∞ ≤ 1. Since βf ≤ ||f ||∞ , for all z ∈ D, Qf ◦ϕ (z) = (1 − |z|2 ) |f (ϕ(z))| |ϕ (z)| = (1 − |ϕ(z)| ) |f (ϕ(z))| 2

≤ βf ≤

(1 − |z| ) |ϕ (z)| 2

2

1 − |ϕ(z)|

(1 − |z| ) |ϕ (z)| 2

1 − |ϕ(z)|2

(1 − |z| ) |ϕ (z)| 2

2

1 − |ϕ(z)|

.

Thus θϕ (z) ≤

(1 − |z| ) |ϕ (z)| 2

2

1 − |ϕ(z)|

.

On the other hand, ﬁxing z ∈ D, the function f (w) =

ϕ(z) − w 1 − ϕ(z)w

, w ∈ D,

is an automorphism of D, and thus ||f ||∞ = 1. Moreover, Qf ◦ϕ (z) = and so θϕ (z) =

(1−|z|2 )|ϕ (z)| , 1−|ϕ(z)|2

(1 − |z|2 ) |ϕ (z)| 2

1 − |ϕ(z)|

(8)

which yields (7).

Theorem 3.4. Let Wψ,ϕ be bounded from H ∞ (D) to B(D) or B0 (D). Then max{||ψ||B , θψ,ϕ } ≤ ||Wψ,ϕ || ≤ ||ψ||B + θψ,ϕ . Proof. By Theorem 3.1, It suﬃces to show that θψ,ϕ ≤ ||Wψ,ϕ ||. Fix λ ∈ D and for z ∈ D, let ϕ(λ) − z . fλ (z) = 1 − ϕ(λ)z Then ||fλ ||∞ = 1 and fλ (ϕ(λ)) = 0. From (8), we obtain ||Wψ,ϕ || ≥ ||Wψ,ϕ fλ ||B ≥ sup(1 − |z| ) |ψ (z)fλ (ϕ(z)) + ψ(z)(fλ ◦ ϕ) (z)| 2

z∈D

≥ |ψ(λ)|Qfλ ◦ϕ (λ) (1 − |λ| ) |ψ(λ)| |ϕ (λ)| 2

=

1 − |ϕ(λ)|

2

.

Taking the supremum over all λ ∈ D, we get ||Wψ,ϕ || ≥ θψ,ϕ .

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4. Compactness The following lemma will be used to prove a suﬃcient condition for the compactness of Wψ,ϕ . Lemma 4.1. Wψ,ϕ is compact from H ∞ (D) into B(D) if and only if for every bounded sequence {fk } in H ∞ (D) converging to 0 locally uniformly in D, ||ψ(fk ◦ ϕ)||B → 0 as k → ∞. Proof. Assume that Wψ,ϕ : H ∞ (D) → B(D) is compact. Let {fk } be a bounded sequence in H ∞ (D) which converges to 0 locally uniformly in D. By rescaling fk , we may assume ||fk ||∞ ≤ 1 for all k ∈ N. We need to show that ||ψ(fk ◦ ϕ)||B → 0 as k → ∞. Since Wψ,ϕ is compact, the sequence {ψ(fk ◦ϕ)} has a subsequence (which for convenience we reindex as the original sequence) converging in the Bloch norm to some function f ∈ B(D). We are going to show that f is identically 0 by proving that ψ(fk ◦ ϕ) → 0 locally uniformly. Fix z0 ∈ D and, subtracting from the elements of this sequence the value of f at z0 , we may assume f (z0 ) = 0. Then ψ(z0 )fk (ϕ(z0 )) → 0 as k → ∞. For z ∈ D, using (2), we ﬁnd |ψ(z)fk (ϕ(z)) − f (z)| ≤ |ψ(z)fk (ϕ(z)) − f (z) − (ψ(z0 )fk (ϕ(z0 )) − f (z0 ))| + |ψ(z0 )fk (ϕ(z0 ))| ≤ ||ψ(fk ◦ ϕ) − f ||B ρ(z, z0 ) + |ψ(z0 )fk (ϕ(z0 ))| . The right-hand side converges to 0 locally uniformly as k → ∞ since ψ(fk ◦ ϕ) − f → 0 in B(D). On the other hand, ψ(fk ◦ ϕ) → 0 locally uniformly, so f is identically 0. Conversely, suppose that whenever {gk } is a bounded sequence in H ∞ (D) converging to 0 locally uniformly in D, ||ψ(gk ◦ ϕ)||B → 0 as k → ∞. To prove the compactness of Wψ,ϕ , it suﬃces to show that if {fk } is a sequence in H ∞ (D) with ||fk ||∞ ≤ 1 for all k ∈ N, there exists a subsequence {fkj } such that ψ(fkj ◦ ϕ) converges in norm in B(D). Fix z0 ∈ D. Replacing fk by fk − fk (z0 ), we may assume that fk (z0 ) = 0 for all k ∈ N. Since {fk } is uniformly bounded on D, by Montel’s theorem some subsequence {fkj } converges locally uniformly to some holomorphic function f on D such that ||f ||∞ ≤ 1. Then, letting gkj = fkj − f , we obtain a bounded ∞ sequence converging to 0 locally uniformly in D. By the hypoth in H (D) esis, ψ(gkj ◦ ϕ)B → 0 as j → ∞. Therefore, ψ(fkj ◦ ϕ) converges in norm to ψ(f ◦ ϕ), completing the proof. Theorem 4.2. (a) Let ψ ∈ B(D). Then Wψ,ϕ : H ∞ (D) → B(D) is compact if lim

ϕ(z)→∂D

Qψ (z) = 0, and

lim

ϕ(z)→∂D

|ψ(z)| θϕ (z) = 0.

(9)

(b) Let ψ ∈ B0∗ (D). Then Wψ,ϕ : H ∞ (D) → B0∗ (D) is compact if lim Qψ (z) = 0, and

z→∂D

lim |ψ(z)| θϕ (z) = 0.

z→∂D

(10)

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Proof. Assume conditions (9) both hold. By Lemma 4.1, to prove that Wψ,ϕ is compact from H ∞ (D) into B(D) it suﬃces to show that for any bounded sequence {fk } such that ||fk ||∞ ≤ 1 and fk → 0 locally uniformly in D, ||ψ(fk ◦ ϕ)||B → 0 as k → ∞. Let {fk } be such a sequence and ﬁx ε > 0. Then, there exists r > 0 such that for all k ∈ N, |ψ(z)| Qfk ◦ϕ (z) < 2ε and Qψ (z) < 2ε whenever ρ(ϕ(z), ∂D) ≤ r. Thus, if ρ(ϕ(z), ∂D) ≤ r, then Qψ(fk ◦ϕ) (z) ≤ |ψ(z)| Qfk ◦ϕ (z) + Qψ (z) < ε. On the other hand, since fk → 0 locally uniformly in D, |fk (ϕ(z))| → 0 uniformly on the compact set Er = {z ∈ D : ρ(ϕ(z), ∂D) ≥ r}. Thus, ∇fk (ϕ(z)) approaches the zero vector, and hence Qfk ◦ϕ (z) → 0, uniformly on Er as k → ∞. Consequently, recalling that ψ ∈ B(D), we see that for all k suﬃciently large, Qψ(fk ◦ϕ) (z) < ε for all z ∈ D. Furthermore, |ψ(0)fk (ϕ(0))| → 0 as k → ∞, so ||ψ(fk ◦ ϕ)||B → 0, completing the proof of (a). The proof of part (b) is analogous. Remark 1. If D is not the unit ball, the multiplicative symbol of the weighted composition operators satisfying conditions (10) reduces to a constant, and hence the compact weighted composition operators of this type have composition component which is compact. The case when D = Bn is discussed in Corollary 4.3. We conjecture that under boundedness assumptions, conditions (9) and (10) are necessary as well. Conjecture. If D is a bounded homogeneous domain, then the bounded operator Wψ,ϕ : H ∞ (D) → B(D) is compact if and only if lim

ϕ(z)→∂D

Qψ (z) = 0, and

lim

ϕ(z)→∂D

|ψ(z)| θϕ (z) = 0.

4.1. Compactness from H ∞ (Bn ) to B0 (Bn ) or B(Bn ) We begin this section by extending Theorem 4 in [18] to the unit ball. Corollary 4.3. Let ϕ be a holomorphic self-map of Bn and ψ ∈ H(Bn ). Then the following are equivalent: (a) Wψ,ϕ : H ∞ (Bn ) → B0 (Bn ) is bounded. (b) Wψ,ϕ : H ∞ (Bn ) → B0 (Bn ) is compact. (c) ψ ∈ B0 (Bn ) and lim |ψ(z)| θϕ (z) = 0. ||z||→1

Proof. The implication (b) =⇒ (a) is obvious. The implication (a) =⇒ (c) follows from Corollary 3.2. Finally, (c) =⇒ (b) follows from part (b) of Theorem 4.2. We now prove the above conjecture in the case of the unit ball. The following result is equivalent to a characterization of the compactness obtained by Zhang and Chen ([26], Theorem 2).

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Theorem 4.4. Let ϕ be a holomorphic self-map of Bn , and ψ ∈ H(Bn ). Then Wψ,ϕ : H ∞ (Bn ) → B(Bn ) is compact if and only if it is bounded, lim

Qψ (z) = 0, and

(11)

lim

|ψ(z)| θϕ (z) = 0.

(12)

||ϕ(z)||→1 ||ϕ(z)||→1

Proof. If Wψ,ϕ is bounded and conditions (11) and (12) hold, then ψ ∈ B(Bn ), so by Theorem 4.2, Wψ,ϕ is compact. Conversely, suppose Wψ,ϕ is compact. Then Wψ,ϕ is bounded and by Theorem 2 in [26], condition (11) holds, and lim

||ϕ(z)||→1

|ψ(z)| Bϕ (z) = 0.

Condition (12) now follows from the inequality θϕ (z) ≤ Bϕ (z) for each z ∈ Bn . 4.2. Compactness from H ∞ (Dn ) to B(Dn ) In [15], Li and Stevi´c characterized the compact weighted composition operators from H ∞ (Dn ) into B(Dn ) in the following result. Theorem 4.5 ([15], Theorem 1.2). Let ϕ = (ϕ1 , . . . , ϕn ) be a holomorphic selfmap of Dn and ψ(z) a holomorphic function on Dn . Then Wψ,ϕ : H ∞ (Dn ) → B(Dn ) is compact if and only if the following conditions are satisﬁed: (a) Wψ,ϕ : H ∞ (Dn ) → B(Dn ) is bounded; n 2 ∂ψ (b) lim (1 − |zk | ) (z) = 0; ∂zk ϕ(z)→∂Dn k=1 n ∂ϕj 1 − |zk |2 (c) lim n |ψ(z)| (z) ∂zk 1 − |ϕ (z)|2 = 0. ϕ(z)→∂D j k,j=1 We will prove the conjecture posed in the previous section in the setting of the polydisk Dn . To do this, we need the following lemma. Lemma 4.6. Let ϕ = (ϕ1 , . . . , ϕn ) be a holomorphic self-map of Dn . For z ∈ Dn , n ∂ϕj 1 − |zk |2 |ψ(z)| θϕ (z) ≤ |ψ(z)| ∂zk (z) 1 − |ϕ (z)|2 . j j,k=1

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Proof. Observe that by (1.2) of [9], for all z ∈ Dn , |ψ(z)| θϕ (z) ≤ |ψ(z)| Bϕ (z)

⎛ 2 ⎞1/2 n n 2 ∂ϕ | )w (1 − |z j k k ⎠ = |ψ(z)| max ⎝ (z) 2 ||w||=1 ∂zk 1 − |ϕj (z)| j=1 k=1

⎛

n 2 ⎞1/2 n ∂ϕj (1 − |zk |2 ) |wk | ⎠ ≤ |ψ(z)| max ⎝ ∂zk (z) 1 − |ϕ (z)|2 ||w||=1 j=1

j

k=1

n ∂ϕj 1 − |zk |2 ≤ |ψ(z)| ∂zk (z) 1 − |ϕ (z)|2 . j

k,j=1

Theorem 4.7. Let ϕ be a holomorphic self-map of Dn and ψ ∈ B(Dn ). Then Wψ,ϕ : H ∞ (Dn ) → B(Dn ) is compact if and only if lim

ϕ(z)→∂Dn

Qψ (z) = 0 and

lim

ϕ(z)→∂Dn

|ψ(z)| θϕ (z) = 0.

(13)

Proof. By Theorem 4.2, it suﬃces to show that if Wψ,ϕ is compact from H ∞ (Dn ) into B(Dn ), then conditions (13) hold. First, observe that by Theorem 3.3 of [9], for z ∈ Dn ,

∂ψ ∂ψ (z), . . . , (1 − |zn |2 ) (z) Qψ (z) = (1 − |z1 |2 ) ∂z1 ∂zn n 2 ∂ψ ≤ (1 − |zk | ) (z) . ∂zk k=1

Since Wψ,ϕ is compact, Theorem 4.5(b) implies that lim

ϕ(z)→∂Dn

Qψ (z) = 0.

In addition, by Lemma 4.6, we have n ∂ϕj 1 − |zk |2 (z) |ψ(z)| θϕ (z) ≤ |ψ(z)| ∂zk 1 − |ϕ (z)|2 j k,j=1

for all z ∈ Dn . Thus, by part (c) of Theorem 4.5, we obtain lim

ϕ(z)→∂Dn

completing the proof.

|ψ(z)| θϕ (z) = 0,

5. Component Operators In this section, we look at the issues of boundedness and compactness of the multiplication and the composition operators separately.

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5.1. Multiplication Operators from H ∞ into the Bloch Space Let us now consider the implications of Theorem 3.1 for the case when ϕ is the identity and D is a bounded symmetric domain. Theorem 5.1. Let D be a bounded symmetric domain in standard form and let ψ ∈ H(D). Then (a) Mψ : H ∞ (D) → B(D) is bounded if and only if ψ ∈ H ∞ (D). (b) Mψ : H ∞ (D) → B0∗ (D) is bounded if and only if ψ ∈ B0∗ (D)∩H ∞ (D). Furthermore, if Mψ is bounded as an operator from H ∞ (D) into the Bloch space or the ∗-little Bloch space, then max{||ψ||B , cD ||ψ||∞ } ≤ ||Mψ || ≤ ||ψ||B + cD ||ψ||∞ , where cD is the Bloch constant of D. Proof. Suppose ﬁrst D is an irreducible domain. In [8] and [25], it was shown that there exists a polynomial p on D and ξ ∈ ∂D such that p(0) = 0, ||p||∞ = 1, |∇p(0)ξ| = 1, and Qp (0) =

1 = cD . H0 (ξ, ξ)1/2

(14)

If D = D1 × · · · × Dk , a product of irreducible domains, then by (1), there exists j = 1, . . . , k such that cD = cDj , and so there exists a polynomial pj on Dj and ξj ∈ ∂Dj such that ||pj ||∞ = 1, |∇(pj )(0)ξj | = 1, and Qpj (0) = cD . Then letting p be the polynomial on D such that p(z) = pj (zj ) (where zj denotes the component of z in Dj ), and ξ the vector whose component in Dj equals ξj and whose components in each irreducible factor other than Dj are 0, we obtain a function on D with supremum norm 1 and a vector in ∂D such that |∇(p)(0)ξ| = |∇(pj )(0)ξj | = 1, p(0) = 0, and Qp (0) = Qpj (0) = cD . Fix a ∈ D and let S ∈ Aut(D) be such that S(a) = 0. Since the Jacobian matrix JS(a) is invertible, there exists a nonzero v ∈ Cn such that JS(a)v = ξ. Composing p with S, we obtain a function g ∈ B0∗ (D) such that ||g||∞ = 1 and Qg (a) = cD . In particular, θψ,id = sup |ψ(a)| sup{Qf (a) : ||f ||∞ ≤ 1} = cD ||ψ||∞ . a∈D

Moreover, by the boundedness of Mψ , the invariance of the Bergman metric under the action of Aut(D), recalling that |∇p(0)ξ| = 1 and using (14), we obtain |∇(ψg)(a)u| ||Mψ g||B ≥ Qψg (a) = sup 1/2 u∈Cn \{0} Ha (u, u) =

sup

≥ =

|ψ(a)∇(p)(0)JS(a)u|

H0 (JS(a)u, JS(a)u)1/2 |∇(p)(0)ξ| |ψ(a)| = |ψ(a)|Qp (0) H0 (ξ, ξ)1/2 |ψ(a)|cD .

u∈Cn \{0}

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Taking the supremum over all a ∈ D, we deduce ||Mψ || ≥ cD ||ψ||∞ . The result now follows at once from Theorem 3.1. Theorem 5.2 ([8],[25]). If D is an irreducible bounded symmetric domain in Cn , then ⎧ 2 ⎪ if D is of type RI , ⎪ m+n ⎪ ⎪ ⎪ ⎪ 2 ⎪ if D is of type RII , ⎪ n+1 ⎪ ⎪ ⎨√ 1 if D is of type RIII , cD = n−1 ⎪ 2 ⎪ if D is of type RIV , ⎪ n ⎪ ⎪ ⎪ 1 ⎪ √ ⎪ if D = RV , ⎪ ⎪ ⎩1 6 if D = RV I . 3 Recalling that the Bloch seminorm of a bounded analytic function is no greater than its supremum norm and observing from Theorem 5.2 that cD ≤ 1 for any bounded symmetric domain D in standard form, and the only domains D for which cD = 1 are those which contain the unit disk as a factor, we obtain the following result. Corollary 5.3. Let D be a bounded symmetric domain in standard form with D as a factor and let ψ ∈ H(D). If Mψ is bounded from H ∞ (D) into either B(D) or B(D)0∗ , then max{||ψ||B , ||ψ||∞ } ≤ ||Mψ || ≤ ||ψ||B + ||ψ||∞ . Furthermore, if ψ(0) = 0, then ||ψ||∞ ≤ ||Mψ || ≤ ||ψ||B + ||ψ||∞ . We now characterize the compact multiplication operators from H ∞ to B when the underlying space is the ball or the polydisk. Theorem 5.4. For D = Bn or Dn , the following statements are equivalent: (a) Mψ : H ∞ (D) → B(D) is compact. (b) ψ is identically zero. Proof. (a) =⇒ (b): By Theorems 4.4 and 4.7, the compactness of Mψ implies that lim |ψ(z)| sup{Qf (z) : ||f ||∞ ≤ 1} = 0. z→∂D

Since sup{Qf (z) : ||f ||∞ ≤ 1} = cD , it follows that lim |ψ(z)| = 0, hence ψ z→∂D

is identically 0. (b) =⇒ (a) is obvious.

Using Corollary 4.3, we deduce that there are no nontrivial bounded multiplication operators from H ∞ (Bn ) to B0 (Bn ). Corollary 5.5. The following statements are equivalent: (a) Mψ : H ∞ (Bn ) → B0 (Bn ) is bounded. (b) Mψ : H ∞ (Bn ) → B0 (Bn ) is compact.

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(c) ψ is identically zero. We next look at the case when ψ is identically 1, that is, the weighted composition operator reduces to the composition operator from H ∞ (D) into B(D). 5.2. Composition Operators from H ∞ into the Bloch Space From (3) we deduce that for any holomorphic self-map of a bounded homogeneous domain D, the supremum θϕ of θϕ (z), over all z ∈ D, is ﬁnite. Indeed, θϕ ≤ cD Bϕ . Thus, Theorem 3.1 yields the following result. Corollary 5.6. Let D be a bounded homogeneous domain and let ϕ be a holomorphic self-map of D. Then (a) Cϕ : H ∞ (D) → B(D) is bounded. (b) Cϕ : H ∞ (D) → B0∗ (D) is bounded if and only if lim θϕ (z) = 0.

z→∂ ∗ D

Furthermore, if Cϕ is bounded as an operator into B(D) or B0∗ (D), then 1 ≤ ||Cϕ || ≤ 1 + θϕ . Remark 2. By Corollary 4.3, all bounded composition operators from H ∞ (D) to B0 (D) are compact and the corresponding symbol ϕ must satisfy the condition 2 (1 − |z| ) |ϕ (z)| = 0. (15) lim 2 |z|→1 1 − |ϕ(z)| Besides the symbols whose range is relatively compact in D, examples of b , symbols satisfying (15) include the functions of the form ϕ(z) = 1−λz 2 where |λ| = 1 and 0 < b < 1.

6. Isometries 6.1. Isometric multiplication operators In [1] and [3], we proved that the only isometric multiplication operators from the Bloch space of the unit disk or of a bounded symmetric domain that does not have the disk as a factor to itself are those induced by constant functions of modulus one. We now show that there are no isometric multiplication operators from H ∞ (D) into the Bloch space B(D) (and hence into the little Bloch space as well). Lemma 6.1. If Mψ : H ∞ (D) → B(D) is an isometry, then ψ(0) = 0. Proof. Arguing by contradiction, assume ψ(0) = 0. Since Mψ is an isometry, ||ψ||B = βψ = ||Mψ 1||B = 1. For a ∈ D deﬁne the automorphism of D a−z , z ∈ D. La (z) = 1 − az

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Then, La ∈ H ∞ (D) with ||La ||∞ = 1. Again, since Mψ is an isometry, we obtain ||ψLa ||B = ||La ||∞ = 1. Noting that |ψ(a)| = (1 − |a| ) |(ψLa ) (a)| ≤ βψLa = ||ψLa ||B = 1, 2

taking the supremum over all a ∈ D, it follows that ||ψ||∞ ≤ 1. Since βψ ≤ ||ψ||∞ , we have 1 = βψ ≤ ||ψ||∞ ≤ 1. Thus 1 = ||ψ||∞ = ||Mψ (ψ)||B = βψ2 .

(16)

On the other hand, by the Schwarz-Pick lemma, we get βψ2 = 2 sup(1 − |z|2 ) |ψ(z)| |ψ (z)| ≤ 2 sup |ψ(z)| (1 − |ψ(z)|2 ) z∈D

z∈D

4 ≤ 2 max (x − x ) = √ < 1, x∈[0,1] 3 3 3

which contradicts (16).

Theorem 6.2. There are no isometric multiplication operators Mψ from H ∞ (D) to B(D). Proof. Assume Mψ is an isometry from H ∞ (D) into B(D). By Lemma 6.1, the symbol ψ cannot ﬁx the origin. Since the identity function has supremum norm 1, the function f deﬁned by f (z) = zψ(z) has Bloch semi-norm 1. Thus, by Theorem 2.1 of [9], either f is a rotation or the zeros of f form an inﬁnite sequence {zk } satisfying the condition lim sup(1 − |zk |2 )|f (zk )| = 1.

(17)

k→∞

If f is a rotation, then ψ is a constant of modulus 1. Observe that constants cannot induce isometric multiplication operators since there exist functions in H ∞ (D) with supremum norm 1 which ﬁx the origin and have Bloch seminorm strictly less than 1 (e.g. the function z → z 2 ). Thus, ψ cannot be a constant of modulus 1. If f is not a rotation, then ψ has the same non-zero zeros of f and |zk | → 1 as k → ∞. Since f (zk ) = zk ψ (zk ) + ψ(zk ) = zk ψ (zk ), by (17) we obtain 1 = lim sup(1 − |zk |2 )|zk ψ (zk )| = lim sup(1 − |zk |2 )|ψ (zk )|. k→∞

k→∞

Hence βψ = 1. On the other hand, since ψ does not ﬁx the origin, 1 = ||ψ||B = |ψ(0)| + βψ > 1, which yields a contradiction. Therefore, no isometric multiplication operators from H ∞ (D) into B(D) can exist. 6.2. Isometric composition operators In [10] and [2], it was shown that there is a large class of isometric composition operators on the Bloch space of the disk and more generally on bounded homogeneous domains that have the disk as a factor. We shall now prove that there are no isometries among the composition operators between the Hardy space H ∞ and the Bloch space of the disk.

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Lemma 6.3. If ϕ is an analytic self-map of D inducing an isometric composition operator, then ϕ(0) = 0. Proof. Assume Cϕ is an isometry from H ∞ (D) into B(D). Then letting f be the identity, we have |ϕ(0)| + βϕ = ||ϕ||B = ||Cϕ f ||B = ||f ||∞ = 1.

(18)

Furthermore, letting, for z ∈ D, g+ (z) =

1+z 1−z , and g− (z) = , 2 2

we see that ||g+ ||∞ = ||g− ||∞ = 1. Thus ||Cϕ g+ ||B = ||Cϕ g− ||B = 1, and hence |1 + ϕ(0)| + βϕ = 2 = |1 − ϕ(0)| + βϕ . This, combined with (18), yields |1 + ϕ(0)| = 1 + |ϕ(0)| = |1 − ϕ(0)|, which implies ϕ(0) = 0. Theorem 6.4. There are no isometric composition operators from H ∞ (D) to B(D). Proof. Suppose Cϕ is an isometry from H ∞ (D) to B(D). Consider the function f (z) = z 2 for z ∈ D. Since Cϕ is an isometry, we have ||Cϕ f ||B = ||f ||∞ = 1. On the other hand, arguing as in the proof of Lemma 6.1, by Lemma 6.3 and the Schwarz-Pick Lemma, we obtain ||Cϕ f ||B

=

sup 2(1 − |z| ) |ϕ(z)| |ϕ (z)| 2

z∈D

≤

4 2 sup 2 |ϕ(z)| (1 − |ϕ(z)| ) ≤ √ < 1, 3 3 z∈D

reaching a contradiction.

6.3. Isometric weighted composition operators from B to H ∞ in the polydisk Let D be a bounded homogeneous domain. In [4], we characterized the bounded weighted composition operators from B(D) to H ∞ (D). In particular, for D = Dn , we obtained a characterization of boundedness equivalent to the following result proved by Li and Stevi´c in [15]. Theorem 6.5 ([15]). Wψ,ϕ : B(Dn ) → H ∞ (Dn ) is bounded if and only if ψ ∈ H ∞ (Dn ) and sup |ψ(z)|

z∈D

n k=1

log

2 < ∞. 1 − |ϕj (z)|

(19)

As a consequence we obtain the following result. Theorem 6.6. There are no isometric weighted composition operators from B(Dn ) to H ∞ (Dn ).

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Proof. Assume Wψ,ϕ : B(Dn ) → H ∞ (Dn ) is an isometry. Then ||ψ||∞ = ||Wψ,ϕ 1||∞ = 1 and, ﬁxing j = 1, . . . , n, ||ψϕj ||∞ = ||Wψ,ϕ pj ||∞ = ||pj ||B = 1, where pj is the z → zj . Thus, there exists a sequence {z (m) } in projection n (m) (m) D such that ψ(z )ϕj (z ) → 1 as m → ∞. Since both ψ and ϕj map Dn into D, it follows that ψ(z (m) ) → 1 and ϕj (z (m) ) → 1 as m → ∞. On the other hand, by the boundedness of Wψ,ϕ , (19) implies that ψ(z (m) ) → 0, a contradiction. 6.4. Final remarks By Corollary 4.3, the bounded weighted composition operators from H ∞ (Bn ) to B0 (Bn ) are necessarily compact, so there exist no isometric weighted composition operators from H ∞ (Bn ) to B0 (Bn ). We have not been able to prove or disprove the existence of isometric weighted composition operators from H ∞ to the Bloch space, even in the case of the unit disk. We conjecture that there are no isometric weighted composition operators from H ∞ (D) to B(D) for any bounded homogeneous domain D. Acknowledgments We wish to thank the referees for their thoughtful comments and helpful suggestions, and for bringing to our attention important references.

References [1] R. F. Allen and F. Colonna, Isometries and spectra of multiplication operators on the Bloch space, Bull. Austral. Math. Soc. 79 (2009), 147–160. [2] R. F. Allen and F. Colonna, On the isometric composition operators on the Bloch space in Cn , J. Math. Anal. Appl. 355 (2009), 675–688. [3] R. F. Allen and F. Colonna, Multiplication operators on the Bloch space of bounded homogeneous domains, Comput. Methods Function Theory 9(2) (2009), 679–693. [4] R. F. Allen and F. Colonna, Weighted composition operators on the Bloch space of a bounded homogeneous domain, Proc. IWOTA (to appear). [5] E. Cartan, Sur les domains bourn´ es de l’espace de n variable complexes (French), Abh. Math. Sem. Univ. Hamburg 11 (1935), 116–162. [6] R. Chen, S. Stevi´c and Z. Zhou, Weighted composition operators between Bloch type spaces in the polydisk, Mat. Sb. (to appear). [7] D. Clahane, S. Stevi´c and Z. Zhou, On composition operators between Bloch type spaces in polydisc, arXiv:math\0507339v1.

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[8] J. M. Cohen and F. Colonna, Bounded holomorphic functions on bounded symmetric domains, Trans. Amer. Math. Soc. 343 (1994), 135–156. [9] J. M. Cohen and F. Colonna, Isometric composition operators on the Bloch space in the polydisk, Contemp. Math., 454 (2008), 9–21. [10] F. Colonna, Characterisation of the isometric composition operators on the Bloch space, Bull. Austral. Math. Soc. 72 (2005), 283–290. [11] D. Drucker, Exception Lie algebras and the structure of Hermitian symmetric spaces, Mem. Amer. Math. Soc. 208 (1978), 1–207. [12] K. T. Hahn, Holomorphic mappings of the hyperbolic space in the complex Euclidean space and the Bloch theorem, Canad. J. Math 27 (1975), 446–458. [13] S. Helgason, Diﬀerential Geometry and Symmetric Spaces, Academic Press, New York (1962). [14] T. Hosokawa, K. Izuchi and S. Ohno, Topological structure of the space of weighted composition operators on H ∞ , Integr. Equ. Oper. Theory 53 (2005), 509–526. [15] S. Li and S. Stevi´c, Weighted composition operators from H ∞ to the Bloch space on the polydisc, Abstr. Appl. Anal. Vol. 2007, Article ID 48478, (2007), 13 pp. [16] S. Li and S. Stevi´c, Weighted composition operators from α-Bloch spaces to H ∞ on the polydisk, Numer. Funct. Anal. Optimization 28(7) (2007), 911–925. [17] S. Li and S. Stevi´c, Weighted composition operators between H ∞ and α-Bloch spaces in the unit ball, Taiwanese J. Math. 12 (2008), 1625–1639. [18] S. Ohno, Weighted composition operators between H ∞ and the Bloch space, Taiwanese J. Math. 5(3) (2001), 555563. [19] S. Ohno and R. Zhao, Weighted composition operators on the Bloch space, Bull. Austral. Math. Soc 63 (2001), 177–185. ˇ [20] I. I. Pjatecki˘ı-Sapiro, On a problem proposed by E. Cartan (Russian), Dokl. Akad. Nauk SSSR 124 (1959), 272–273. [21] S. Stevi´c, Composition operators between H ∞ and the α-Bloch spaces on the polydisc, Z. Anal. Anwendungen, 25(4) (2006), 457–466. [22] S. Stevi´c, Norm of weighted composition operators from Bloch space to H ∞ on the unit ball, Ars. Combin. 88 (2008), 125–127. [23] R. M. Timoney, Bloch functions in several complex variables I, Bull. London Math. Soc. 12 (1980), 241–267. [24] R. M. Timoney, Bloch functions in several complex variables II, J. Reine Angew. Math. 319 (1980), 1–22. [25] G. Zhang, Bloch constants of bounded symmetric domains, Trans. Amer. Math. Soc. 349 (1997), 2941–2949. [26] M. Zhang and H. Chen, Weighted composition operators of H ∞ into α-Bloch spaces on the unit ball, Acta Math. Sinica (English) 25 (2009), 265–278. [27] Z. Zhou and R. Chen, Weighted composition operators from F (p, q, s) to Bloch type spaces on the unit ball (preprint) (http://arxiv.org/abs/math/0503614v9).

40

R. F. Allen and F. Colonna

Robert F. Allen Department of Mathematics University of Wisconsin–La Crosse La Crosse, WI 54601 USA e-mail: [email protected] Flavia Colonna Department of Mathematical Sciences George Mason University Fairfax, VA 22030 USA e-mail: [email protected] Submitted: March 24, 2009. Revised: September 1, 2009.

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Integr. Equ. Oper. Theory 66 (2010), 41–52 DOI 10.1007/s00020-009-1737-3 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

The Heat Equation for the Generalized Hermite and the Generalized Landau Operators Viorel Catan˘a

Abstract. We give a formula for the one-parameter strongly continuous λ ˜ semigroups e−tL and e−tA , t > 0 generated by the generalized Hermite λ operator L , λ ∈ R\{0} respectively by the generalized Landau operator ˜ These formula are derived by means of pseudo-diﬀerential operators A. of the Weyl type, i.e. Weyl transforms, Fourier-Wigner transforms and Wigner transforms of some orthonormal basis for L2 (R2n ) which consist of the eigenfunctions of the generalized Hermite operator and of the generalized Landau operator. Applications to an L2 estimate for the solutions of initial value problems for the heat equations governed by ˜ in terms of Lp norm, 1 ≤ p ≤ ∞ of the initial data Lλ respectively A, are given. Mathematics Subject Classiﬁcation (2010). Primary 47G10, 47G30; Secondary 35S10. Keywords. Weyl transform, Fourier-Wigner transform, Wigner transform, one-parameter strongly continuous semigroup.

Following Wong’s point of view (see [6], by Wong), we give a formula for λ the one-parameter strongly continuous semigroup e−tL , t > 0, generated by the generalized Hermite operator Lλ , for a ﬁxed λ ∈ R \ {0}, in terms of the Weyl transforms. Then we use it to obtain an L2 estimate for the solution of the initial value problem for the heat equation governed by Lλ , in terms of the Lp norm, 1 ≤ p ≤ ∞, of the initial data. Similar results have also been derived for the generalized Landau operator A˜ which was ﬁrstly introduced by M.A. de Gosson (see [3] by de Gosson) who has studied its spectral properties.

This work was completed with the support of our TEX-pert.

42

V. Catana˘

For a ﬁxed λ ∈ R \ {0} let

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∂ ∂ , be the linear partial diﬀerential ∂z λ ∂ z¯λ

operators on R2 , given by ∂ ∂ ∂ − i|λ|1/2 , = |λ|−1/2 λ ∂z ∂x ∂y

and

∂ ∂ ∂ + i|λ|1/2 . = |λ|−1/2 λ ∂ z¯ ∂x ∂y

Then we deﬁne the linear partial diﬀerential operator Lλ on R2 by 1 Lλ = − (Z λ Z¯ λ + Z¯ λ Z λ ), (1) 2 where ∂ 1 + z¯λ , z¯λ = |λ|1/2 x − i|λ|−1/2 y Zλ = ∂z λ 2 and ∂ 1 Z¯ λ = λ − z λ , z λ = |λ|1/2 x + i|λ|−1/2 y. ∂ z¯ 2 In fact −Z¯ λ is the formal adjoint of Z λ and Lλ is an elliptic partial diﬀerential operator on R2 , given by ∂2 ∂ ∂2 1 ∂ Lλ = −|λ|−1 2 − |λ| 2 + (|λ|x2 + |λ|−1 y 2 ) − i x|λ| − y|λ|−1 . ∂x ∂y 4 ∂y ∂x (2) We call the partial diﬀerential operator deﬁned by (1) or (2) the generalized Hermite operator. If λ ∈ R, |λ| = 1 in (2), then the operator Lλ becomes 1 the ordinary Hermite operator −∆ + (x2 + y 2 ) perturbed by the partial 4 ∂ ∂ −y . diﬀerential operator −iN , where N = x ∂y ∂x First we need some basic deﬁnitions (see the book [5], by Wong) and preliminary results. Let f, g be functions in the Schwartz space S(R) on R. Then we deﬁne the Fourier-Wigner transform V (f, g) of f and g by +∞ p p dy (3) V (f, g)(q, p) = (2π)−1/2 eiqy f y + g y− 2 2 −∞ for all q, p ∈ R. The Wigner transform W (f, g) of f and g is deﬁned by W (f, g) = V (f, g)∧ , where ∧ is the Fourier transform. Now, let σ ∈ Lp (R2 ), 1 ≤ p ≤ ∞. Then we deﬁne Wσ f to be the temperate distribution on R by +∞ +∞ −1/2 σ(x, ξ)W (f, g)(x, ξ)dxdξ, (4) (Wσ f, g) = (2π) −∞

−∞

for all g ∈ S(R). We call Wσ the Weyl transform associated to the symbol σ. For k ∈ N, let ek be the Hermite function on R of order k deﬁned by √ −x2 ek (x) = (2k · k! π)−1/2 e 2 Hk (x), where Hk is the Hermite polynomial of degree k. Now, for k ∈ N and a ﬁxed λ ∈ R \ {0}, we introduce the generalized Hermite function on R, eλk , deﬁned by eλk (x) = |λ|−1/4 ek (|λ|−1/2 x), for all

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The Heat Equation for the Generalized Hermite

43

x ∈ R (see [1]). For j, k ∈ R and a ﬁxed λ ∈ R \ {0} we deﬁne the generalized Hermite function eλj,k on R2 by eλj,k (x, y) = V (eλj , eλk )(x, y), for all x, y ∈ R. Theorem 1. {eλj ; j = 0, 1, 2, . . .} is an orthonormal basis for L2 (R). Proof. Firstly let us observe that (eλj , eλk ) = δjk , for all j, k = 0, 1, 2, . . . . Thus (eλj )j∈N is an orthonormal system. To prove that (eλj )j∈N is an orthonormal basis for L2 (R) we need to verify that for f ∈ L2 (R) such that (f, eλj ) = 0, for all j ∈ N, it follows f = 0. To this end we use the fact that (ej )j∈N is an orthonormal basis for L2 (R) and that (f, ejλ ) = (g, ej ), where f ∈ L2 (R) and g = |λ|1/4 f (|λ|1/2 .) ∈ L2 (R), j = 0, 1, 2, . . .. So, if (f, eλj ) = 0 for all j ∈ N, it follows that (g, ej ) = 0, for all j ∈ N. This implies that g = 0 and a fortiori f = 0. Thus the proof is complete. Theorem 2. {eλj,k ; j, k = 0, 1, 2, . . .} is an orthonormal basis for L2 (R2 ). The proof of Theorem 2 is based on Theorem 1 and is similar with the proof of Theorem 21.2 in the book [5] by Wong, so we omit this. The spectral analysis of the generalized Hermite operator Lλ on R2 is based on the following theorem. Theorem 3. Lλ eλj,k = (2k + 1)eλj,k , for all j and k in N . Proof. Following the same reasoning as in Theorem 22.1 in the book [5] by Wong we can prove that for all x and y ∈ R, (Z λ eλj,k )(x, y) = i(2k)1/2 eλj,k−1 (x, y), j = 0, 1 . . . , k = 1, 2, . . .

(5)

and (Z¯ λ eλj,k )(x, y) = i(2k + 2)1/2 eλj,k+1 (x, y), j = 0, 1, 2, . . . , k = 0, 1, 2, . . . . (6) Hence by (1), (5) and (6) the conclusion in Theorem 3 follows.

Remark. Theorem 3 says that for k = 0, 1, 2, . . . the number 2k+1 is an eigenvalue of the generalized Hermite operator Lλ , and the generalized Hermite functions eλj,k , j = 0, 1, 2, . . . on R2 are eigenfunctions of Lλ corresponding to the eigenvalue 2k + 1. λ

A consequence of Theorem 3 is a formula for Hermite semigroup e−tL , t > 0 which is given in the following theorem. Theorem 4. For f ∈ S(R2 ) and for all t > 0, λ

e−tL f = (2π)1/2

∞

e−(2k+1)t V (Wfˆeλk , eλk ),

k=0

where the convergence is uniform and absolute on R2 . Proof. Let f ∈ S(R2 ). Then for t > 0, we use the Theorem 3 to get λ

e−tL f =

∞ ∞ k=0 j=0

e−(2k+1)t (f, eλj,k )eλj,k ,

(7)

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where the series is convergent in L2 (R2 ) and is also uniform and absolute convergent on R2 . Now by the deﬁnitions of Fourier-Wigner transform and Wigner transform respectively and Plancherel’s theorem, f (z)V (eλj , eλk )(z)dz = fˆ(ζ)V (eλj , eλk )∧ (ζ)dζ (f, eλj,k ) = R2

R2

1/2

(Wfˆeλk , eλj ),

= (2π)

j, k = 0, 1, 2, . . .

(8)

2

Similarly, for j, k = 0, 1, 2, . . . and g ∈ S(R ), we get (eλj,k , g) = (g, eλj,k ) = (2π)1/2 (Wgˆ eλk , eλj ) = (2π)1/2 (eλj , Wgˆ eλk ).

(9)

So by the relations (7)-(9), Fubini’s theorem and Parseval’s identity, ∞ ∞ λ e−(2k+1)t (f, eλj,k )(eλj,k , g) (e−tL f, g) = k=0 j=0 ∞

e

= (2π)

= (2π)

k=0 ∞

−(2k+1)t

∞

(Wfˆeλk , eλj )(eλj , Wgˆ eλk )

j=0

e−(2k+1)t (Wfˆeλk , Wgˆ eλk ),

for

t > 0,

(10)

k=0

where the series is absolutely convergent on R. But, by the deﬁnition and proprieties of the Wigner transform and Plancherel’s theorem, (Wfˆeλk , Wgˆ eλk )

=

(Wgˆ eλk , Wfˆeλk )

= (2π)−1/2

R2

−1/2

= (2π)

R2

gˆ(z)W (eλk , Wfˆeλk )(z)dz

W (Wfˆeλk , eλk )(z)ˆ g(x)dz,

for k = 0, 1, 2, . . . . (11)

Thus, by the relations (10)-(11) and Fubini’s theorem, ∞ λ e−(2k+1)t V (Wfˆeλk , eλk )∧ (ζ)ˆ g (ζ)dζ (e−tL f, g) = (2π)1/2 = (2π)1/2 1/2

= (2π)

k=0 ∞

R2

e−(2k+1)t (V (Wfˆeλk , eλk ), g)

k=0 ∞

e

−(2k+1)t

V

(Wfˆeλk , eλk ), g

,

(12)

k=0

for all f, g ∈ S(R2 ) and t > 0. Thus by the relation (12), ∞ −tLλ 1/2 e f = (2π) e−(2k+1)t V (Wfˆeλk , eλk ),

(13)

k=0

for all f ∈ S(R2 ) and t > 0, where the uniform and absolute convergence of the series follows from the properties of the Fourier-Wigner transform and the following theorem (see the Theorem 14.3 in the book [5] by Wong).

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Theorem 5. Let σ ∈ Lp (R2 ), 1 ≤ p ≤ 2. Then Wσˆ is a bounded linear operator from L2 (R2 ) into L2 (R) and ||Wσˆ ||∗ ≤ (2π)−1/p ||σ||Lp (R2 ) ,

(14)

where ||Wσˆ ||∗ is the operator norm of Wσˆ : L2 (R) → L2 (R). As an application of the formula for the Hermite semigroup given in Theorem 4 we can now prove the following theorem. λ

Theorem 6. For t > 0, the Hermite semigroup e−tL , initially deﬁned on S(R2 ), can be extended to a unique bounded linear operator from Lp (R2 ) λ into L2 (R2 ) which we again denote by e−tL and λ

||e−tL f ||L2 (R2 ) ≤ (2π)1/2−1/p

1 ||f ||Lp (R2 ) , for all f ∈ Lp (R2 ), 1 ≤ p ≤ 2. 2sht

We need the following result, which is known as the Moyal identity and can be found in the book [5] by Wong. Theorem 7. For all f, g ∈ S(R), ||V (f, g)||L2 (R2 ) = ||f ||L2 (R) ||g||L2 (R) .

(15)

Proof of Theorem 6. Let f ∈ S(R2 ). Then by Theorem 4 and Theorem 7, and Minkowski’s inequality, λ

||e−tL f ||L2 (R2 ) ≤ (2π)1/2 = (2π)1/2 = (2π)1/2

∞ k=0 ∞ k=0 ∞

e−(2k+1)t ||V (Wfˆeλk , eλk )||L2 (R2 ) e−(2k+1)t ||Wfˆeλk ||L2 (R) ||eλk ||L2 (R) e−(2k+1)t ||Wfˆeλk ||L2 (R) , for t > 0.

(16)

k=0

So by the relation (16) and Theorem 5, we get for t > 0, λ

||e−tL f ||L2 (R2 ) ≤ (2π)1/2 ≤ (2π)1/2

∞ k=0 ∞

e−(2k+1)t ||Wfˆ||∗ ||eλk ||L2 (R) e−(2k+1)t (2π)−1/p ||f ||Lp (R2 )

k=0

= (2π)1/2−1/p

1 ||f ||Lp (R2 ) , 2sh t

(17)

for all f ∈ S(R2 ). Now, by the relation (17) and a density argument, the proof is complete.

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Remark. Theorem 6 gives an L2 estimate for the solution of the initial value problem ∂u (z, t) = (−Lλ u)(z, t), z ∈ R2 , t > 0 ∂t u(z, 0) = f (z), z ∈ R2 , in terms of the Lp norm of the initial data f , 1 ≤ p ≤ 2. 2

Theorem 8. Let Λ ∈ S(R2 ) be a function on R2 deﬁned by Λ(z) = π −1 e−|z| , z ∈ R2 , let g ∈ Lp (R2 ), 1 ≤ p ≤ ∞, and let uλ be the solution of the initial value problem considered in Remark 2, with initial data (g ∗ Λ)V , where V is the inverse Fourier transform. Then ||uλ ||L2 (R2 ) ≤ (2π)1/2−1/p

1 ||g||Lp (R2 ) . 2sht

For the proof of Theorem 8, we need the following results (see Chapter 17 of the book [7] by Wong and Theorem 15.4 and Theorem 17.1 in the book [5] by Wong). Theorem 9. Let Λ : C → R, be a function deﬁned by 2

Λ(z) = π −1 e−|z| , z ∈ C. Then for all F ∈ Lp (C), 1 ≤ p ≤ ∞, WF ∗Λ = LF , where LF is the localization operator on the Weyl-Heisenberg group with symbol F . Theorem 10. Let F ∈ Lp (C), 1 ≤ p ≤ ∞. Then ||LF ||∗ ≤ (2π)−1/p ||F ||Lp (C) ,

(18)

where LF is the localization operator on the Weyl-Heisenberg group with symbol F . The proof of Theorem 8 is the same as that of Theorem 6, if we note that, by Theorem 9, WFˆ = Wg∗Λ = Lg , and hence the estimate follows from Theorem 10. Remark. For λ ∈ R, |λ| = 1, we obtain from our statements the results obtained in [6], by Wong. Remark. The above results are valid for the Hermite operator Lλ on R2n . Now we want to give similar results as the preceding ones in the case of a class of generalized Landau operators. This class of operators has been ﬁrstly introduced and studied by de Gosson in [3]. To this end let us ﬁrst recall the following deﬁnitions and results from Shubin [4] (Chapter 4). Let m0 , m1 and ρ be real numbers such that m0 ≤ m1 and 0 < ρ ≤ 1.

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1 ,m0 We shall write a ∈ HΓm (R2n ), if a ∈ C ∞ (R2n ) and there exist ρ R, C0 , C1 ≥ 0, and for every α ∈ Nn , there exists Cα ≥ 0 such that the following estimates hold for |z| ≥ R

C0 |z|m0 ≤ |a(z)| ≤ C1 |z|m1 |∂zα a(z)| ≤ Cα |a(z)||z|−ρ|α| . 1 ,m0 (R2n ) we denote the class of operators A, given by the By HGm ρ 1 ,m0 (R2n ); this means that for every τ ∈ R, there exists τ -symbols aτ ∈ HΓm ρ 1 ,m0 aτ ∈ HΓm (R2n ) such that ρ ei(x−y)·ξ aτ ((1 − τ )x + τ y, ξ)u(y)dydξ, u ∈ S(Rn ). Au(x) = (2π)−n

R2n

1 , in particular, we observe that every Weyl operator 2 m1 ,m0 1 ,m0 with symbol a ∈ HΓρ (R2n ) is in HGm (R2n ). Indeed, by Proposition ρ 1 ,m0 (R2n ) is true for some τ ∈ R, 25.1 in Shubin [4] (Chapter 4), if aτ ∈ HΓm ρ 1 ,m0 (R2n ) is also then it is true for all τ ∈ R. So, the condition a ∈ HΓm ρ m1 ,m0 2n suﬃcient to write A ∈ HGρ (R ). The following result tell us about discretness of the spectrum of the operators of Shubin’s classes. If we choose τ =

1 ,m0 (R2n ), where Theorem 11. (see Theorem 26.3 in [4]). Let A ∈ HGm ρ m0 > 0. Suppose that A is formally self-adjoint, that is (Au, v) = (u, Av) for all u, v ∈ C0∞ (Rn ). Then A has discret spectrum in L2 (Rn ). More precisely, there exists an orthonormal basis of eigenfunctions ϕj ∈ S(Rn ), j = 0, 1, 2, . . . , with eigenvalues λj ∈ R such that |λj | → ∞ as j → ∞.

In order to deﬁne the Landau operators A˜ which have been introduced by de Gosson in [3] and to state the main result about these, we need some preliminary. Let ϕ ∈ S(Rn ), ||ϕ|| = 1 and let the mapping Wϕ : S(Rn ) → S(R2n ) deﬁned by π n/2 1 ˜ W (u, ϕ) z , (19) Wϕ u(z) = 2 2 ˜ (u, ϕ) = (2π)−n/2 W (u, ϕ) and W (u, ϕ) is the Wigner transform of where W the pair (u, ϕ). Explicitely: −n/2 2i ξx e e−iξy u(y)ϕ(x ¯ − y)dy. (20) Wϕ u(z) = (2π) Rn

We can extend it by continuity into a linear mapping Wϕ : L2 (Rn ) → L2 (R2n ). The integral operator Wϕ deﬁned by (20) has been introduced by de Gosson in [2] and it has been called the ”Wigner wave-packet transform” associated with ϕ.

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Let A = aw (x, D) be the Weyl operator corresponding to the symbol a ∈ C ∞ (R2n ) (or generally a ∈ S (R2n )) 1 (x + y), ξ u(y)dydξ. (21) Au(x) = (2π)−n ei(x−y)ξ a 2 R2n Then we deﬁne a ˜ ∈ C ∞ (R4n ) by a ˜(z, ζ) = a

1 z − Jζ , 2

(22)

where Jζ = (ζ2 , −ζ1 ) if ζ = (ζ1 , ζ2 ) and we consider the operator A˜ = a ˜w (z, −i∂z ). 2

(23)

2

Let us observe that if we take a(x, ξ) = x + ξ in (23) then the operator A˜ coincide with the ”Landau Hamiltonian” ˜ = −(∂x2 + ∂y2 ) + i(x∂y − y∂x ) + 1 (x2 + y 2 ) H 4 ˜ (see [3]). So, the operator A generalize the ”Landau Hamiltonian”. Then we will call the operator A˜ deﬁned by (23) the generalized Landau operator in the same manner as it has been called by de Gosson in [3]. Now we can state the main result in the article [3], by de Gosson, concerning the generalized Landau operators. 1 ,m0 (R2n ). Then the Theorem 12. (see Theorem 6 in [3]). Let A ∈ HGm ρ following statements are true: (i) The operators A and A˜ have the same eigenvalues. If u is an eigenfunction of A corresponding to the eigenvalue λ, then Uϕ = Wϕ u is an eigenfunction of A˜ corresponding to the eigenvalue λ, for every ϕ, and we have Uϕ ∈ S(R2n ). (ii) Suppose that m0 > 0 and that A is formally self-adjoint. Then A˜ has discrete spectrum (λj )j≥0 and |λj | → ∞ as j → ∞. (iii) In the context of (ii) the eigenfunctions of A˜ are given by Φjk = Wϕj ϕk , where ϕj , j = 0, 1, 2, . . . are the eigenfunctions of A. (iv) Φjk ∈ S(R2n ) and Φjk , j, k = 0, 1, 2, . . . form an orthonormal basis of L2 (R2n ).

As a consequence of Theorem 12 and by aditional hypothesis we can give ˜ a formula for the Landau semigroup e−tA , t > 0 in the following theorem. Theorem 13. Let A˜ be a positive generalized Landau operator in the context of (ii) of Theorem 12. Suppose that its symbol a ˜ is a real function and its spectrum (λ ) is a faster increasing sequence (this means that the series j j≥0 −λj t e is convergent; for example λj > j, (∀)j ≥ j0 , for some j0 ∈ N). j≥0

Then, for f ∈ S(R2n ) and for all t > 0, ∞ ˜ e−tA f = (2π)n/2 e−λj t θ1/2 W (Wθ2 f ϕj , ϕj ), j=0

(24)

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49

where the convergence is uniform and absolute on R2n and (θa f )(z) = f (az), for all a ∈ R \ {0}, z ∈ R2n . Proof. Let f ∈ S(R2n ). Then for t > 0, we use Theorem 12 ((ii), (iii)) to get ∞ ∞ ˜ −tA e f= e−λj t (f, Φjk )Φjk , (25) j=0 k=0

where the series is convergent in L2 (R2n ) and is also uniform and absolute convergent on R2n . Now by the deﬁnitions of Φjk and Weyl transform, π n/2 1 ˜ z dz f (z)W (ϕk , ϕj ) (f, Φjk ) = 2 2 2n R ˜ (ϕj , ϕk )(w)dw = (8π)n/2 f (2w)W R2n = 2n (θ2 f )(w)W (ϕj , ϕk )(w)dw R2n n/2

= (8π)

(Wθ2 f ϕj , ϕk ), j, k = 0, 1, 2, . . .

(26)

2n

Similarly, for j, k = 0, 1, 2, . . . and g ∈ S(R ), we get (Φjk , g) = (g, Φjk ) = (8π)n/2 (Wθ2 g ϕj , ϕk ) = (8π)n/2 (ϕk , Wθ2 g ϕj ).

(27)

Thus by (25)-(27) and Fubini’s theorem and Parseval’s identity ∞ ∞ ˜ (e−tA f, g) = e−λj t (f, Φjk )(Φjk , g) j=0 k=0

= (8π)n

∞

e−λj t

j=0

= (8π)n

∞

∞

(Wθ2 f ϕj , ϕk )(ϕk , Wθ2 g ϕj )

k=0

e−λj t (Wθ2 f ϕj , Wθ2 g ϕj ), for t > 0,

(28)

j=0

where the series is absolutely convergent on R. By the deﬁnition of Weyl transform, and by the appropriate changes of variables, we get (Wθ2 f ϕj , Wθ2 g ϕj ) = (Wθ2 g ϕj , Wθ2 f ϕj ) −n/2 = (2π) (θ2 g)(z)W (ϕj , Wθ2 f ϕj )(z)dz 2n R W (Wθ2 f ϕj , ϕj )(z)(θ2 g)(z)dz = (2π)−n/2 R2n = (32π)−n/2 W (Wθ2 f ϕj , ϕj ) (1/2w) g(w)dw R2n

−n/2

= (32π) 2n

(θ1/2 W (Wθ2 f ϕj , ϕj ), g),

for all f, g ∈ S(R ) and t > 0.

(29)

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Thus by (28), (29) and Fubini’s theorem ˜

(e−tA f, g) = (2π)n/2

∞

e−λj t (θ1/2 W (Wθ2 f ϕj , ϕj ), g)

j=0

⎛ = (2π)n/2 ⎝

∞

⎞ e−λj t θ1/2 W (Wθ2 f ϕj , ϕj ), g ⎠ ,

(30)

j=0

for all f, g ∈ S(R2n ) and t > 0. Thus by (30), ˜

e−tA f = (2π)n/2

∞

e−λj t θ1/2 W (Wθ2 f ϕj , ϕj ),

(31)

j=0

for all f ∈ S(R2n ) and t > 0, where the uniform and absolute convergence of the series on R2n follows from the properties of Wigner transform and the following theorem. Theorem 14. (see the book [5] by Wong). Let σ ∈ Lp∗ (R2n ), 2 ≤ p ≤ ∞. Then Wσ : L2 (Rn ) → L2 (Rn ) is a bounded linear operator and

σ ||Lp (R2n ) , ||Wσ ||∗ ≤ (2π)−n/p ||ˆ

σ ∈ Lp∗ (R2n )

(32)

2 ≤ p ≤ ∞, where Lp∗ (R2n ) = {f ∈ Lp (R2n ); fˆ ∈ Lp (R2n )}. Now we can give an application of the formula for the Landau semigroup which was proved in Theorem 13. To this end let us recall the following result which is known as the Moyal identity for the Wigner transform (see the book [5] by Wong). Theorem 15. W : S(Rn ) × S(Rn ) → S(R2n ) can be extended uniquely to a bilinear operator W : L2 (Rn ) × L2 (Rn ) → L2 (R2n ) such that ||W (f, g)||L2 (R2n ) = ||f ||L2 (Rn ) ||g||L2 (Rn ) , 2

(33)

n

for all f and g in L (R ). ˜

Theorem 16. For t > 0, the Landau semigroup e−tA initially deﬁned on S(R2n ) can be extended to a unique bounded linear operator from Lp∗ (R2n ) ˜ into L2 (R2n ) which we again denote by e−tA and ˜

||e−tA f ||L2 (R2n ) ≤ 2(1/2−1/p)n π (1/2−1/p )n

∞

e−λj t ||f ||Lp∗ (R2n ) ,

2 ≤ p ≤ ∞,

j=0

(34) for all f ∈ Lp∗ (R2n ), where Lp∗ (R2n ) = {g ∈ Lp (R2n ); gˆ ∈ Lp (R2n )}, and g||Lp (R2n ) is a norm on Lp∗ (R2n ). ||g||Lp∗ (R2n ) = ||ˆ

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Proof. Let f ∈ S(R2n ). Then by Theorem 13, Theorem 15, Minkowski’s inequality and Theorem 14, ˜

||e−tA f ||L2 (R2n ) ≤ (2π)n/2

∞

e−λj t ||θ1/2 W (Wθ2 f ϕj , ϕj )||L2 (R2n )

j=0

= (8π)n/2

∞

e−λj t ||W (Wθ2 f ϕj , ϕj )||L2 (R2n )

j=0

= (8π)n/2

∞

e−λj t ||Wθ2 f ϕj ||L2 (Rn ) ||ϕj ||L2 (Rn )

j=0

≤ (8π)n/2

∞

e−λj t ||Wθ2 f ||∗ ||ϕj ||L2 (Rn )

j=0

≤ (8π)n/2 (2π)−n/p

∞ j=0

=2

e−λj t ||θ 2 f ||Lp (R2n )

(1/2−1/p)n (1/2−1/p )n

π

∞

e−λj t ||f ||Lp∗ (R2n )

(35)

j=0

Now, by (35) and a density argument, the proof is complete.

Remark. Theorem 14 gives an L2 estimate for the solution of the initial value problem ∂u ˜ (z, t) = (−Au)(z, t), z ∈ R2n , t > 0 ∂t u(z, 0) = f (z), in terms of the

Lp∗

z ∈ R2n ,

(36)

norm of the initial data f , 2 ≤ p ≤ ∞.

Theorem 17. Let Λ ∈ S(R2n ) be a function on R2n deﬁned by Λ(z) = 2 π −n e−|z| , z ∈ R2n , let g ∈ Lp (R2n ), 1 ≤ p ≤ ∞ and let u be the solution of the initial value problem (36), with initial data f = θ1/2 (g ∗ Λ). Then ||u||L2 (R2n ) ≤ 2(3/2−1/p)n π (1/2−1/p)n

∞

e−λj t ||g||Lp (R2n ) .

(37)

j=0

For the proof, we need the following results (see Theorem 15.4 and Theorem 17.1 in the book [5] by Wong and Chapter 17 of the book [7] by Wong). Theorem 18. Let Λ : Cn → R be a function deﬁned by 2

Λ(z) = π −n e−|z| ,

z ∈ Cn .

Then for all F ∈ Lp (Cn ), 1 ≤ p ≤ ∞, WF ∗Λ = LF , where LF : L2 (Rn ) → L2 (Rn ) is the localization operator on the Weyl-Heisenberg group with symbol F .

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Theorem 19. Let F ∈ Lp (Cn ), 1 ≤ p ≤ ∞. Then ||LF ||∗ ≤ (2π)−n/p ||F ||Lp (Cn ) , 2

n

2

(38)

n

where LF : L (R ) → L (R ) is the localization operator on the WeylHeisenberg group with symbol F and ||LF ||∗ is its operator norm. Proof of Theorem 17. Now, we use the same reasoning as that of Theorem 16, noting that by Theorem 18, Wθ2 f = Wg∗Λ = Lg , and hence the estimate (37) follows from (38). Thus the proof is complete. Acknowledgement The author is grateful to the referee for some useful observations on the presentation of this paper.

References [1] A. Hulanicki, The distribution of energy in the Brownian motion in the Gaussian ﬁeld and analytic-hypoellipticity of certain subelliptic operators on the Heisenberg group. Studia Math. 56 (1976), 165-173. [2] M.A. de Gosson, Symplectic Geometry and Quantum Mechanics. Basel, Birkh¨ auser, 2006. [3] M.A. de Gosson, Spectral properties of a class of generalized Landau operators. Comm. Part. Diﬀ. Equ., 33 (2008), 2096-2104. [4] M.A. Shubin, Pseudodiﬀerential Operators and Spectral Theory. Berlin, Springer Verlag, 1987. [5] M.W. Wong, Weyl Transforms. Springer-Verlag, 1998. [6] M.W. Wong, The heat equation for the Hermite operator on the Heisenberg group. Hokkaido Math. Journal, 34 (2005), 393-404. [7] M.W. Wong, Wavelet Transforms and Localization Operators. Birkh¨ auser, 2002. Viorel Catan˘ a University Politehnica of Bucharest Department of Mathematics I Splaiul Independent¸ei 313 060042 Bucharest Romania e-mail: catana [email protected] Submitted: April 9, 2009. Revised: July 10, 2009.

Integr. Equ. Oper. Theory 66 (2010), 53–77 DOI 10.1007/s00020-009-1734-6 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

Toeplitz Operators on Generalized Bergman Spaces Kamthorn Chailuek and Brian C. Hall Abstract. We consider the weighted Bergman spaces HL2 (Bd, µλ ), where we set dµλ (z) = cλ (1 − |z|2 )λ dτ (z), with τ being the hyperbolic volume measure. These spaces are nonzero if and only if λ > d. For 0 < λ ≤ d, spaces with the same formula for the reproducing kernel can be deﬁned using a Sobolev-type norm. We deﬁne Toeplitz operators on these generalized Bergman spaces and investigate their properties. Speciﬁcally, we describe classes of symbols for which the corresponding Toeplitz operators can be deﬁned as bounded operators or as a Hilbert–Schmidt operators on the generalized Bergman spaces. Mathematics Subject Classiﬁcation (2010). Primary 47B35; Secondary 32A36, 81S10. Keywords. Bergman space, Toeplitz operator, quantization, holomorphic Sobolev space, Berezin transform.

1. Introduction 1.1. Generalized Bergman spaces Let Bd denote the (open) unit ball in Cd and let τ denote the hyperbolic volume measure on Bd , given by dτ (z) = (1 − |z|2 )−(d+1) dz,

(1.1)

where dz denotes the 2d-dimensional Lebesgue measure. The measure τ is natural because it is invariant under all of the automorphisms (biholomorphic mappings) of Bd . For λ > 0, let µλ denote the measure dµλ (z) = cλ (1 − |z|2 )λ dτ (z), where cλ is a positive constant whose value will be speciﬁed shortly. Finally, let HL2 (Bd , µλ ) denote the (weighted) Bergman space, consisting of those holomorphic functions on Bd that are square-integrable with respect to µλ . Supported in part by a grant from Prince of Songkla University. Supported in part by NSF Grant DMS-0555862.

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(Often these are deﬁned using the Lebesgue measure as the reference measure, but all the formulas look nicer if we use the hyperbolic volume measure instead.) These spaces carry a projective unitary representation of the group SU (d, 1). If λ > d, then the measure µλ is ﬁnite, so that all bounded holomorphic functions are square-integrable. For λ > d, we choose cλ so that µλ is a probability measure. Calculation shows that Γ(λ) , λ > d. (1.2) cλ = d π Γ(λ − d) (This diﬀers from the value in Zhu’s book [33] by a factor of π d /d!, because Zhu uses normalized Lebesgue whereas we use un-normalized Lebesgue measure in (1.1).) On the other hand, if λ ≤ d, then µλ is an inﬁnite measure. In this case, it is not hard to show that there are no nonzero holomorphic functions that are square-integrable with respect to µλ (no matter which nonzero value for cλ we choose). Although the holomorphic L2 space with respect to µλ is trivial (zero dimensional) when λ ≤ d, there are indications that life does not end at λ = d. First, the reproducing kernel for HL2 (Bd , µλ ) is given by 1 Kλ (z, w) = (1 − z · w) ¯ λ for λ > d. The reproducing kernel is deﬁned by the property that it is antiholomorphic in w and satisﬁes Kλ (z, w)f (w) dµλ (w) = f (z) Bd

2

for all f ∈ HL (B , µλ ). Nothing unusual happens to Kλ as λ approaches d. In fact, Kλ (z, w) := (1 − z · w) ¯ −λ is a “positive deﬁnite reproducing kernel” for all λ > 0. Thus, it is possible to deﬁne a reproducing kernel Hilbert space for all λ > 0 that agrees with HL2 (Bd , µλ ) for λ > d. Second, in representation theory, one is sometimes led to consider spaces like HL2 (Bd , µλ ) but with λ < d. Consider, for example, the much-studied metaplectic representation of the connected double cover of SU (1, 1) ∼ = Sp(1, R). This representation is a direct sum of two irreducible representations, one of which can be realized in the Bergman space HL2 (B1 , µ3/2 ) and the other of which can be realized in (a suitably deﬁned version of) the Bergman space HL2 (B1 , µ1/2 ). To be precise, we can say that the second summand of the metaplectic representation is realized in a Hilbert space of holomorphic functions having Kλ , λ = 1/2, as its reproducing kernel. See [14, Sect. 4.6]. Last, one often wants to consider the inﬁnite-dimensional (d → ∞) limit of the spaces HL2 (Bd , µλ ). (See, for example, [25] and [23].) To do this, one wishes to embed each space HL2 (Bd , µλ ) isometrically into a space of functions on Bd+1 , as functions that are independent of zn+1 . It turns out that if one uses (as we do) hyperbolic volume measure as the reference measure, then the desired isometric embedding is achieved by embedding d

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HL2 (Bd , µλ ) into HL2 (Bd+1 , µλ ). That is, if we use the same value of λ on Bd+1 as on Bd , then the norm of a function f (z1 , . . . , zd ) is the same whether we view it as a function on Bd or as a function on Bd+1 that is independent of zd+1 . (See, for example, Theorem 4, where the inner product of z m with z n is independent of d.) If, however, we keep λ constant as d tends to inﬁnity, then we will eventually violate the condition λ > d. Although it is possible to describe the Bergman spaces for λ ≤ d as reproducing kernel Hilbert spaces, this is not the most convenient description for calculation. Instead, drawing on several inter-related results in the literature, we describe these spaces as “holomorphic Sobolev spaces,” also called Besov spaces. The inner product on these spaces, which we denote as H(Bd , λ), is an L2 inner product involving both the functions and derivatives of the functions. For λ > d, H(Bd , λ) is identical to HL2 (Bd , µλ ) (the same space of functions with the same inner product), but H(Bd , λ) is deﬁned for all λ > 0. It is worth mentioning that in the borderline case λ = d, the space H(Bd , λ) can be identiﬁed with the Hardy space of holomorphic functions that are square-integrable over the boundary. To see this, note that the normalization constant cλ tends to zero as λ approaches d from above. Thus, the measure of any compact subset of Bd tends to zero as λ → d+ , meaning that most of the mass of µλ is concentrated near the boundary. As λ → d+ , µλ converges, in the weak-∗ topology on Bd , to the unique rotationally invariant probability measure on the boundary. Alternatively, we may observe that the formula for the inner product of monomials in H(Bd , d) (Theorem 4 with λ = d) is the same as in the Hardy space. 1.2. Toeplitz operators One important aspect of Bergman spaces is the theory of Toeplitz operators on them. If φ is a bounded measurable function, the we can deﬁne the Toeplitz operator Tφ on HL2 (Bd , µλ ) by Tφ f = Pλ (φf ), where Pλ is the orthogonal projection from L2 (Bd , µλ ) onto the holomorphic subspace. That is, Tφ consists of multiplying a holomorphic function by φ, followed by projection back into the holomorphic subspace. Of course, Tφ depends on λ, but we suppress this dependence in the notation. The function φ is called the (Toeplitz) symbol of the operator Tφ . The map sending φ to Tφ is known as the Berezin–Toeplitz quantization map and it (and various generalizations) have been much studied. See, for example, the early work of Berezin [5, 6], which was put into a general framework in [26, 27], along with [22, 8, 7, 10], to mention just a few works. The Berezin–Toeplitz quantization may be thought of as a generalization of the anti-Wick-ordered quantization on Cd (see [15]). When λ < d, the inner product on H(Bd , λ) is not an L2 inner product, and so the “multiply and project” deﬁnition of Tφ no longer makes sense. Our strategy is to ﬁnd alternative formulas for computing Tφ in the case λ > d, with the hope that these formulas will continue to make sense (for certain classes of symbols φ) for λ ≤ d. Speciﬁcally, we will identify classes of symbols φ for which Tφ can be deﬁned as:

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• A bounded operator on H(Bd , λ) (Section 4) • A Hilbert–Schmidt operator on H(Bd , λ) (Section 5). We also consider in Section 3 Toeplitz operators whose symbols are polynomials in z and z¯ and observe some unusual properties of such operators in the case λ < d.

2. H(Bd , λ) as a holomorphic Sobolev space In this section, we construct a Hilbert space of holomorphic functions on Bd with reproducing kernel (1 − z · w) ¯ −λ , for an arbitrary λ > 0. We denote this d space as H(B , λ). The inner product on this space is an L2 inner product with respect to the measure µλ+2n , where n is chosen so that λ + 2n > d. The inner product, however, involves not only the holomorphic functions but also their derivatives. That is, H(Bd , λ) is a sort of holomorphic Sobolev space (or Besov space) with respect to the measure µλ+2n . When λ > d, our space is identical to HL2 (Bd , µλ )—not just the same space of functions, but also the same inner product. When λ ≤ d, the Hilbert space H(Bd , λ), with the associated projective unitary action of SU (d, 1), is sometimes referred to as the analytic continuation (with respect to λ) of the holomorphic discrete series. Results in the same spirit as—and in some cases almost identical to— the results of this section have appeared in several earlier works, some of which treat arbitrary bounded symmetric domains and not just the ball in Cd . For example, in the case of the unit ball in Cd , Theorem 3.13 of [30] would presumably reduce to almost the same expression as in our Theorem 4, except that Yan has all the derivatives on one side, in which case the inner product has to be interpreted as a limit of integrals over a ball of radius 1 − ε. (Compare the formula for Dλk on p. 13 of [30] to the formula for A and B in Theorem 4.) See also [2, 4, 21, 31, 32]. Note, however, a number of these references give a construction that yields, for λ > d, the same space of functions as HL2 (Bd , µλ ) with a diﬀerent but equivalent norm. Such an approach is not suﬃcient for our needs; we require the same inner product as well as the same space of functions. Although our results in this section are not really new, we include proofs to make the paper self-contained and to get the precise form of the results that we want. The integration-by-parts argument we use also serves to prepare for our deﬁnition of Toeplitz operators on H(Bd , λ) in Section 4. We ourselves were introduced to this sort of reasoning by the treatment in Folland’s book [14] of the disk model for the metaplectic representation. The paper [16] obtains results in the same spirit as those of this section, but in the context of a complex semisimple Lie group. We begin by showing that for λ > d, the space HL2 (Bd , µλ ) can be expressed as a subspace of HL2 (Bd , µλ+2n ), with a Sobolev-type norm, for

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any positive integer n. Let N denote the “number operator,” deﬁned by N=

d j=1

zj

∂ . ∂zj

This operator satisﬁes N z m = |m|z m for all multi-indices m. If f is holomorphic, then N f coincides with the “radial derivative” df (rz)/dr|r=1 . We use ¯ = d z¯j ∂/∂ z¯j . also the operator N j=1 A simple computation shows that ¯ N N 2 α 2 α+1 (1 − |z| ) = I − = I− (1 − |z| ) (1 − |z|2 )α+1 . (2.1) α+1 α+1 We will use (2.1) and the following integration by parts result, which will also be used in Section 4. Lemma 1. If λ > d and ψ is a continuously diﬀerentiable function for which ψ and N ψ are bounded, then N 2 λ−d−1 ψ(z)(1 − |z| ) dz = cλ+1 cλ I+ ψ (z)(1 − |z|2 )λ−d dz λ d d B B ¯ N = cλ+1 I+ ψ (z)(1 − |z|2 )λ−d dz. λ Bd Here dz is the 2d-dimensional Lebesgue measure on Bd . Proof. We start by applying (2.1) and then think of the integral over Bd as the limit as r approaches 1 of the integral over a ball of radius r < 1. On the ball of radius r, we write out ∂/∂zj in terms of ∂/∂xj and ∂/∂yj . For, say, the ∂/∂xj term we express the integral as a one-dimensional integral with respect to xj (with limits of integration depending on the other variables) followed by an integral with respect to the other variables. We then use ordinary integration by parts in the xj integral, and similarly for the ∂/∂yj term. The integration by parts will yield a boundary term involving zj ψ(z)(1− |z|2 )λ−d ; this boundary term will vanish as r tends to 1, because we assume λ > d. In the nonboundary term, the operator N applied to (1 − |z|2 )λ−d d will turn into the operator − j=1 ∂/∂zj ◦ zj = −(dI + N ) applied to ψ. Computing from (1.2) that cλ /cλ+1 = (λ − d)/λ, we may simplify and let r tend to 1 to obtain the desired result involving N. The same reasoning gives ¯ as well. the result involving N We now state the key result, obtained from (2.1) and Lemma 1, relating the inner product in HL2 (Bd , µλ ) to the inner product in HL2 (Bd , µλ+1 ) (compare [14, p. 215] in the case d = 1).

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Proposition 2. Suppose that λ > d and f and g are holomorphic functions on Bd for which f, g, N f, and N g are all bounded. Then N f, g L2 (Bd ,µλ ) = f, I + g λ L2 (Bd ,µλ+1 ) (2.2) N = I+ . f, g λ L2 (Bd ,µλ+1 ) Proof. Recalling the formula (1.1) for the measure τ, we apply Lemma 1 with ψ(z) = f (z)g(z) with f and g holomorphic. Observing that N (f¯g) = f¯N g ¯ (f¯g) = (N f )g gives the second gives the ﬁrst equality and observing that N equality. Now, a general function in HL2 (Bd , µλ ) is not bounded. Indeed, the pointwise bounds on elements of HL2 (Bd , µλ ), coming from the reproducing kernel, are not suﬃcient to give a direct proof of the vanishing of the boundary terms in the integration by parts in Proposition 2. Nevertheless, (2.2) does hold for all f and g in HL2 (Bd , µλ ), provided that one interprets the inner product as the limit as r approaches 1 of integration over a ball of radius r. (See [14, p. 215] or [30, Thm. 3.13].) We are going to iterate (2.2) to obtain an expression for the inner product on HL2 (Bd , µλ ) involving equal numbers of derivatives on f and g. This leads to the following result. Theorem 3. Fix λ > d and a non-negative integer n. Then a holomorphic function f on Bd belongs to HL2 (Bd , µλ ) if and only if N l f belongs to HL2 (Bd , µλ+2n ) for 0 ≤ l ≤ n. Furthermore, f, g HL2 (Bd ,µλ ) = Af, Bg HL2 (Bd ,µλ+2n )

(2.3)

for all f, g ∈ HL2 (Bd , µλ ), where N N N A= I+ I+ ··· I + λ+n λ+n+1 λ + 2n − 1 N N N I+ ··· I + . B= I+ λ λ+1 λ+n−1 Let us make a few remarks about this result before turning to the proof. Let σ = λ + 2n. It is not hard to see that N k f belongs to HL2 (Bd , µσ ) for 0 ≤ k ≤ n if and only if all the partial derivatives of f up to order n belong to HL2 (Bd , µµ ), so we may describe this condition as “f has n derivatives in HL2 (Bd , µσ ).” This condition then implies that f belongs to HL2 (Bd , µσ−2n ), 2 which in turn means that f (z)/(1 − |z| )n belongs to L2 (Bd , µσ ). Since 1/(1 − |z|2 )n blows up at the boundary of Bd , saying that f (z)/(1 − |z|2 )n belongs to L2 (Bd , µσ ) says that f (z) has better behavior at the boundary than a typical element of HL2 (Bd , µσ ). We may summarize this discussion by saying that each derivative that f ∈ HL2 (Bd , µσ ) has in HL2 (Bd , µσ ) results, roughly speaking, in an improvement by a factor of (1 − |z|2 ) in the behavior of f near the boundary.

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This improvement is also reﬂected in the pointwise bounds on f coming from the reproducing kernel. If f has n derivatives in HL2 (Bd , µσ ), then f belongs to HL2 (Bd , µσ−2n ), which means that f satisﬁes the pointwise bounds 1/2

|f (z)| ≤ f L2 (Bd ,µσ−2n ) (Kσ−2n (z, z))

σ2 −n 1 = f L2 (Bd ,µσ−2n ) . 2 1 − |z|

(2.4)

2

These bounds are better by a factor of (1 − |z| )n than the bounds on a typical element of HL2 (Bd , µσ ). See also [16] for another setting in which the existence of derivatives in a holomorphic L2 space can be related in a precise way to improved pointwise behavior of the functions. The results of the two previous paragraphs were derived under the assumption that λ = σ − 2n > d. However, Theorem 4 will show that (2.4) still holds under the assumption λ = σ − 2n > 0. Proof. If f and g are polynomials, then (2.3) follows from iteration of Proposition 2. Note that N is a non-negative operator on polynomials, because the monomials form an orthogonal basis of eigenvectors with non-negative eigenvalues. It is well known and easily veriﬁed that for any f in HL2 (Bd , µλ ), the partial sums of the Taylor series of f converge to f in norm. We can therefore choose polynomials fj converging in norm to f . If we apply (2.3) with f = g = (fj − fk ) and expand out the expressions for A and B, then the positivity of N will force each of the terms on the right-hand side to tend to zero. In particular, N l fj is a Cauchy sequences in HL2 (Bd , µλ+2n ), for all 0 ≤ l ≤ n. It is easily seen that the limit of this sequence is N l f ; for holomorphic functions, L2 convergence implies locally uniform convergence of the derivatives to the corresponding derivatives of the limit function. This shows that N l f is in HL2 (Bd , µλ+2n ). For any f, g ∈ HL2 (Bd , µλ ), choose sequences fj and gk of polynomials converging to f, g. Since N l fj and N l gj converge to N l f and N l g, respectively, plugging fj and gj into (2.3) and taking a limit gives (2.3) in general. In the other direction, suppose that N l f belongs to HL2 (Bd , µλ+2n ) for all 0 ≤ l ≤ n. Let fj denote the jth partial sum of the Taylor series of f . Then since N z m = |m|z m for all multi-indices m, the functions N l fj form the partial sums of a Taylor series converging to N l fj , and so these must be the partial sums of the Taylor series of N l f. Thus, for each l, we have that N l fj converges to N l f in HL2 (Bd , µλ+2n ). If we then apply (2.3) with f = g = fj − fk , convergence of each N l fj implies that all the terms on the right-hand side tend to zero. We conclude that fj is a Cauchy sequence in HL2 (Bd , µλ ), which converges to some fˆ. But L2 convergence of holomorphic functions implies pointwise convergence, so the limit in HL2 (Bd , µλ ) (i.e., fˆ) coincides with the limit in HL2 (Bd , µλ+2n ) (i.e., f ). This shows that f is in HL2 (Bd , µλ ).

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Now, when λ ≤ d, Proposition 2.2 no longer holds. This is because the boundary terms, which involve (1 − |z|2 )λ−d , no longer vanish. This failure of equality is actually a good thing, because if we take f = g, then |f (z)|2 (1 − |z|2 )λ dτ (z) = +∞ cλ Bd

for all nonzero holomorphic functions, no matter what positive value we assign to cλ . (Recall that when λ > d, cλ is chosen to make µλ a probability measure, but this prescription does not make sense for λ ≤ d.) Although the left-hand side of (2.2) is inﬁnite when f = g and λ ≤ d, the right-hand side is ﬁnite if λ + 1 > d and, say, f is a polynomial. More generally, for any λ ≤ d, we can choose n big enough such that λ + 2n > d. We then take the right-hand side of (2.3) as a deﬁnition. Theorem 4. For all λ > 0, choose a non-negative integer n so that λ + 2n > d and deﬁne

H(Bd , λ) = f ∈ H(Bd ) N k f ∈ HL2 (Bd , µλ+2n ), 0 ≤ k ≤ n . Then the formula f, g λ = Af, Bg HL2 (Bd ,µλ+2n ) where

N N N I+ I+ ··· I + λ+n λ+n+1 λ + 2n − 1 N N N B= I+ I+ ··· I + λ λ+1 λ+n−1 A=

deﬁnes an inner product on H(Bd , λ) and H(Bd , λ) is complete with respect to this inner product. The monomials z m form an orthogonal basis for H(Bd , λ) and for all multi-indices l and m we have l m m!Γ(λ) . z , z λ = δl,m Γ(λ + |m|) Furthermore, H(Bd , λ) has a reproducing kernel given by Kλ (z, w) =

1 . (1 − z · w) ¯ λ

Using power series, it is easily seen that for any holomorphic function f, if N n f belongs to HL2 (Bd , µλ+2n ), then N k f belongs to HL2 (Bd , µλ+2n ) for 0 ≤ k < n. Note that the reproducing kernel and the inner product of the monomials are independent of n. Thus, we obtain the same space of functions with the same inner product, no matter which n we use, so long as λ + 2n > d. From the reproducing kernel we obtain the pointwise bounds given by 2 2 2 |f (z)| ≤ f λ (1 − |z| )−λ .

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Proof. Using a power series argument, it is easily seen that if f and N k f 2 d k belong to HL (B , µλ+2n ), then f, N f L2 (Bd ,µ ≥ 0. From this, we λ+2n ) obtain positivity of the inner product ·, · λ . If fj is a Cauchy sequence in H(Bd , λ), then positivity of the coeﬃcients in the expressions for A and B imply that for 0 ≤ k ≤ n, N k fj is a Cauchy sequence in HL2 (Bd , µλ+2n ), which converges (as in the proof of Theorem 3) to N k f. This shows that N k f is in HL2 (Bd , µλ+2n ) for each 0 ≤ k ≤ n, and so f ∈ H(Bd , λ). Further, convergence of each N k fj to N k f implies that fj converges to f in H(Bd , λ). To compute the inner product of two monomials in H(Bd , λ), we apply the deﬁnition. Since N z m = |m|z m , we obtain l m z ,z λ λ + |m| λ + 1 + |m| λ + 2n − 1 + |m| m!Γ(λ + 2n) = δl,m ··· λ λ+1 λ + 2n − 1 Γ(λ + 2n + |m|) m!Γ(λ) = δl,m , Γ(λ + |m|) where we have used the known formula for the inner product of monomials in HL2 (Bd , µλ+2n ) (e.g., [33]). Completeness of the monomials holds in H(Bd , λ) for essentially the same reason it holds in the ordinary Bergman spaces. For f ∈ HL2 (Bd , µλ ), expand f in a Taylor series and then consider z m , f λ . Each term in the inner product is an integral over Bd with respect to µλ+2n , and each of these integrals can be computed as the limit as r tends to 1 of integrals over a ball of radius r < 1. On the ball of radius r, we may interchange the integral with the sum in the Taylor series. But distinct monomials are orthogonal not just over Bd but also over the ball of radius r, as is easily veriﬁed. The upshot of all of this is that z m , f λ is a nonzero multiple of the mth Taylor coeﬃcient of f. Thus if z m , f λ = 0 for all m, f is identically zero. Finally, we address the reproducing kernel. Although one can use essentially the same argument as in the case λ > d, using the orthogonal basis of monomials and a binomial expansion (see the proof of Theorem 12), it is more enlightening to relate the reproducing kernel in H(Bd , λ) to that in HL2 (Bd , µλ+2n ). We require some elementary properties of the operators A and B; since the monomials form an orthogonal basis of eigenvectors for these operators, these properties are easily obtained. We need that A is self-adjoint on its natural domain and that A and B have bounded inverses. be the unique element of HL2 (Bd , µλ+2n ) for which Let χλ+2n z λ+2n χz , f L2 (Bd ,µ = f (z) ) λ+2n

2

for all f in HL (B , µλ+2n ). Explicitly, χλ+2n (w) = (1 − z¯ · w)−(λ+2n) . (This z is Theorem 2.2 of [33] with our λ corresponding to n + α + 1 in [33].) Now, a simple calculation shows that d

(I + N/a)(1 − z¯ · w)−a = (1 − z¯ · w)−(a+1) ,

(2.5)

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where N acts on the w variable with z ﬁxed. From this, we see that N k χλ+2n z is a bounded function for each ﬁxed z ∈ Bd and k ∈ N, so that χλ+2n is in z H(Bd , λ). For any f ∈ H(Bd , λ) we compute that f, (AB)−1 χλ+2n = Af, B(AB)−1 χλ+2n z z λ L2 (Bd ,µλ+2n) λ+2n = f, χz = f (z). L2 (Bd ,µ λ+2n)

d

This shows that the reproducing kernel for H(B , λ) is given by Kλ (z, w) = [(AB)−1 χλ+2n ](w). Using (2.5) repeatedly gives the desired result. z We conclude this section with a simple lemma that will be useful in Section 5. Lemma 5. For all λ1 , λ2 > 0, if f is in H(Bd , λ1 ) and g is in H(Bd , λ2 ) then f g is in H(Bd , λ1 + λ2 ). Proof. If, say, λ1 > d, then we have the following simple argument: 2 2 2 2

f g λ1 +λ2 = cλ1 +λ2 |f (z)| |g(z)| (1 − |z| )λ1 +λ2 dτ (z) Bd 2 2 2 2 ≤ cλ1 +λ2 g λ2 |f (z)| (1 − |z| )−λ2 (1 − |z| )λ1 +λ2 dτ (z) Bd

cλ +λ 2 2 = 1 2 f λ1 g λ2 . cλ1 Unfortunately, cλ1 +λ2 /cλ1 tends to inﬁnity as λ1 approaches d from above, so we cannot expect this simple inequality to hold for λ1 < d. For any λ1 , λ2 > 0, choose n so that λ1 + n > d and λ2 + n > d. Then f g belongs to H(Bd , λ1 + λ2 ) provided that N n (f g) belongs to HL2 (Bd , λ1 + λ2 + 2n). But n n n N (f g) = (2.6) N k f N n−k g. k k=0

Using Theorem 4, it is easy to see that if f belongs to H(Bd , λ1 ) then N k f belongs to H(Bd , λ1 + 2k). Thus,

k

N f (z) 2 ≤ ak (1 − |z|2 )−(λ1 +2k) . Now, for each term in (2.6) with k ≤ n/2, we then obtain the following norm estimate:

k

N f (z)N n−k g(z) 2 (1 − |z|2 )λ1 +λ2 +2n dτ (z) cλ1 +λ2 +2n d B

2

n−k 2

N ≤ cλ1 +λ2 +2n ak g(z) (1 − |z| )λ2 +2n−2k dτ (z). (2.7) Bd

Since k ≤ n/2, we have λ2 + 2n − 2k ≥ λ2 + n > d. We are assuming that g is in H(Bd , λ2 ), so that N n−k g is in H(Bd , λ2 + 2n − 2k), which coincides with HL2 (Bd , µλ2 +2n−2k ). Thus, under our assumptions on f and g, each term in

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(2.6) with k ≤ n/2 belongs to HL2 (Bd , λ1 + λ2 + 2n). A similar argument with the roles of f and g reversed takes care of the terms with k ≥ n/2.

3. Toeplitz operators with polynomial symbols In this section, we will consider our ﬁrst examples of Toeplitz operators on generalized Bergman spaces, those whose symbols are (not necessarily holomorphic) polynomials. Such examples are suﬃcient to see some interesting new phenomena, that is, properties of ordinary Toeplitz operator that fail when extended to these generalized Bergman spaces. The deﬁnition of Toeplitz operators for the case of polynomial symbols is consistent with the deﬁnition we use in Section 4 for a larger class of symbols. For λ > d, we deﬁne the Toeplitz operator Tφ by Tφ f = Pλ (φf ) for all f in HL2 (Bd , µλ ) and all bounded measurable functions φ. Recall that Pλ is the orthogonal projection from L2 (Bd , τ ) onto the holomorphic subspace. Because Pλ is a self-adjoint operator on L2 (Bd , µλ ), the matrix entries of Tφ may be calculated as f1 , Tφ f2 HL2 (Bd ,µλ ) = f1 , φf2 L2 (Bd ,µλ ) ,

λ > d,

(3.1)

for all f1 , f2 ∈ HL2 (Bd , µλ ). From this formula, it is easy to see that Tφ¯ = (Tφ )∗ . If ψ is a bounded holomorphic function and φ is any bounded measurable function, then it is easy to see that Tφψ = Tφ Mψ . Thus, for any two multiindices m and n, we have Tz¯m zn = (Mzm )∗ (Mzn ).

(3.2)

We will take (3.2) as a deﬁnition for 0 < λ ≤ d. Our ﬁrst task, then, is to show that Mzn is a bounded operator on HL2 (Bd , µλ ) for all λ > 0. Proposition 6. For all λ > 0 and all multi-indices n, the multiplication operator Mzn is a bounded operator on H(Bd , λ). Thus, for any polynomial φ, the Toeplitz operator Tφ deﬁned in (3.2) is a bounded operator on H(Bd , λ). Proof. The result is a is a special case of a result of Arazy and Zhang [3] and also of the results of Section 4, but it is easy to give a direct proof. It suﬃces to show that Mzj is bounded for each j. Since Mzj preserves the orthogonality of the monomials, we obtain m Mzj = sup zj z λ = sup mj + 1 .

z m λ m m |m| + λ

Note that mj ≤ |m| with equality when mk = 0 for k = j. Thus the supremum is ﬁnite and is easily seen to have the value of 1 if λ ≥ 1 and 1/λ if λ < 1.

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We now record some standard properties of Toeplitz operators on (ordinary) Bergman spaces. These properties hold for Toeplitz operators (deﬁned by the “multiply and project” recipe) on any holomorphic L2 space. We will show that these properties do not hold for Toeplitz operators with polynomial symbols on the generalized Bergman spaces H(Bd , λ), λ < d. Proposition 7. For λ > d and φ(z) bounded, the Toeplitz operator Tφ on the space HL2 (Bd , dµλ ), which is deﬁned by Tφ f = Pλ (φf ), has the following properties. 1. Tφ ≤ supz |φ(z)| 2. If φ(z) ≥ 0 for all z, then Tφ is a positive operator. Both of these properties fail when λ < d. In fact, for λ < d, there is no constant C such that Tφ ≤ C supz |φ(z)| for all polynomials φ. As we remarked in the introduction, when λ = d, the space H(Bd , λ) may be identiﬁed with the Hardy space. Thus Properties 1 and 2 in the proposition still hold when λ = d, if, say, φ is continuous up to the boundary of Bd (or otherwise has a reasonable extension to the closure of Bd ). Proof. When λ > d, the projection operator Pλ has norm 1 and the multiplication operator Mφ has norm equal to supz |φ(z)| as an operator on L2 (Bd , µλ ). Thus, the restriction to HL2 (Bd , µλ ) of Pλ Mφ has norm at most supz |φ(z)|. Meanwhile, if φ is non-negative, then from (3.1) we see that f, Tφ f ≥ 0 for all f ∈ HL2 (Bd , µλ ). Let us now assume that 0 < λ < d. Computing on the orthogonal basis in Theorem 4, it is a simple exercise to show that Tz¯j zj (z m ) =

Γ(λ + |m|) (m + ej )! 1 + mj m zm = z . m! Γ(λ + |m| + 1) λ + |m|

(3.3)

If we take φ(z) = |z|2 , then summing (3.3) on j gives Tφ z m =

d + |m| m z . λ + |m|

Since λ < d, this shows that Tφ > 1, even though |φ(z)| < 1 for all z ∈ Bd . Thus, Property 1 fails for λ < d. (From this calculation it easily follows that 2 if φ(z) = (1 − |z| )/(λ − d), then Tφ is the bounded operator (λI + N )−1 , for all λ = d.) For the second property, we let ψ(z) = 1 − φ(z) = 1 − |z|2 which is positive. From the above calculation we obtain d + |m| Tψ z m , z m Hλ = z m 2Hλ −

z m 2H(Bd ,λ) , λ + |m| which is negative if 0 < λ < d.

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We now show that there is no constant C such that Tφ ≤ C supz |φ(z)|. Consider k

d 2 k 2 φk (z) := (|z| ) = |zi | i=1

k! k! (|z1 |2 )i1 (|z2 |2 )i2 · · · (|zd |2 )id = = zi zi. i! i! |i|=k

|i|=k

Computing on the orthogonal basis in Theorem 4 we obtain k! k! i!Γ(λ) k!Γ(λ) (Tzi zi 1) = 1=I 1, T φk 1 = i! i! Γ(λ + k) Γ(λ + k) |i|=k

|i|=k

where 1 is the constant function. Here, I is the of multi-indices i of number length d such that |i| = k, which is equal to k+d−1 . Thus d−1 T φk 1 =

k−1 d+j (d + k − 1) · · · (d) (k + d − 1)! Γ(λ) 1= 1= 1. (d − 1)! Γ(λ + k) (λ + k − 1) · · · (λ) λ+j j=0

. Since d > λ, the terms d−λ λ+j ∞ d+j are positive diverges. This implies j=0 λ+j = ∞. Since supz |φk (z)| = 1 for all k, there is no a constant C such that Tφ ≤ C supz |φ(z)|. k−1

d+j j=0 λ+j = ∞ d−λ and j=0 λ+j

Consider

k−1 j=0 1 +

d−λ λ+j

Remark 8. For λ < d, there does not exist any positive measure ν on Bd such that f λ = f L2 (Bd ,ν) for all f in H(Bd , λ). If such a ν did exist, then the argument in the ﬁrst part of the proof of Proposition 7 would show that Properties 1 and 2 in the proposition hold.

4. Bounded Toeplitz operators In this section, we will consider a class of symbols φ for which we will be able to deﬁne a Toeplitz operator Tφ as a bounded operator on H(Bd , λ) for all λ > 0. Our deﬁnition of Tφ will agree (for the relevant class of symbols) with the usual “multiply and project” deﬁnition for λ > d. In light of the examples in the previous section, we cannot expect boundedness of φ to be suﬃcient to deﬁne Tφ as a bounded operator. Instead, we will consider functions φ for which φ and a certain number of derivatives of φ are bounded. Our strategy is to use integration by parts to give an alternative expression for the matrix entries of a Toeplitz operator with suﬃciently regular symbol, in the case λ > d. We then take this expression as our deﬁnition of Toeplitz operator in the case 0 < λ ≤ d. Theorem 9. Assume λ > d and ﬁx a positive integer n. Let φ be a function ¯ k N l φ is bounded that is 2n times continuously diﬀerentiable and for which N

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for all 0 ≤ k, l ≤ n. Then

f, Tφ g HL2 (Bd ,µλ ) = cλ+2n

C

Bd

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λ+2n f (z)φ(z)g(z) 1 − |z|2 dτ (z)

for all f, g ∈ HL2 (Bd , µλ ), where C is the operator given by ¯ ¯ N N N N C= I+ ··· I + I+ ··· I + . (4.1) λ + 2n − 1 λ+n λ+n−1 λ Thus, there exist constants Ajklm (depending on n and λ) such that f, Tφ g HL2 (Bd ,µλ ) =

n

k l m ¯ N φ N g 2 d Ajklm N j f, N L (B ,µ

λ+2n )

. (4.2)

j,k,l,m=1

Proof. Assume at ﬁrst that f and g are polynomials, so that f and g and all of their derivatives are bounded. We use (3.1) and apply the ﬁrst equality in Lemma 1 with ψ = f¯φg. We then apply the ﬁrst equality in the lemma again with ψ = (I +N/λ)[f¯φg]. We continue on in this fashion until we have applied the ﬁrst equality in Lemma 1 n times and the second equality n times. This establishes the desired equality in the case that f and g are polynomials. For general f and g in HL2 (Bd , µλ ), we approximate by sequences fa and ga of polynomials. From Theorem 3 we can see that convergence of fa and ga in HL2 (Bd , µλ ) implies convergence of N j fa and N k ga to N j f and N k g, so that applying (4.2) to fa and ga and taking a limit establishes the desired result for f and g. Deﬁnition 10. Assume 0 < λ ≤ d and ﬁx a positive integer n such that λ + 2n > d. Let φ be a function that is 2n times continuously diﬀerentiable ¯ k N l φ is bounded for all 0 ≤ k, l ≤ n. Then we deﬁne the and for which N Toeplitz operator Tφ to be the unique bounded operator on H(Bd , λ) whose matrix entries are given by λ+2n f, Tφ g H(Bd ,λ) = cλ+2n C f (z)φ(z)g(z) 1 − |z|2 dz, (4.3) Bd

where C is given by (4.1). Note that from Theorem 4, N j f and N m g belong to L2 (Bd , µλ+2n ) for all 0 ≤ j, m ≤ n, for all f and g in HL2 (Bd , µλ ). Furthermore, N j f 2 d and N m g 2 d are bounded by constants times L (B ,µλ+2n )

L (B ,µλ+2n )

f λ and g λ , respectively. Thus, the right-hand side of (4.3) is a continuous sesquilinear form on H(Bd , λ), which means that there is a unique bounded operator Tφ whose matrix entries are given by (4.3). If λ = d, then (as discussed in the introduction) the Hilbert space H(Bd , λ) is the Hardy space of holomorphic functions that are square-integrable over the boundary. In that case, the Toeplitz operator Tφ will be the zero operator whenever φ is identically zero on the boundary of Bd . If λ = d− 1, d− 2, . . . , then the inner product on H(Bd , λ) can be related to the inner product on the Hardy space. It is not hard to see that in these cases,

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Tφ will be the zero operator if φ and enough of its derivatives vanish on the boundary of Bd . Let us consider the case in which φ(z) = ψ1 (z)ψ2 (z), where ψ1 and ψ2 are holomorphic functions such that the function and the ﬁrst n derivatives are bounded. Then when applying C to f (z)φ(z)g(z), all the N -factors go ¯ -factors go onto f (z)ψ1 (z). Reonto the expression ψ2 (z)g(z) and all the N calling from Theorem 4 the formula for the inner product on H(Bd , λ), we see that f, Tφ g HL2 (Bd ,µλ ) = ψ1 f, ψ2 g H(Bd ,λ) , as expected. This means that in this case, Tψ¯1 ψ2 = (Mψ1 )∗ (Mψ2 ), as in the case λ > d. In particular, Deﬁnition 10 agrees with the deﬁnition we used in Section 3 in the case that φ is a polynomial in z and z¯.

5. Hilbert–Schmidt Toeplitz operators 5.1. Statement of results In this section, we will give suﬃcient conditions under which a Toeplitz operator Tφ can be deﬁned as a Hilbert–Schmidt operator on H(Bd , λ). Speciﬁcally, if φ belongs to L2 (Bd , τ ) then Tφ can be deﬁned as a Hilbert–Schmidt operator, provided that λ > d/2. Meanwhile, if φ belongs to L1 (Bd , τ ), then Tφ can be deﬁned as a Hilbert–Schmidt operator for all λ > 0. In both cases, we deﬁne Tφ in such a way that for all bounded functions f and g in H(Bd , λ), we have f, Tφ g λ = cλ

Bd

f (z)φ(z)g(z)(1 − |z|2 )λ dτ (z),

(5.1)

where cλ is deﬁned by cλ = Γ(λ)/(π d Γ(λ − d)). This expression is identical to (3.1) in the case λ > d. The value of cλ should be interpreted as 0 when λ−d = 0, −1, −2, . . .. This means that for φ in L2 (Bd , τ ) or L1 (Bd , τ ) (but not for other classes of symbols!), Tφ is the zero operator when λ = d, d − 1, . . . . This strange phenomenon is discussed in the next subsection. Note that we are not claiming Tφ = 0 for arbitrary symbols when λ = d, d − 1, . . . , but only for symbols that are integrable or square-integrable with respect to the hyperbolic volume measure τ. Such functions must have reasonable rapid decay (in an average sense) near the boundary of Bd . In the case φ ∈ L2 (Bd , τ ), the restriction λ > d/2 is easy to explain: the function (1 − |z|2 )λ belongs to L2 (Bd , τ ) if and only if λ > d/2. Thus, if f and g are bounded and φ is in L2 (Bd , τ ), then (5.1) is absolutely convergent for λ > d/2. In this subsection, we state our results; in the next subsection, we discuss some unusual properties of Tφ for λ < d; and in the last subsection of this section we give the proofs. We begin by considering symbols φ in L2 (Bd , τ ).

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Theorem 11. Fix λ > d/2 and let cλ = Γ(λ)/(π d Γ(λ − d)). (We interpret cλ to be zero if λ is an integer and λ ≤ d.) Then the operator Aλ given by λ (1 − |z|2 )(1 − |w|2 ) φ(w) dτ (w) Aλ φ(z) = c2λ ¯)(1 − w ¯ · z) Bd (1 − w · z is a bounded operator from L2 (Bd , τ ) to itself. Theorem 12. Fix λ > d/2. Then for each φ ∈ L2 (Bd , τ ), there is a unique Hilbert–Schmidt operator on H(Bd , λ), denoted Tφ , with the property that f, Tφ g λ = cλ f (z)φ(z)g(z)(1 − |z|2 )λ dτ (z) (5.2) Bd

for all bounded holomorphic functions f and g in H(Bd , λ). The Hilbert– Schmidt norm of Tφ is given by 2

Tφ HS = φ, Aλ φ L2 (Bd ,τ ) . If λ > d and φ ∈ L2 (Bd , τ ) ∩ L∞ (Bd , τ ), then the deﬁnition of Tφ in Theorem 12 agrees with the “multiply and project” deﬁnition; compare (3.1). Applying Lemma 5 with λ1 = λ2 = λ and λ > d/2, we see that for all 2 f and g in H(Bd , λ), the function z → f (z)g(z)(1 − |z| )λ is in L2 (Bd , τ ). This means that the integral on the right-hand side of (5.2) is absolutely convergent for all f, g ∈ H(Bd , λ). It is then not hard to show that (5.2) holds for all f, g ∈ H(Bd , λ). The operator Aλ coincides, up to a constant, with the Berezin transform. Let χλz (w) := Kλ (z, w) be the coherent state at the point z, which satisﬁes f (z) = χλz , f λ for all f ∈ H(Bd , λ). Then one standard deﬁnition of the Berezin transform Bλ is λ χz , Tφ χλz λ . Bλ φ = χλz , χλz λ The function Bλ φ may be thought of as the Wick-ordered symbol of Tφ , where Tφ is thought of as the anti-Wick-ordered quantization of φ. Using the formula (Theorem 4) for the reproducing kernel along with (5.2), we see χλz (w) is a bounded function of w for each ﬁxed that Aλ = cλ Bλ . (Note that d λ λ z ∈ B and that χz , χz λ = Kλ (z, z).) Note that τ is an inﬁnite measure, which means that if φ is in L2 (Bd , τ ) 1 or L (Bd , τ ), then φ must tend to zero at the boundary of Bd , at least in an average sense. This decay of φ is what allows (5.2) to be a convergent integral. If, for example, we want to take φ(z) ≡ 1, then we cannot use (5.2) to deﬁne Tφ , but must instead use the deﬁnition from Section 3 or Section 4. Note also that the space of Hilbert–Schmidt operators on H(Bd , λ) may be viewed as the quantum counterpart of L2 (Bd , τ ). It is thus natural to investigate the question of when the Berezin–Toeplitz quantization maps L2 (Bd , τ ) into the Hilbert–Schmidt operators. We now show that if one considers a symbol φ in L1 (Bd , τ ), then one obtains a Hilbert–Schmidt Toeplitz operator Tφ for all λ > 0.

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Theorem 13. Fix λ > 0 and let cλ be as in Theorem 12. Then for each φ ∈ L1 (Bd , τ ), there exists a unique Hilbert–Schmidt operator on H(Bd , λ), denoted Tφ , with the property that f (z)φ(z)g(z)(1 − |z|2 )λ dτ (z) (5.3) f, Tφ g λ = cλ Bd

for all bounded holomorphic functions f and g in H(Bd , λ). The Hilbert– Schmidt norm of Tφ satisﬁes

Tφ HS ≤ cλ φ L1 (Bd ,τ ) . Using the pointwise bounds on elements of H(Bd , λ) coming from the reproducing kernel, we see immediately that for all f, g ∈ H(Bd , λ), the function z → f (z)g(z)(1−|z|2 )λ is bounded. It is then not hard to show that (5.3) holds for all f, g ∈ H(Bd , λ). We have already remarked that the deﬁnition of Tφ given in this section agrees with the “multiply and project” deﬁnition when λ > d (and φ is bounded). It is also easy to see that the deﬁnition of Tφ given in this section agrees with the one in Section 4, when φ falls under the hypotheses of both Deﬁnition 10 and either Theorem 12 or Theorem 13. For some positive integer n, consider the set of λ’s for which λ + 2n > d and λ > d/2, i.e., λ > ¯ l φ is max(d−2n, d/2). Now suppose that φ belongs to L2 (Bd , τ ) and that N k N bounded for all 0 ≤ k, l ≤ n. It is easy to see that the matrix entries f, Tφ g λ depend real-analytically on λ for ﬁxed polynomials f and g, whether Tφ is deﬁned by Deﬁnition 10 or by Theorem 12. For λ > d, the two matrix entries agree because both deﬁnitions of Tφ agree with the “multiply and project” deﬁnition. The matrix entries therefore must agree for all λ > max(d − 2n, d/2). Since polynomials are dense in H(Bd , λ) and both deﬁnitions of Tφ give bounded operators, the two deﬁnitions of Tφ agree. The same reasoning shows agreement of Deﬁnition 10 and Theorem 13. 5.2. Discussion Before proceeding on with the proof, let us make a few remarks about the way we are deﬁning Toeplitz operators in this section. For λ > d, cλ is the normalization constant that makes the measure µλ a probability measure, which can be computed to have the value Γ(λ)/(π d Γ(λ − d)). For λ ≤ d, although the measure (1 − |z|2 )λ dτ (z) is an inﬁnite measure, we simply use the same formula for cλ in terms of the gamma function. We understand this to mean that cλ = 0 whenever λ is an integer in the range (0, d]. It also means that cλ is negative when d − 1 < λ < d and when d − 3 < λ < d − 2, etc. In the cases where cλ = 0, we have that Tφ = 0 for all φ in L1 (Bd , τ ) 2 or L (Bd , τ ). This ﬁrst occurs when λ = d. Recall that for λ = d, the space H(Bd , λ) can be identiﬁed with the Hardy space of holomorphic functions square-integrable over the boundary. Meanwhile, having φ being integrable or square-integrable with respect to τ means that φ tends to zero (in an average sense) at the boundary, in which case it is reasonable that Tφ should be zero as an operator on the Hardy space. For other integer values of λ ≤ d, the

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inner product on H(Bd , λ) can be expressed using the methods of Section 2 in terms of integration over the boundary, but involving the functions and their derivatives. In that case, we expect Tφ to be zero if φ has suﬃciently rapid decay at the boundary, and it is reasonable to think that having φ in L1 or L2 with respect to τ constitutes suﬃciently rapid decay. Note, however, that the conclusion that Tφ = 0 when cλ = 0 applies only when φ is in L1 or L2 ; for other classes of symbols, such as polynomials, Tφ is not necessarily zero. For example, Tzm is equal to Mzm , which is certainly a nonzero operator on H(Bd , λ), for all λ > 0. Meanwhile, if cλ < 0, then we have the curious situation that if φ is positive and in L1 or L2 with respect to τ, then the operator Tφ is actually a negative operator. This is merely a dramatic example of a phenomenon we have already noted: for λ < d, non-negative symbols do not necessarily give rise to non-negative Toeplitz operators. Again, though, the conclusion that Tφ is negative for φ positive applies only when φ belongs to L1 or L2 . For example, the constant function 1 always maps to the (positive!) identity operator, regardless of the value of λ.

5.3. Proofs As motivation, we begin by computing the Hilbert–Schmidt norm of Toeplitz operators in the case λ > d. For any bounded measurable φ, we extend the Toeplitz operator Tφ to all of L2 (Bd , µλ ) by making it zero on the orthogonal complement of the holomorphic subspace. This extension is given by the formula Pλ Mφ Pλ . Then the Hilbert–Schmidt norm of the operator Tφ on HL2 (Bd , µλ ) is the same as the Hilbert–Schmidt norm of the operator Pλ Mφ Pλ on L2 (Bd , µλ ). Since Pλ is computed as integration against the reproducing kernel, we may compute that Pλ Mφ Pλ f (z) = where

Bd

Kφ (z, w)f (w) dµλ (w),

Kφ (z, w) =

Bd

K(z, u)φ(u)K(u, w) dµλ (u).

If we can show that Kφ is in L2 (Bd × Bd , µλ × µλ ), then it will follow by a standard result that Tφ is Hilbert–Schmidt, with Hilbert–Schmidt norm equal to the L2 norm of Kφ . For suﬃciently nice φ, we can compute the L2 norm of Kφ by rearranging the order of integration and using twice the repro ducing identity K(z, w)K(w, u) dµλ (w) = K(z, u). (This identity reﬂects that Pλ2 = Pλ .) This yields Bd ×Bd

2

|Kφ (z, w)| dµλ (z) dµλ = φ, Aφ L2 (Bd ,τ ) ,

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where Aλ is the integral operator given by 2 |K(z, w)| (1 − |z|2 )λ (1 − |w|2 )λ φ(w) dτ (w) Aλ φ(z) = c2λ Bd

=

c2λ

Bd

(1 − |z|2 )(1 − |w|2 ) (1 − w ¯ · z)(1 − z¯ · w)

λ φ(w) dτ (w).

(5.4)

In the case d/2 < λ ≤ d, it no longer makes sense to express Tφ as Pλ Mφ Pλ . Nevertheless, we can consider an operator Aλ deﬁned by (5.4). Our goal is to show that for all λ > d/2, (1) Aλ is a bounded operator on L2 (Bd , τ ) and (2) if we deﬁne Tφ by (5.1), then the Hilbert–Schmidt norm of Tφ is given by φ, Aλ φ L2 (Bd ,τ ) . We will obtain similar results for all λ > 0 if φ ∈ L1 (Bd , τ ). Proof of Theorem 11. We give two proofs of this result; the ﬁrst generalizes more easily to other bounded symmetric domains, whereas the second relates Aλ to the Laplacian for Bd (compare [13]). First Proof. We let λ (1 − |z|2 )(1 − |w|2 ) Fλ (z, w) = c2λ ; (1 − w ¯ · z)(1 − z¯ · w) i.e., Fλ is the integral kernel of the operator Aλ . A key property of Fλ is its invariance under automorphisms: Fλ (ψ(z), ψ(w)) = Fλ (z, w) for each automorphism (biholomorphism) ψ of Bd and all z, w ∈ Bd . To establish the invariance of Fλ , let (5.5) fλ (z) = c2λ (1 − |z|2 )λ . According to Lemma 1.2 of [33], Fλ (z, w) = fλ (φw (z)), where φw is an automorphism of Bd taking 0 to w and satisfying φ2w = I. Now, if ψ is any automorphism, the classiﬁcation of automorphisms (Theorem 1.4 of [33]) implies that ψ ◦ φw = φψ(w) ◦ U for some unitary matrix U. From this we can obtain φψ(w) = U ◦ φw ◦ ψ −1 , and so fλ (φψ(w) (ψ(z))) = fλ (U (φw (ψ −1 (ψ(z)))) = fλ (φw (z)), i.e., Fλ (ψ(z), ψ(w)) = Fλ (z, w). The invariance of Fλ under automorphisms means that Aλ φ can be thought of as a convolution (over the automorphism group P SU (d, 1)) of φ with the function fλ . What this means is that Aλ φ(z) = fλ (gh−1 · 0)φ(h · 0) dh, G

where g ∈ G is chosen so that g · 0 = z. Here G = P SU (d, 1) is the group of automorphisms of Bd (given by fractional linear transformations) and dh is an appropriately normalized Haar measure on G. Furthermore, L2 (Bd , τ ) can be identiﬁed with the right-K-invariant subspace of L2 (G, dg), where K := U (d) is the stabilizer of 0. If λ > d, then fλ is in L1 (Bd , τ ), in which case it is easy to prove that Aλ is bounded; see, for example, Theorem 2.4 in [5]. This argument does not

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work if λ ≤ d. Nevertheless, if λ > d/2, an easy computation shows that fλ belongs to L2 (Bd , τ ) and also to Lp (Bd , τ ) for some p < 2. We could at this point appeal to a general result known as the Kunze–Stein phenomenon [24]. The result states that on connected semisimple Lie groups G with ﬁnite center (including P SU (d, 1)), convolution with a function in Lp (G, dg), p < 2, is a bounded operator from L2 (G, dg) to itself. (See [11] for a proof in this generality.) However, the proof of this result is simpler in the case we are considering, where the function in Lp (G, dg) is bi-K-invariant and the other function is right-K-invariant. (In our case, the function in Lp (G, dg) is the function g → fλ (g · 0) and the function in L2 (G, dg) is g → φ(g · 0).) Using the Helgason Fourier transform along with its behavior under convolution with a bi-K-invariant function ([19, Lemma III.1.4]), we need only show that the spherical Fourier transform of fλ is bounded. (Helgason proves Lemma III.1.4 under the assumption that the functions are continuous and of compact support, but the proof also applies more generally.) Meanwhile, standard estimates show that for every ε > 0, the spherical functions are in L2+ε (G/K), with L2+ε (G/K) norm bounded independent of the spherical function. (Speciﬁcally, in the notation of [18, Sect. IV.4], for all λ ∈ a∗ , we have |φλ (g)| ≤ φ0 (g), and estimates on φ0 (e.g., [1, Prop. 2.2.12]) show that φ0 is in L2+ε for all ε > 0.) Choosing ε so that 1/p + 1/(2 + ε) = 1 establishes the desired boundedness. Second proof. If cλ = 0 (i.e., if λ ∈ Z and λ ≤ d), then there is nothing to prove. Thus we assume cλ is nonzero, in which case cλ+1 is also nonzero. The invariance of Fλ under automorphisms together with the square-integrability of the function (1 − |z|2 )λ for λ > d/2 show that the integral deﬁning Aλ f (z) is absolutely convergent for all z. We introduce the (hyperbolic) Laplacian ∆ for Bd , given by ∆ = (1 − |z|2 )

d

(δjk − z¯j zk )

j,k=1

∂2 . ∂ z¯j ∂zk

(5.6)

(This is a negative operator.) This operator commutes with the automorphisms of Bd . It is known (e.g., [28]) that ∆ is an unbounded self-adjoint operator on L2 (Bd , τ ), on the domain consisting of those f ’s in L2 (Bd , τ ) for which ∆f in the distribution sense belongs to L2 (Bd , τ ). In particular, if f ∈ L2 (Bd , τ ) is C 2 and ∆f in the ordinary sense belongs to L2 (Bd , τ ), then f ∈ Dom(∆). We now claim that ∆z Fλ (z, w) = λ(λ − d)(Fλ (z, w) − Fλ+1 (z, w)),

(5.7)

where ∆z indicates that ∆ is acting on the variable z with w ﬁxed. Since ∆ commutes with automorphisms, it again suﬃces to check this when w = 0, in which case it is a straightforward algebraic calculation. Suppose, then, that φ is a C ∞ function of compact support. In that case, we are free to diﬀerentiate

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under the integral to obtain ∆Aλ φ = λ(λ − d)Aλ φ − λ(λ − d)Aλ+1 φ. 2

(5.8)

d

Now, the invariance of Fλ tells us that L (B , τ ) norm of Fλ (z, w) as a function of z is ﬁnite for all w and independent of w. Putting the L2 norm inside the integral then shows that Aλ φ and Aλ+1 φ are in L2 (Bd , τ ). This shows that Aλ φ is in Dom(∆). Furthermore, the condition λ > d/2 implies that λ(λ − d/2) > −d2 /4. It is known that the L2 spectrum of ∆ is (−∞, −d2 /4]. For general symmetric space of the noncompact type, the 2 L2 spectrum of the Laplacian is (−∞, − ρ ], where ρ is half the sum of the positive (restricted) roots for G/K, counted with their multiplicity. In our case, there is one positive root α with multiplicity (2d − 2) and another positive root 2α with multiplicity 1. (See the entry for “A IV” in Table VI of Chapter X of [17].) Thus, ρ = dα. It remains only to check that if the metric is normalized so that the Laplacian comes out as in (5.6), then α 2 = 1/4. This is a straightforward but unilluminating computation, which we omit. Since λ(λ − d) is in the resolvent set of ∆, we may rewrite (5.8) as Aλ φ = −λ(λ − d)[∆ − λ(λ − d)I]−1 Aλ+1 φ. Suppose now that λ + 1 > d, so that (as remarked above) Aλ+1 is bounded. Since [∆ − cI]−1 is a bounded operator for all c in the resolvent of ∆, we see that Aλ has a bounded extension from Cc∞ (Bd ) to L2 (Bd , τ ). Since the integral computing Aλ φ(z) is a continuous linear functional on L2 (Bd , τ ) (integration against an element of L2 (Bd , τ )), it is easily seen that this bounded extension coincides with the original deﬁnition of Aλ . The above argument shows that Aλ is bounded if λ > d/2 and λ+1 > d. Iteration of the argument then shows boundedness for all λ > d/2. Proof of Theorem 12. We wish to show that for all λ > d/2, if φ is in L2 (Bd , τ ), then there is a unique Hilbert–Schmidt operator Tφ with matrix entries given in (5.1) for all polynomials, and furthermore, Tφ 2HS = φ, Aλ φ λ . At the beginning of this section, we had an calculation of Tφ in terms of Aλ , but this argument relied on writing Tφ as Pλ Mφ Pλ , which does not make sense for λ ≤ d. We work with an orthonormal basis for H(Bd , λ) consisting of normalized monomials, namely, Γ(λ + |m|) em (z) = z m , m!Γ(λ) for each multi-index m. Then we want to establish the existence of a Hilbert– Schmidt operator whose matrix entries in this basis are given by el (z)φ(z)em (z)(1 − |z|2 )λ dτ (z). (5.9) alm := cλ Bd

There will exist a unique such operator provided that

l,m

|alm |2 < ∞.

K. Chailuek and B. C. Hall

74

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If we assume, for the moment, that Fubini’s Theorem applies, we obtain Γ(λ + |l|) Γ(λ + |m|) l l m m 2 2 z¯ w z w |alm | = cλ ¯ l!Γ(λ) m!Γ(λ) d d B B l,m

l,m

× φ(z)φ(w)(1 − |z|2 )λ (1 − |w|2 )λ dτ (z) dτ (w),

(5.10)

where l and m range over all multi-indices of length d. We now apply the binomial series ∞ 1 λ+k−1 k = r k (1 − r)λ k=0

for r ∈ C with |r| < 1, where λ+k−1 Γ(λ + k) . = k k!Γ(λ) (This is the so-called negative binomial series.) We apply this with r = z ¯ w j j , and we then apply the (ﬁnite) multinomial series to the compuj tation of (¯ z · w)k . The result is that Γ(λ + |l|) 1 z¯l wl = , (5.11) l!Γ(λ) (1 − z¯ · w)λ l

where the sum is over all multi-indices l. Applying this result, (5.10) becomes 2 |alm | = φ, Aλ φ λ , (5.12) l,m

which is what we want to show. Assume at ﬁrst that φ is “nice,” say, continuous and supported in a ball of radius r < 1. This ball has ﬁnite measure and φ is bounded on it. Thus, if we put absolute values inside the sum and integral on the right-hand side of (5.10), ﬁniteness of the result follows from the absolute convergence of the series (5.11). Thus, Fubini’s Theorem applies in this case. Now for a general φ ∈ L2 (Bd , τ ), choose φj converging to φ with φj “nice.” Then (5.12) tells us that Tφj is a Cauchy sequence in the space of Hilbert–Schmidt operators, which therefore converges in the Hilbert–Schmidt norm to some operator T. The matrix entries of Tφj in the basis {em } are by construction given by the integral in (5.9). The matrix entries of T are the limit of the matrix entries of Tφj , hence also given by (5.9), because el and em are bounded and (1 − |z|2 )λ belongs to L2 (Bd , τ ) for λ > d/2. We can now establish that (5.2) in Theorem 12 holds for all bounded holomorphic functions f and g in H(Bd , λ) by approximating these functions by polynomials. Proof of Theorem 13. In the proof of Theorem 12, we did not use the assumption λ > d/2 until the step in which we approximated arbitrary functions in L2 (Bd , τ ) by “nice” functions. In particular, if φ is nice, then (5.9) makes

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sense for all λ > 0, and (5.12) still holds. Now, since Fλ (z, w) = fλ (φw (z)), where fλ is given by (5.5), we see that |Fλ (z, w)| ≤ c2λ for all z, w ∈ Bd . Thus, 2

φ, Aλ φ λ ≤ c2λ φ L1 (Bd ,τ ) for all nice φ. An easy approximation argument then establishes the existence of a Hilbert–Schmidt operator with the desired matrix entries for all φ ∈ L1 (Bd , τ ), with the desired estimate on the Hilbert–Schmidt norm.

Acknowledgment The authors thank M. Engliˇs for pointing out to them several useful references and B. Driver for useful suggestions regarding the results in Section 4. This article is an expansion of the Ph.D. thesis of the ﬁrst author, written under the supervision of the second author. We also thank the referee for helpful comments and corrections.

References [1] J.-P. Anker and L. Ji, Heat kernel and Green function estimates on noncompact symmetric spaces. Geom. Funct. Anal. 9 (1999), 1035-1091. [2] J. Arazy, Integral formulas for the invariant inner products in spaces of analytic functions on the unit ball. Function spaces (Edwardsville, IL, 1990), 9–23, Lecture Notes in Pure and Appl. Math., 136, Dekker, New York, 1992. [3] J. Arazy and G. Zhang, Homogeneous multiplication operators on bounded symmetric domains. J. Funct. Anal. 202 (2003), 44–66. [4] F. Beatrous Jr. and J. Burbea, Holomorphic Sobolev spaces on the ball. Dissertationes Math. (Rozprawy Mat.) 276 (1989), 60 pp. [5] F. A. Berezin, Quantization. Math. USSR Izvestija, 8 (1974), 1109-1165. [6] F. A. Berezin, Quantization in complex symmetric spaces. Math. USSR Izvestija 9 (1976), 341-379. [7] M. Bordemann, E. Meinrenken, and M. Schlichenmaier, Toeplitz quantization of K¨ ahler manifolds and gl(N ), N → ∞ limits. Comm. Math. Phys. 165 (1994), 281–296. [8] D. Borthwick, A. Lesniewski, and H. Upmeier, Nonperturbative deformation quantization of Cartan domains. J. Funct. Anal. 113 (1993), 153–176. [9] D. Borthwick, T. Paul, and A. Uribe, Legendrian distributions with applications to relative Poincar´e series. Invent. Math. 122 (1995), 359–402. [10] L. A. Coburn, Deformation estimates for the Berezin-Toeplitz quantization. Comm. Math. Phys. 149 (1992), 415–424. [11] M. Cowling, The Kunze–Stein phenomenon. Ann. Math. 107 (1978), 209-234. [12] P. Duren and A. Schuster, Bergman spaces. Mathematical surveys and monographs; no.100, American Mathematical Society, 2004.

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[13] M. Engliˇs, Berezin transform and the Laplace-Beltrami operator. Algebra i Analiz 7 (1995), 176–195; translation in St. Petersburg Math. J. 7 (1996), 633–647. [14] G. Folland, Harmonic analysis in phase space. Princeton University Press, 1989. [15] B. C. Hall, Holomorphic methods in analysis and mathematical physics. In: First Summer School in Analysis and Mathematical Physics (S. P´erez-Esteva and C. Villegas-Blas, Eds.), 1–59, Contemp. Math., 260, Amer. Math. Soc., 2000. [16] B. C. Hall and W. Lewkeeratiyutkul, Holomorphic Sobolev spaces and the generalized Segal–Bargmann transform. J. Funct. Anal. 217 (2004), 192–220. [17] S. Helgason, Diﬀerential geometry, Lie groups, and symmetric spaces. Academic Press, 1978. [18] S. Helgason, Groups and geometric analysis: Integral geometry, invariant diﬀerential operators, and spherical functions, corrected reprint of the 1984 edition. Amer. Math. Soc., 2000. [19] S. Helgason, Geometric analysis on symmetric spaces. Amer. Math. Soc., 1994. [20] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman spaces. Springer-Verlag, 2000. [21] H. T. Kaptano˘ glu, Besov spaces and Bergman projections on the ball. C. R. Math. Acad. Sci. Paris 335 (2002), 729–732. [22] S. Klimek and A. Lesniewski, Quantum Riemann surfaces I. The unit disc. Comm. Math. Phys., 46 (1976) 103-122. [23] A. Konechny, S. G. Rajeev, and O. T. Turgut, Classical mechanics on Grassmannian and disc. In: Geometry, integrability and quantization (Varna, 2000), 181–207, Coral Press Sci. Publ., Soﬁa, 2001. [24] R. A. Kunze and E. M. Stein, Uniformly bounded representations and harmonic analysis of the 2 × 2 unimodular group. Amer. J. Math. 82 (1960), 1-62. [25] S. G. Rajeev and O. T. Turgut, Geometric quantization and two-dimensional QCD. Comm. Math. Phys. 192 (1998), 493–517. [26] J. H. Rawnsley, Coherent states and K¨ ahler manifolds. Quart. J. Math. Oxford Ser. (2) 28 (1977), 403–415. [27] J. H. Rawnsley, M. Cahen, S. Gutt, Quantization of K¨ ahler manifolds. I. Geometric interpretation of Berezin’s quantization. J. Geom. Phys. 7 (1990), 45–62. [28] R. Strichartz, Harmonic analysis as spectral theory of Laplacians. J. Funct. Anal. 87 (1989), 51–148. [29] A. Unterberger and H. Upmeier The Berezin transform and invariant diﬀerential operators. Comm. Math. Phys, 164 (1994), 563-597. [30] Z. Yan, Invariant diﬀerential operators and holomorphic function spaces. J. Lie Theory 10 (2000), 31 pp. [31] R. Zhao and K. H. Zhu, Theory of Bergman spaces in the unit ball of Cn . Preprint, arxiv.org/abs/math/0611093.

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[32] K. Zhu, Holomorphic Besov spaces on bounded symmetric domains. Quart. J. Math. Oxford Ser. (2) 46 (1995), 239–256. [33] K. Zhu, Spaces of holomorphic functions in the unit ball. Springer-Verlag, 2004. Kamthorn Chailuek Department of Mathematics Prince of Songkla University Hatyai Songkhla 90112 Thailand e-mail: [email protected] Brian C. Hall Department of Mathematics University of Notre Dame 255 Hurley Building Notre Dame, IN 46556-4618 USA e-mail: [email protected] Submitted: January 21, 2009. Revised: July 10, 2009.

Integr. Equ. Oper. Theory 66 (2010), 79–112 DOI 10.1007/s00020-009-1733-7 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

Interpolation of Banach Lattices and Factorization of p-Convex and q-Concave Operators Yves Raynaud and Pedro Tradacete ˇ Abstract. We extend a result of Sestakov to compare the complex inon-Lozanovskii’s construction terpolation method [X0 , X1 ]θ with Calder´ X01−θ X1θ , in the context of abstract Banach lattices. This allows us to prove that an operator between Banach lattices T : E → F which is p-convex and q-concave, factors, for any θ ∈ (0, 1), as T = T2 T1 , where p q )-convex and T1 is ( 1−θ )-concave. T2 is ( θ+(1−θ)p Mathematics Subject Classiﬁcation (2010). Primary 47B60; Secondary 46B42, 46B70. Keywords. Interpolation of Banach lattices, method of Calder´ on-Lozanovskii, factorization, p-convex operator, q-concave operator.

1. Introduction In [10], J. L. Krivine showed that the composition T2 T1 of a p-convex operator T1 : X → E and a p-concave operator T2 : E → Y , where X, Y are Banach spaces and E is a Banach lattice, factors always through a space Lp (µ). Motivated by this fact, in this note we study factorization properties of pconvex and q-concave operators. More precisely, we consider the following question: if an operator between Banach lattices T : E → F is both p-convex and q-concave, does it necessarily factor as T = T2 T1 where T2 is p-convex and T1 q-concave? Note that such a product is always p-convex and q-concave, hence we are interested in a converse statement. In general, the answer to this question is negative (see Examples 4 and 5). However, we show that for every θ ∈ (0, 1), the operator T can be

The second author was partially supported by grants MICINN MTM2008-02652, AP-20044841, Grupo UCM 910346 and Santander/Complutense PR34/07-15837.

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p q written as T = T2 T1 where T2 is ( θ+(1−θ)p )-convex and T1 is ( 1−θ )-concave (see Theorem 15). To prove this fact, we exhibit ﬁrst a canonical way in which a p-convex (respectively q-concave) operator factors through a p-convex (resp. q-concave) Banach lattice. Afterwards, we present some interpolation results regarding the complex interpolation method and the Calder´ on-Lozanovskii construction for Banach lattices. In particular, we prove a comparison theorem between these two constructions that had been apparently known in the literature only in the case of Banach lattices of measurable functions. ˇ Thus, we extend this comparison theorem due to Sestakov (see [19]) to the more general setting of compatible pairs of Banach lattices which need not be function spaces (that is, ideals in the space of measurable functions on some measure space). This will constitute a key ingredient in our proof of the main factorization result.

The problem of factoring an operator through p-convex and q-concave operators had also been considered, although in a quite diﬀerent manner, by S. Reisner in [17]; in particular, Theorem 1 was essentially proved in [17, Sec. 2, Lemma 6]. Moreover, this author showed that for ﬁxed p, q, the class of operators between Banach spaces T : E → F such that the composition with the canonical inclusion jF : F → F ∗∗ factors as jF T = U V with V pconvex and U q-concave, forms an operator ideal. However, his approach to an analogous statement of Theorem 1 for p-convex operators is not satisfactory for our interests, because in [17] this is only considered as a dual fact to that of factoring q-concave operators, and, as we will show in Section 3, these factorizations do not behave in an entirely dual way. Moreover, from Theorem 1 we can only get that for a p-convex operator T : E → F , the bi-adjoint T ∗∗ : E ∗∗ → F ∗∗ factors through a p-convex Banach lattice, which suﬃces for the purposes in [17], but are not enough to prove our main result on factorization (Theorem 15). We mention that our proofs of Theorems 1 and 3 have been inspired in fact by the work of P. Meyer-Nieberg in [14] on factorization of cone p-summing and p-majorizing operators (see also [15, 2.8]). Then we realized that some of the main ideas of our work were already present in the paper [17]. The organization of the paper goes as follows. Section 2 contains the proofs of the basic factorizations for p-convex (resp. q-concave) operators. It is also shown that these constructions can be equivalently obtained by means of maximality properties of factorization diagrams. The next section, Section 3, is devoted to the study of the duality relation between the factorization spaces for p-convex and q-concave operators. Next, Section 4 is ˇ mainly devoted to the proof of the extension of Sestakov’s result to compatible pairs of Banach lattices. Then, in Section 5 we prove the main theorem on factorization of operators which are both p-convex and q-concave. Here ˇ the extension of Sestakov’s result is used for interpolating operators which are not necessarily positive (at the diﬀerence of the situation in [17]) between Banach lattices which are perhaps not representable as ideal function spaces. In this section we show also how some examples can be used to see that in

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general the factorization cannot be improved much further. Finally, in Section 6 we show the connection between the constructions of the ﬁrst section and the factorization theorem of Krivine. We refer the reader to [11], [15] and [18] for any unexplained terminology on Banach lattice theory, and to [4] and [9] for those of interpolation theory.

2. Two basic constructions Let E be a Banach lattice and X a Banach space. Recall that an operator T : E → X is q-concave for 1 ≤ q ≤ ∞, if there exists a constant M < ∞ so that 1q 1q n n q q , T xi ≤ M |xi | if 1 ≤ q < ∞, i=1

or

i=1

n |xi | max T xi ≤ M , 1≤i≤n

if q = ∞,

i=1

for every choice of vectors (xi )ni=1 in E (cf. [11, 1.d]). The smallest possible value of M is denoted by M(q) (T ). Similarly, an operator T : X → E is p-convex for 1 ≤ p ≤ ∞, if there exists a constant M < ∞ such that p1 p1 n n p p |T xi | xi , if 1 ≤ p < ∞, ≤M i=1

or

i=1

n |T xi | max xi , ≤ M 1≤i≤n

if p = ∞,

i=1

for every choice of vectors (xi )ni=1 in X. The smallest possible value of M is denoted by M (p) (T ). Recall that a Banach lattice is q-concave (resp. pconvex) whenever the identity operator is q-concave (resp. p-convex). The following result was essentially proved in [17, Sec. 2, Lemma 6]. However, we include a similar proof for completeness, since we will be using the explicit construction throughout. Theorem 1. Let E be a Banach lattice, X a Banach space and 1 ≤ q ≤ ∞. An operator T : E → X is q-concave if and only if there exist a q-concave Banach lattice V , a positive operator φ : E → V (in fact, a lattice homomorphism with dense image), and another operator S : V → X such that T = Sφ. E@ @@ @@ φ @@

T

V

/X > ~ ~~ ~ ~ ~~ S

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Proof. Let us suppose q < ∞. The proof for the case q = ∞ is trivial because every Banach lattice is ∞-concave. However, the precise construction carried out here for q < ∞ has its analogue for q = ∞. For the “if” part, let (xi )ni=1 in E. Since V is q-concave and φ is positive, by [11, Prop. 1.d.9] we have n 1q 1 n q q , T xi q ≤ SM(q) (IV )φ |x | i i=1

i=1

which yields that T is q-concave. Now, for the other implication, given x ∈ E, let us consider q1 1q n n ρ(x) = sup T xi q : |xi |q ≤ |x| . i=1

i=1

If M(q) (T ) denotes the q-concavity constant of T , then for (xi )ni=1 in E, we have n q1 1q n q q . T xi ≤ M(q) (T ) |xi | i=1

i=1

In particular, for all x ∈ E

T x ≤ ρ(x) ≤ M(q) (T )x. Moreover, ρ is a lattice semi-norm on E. Indeed, for any x ∈ E and λ ≥ 0 it is clear that ρ(λx) = λρ(x). In order to prove the triangle inequality, let x, y ∈ E and z = |x| + |y|, and denote Iz ⊂ E the ideal generated by z in E, which is identiﬁed with a space C(K) in which z corresponds to the function identically one [18, II.7]. Now, for every ε > 0 let z1 , . . . , zn ∈ E n

1 |zi |q q ≤ |z| and such that i=1

ρ(z) ≤

n

T zi q

1q + ε.

i=1

Since x, y ∈ Iz , they correspond to functions f, g ∈ C(K) such that |f (t)| + |g(t)| = 1 for every t ∈ K. Similarly, zi corresponds to hi ∈ C(K) with n

1 q q ≤ 1 for every t ∈ K. Hence we can consider i=1 |hi (t)| fi (t) = hi (t)f (t), gi (t) = hi (t)g(t), which belong to C(K) and satisfy 1q 1q n n q q |fi (t)| ≤ |f (t)| and |gi (t)| ≤ |g(t)|. i=1

i=1

This means that we can consider (xi )ni=1 and (yi )ni=1 such that q1 q1 n n q q |xi | ≤ |x| and |yi | ≤ |y| i=1

i=1

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in E, with xi + yi = zi . Thus, it follows that ρ(x + y) ≤ ρ(x) + ρ(y) + ε, and since this holds for every ε > 0, the triangle inequality is proved. n q1 n q |xi | ≤ |y|, it Now, if |y| ≤ |x|, then for any (xi )i=1 such that i=1 1q 1q n n |xi |q ≤ |x|, hence for any such {xi }ni=1 , T xi q ≤ holds that i=1

i=1

ρ(x). This implies that ρ(y) ≤ ρ(x). Let V denote the Banach lattice obtained by completing E/ρ−1 (0) with the norm induced by ρ. Let φ denote the quotient map from E to E/ρ−1 (0), seen as a map to V . Now, for x ∈ E let us deﬁne S(φ(x)) = T (x). Since T x ≤ ρ(x), S is well deﬁned and extends to a bounded operator S : V → X, such that T = Sφ. Now, let (xi )ni=1 in E. For every ε > 0 and for every i = 1, . . . , n there

1 ki i q q i exist {yji }kj=1 in E such that ≤ |xi | and j=1 |yj | ρ(xi )q = sup

k

T yj q :

k

j=1

|yj |q

1q

j=1

ki εq ≤ |xi | ≤ T yji q + , n j=1

for every i = 1, . . . , n. Therefore, we have n i=1

q

ρ(xi )

1q

≤ρ

n

|xi |

q

1q + ε,

i=1

and since this holds for every ε > 0, the normed lattice E/ρ−1 (0) is q-concave; hence, the same holds for its completion V . Since the lattice V constructed in the proof depends on the operator T : E → X and q, we will denote it by VT,q whenever needed. Similarly we will denote ρT for the expression deﬁning the norm of VT,q . Remark 1. Note that VT,q has q-concavity constant one. In particular if E is q-concave and T = idE is the identity, then VT,q is the usual lattice renorming of E with q-concavity constant one. Remark 2. In [8], it was proved that an order weakly compact operator T : E → Y (i.e. T [−x, x] is relatively weakly compact for every x ∈ E+ ) always factors through an order continuous Banach lattice F . The Banach lattice F is constructed by means of the expression xF = sup{T y : |y| ≤ |x|}, for x ∈ E, which yields a Banach lattice in the usual way. Notice that if T : E → Y is q-concave, which implies being order weakly compact, then i xF ≤ ρT (x), hence we can consider a natural map VT,q → F such that we

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can factor T as follows:

φ

VT,q

/Y O

T

E

T

/F

i

Moreover, F coincides with VT,∞ , so in a sense the previous Theorem is an extension of [8, Thm. I.2]. The factorization given in Theorem 1 is in a certain sense maximal, as the following Proposition shows. Proposition 2. Let T : E → X be a q-concave operator. Suppose that T factors through a q-concave Banach lattice V with factors A : E → V and B : V → X, such that A is a lattice homomorphism whose image is dense in V , and T = B ◦ A. Then there is a lattice homomorphism u : V → VT,q such that φ = u ◦ A and S ◦ u = B. T / E =NNN p7 @ X p == NNN p B ppp == NNAN ppp == NNN p p NN' ppp == == V S φ == == == u = VT,q

Proof. Let us deﬁne for x ∈ E, u(A(x)) = φ(x). Notice that, since A is a n 1q |xi |q ≤ |x|, we lattice homomorphism, for {xi }ni=1 in E, such that i=1

have n i=1

T xi

q

q1 =

n

BAxi

q

1q

≤ B

i=1

n

Axi

q

1q

i=1

1q n q ≤ BM(q) (IV ) |A(x )| i i=1

n q1 q = BM(q) (IV )A |xi | i=1

≤ BM(q) (IV )A(x). Therefore, u(A(x)) = φ(x) = ρT (x) ≤ BM(q) (IV )A(x). Since A has dense image, the preceding inequality implies that u can be extended to a bounded operator u : V → V(T,q) , which is clearly a lattice homomorphism and satisﬁes the required properties.

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There is an analogous version of Theorem 1 for p-convex operators, which could be considered, in a sense, as a predual construction to that given in Theorem 1 (see Section 3). Theorem 3. Let E be a Banach lattice, X a Banach space and 1 ≤ p ≤ ∞. An operator T : X → E is p-convex if and only if there exist a p-convex Banach lattice W , a positive operator (in fact, an injective interval preserving lattice homomorphism) ϕ : W → E and another operator R : X → W such that T = ϕR. T /E XB > BB } BB }} } B }} ϕ R BB }} W Proof. Let us suppose p < ∞. The proof for the case p = ∞ is analogous, with the usual changes. As in the proof of Theorem 1, [11, 1.d.9] yields one implication. For the non-trivial one, let T : X → E be p-convex. Let us consider the following set p1 k k |T xi |p , where xi p ≤ 1 and k ∈ N}. S = {y ∈ E : |y| ≤ i=1

i=1

We can consider the Minkowski functional deﬁned by its closure S in E zW = inf{λ > 0 : z ∈ λS}. Clearly S is solid, and since T is p-convex, it is also a bounded set of E. Let us consider the space W = {z ∈ E : zW < ∞}. We claim that for any k

1/p z1 , . . . , zn in W , it holds that |zi |p belongs to W and i=1

p1 k p |z | i

≤

W

i=1

p1 n zi pW . i=1

Indeed, given z1 , . . . , zn in W , for every ε > 0 and for every i = 1, . . . , n there exist λi with zi ∈ λi S, such that εp p p λi ≤ inf µ : zi ∈ µS + , n for each i = 1, . . . , n. This means that for every i = 1, . . . , n, and for every δ > 0 there exists yiδ in E with zi − yiδ E ≤ δ, and m p1 i,δ i,δ p δ |T xj | , |yi | ≤ j=1

where

mi,δ {xi,δ j }j=1

satisfy m i,δ j=1

p xi,δ j

p1

≤ λi ,

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for each i = 1, . . . , n, and each δ > 0. Now, for each δ > 0 let p1 n δ p wδ = |yi | . i=1

Notice that n n p1 p1 n p δ p |z | − w ≤ |z − y | ≤ zi − yiδ E ≤ nδ. i δ i i E

i=1

E

i=1

Moreover, note that for every δ > 0, wδ belongs to |wδ | =

n

|yiδ |p

p1

≤

i,δ n m

i=1

and

Hence,

n

|zi |

p

p1

i=1

n

p1 |zi |p i=1

p xi,δ j

∈

n

λpi i=1

≤

W n i=1

λpi

p1 S. Indeed,

i=1 p |T xi,δ j |

p1 ,

n

λpi

p1 .

i=1

p1

= inf{µ > 0 : ≤

p1

i=1 j=1

n

i=1 j=1

i,δ n m

i=1

S. Therefore, it follows that n

|zi |

p

p1

∈ µS} ≤

i=1

n

p

1

ε inf µp : zi ∈ µS + n

p

λpi

p1

i=1

≤

n

zi pW

p1

+ ε.

i=1

Since this holds for every ε > 0, we ﬁnally have n 1 p1 n p p p |z | ≤ z . i i W i=1

W

i=1

It follows that the Minkowski functional .W is a norm on W . Indeed since S is bounded, xW = 0 implies x = 0. Moreover if x, y ∈ W are |x| |y| xW yW , v = y , α = xW non zero, set u = x +yW , β = xW +yW . Since W W uW = vW = 1, α, β ≥ 0 and α + β = 1 we have x + yW ≤ |x| + |y| W = (xW + yW )αu + βvW ≤ (xW + yW )(αup + βv p )1/p W 1/p

≤ (xW + yW )(αupW + βvp )W = xW + yW Therefore, (W, · W ) is a p-convex normed lattice. We claim that W is complete, and hence a p-convex Banach lattice. Indeed, let (wi )∞ i=1 be a Cauchy sequence in W . Since for every z ∈ E it holds that zE ≤ M (p) (T )zW ,

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it follows that (wi )∞ i=1 is also a Cauchy sequence in E. Let w ∈ E be its limit. Notice that since wi are bounded in W , there exists some λ < ∞ such that wi ∈ λS for every i = 1, 2, . . . and since S is closed in E, we must have w ∈ λS. Thus, w belongs to W , and we will show that (wi )∞ i=1 converges to w also in W . To this end, let ε > 0. Since (wi )∞ is a Cauchy sequence, there i=1 exists N such that wi − wj ∈ εS when i, j ≥ N . Thus, if i ≥ N we can write w − wi = (w − wj ) + (wj − wi ) for every j ∈ N, and letting j → ∞ we obtain that w − wi ∈ εS. This shows that wi → w in W , and hence W is complete, as claimed. Clearly, by the deﬁnition of W we have T xW ≤ xX for every x ∈ X. Moreover, as noticed above it also holds that zE ≤ M (p) (T )zW for each z ∈ E, therefore the formal inclusion ϕ : W → E is clearly an injective interval preserving lattice homomorphism, and we have the following diagram T /E XB > BB } BB }} } B } R BB }} ϕ .} W

where R is deﬁned by Rx = T x for x ∈ X. This ﬁnishes the proof.

As with the Banach lattice constructed in Theorem 1, we will denote by WT,p the Banach lattice obtained in the proof of Theorem 3. Remark 3. The operator ϕ : WT,p → E constructed in the proof is an injective, interval preserving lattice homomorphism. Moreover, it satisﬁes that the image of the closed unit ball ϕ(BWT ,p ) is a closed set in E. This let us introduce the class C consisting of operators T : E → F between Banach lattices which are injective, interval preserving lattice homomorphisms, such that the image of the closed unit ball T (BE ) is closed in F . The importance of this class will become clear next. Remark 4. Note that if T : X → E is p-convex, then it is also p -convex for every 1 ≤ p ≤ p. Hence, if we consider the factorization spaces WT,p and WT,p it always holds that WT,p → WT,p , with norm smaller than or equal to one. Indeed, this follows from the following two facts. First, the set p1 k k p S = {y ∈ E : |y| ≤ |T xi | , with xi p ≤ 1} i=1

i=1

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can be equivalently described by p1 k k p S = {y ∈ E : |y| ≤ ai |T wi | , with wi ≤ 1, ai ≥ 0, ai = 1}. i=1

i=1

Furthermore, for 1 ≤ p ≤ p, and ai ≥ 0 with

k

ai = 1 it always holds that

i=1

k

ai |T wi |p

1 p

≤

k

i=1

ai |T wi |p

p1 .

i=1

Hence the unit ball of WT,p is contained in that of WT,p . Remark 5. WT,p has p-convexity constant equal to one. If E is already pconvex and T : E → E is the identity then WT,p is a renorming of E with p-convexity constant one. As for Proposition 2, the construction of Theorem 3 is in a sense minimal. Proposition 4. Let T : X → E be a p-convex operator, such that there exist and operators A : X → W and B : W →E a p-convex Banach lattice W with T = BA and B belonging to the class C. Then there exists an operator such that vR = A and Bv = ϕ. v : WT,p → W T / X >OOO oo7 @ E >> OOO o o B oo >> OOAO ooo >> OOO o o OO' >> ooo >> W ϕ O R >> >> >> v > WT,p

Proof. Let us deﬁne v. Let y ∈ WT,p with yWT ,p ≤ 1. By deﬁnition, there exists a sequence (yn )∞ n=1 in E such that yn → ϕ(y) in E, and for each n ∈ N, |yn | ≤ with

|T xni |p

p1

i=1

kn

n p i=1 xi X kn

≤ 1. Notice that since B is a lattice homomorphism

|T xni |p

p1

i=1

kn

=

kn i=1

p1

|BAxni |p

p1

=B

kn

|Axni |p

p1

,

i=1

. Hence, since B is interval preserving belongs to W

1 with |wn | ≤ kn |Axn |p p such that B(wn ) = yn . there exists wn ∈ W i i=1 where

kn i=1

|Axni |p

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is p-convex and kn xi p ≤ 1, for every n we have Notice that since W i=1 X kn

p1 |Axni |p wn W ≤ i=1

W

kn

p1 ≤ M (p) (IW Axni p ≤ M (p) (IW ) )A. i=1

under B is closed, and B(wn ) = yn Now, since the image of the unit ball of W with w ≤ M (p) (I )A such converge to ϕ(y) ∈ E, there exists w ∈ W W W that B(w) = ϕ(y). Moreover, this element is unique because B is injective. Let us deﬁne v(y) = w. is linear. It is clear, because of the injectivity of B, that v : WT,p → W Moreover, by the previous argument v is a bounded operator of norm less than or equal to M (p) (IW )A. It is clear by construction that B ◦ v = ϕ. Moreover, since B ◦ A = T = ϕ ◦ R = B ◦ v ◦ R and B is injective, we also get that A = v ◦ R as desired. This ﬁnishes the proof. Remark 6. Notice that the factorizations given in Theorems 1 and 3 also make sense in the more general context of quasi-Banach lattices and for pconvex or q-concave operators with p, q ∈ (0, ∞) (not necessarily p, q ≥ 1). It can be seen that in these cases, the factorization spaces are quasi-Banach lattices which need not be locally convex, except in the case when p ≥ 1.

3. Duality relations In this section we show the precise relation between the Banach lattices constructed in the proofs of Theorems 1 and 3. Namely we will prove the following Theorem 5. Let T : X → E be p-convex. By Theorem 3, T can be factored through WT,p ; moreover, since T ∗ : E ∗ → X ∗ is q-concave for 1p + 1q = 1 (see [11, Prop. 1.d.4]), T ∗ can also be factored through VT ∗ ,q . It holds that: 1. VT ∗ ,q is lattice isometric to a sublattice of (WT,p )∗ , 2. WT,p is lattice isometric to a sublattice of (VT ∗ ,q )∗ . Moreover, under this identiﬁcations VT ∗ ,q is always an ideal in (WT,p )∗ , and if E is order continuous WT,p is an ideal of (VT ∗ ,q )∗ . We need some preliminary lemmas ﬁrst. ∗ . Lemma 6. Let E be a Banach lattice, x, y ∈ E+ , x ∧ y = 0, and z ∗ ∈ E+ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ There exist u , v in E+ such that z = u + v , u ∧ v = 0 and ∗

z , x = u∗ , x

z ∗ , y = v ∗ , y

Proof. By [15, Lemma 1.4.3], there exist ∗

z (x), u = z ∗ , u

z ∗ (x), u = 0 ∗

z (y), u = z ∗ , u

z ∗ (y), u = 0

∗ z ∗ (x) and z ∗ (y) in E+ such that

for all u ∈ Ex for all u ∈ {x}⊥ for all u ∈ Ey for all u ∈ {y}⊥

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where Ex denotes the principal ideal generated by x in E, and {x}⊥ denotes the orthogonal complement of x (i.e. {x}⊥ = {u ∈ E : u ∧ x = 0}). Moreover, without loss of generality we can assume that z ∗ (x), z ∗ (y) ≤ ∗ z (simply consider z ∗ (x) ∧ z ∗ and z ∗ (y) ∧ z ∗ ), and that z ∗ (x) ∧ z ∗ (y) = 0 (consider z ∗ (x) − z ∗ (x) ∧ z ∗ (y) and z ∗ (y) − z ∗ (x) ∧ z ∗ (y)). Let then P be the band projection onto the band generated by z ∗ (x) in the Dedekind complete Banach lattice E ∗ and Q be the complementary band projection. Then set u∗ = P z ∗ , v ∗ = Qz ∗ . ∗ Lemma 7. Let E be a Banach lattice. For any z ∗ ∈ E+ , and x1 , . . . , xn ∈ E+ , n ∗ , such that z ∗ = x∗i , x∗i ∧ x∗j = 0 for i = j, and there exist x∗1 , . . . , x∗n in E+ i=1

z ∗ ,

n

xi =

i=1

n

x∗i , xi .

i=1

Proof. Given x, y ∈ E+ , Lemma 6 applied to x − x ∧ y and y − x ∧ y yields the result for n = 2. An easy induction on n completes the proof. Recall that given a Banach space X, the polar of a set A in X is the set A0 = {x∗ ∈ X ∗ : | x∗ , x| ≤ 1, ∀x ∈ A}. Similarly, for a set B in X ∗ , the prepolar of B is the set B0 = {x ∈ X : | x∗ , x| ≤ 1, ∀x∗ ∈ B}. Lemma 8. Let T : X → E be p-convex, and let p1 k k p S := {y ∈ E : |y| ≤ |T xi | , with xi p ≤ 1}. i=1

i=1

Since T ∗ : E ∗ → X ∗ is q-concave (with 1p + 1q = 1), we can consider ρT ∗ , the seminorm which induces the norm on VT ∗ ,q (see Theorem 1). Hence, we can also consider the convex set U := {y ∗ ∈ E ∗ : ρT ∗ (y ∗ ) ≤ 1}. Then 0

S = U, where S denotes the closure of S in E. Proof. First of all, we claim that S ⊂ U0 . Indeed, let y ∈ E be such that |y| ≤

n

|T xi |

p

p1 with

i=1

For every y ∗ ∈ E ∗ such that ρT ∗ (y ∗ ) ≤ 1, we have:

n

xi p ≤ 1.

i=1

| y ∗ , y| n p1 |T xi |p ≤ |y ∗ |, i=1 n n ∗ = |y |, sup ai T xi : |ai |q ≤ 1 i=1 i=1 N n n ∗ m m q |y |, : = sup ai T xi |ai | ≤ 1, m = 1, . . . , N, N ∈ N . m=1

i=1

i=1

Where we have made use of [15, Cor. 1.3.4.ii)] in the last step.

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∗ N Now, by Lemma 7, there exist (ym )m=1 pairwise disjoint elements of N ∗ ∗ such that |y | = m=1 ym and N n N n ∗ m = y |y ∗ |, am T x , a T x i i . i m i

∗ E+

m=1

m=1

i=1

Therefore, setting zi∗ =

N

i=1

∗ am i ym , we have

m=1

q1 p1 N n n n ∗ ∗ q p ≤ |y ∗ |, am T x T z x . i i i i m=1

i=1

Note that, since

i=1

∗ N (ym )m=1 n

i=1

are pairwise disjoint we have that

|zi∗ |q

1q

N

≤

∗ ym = |y ∗ |.

m=1

i=1

Since ρT ∗ (y ∗ ) ≤ 1, this implies that any y ∗ with ρT ∗ (y ∗ ) ≤ 1,

n i=1

T ∗ zi∗ q

1q

≤ 1. Therefore, for

| y ∗ , y| N q q1 n n ∗ m ∗ m q ai y m : |ai | ≤ 1, m ≤ N, N ∈ N ≤ sup T ≤ 1.

i=1

m=1

i=1

This means that y ∈ U0 . Since U0 is closed, this proves that S ⊆ U0 as claimed. 0 0 Therefore, it follows that (U0 )0 ⊆ S . So in particular, U ⊆ S . 0 0 Let us prove now the converse inclusion (S ⊆ U ). Given y ∗ ∈ S , we ∗ ∗ ∗ want to show that ρT ∗ (y ) ≤ 1. To this end, let y1 , . . . , yk be elements in E ∗ , such that 1q k |yi∗ |q ≤ |y ∗ |. i=1 0

0

Notice that since S is solid, then so is S . In particular, |y ∗ | ∈ S whenever 0 y∗ ∈ S . k Now, for every ε > 0 there exist x1 , . . . , xk in X, such that xi p ≤ 1, i=1

and

k

T ∗ yi∗ q

1 q

k ≤ | T ∗ yi∗ , xi | + ε.

i=1

i=1

Moreover, by [11, Prop. 1.d.2] we have 1q p1 k k k k ∗ ∗ ∗ ∗ q p | T yi , xi | = | yi , T xi | ≤

|yi | , |T xi | ≤1 i=1

i=1

i=1

i=1

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0

0

because |y ∗ | ∈ S . Therefore, ρT ∗ (y ∗ ) ≤ 1 for every y ∗ ∈ S . This ﬁnishes the proof. 0

Remark 7. Note that the equality S = U proved above, yields in particular that U is weak*-closed. Hence, by the bipolar theorem it also holds that S = U0 . Remark 8. Moreover, notice that Lemma 8 also provides an alternative proof of the fact that the norm · W deﬁned by S in Theorem 3, is a p-convex lattice norm. Now we can give the proof of Theorem 5. Proof of Theorem 5. We stick to the notation of Theorems 1 and 3. Let us consider the inclusion ϕ : WT,p → E. Hence, we also have ϕ∗ : E ∗ → (WT,p )∗ . Notice that for every y ∗ ∈ E ∗ we have ϕ∗ (y ∗ )(WT ,p )∗ = sup{ ϕ∗ (y ∗ ), y : yWT ,p ≤ 1} = sup{ y ∗ , ϕ(y) : y ∈ S} 0

= inf{λ > 0 : y ∗ ∈ λS } = ρT ∗ (y ∗ ), by Lemma 8. Thus ker ϕ∗ ⊃ ρ−1 T (0), which allows us to deﬁne A:

−→ (WT,p )∗ E ∗ /ρ−1 T ∗ (0) −1 ∗ y + ρT ∗ (0) −→ ϕ∗ (y ∗ )

Moreover, A can be extended to an isometry from VT ∗ ,q into (WT,p )∗ . Furthermore, since the unit ball of WT,p is a solid subset of E, then ϕ + is interval preserving (i.e. ϕ([0, x]) = [0, ϕ(x)] for x ∈ WT,p ). Thus, ϕ∗ is a lattice homomorphism (cf. [1, Theorem 1.35]). Now, for v ∈ VT ∗ ,q , we can consider a sequence (yn∗ ) in E ∗ such that lim yn∗ + ρ−1 T ∗ (0) = v in VT ∗ ,q . Hence, n

we have

∗ ∗ A(|v|) = lim A(|yn∗ | + ρ−1 T ∗ (0)) = lim ϕ (|yn |) n

n

= lim |ϕ∗ (yn∗ )| = lim |A(yn∗ + ρ−1 T ∗ (0))|, n

n

which coincides with |A(v)|. Therefore, A is a lattice homomorphism, which implies that VT ∗ ,q is lattice isometric to a sublattice of (WT,p )∗ . In order to see that VT ∗ ,q is in fact an ideal of (WT,p )∗ , let y ∈ (WT,p )∗ with 0 ≤ y ≤ A(x) for some x ∈ VT ∗ ,q . Notice that x = lim φ(x∗n ) in VT ∗ ,q , where (x∗n ) belong to E ∗ . Thus, A(x) = lim A(φ(x∗n )) = lim ϕ∗ (xn ). If we denote yn = y ∧ ϕ∗ (xn ), then we clearly have that yn tends to y in (WT,p )∗ . Moreover, since ϕ is a lattice homomorphism, by [1, Thm. 1.35], it follows that ϕ∗ is interval preserving. Hence, since 0 ≤ yn ≤ ϕ∗ (x∗n ), for every n ∈ N, there exists u∗n ∈ [0, x∗n ], such that yn = ϕ∗ (u∗n ). Notice that ϕ∗ (u∗n )

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tends to y in (WT,p )∗ . In particular, we have ρT ∗ (u∗n −u∗m ) = ϕ∗ (u∗n −u∗m ) → 0 when n, m → ∞, which yields that φ(u∗n ) tends to some u∗ in VT ∗ ,q . By construction, we obtain that A(u∗ ) = y, which implies that A is interval preserving. This shows that VT ∗ ,q is an ideal of (WT,p )∗ , as claimed. On the other hand, we can also deﬁne a mapping B : WT,p → (VT ∗ ,q )∗ . Indeed, given y ∈ S and y ∗ ∈ E ∗ , since S = U0 , we have y ∗ , ϕ(y) ≤ ρT ∗ (y ∗ ). Therefore, for every y ∈ WT,p and y ∗ ∈ E ∗ we get y ∗ , ϕ(y) ≤ ∗ ρT ∗ (y ∗ )yWT ,p . Hence, there exists a unique element B(y) ∈ (E ∗ /ρ−1 T ∗ (0)) such that ∗

y ∗ + ρ−1 T ∗ (0), B(y) = y , ϕ(y) for every y ∗ ∈ E ∗ . Clearly, B(y) is a linear functional which is continuous for the norm in VT ∗ ,q , thus, it can be extended to an element of (VT ∗ ,q )∗ , with B(y)(VT ∗ ,q )∗ ≤ yWT ,p . Hence, B : WT,p → (VT ∗ ,q )∗ is a linear mapping which is bounded of norm ≤ 1. Moreover, for y ∈ WT,p we have B(y)(VT ∗ ,q )∗ = sup{ v, B(y) : vVT ∗ ,q ≤ 1}

= sup{ y ∗ , ϕ(y) : ρT ∗ (y ∗ ) ≤ 1}

which is the value of the Minkowski functional of U0 = S at ϕ(y). Hence, B(y)(VT ∗ ,q )∗ = inf{λ ≥ 0 : ϕ(y) ∈ λS} = yWT ,p . This means that B is an isometry. ∗ Moreover, for y ∗ ∈ E+ and every y ∈ WT,p we have −1 ∗ ∗ ∗

y ∗ + ρ−1 T ∗ (0), |B(y)| = sup{| x + ρT ∗ (0), B(y)| : |x | ≤ y }

= sup{| x∗ , ϕ(y)| : |x∗ | ≤ y ∗ }

= y ∗ , |ϕ(y)|, and since ϕ is a lattice homomorphism we have −1 ∗ ∗

y ∗ + ρ−1 T ∗ (0), |B(y)| = y , ϕ(|y|) = y + ρT ∗ (0), B(|y|). ∗ Since this holds for every y ∗ ∈ E+ , we have that |B(y)| = B(|y|). Therefore, B is a lattice homomorphism and the claimed result follows. To prove the last statement, let u ∈ (VT ∗ ,q )∗ such that 0 ≤ u ≤ B(y) for some y ∈ WT,p . We consider φ : E ∗ → VT ∗ ,q the operator induced by the quotient map. Since φ is positive, so is φ∗ : (VT ∗ ,q )∗ → E ∗∗ . It holds that

φ∗ (u) ≤ φ∗ (B(y)) = ϕ(y). Indeed, for every y ∗ ∈ E ∗ we have

φ∗ (B(y)), y ∗ = B(y), φ(y ∗ ) = ϕ(y), y ∗ . Hence, φ∗ (u) ∈ [0, ϕ(y)] in E ∗∗ . However, if E is order continuous and ϕ(y) belongs to E, then we have [0, ϕ(y)] ⊂ E. Moreover, since ϕ is interval preserving, there exists x ∈ [0, y] in WT,p , such that φ∗ (u) = ϕ(x). This implies that u = B(x), which means that WT,p is an ideal in (VT ∗ ,q )∗ .

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Notice that the isometries A and B given in the proof of Theorem 5 may not be surjective, as the following examples show. Moreover, if E is not order continuous, WT,p may not be an ideal in (VT ∗ ,q )∗ . Example 1. Let T : L∞ (0, 1) → L1 (0, 1) denote the formal inclusion. Clearly, for every 1 ≤ p ≤ ∞, T is p-convex. First, notice that the set n n

p1 |T fi |p , fi pL∞ ≤ 1}, S = {f ∈ L1 (0, 1) : |f | ≤ i=1

i=1

satisﬁes that S = {f ∈ L1 (0, 1) : f L∞ ≤ 1}. This implies that WT,p = L∞ (0, 1). On the other hand, if we consider the adjoint operator T ∗ : L1 (0, 1)∗ → L∞ (0, 1)∗ , which is p -concave (for p1 + p1 = 1), then for f ∈ L∞ (0, 1) = L1 (0, 1)∗ we clearly have T ∗ f L∗∞ = f L1 . From here, it follows that the expression n n

1

1 p p ∗ p p ρT ∗ ,p (f ) = sup T fi L∗∞ : |fi | ≤ |f | , i=1

i=1

trivially satisﬁes f L1 ≤ ρT ∗ ,p (f ). While on the other hand, for f ∈ L∞ (0, 1) and (fi )ni=1 with 1/p n |fi |p ≤ |f | i=1

we have n

T ∗fi pL∗∞

1 p

=

i=1

n

fi pL1

1 p

n

1 p ≤ |fi |p

i=1

i=1

L1

≤ f L1 .

Thus, ρT ∗ ,p (f ) = f L1 , which implies that VT ∗ ,p = L1 (0, 1). Hence, the isometry A : VT ∗ ,p → (WT,p )∗ given in Theorem 5 is not surjective. Example 2. Let T : 1 → ∞ denote the formal inclusion. Clearly, T is ∞-convex. Moreover, it is easy to see that the set S = {y ∈ ∞ : |y| ≤

n

|yi |,

i=1

n

yi 1 ≤ 1},

i=1

satisﬁes S = Bc0 . Hence, WT,∞ = c0 . On the other 1-convex. It is

∗

hand, let T : ∗∞ well known that ∗∞

→ ∗1 be the adjoint operator, which is = ∗∗ 1 can be decomposed as

⊥ ∗∗ 1 = J(1 ) ⊕ J(1 ) ,

where J(1 ) denotes the canonical image of 1 in its bidual, and J(1 )⊥ its disjoint complement.

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Notice that every y ∈ J(1 )⊥ , viewed as an element of ∗∞ , satisﬁes y|c0 = 0. Indeed, for every n ∈ N, let en denote the sequence formed by zeros except 1 in the nth entry. For y ∈ J(1 )⊥ , by disjointness we have 0 = |y| ∧ J(en ), en = inf{ |y|, x + en , z : x, z ∈ + ∞ , x + z = en } = inf{λ |y|, en + 1 − λ : λ ∈ [0, 1]} = |y|, en , for every n ∈ N, which clearly implies y|c0 = 0. In particular, for y ∈ J(1 )⊥ we have T ∗ (y) = sup{ T ∗ (y), x : x ∈ 1 , x1 ≤ 1} = sup{ y, T x : x ∈ 1 , x1 ≤ 1} = 0, since T x ∈ c0 ⊂ ∞ for every x ∈ 1 . Therefore, for y ∈ J(1 )⊥ , since J(1 )⊥ is solid, we have n n ρT ∗ ,1 (y) = sup T ∗ yi ∞ : |yi | ≤ |y| = 0. i=1

i=1

While for y ∈ J(1 ) we have n n T ∗ yi ∞ : |yi | ≤ |y| = y1 . ρT ∗ (y) = sup i=1

i=1

Hence, VT ∗ ,1 = 1 , which implies that the isometry B : WT,∞ → (VT ∗ ,1 )∗ of Theorem 5 is not surjective. Example 3. Let T : 1 → c be deﬁned by T (x1 , x2 , . . . , xn , . . .) = (x1 , x1 + n x2 , . . . , k=1 xk , . . .), where c denotes the space of convergent sequences of real numbers with the supremum norm. Clearly, T is positive and p-convex for every 1 ≤ p ≤ ∞. Notice that the set n n

1 p p |T yi | , yi p1 ≤ 1}, S = {y ∈ c : |y| ≤ i=1

i=1

contains the constant sequence equal to one, so since S is solid, S coincides with the closed unit ball of c. Hence, WT,p = c. Now, we can consider the adjoint operator T ∗ : c∗ → ∗1 , which is clearly q-concave for every 1 ≤ q ≤ ∞. Recall that c∗ can be identiﬁed with the space 1 (N) in the following way: for an element x = (x0 , x1 , . . .) in 1 (N) and another element y = (y1 , y2 , . . .) in c, we set ∞ xn yn .

x, y = x0 lim yn + n=1

Therefore, for a positive element x ∈ c∗ we have T ∗ x∗1

= sup{ T ∗ x, y : y1 ≤ 1} = sup{ x, T y : y1 ≤ 1} ≥ x, T e1 = ∞ n=0 xn = xc∗ .

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Since T ≤ 1, it holds that T ∗ x∗1 = xc∗ for every positive x ∈ c∗ . This implies that n n

1q

1q T ∗xi q∗ : |xi |q ≤ |x| = xc∗ , ρT ∗ ,q (x) = sup 1

i=1

i=1 ∗

which yields that V =c . Notice that, in particular, the operator ϕ : c → c deﬁned in Theorem 3 coincides with the identity on c, and the operator φ : c∗ → c∗ deﬁned in Theorem 1 coincides as well with the identity on c∗ . Now, by the deﬁnition of the operator B : WT,p → (VT ∗ ,q )∗ in Theorem 5, it follows that for every y ∈ c and y ∗ ∈ c∗ we have T ∗ ,q

B(y), y ∗ = B(y), φ(y ∗ ) = ϕ(y), φ(y ∗ ) = y, y ∗ . Hence, B = J, where J : c → c∗∗ denotes the canonical inclusion of c into its bidual (= ∞ ). Now since c is not order continuous, it follows that B(c) is not an ideal in (VT ∗ ,q )∗ , and this shows that the last statement of Theorem 5 does not hold without the assumption of order continuity on E.

4. Interpolation of Banach lattices Throughout this section we will be using the complex interpolation method for Banach lattices, hence we need to consider complex Banach lattices. However, our ﬁnal results, which are given in the next section, remain true for real Banach lattices by means of “complexifying” and considering the real part after the interpolation constructions. Notice that the results presented in the previous section work equally for both real or complex Banach lattices. We refer to [15, Section 2.2] for the notion of complex Banach lattice. Recall that a compatible pair of Banach spaces (X0 , X1 ) is a pair of Banach spaces X0 , X1 which are continuously included in a topological vector space X. In the context of Banach lattices, we will say that two Banach lattices X0 , X1 form a compatible pair of Banach lattices (X0 , X1 ) if there exists a complete Riesz space X, and inclusions ij : Xj → X which are continuous interval preserving lattice homomorphisms, for j = 0, 1. In this way, the space X0 + X1 = {x ∈ X : x = x0 + x1 , with x0 ∈ X0 , x1 ∈ X1 }, equipped with the norm x = inf{x0 X0 + x1 X1 : x = x0 + x1 } is a Banach lattice which contains X0 and X1 as (non-closed) ideals. Given a compatible pair of Banach lattices, (X0 , X1 ), for each θ ∈ [0, 1] we will consider three diﬀerent constructions: 1. X01−θ X1θ denotes the space of elements x ∈ X0 + X1 such that |x| ≤ λ|x0 |1−θ |x1 |θ , for some λ > 0, x0 ∈ X0 and x1 ∈ X1 , with x0 X0 ≤ 1, x1 X1 ≤ 1. Notice that the expressions of the form |f |1−θ |g|θ can be deﬁned in any

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Banach lattice by means of the functional calculus due to Krivine (see [11, pp. 40-43]). The norm in this space is given by xX 1−θ X θ 0

1

= inf{λ > 0 : |x| ≤ λ|x0 |1−θ |x1 |θ for some x0 X0 ≤ 1, x1 X1 ≤ 1}. 2. [X0 , X1 ]θ denotes the space of elements x ∈ X0 + X1 which can be represented as x = f (θ) for some f ∈ F(X0 , X1 ). Here F (X0 , X1 ) denotes the linear space of functions f (z) deﬁned in the strip Π = {z ∈ C : z = x + iy, 0 ≤ x ≤ 1}, with values in the space X0 + X1 , such that • f (z) is continuous and bounded for the norm of X0 + X1 in Π, • f (z) is analytic for the norm of X0 + X1 in the interior of Π, • f (it) assumes values in X0 and is continuous and bounded for the norm of X0 , while f (1 + it) assumes values in X1 and is continuous and bounded for the norm of X1 . In F (X0 , X1 ) we can consider the norm f F (X0 ,X1 ) = max{sup f (it)X0 , sup f (1 + it)X1 }. t

t

The norm in [X0 , X1 ]θ is given by x[X0 ,X1 ]θ = inf{f F (X0,X1 ) : f (θ) = x}. 3. [X0 , X1 ]θ denotes the space of elements x ∈ X0 + X1 which can be represented as x = f (θ) for some f ∈ F (X0 , X1 ). Now F (X0 , X1 ) denotes the linear space of functions f (z) deﬁned in the strip Π = {z ∈ C : z = x + iy, 0 ≤ x ≤ 1}, with values in the space X0 + X1 , such that • f (z)X0 +X1 ≤ c(1 + |z|) for some constant c > 0 and for every z ∈ Π, • f (z) is continuous in Π and analytic in the interior of Π for the norm of X0 + X1 , • f (it1 ) − f (it2 ) has values in X0 and f (1 + it1 ) − f (1 + it2 ) in X1 for any −∞ < t1 < t2 < ∞, endowed with the norm f F(X0 ,X1 ) f (it2 ) − f (it1 ) , sup f (1 + it2 ) − f (1 + it1 ) = max sup . t2 − t 1 t2 − t 1 t1 ,t2 X0 t1 ,t2 X1 The norm in [X0 , X1 ]θ is given by x[X0 ,X1 ]θ = inf{f F(X0 ,X1 ) : f (θ) = x}. These spaces are Banach lattices provided that (X0 , X1 ) is a compatible pair of Banach lattices. Moreover, [X0 , X1 ]θ and [X0 , X1 ]θ are always interpolation spaces, while X01−θ X1θ is an intermediate space between X0 and X1 which is an interpolation space under certain extra assumptions. We will refer to X01−θ X1θ as the Calder´ on-Lozanovskii construction for (X0 , X1 ). We refer to [6], [9], [12], and [13] for more information on these spaces.

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ˇ Next theorem extends a result of Sestakov [19], which was originally proved only for the case of Banach lattices of measurable functions, showing how these constructions are related to each other. Theorem 9. Let X0 , X1 be a compatible pair of Banach lattices. For every θ ∈ (0, 1) it holds that [X0 , X1 ]θ = X0 ∩ X1

[X0 ,X1 ]θ

= X0 ∩ X1

X01−θ X1θ

,

with equality of norms. Before the proof of Theorem 9 we need the following. Lemma 10. Let F : Π → X0 + X1 be a function in F (X0 , X1 ) of the form F (z) = eδz

2

N

xj eλj z ,

j=1

where δ > 0, the λj are real, and xj ∈ X0 ∩ X1 . It holds that F (θ)X 1−θ X θ ≤ F F (X0 ,X1 ) . 0

1

Proof. Let F : Π → X0 ∩ X1 be a function in F (X0 , X1 ) of the form F (z) = eδz

2

N

xj eλj z ,

j=1

where δ > 0, the λj are real, and xj ∈ X0 ∩ X1 . Let x = N j=1 |xj |. We can consider the principal (non closed) ideal in X0 ∩ X1 generated by x, equipped with the norm that makes it isomorphic to a C(K) space for some compact K (i.e. y = inf{λ > 0 : |y| ≤ λx}, cf. [18, Chapter II. §7]). We clearly have inclusions C(K) → X0 ∩ X1 → X0 + X1 , which are bounded lattice homomorphisms. Moreover, since |xj | ≤ x, we have xj ∈ C(K), so we can consider F (ω, z) = eδz

2

N

xj (ω)eλj z ,

j=1

as a function of ω ∈ K, and z ∈ Π. For each z ∈ Π, F (·, z) belongs to C(K). Hence, applying [6, §9.4, ii)], for any ω ∈ K we have

1 |F (ω, θ)| ≤ 1−θ

+∞

1−θ +∞ θ 1 |F (ω, it)|µ0 (θ, t)dt |F (ω, 1 + it)|µ1 (θ, t)dt , θ

−∞

−∞

where µ0 and µ1 are the Poisson kernels for the strip Π, given by (see [6, §9.4]): µ0 (θ, t) =

e−πt sin πθ , sin2 πθ + [cos πθ − e−πt ]2

µ1 (θ, t) =

e−πt sin πθ . sin2 πθ + [cos πθ + e−πt ]2

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Hence setting 1 g(ω) = 1−θ

+∞

1 |F (ω, it)|µ0 (θ, t)dt and h(ω) = θ

−∞

+∞

|F (ω, 1 + it)|µ1 (θ, t)dt, −∞

we ﬁnd that g and h belong to C(K). Indeed, for any ω1 , ω2 in K, we have 1 |g(ω1 ) − g(ω2 )| ≤ 1−θ 1 ≤ 1−θ 1 ≤ 1−θ ≤

N

+∞

|F (ω1 , it) − F (ω2 , it)|µ0 (θ, t)dt −∞ +∞ N

2 (xj (ω1 ) − xj (ω2 ))eiλj t e−δt µ0 (θ, t)dt

−∞ j=1 +∞ N

2

|xj (ω1 ) − xj (ω2 )|e−δt µ0 (θ, t)dt

−∞ j=1

|xj (ω1 ) − xj (ω2 )|,

j=1

since

+∞

µ0 (θ, t)dt = 1 − θ (see [6, §29.4]). This inequality together with the

−∞

fact that xj belongs to C(K) for j = 1, . . . , N , proves that g ∈ C(K). The proof for h is identical. Moreover, +∞ 1 gX0 = |F (ω, it)|µ (θ, t)dt 0 1 − θ X0 −∞

≤

1 1−θ

+∞

F (ω, it)X0 µ0 (θ, t)dt −∞

1 ≤ F F (X0 ,X1 ) 1−θ

+∞

µ0 (θ, t)dt −∞

= F F (X0 ,X1 ) , and similarly hX1 ≤ F F (X0 ,X1 ) . Since |F (θ)| ≤ g 1−θ hθ (in C(K), and thus in X0 + X1 since Krivine’s calculus is preserved under lattice homomorphisms), we have F (θ)X 1−θ X θ ≤ F F (X0 ,X1 ) . 0

And the proof is ﬁnished.

1

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[X0 ,X1 ]θ

Proof of Theorem 9. If x is an element in X0 ∩ X1 , by the deﬁnition of the norm in [X0 , X1 ]θ , for every ε > 0, we can take F in F (X0 , X1 ), such that F (θ) = x and F F (X0 ,X1 ) ≤ x[X0 ,X1 ]θ + ε. By [9, Chapter IV, Thm. 1.1], we can consider a sequence (Fn )∞ n=1 in F (X0 , X1 ), of elements of the form eδz

2

N

xj eλj z ,

j=1

where xj ∈ X0 ∩ X1 and λj ∈ R, such that F − Fn F (X0 ,X1 ) → 0. Then we have Fn (θ) − x[X0 ,X1 ]θ = Fn (θ) − F (θ)[X0 ,X1 ]θ ≤ Fn − F F (X0 ,X1 ) → 0. By Lemma 10, for n, m ∈ N we have Fn (θ) − Fm (θ)X 1−θ X θ ≤ Fn − Fm F (X0 ,X1 ) → 0, 0

1

when n, m → ∞, and Fn (θ)X 1−θ X θ ≤ Fn F (X0 ,X1 ) → F F (X0 ,X1 ) ≤ x[X0 ,X1 ]θ + ε. 0

1

Therefore, Fn (θ) also converges to a limit in X01−θ X1θ of norm not exceeding x[X0 ,X1 ]θ + ε. However, since X01−θ X1θ and [X0 , X1 ]θ are both continuously embedded in X0 + X1 , it follows that x is also the limit of Fn (θ) for the norm of X01−θ X1θ . Hence, x ∈ X01−θ X1θ and xX 1−θ X θ ≤ x[X0 ,X1 ]θ + ε. 1 0 Since this is true for all ε > 0, we have xX 1−θ X θ ≤ x[X0 ,X1 ]θ . 0

We will show now that

1

X01−θ X1θ

⊂ [X0 , X1 ]θ and the inclusion mapping

X01−θ X1θ → [X0 , X1 ]θ is bounded with norm smaller than or equal to one. Indeed, let x ∈ X01−θ X1θ be such that xX 1−θ X θ ≤ 1. Then for every ε > 0 we have g ∈ X0+ , and 0

1

h ∈ X1+ such that gX0 ≤ 1, hX1 ≤ 1, and |x| ≤ (1 + ε)g 1−θ hθ in X0 + X1 . Now, if I denotes the (non closed) order ideal generated by g ∨ h in X0 +X1 , then I can be viewed as a space C(K) over some compact Hausdorﬀ space K. Since |x| ≤ (1 + ε)g 1−θ hθ in X0 + X1 , we can consider f (t) =

x(t) , g 1−θ (t)hθ (t)

which is well deﬁned for all t ∈ K such that g(t)h(t) = 0. This allows us to deﬁne f (t)g(t)1−z h(t)z if g(t)h(t) = 0, F (t, z) = 0 in any other case.

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Note that, since g, h ≤ g ∨ h, we have gC(K), hC(K) ≤ 1; hence, for every z ∈ Π, sup |F (t, z)| ≤ 1 + ε. t∈K ◦

Clearly, for z ∈ Π we can consider φ(z) ∈ C(K) deﬁned by φ(z)(t) = F (t, z). It is routine to verify that the map ◦

φ : Π → C(K) is continuous. We claim that it is analytic. Indeed, note that for every t ∈ K ◦

ﬁxed, F (t, ·) is analytic on Π. Hence, φ(z)(t) = F (t, z) =

1 2πi

γ

F (t, ξ) dξ ξ−z ◦

for every t ∈ K, and for any circumference γ of center z contained in Π. Since this identity is valid for every t ∈ K, we get φ(ξ) 1 dξ. φ(z) = 2πi γ ξ − z ◦

This means that φ : Π → C(K) is analytic. Now, let us deﬁne F (t, ξ)dξ, F1 (t, z) = γz 1 2

for t ∈ K and z ∈ Π, where γz is any continuous path joining ◦

and z, with ◦

all its points except possibly z inside Π. Note that since F is analytic in Π and sup|F (t, z)| ≤ 1 + ε, for all z ∈ Π, F1 is independent of the path γz , t∈K

so it is well deﬁned. Therefore, we can deﬁne φ1 : Π → B(K), where B(K) denotes the bounded measurable functions on K, by φ(ξ)dξ, φ1 (z) = F1 (·, z) = γz ◦

◦

for z ∈ Π. Since φ : Π → C(K) is analytic, so is φ1 on Π, and clearly ◦

φ1 (Π) ⊆ C(K). Moreover, φ1 (z) − φ1 (z )C(K) ≤ (1 + ε)|z − z | ◦

◦

for z, z ∈ Π. Now, for any z in the border of Π, let zn ∈ Π be such that zn → z. Since φ1 (zn )−φ1 (zm )C(K) ≤ (1+ε)|zn −zm |, we get that φ1 (zn ) is a Cauchy sequence in C(K), hence convergent to some ψ ∈ C(K). In particular, for every t ∈ K, φ1 (zn )(t) → ψ(t) and since φ1 (zn )(t) = γz F (t, ξ)dξ we get n that ψ(t) = γz F (t, ξ)dξ. This implies that φ1 (Π) ⊆ C(K), and φ1 (z) − φ1 (z )C(K) ≤ (1 + ε)|z − z | for z, z ∈ Π. Thus φ1 : Π → C(K) is continuous.

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Now, for u, v ∈ R, and for every α ∈ (0, 1) let γα be the path formed by the rectilinear segments L1 = [iu, α + iu], L2 = [α + iu, α + iv] and L3 = [α + iv, iv]. Hence, for every α ∈ (0, 1) and t ∈ K such that g(t)h(t) = 0 |F1 (t, iu) − F1 (t, iv)| ≤ |F (t, ξ)|dξ γα |F (t, ξ)|dξ + |F (t, ξ)|dξ + intL3 |F (t, ξ)|dξ = L1

L2

≤ α(1 + ε) + (1 + ε)g(t)1−α h(t)α |u − v| + α(1 + ε) ≤ (g(t)1−α h(t)α |u − v| + 2α)(1 + ε). Thus, letting α → 0+ , we get |F1 (t, iu) − F1 (t, iv)| ≤ (1 + ε)g(t) |u − v| for t ∈ K with g(t)h(t) = 0. Since the same inequality holds trivially if g(t)h(t) = 0, we have that |φ1 (iu) − φ1 (iv)| ≤ (1 + ε)g |u − v| in X0 . Analogously we have |φ1 (1 + iu) − φ1 (1 + iv)| ≤ (1 + ε)h |u − v| in X1 . Since X0 and X1 are order ideals, it clearly follows that 1 (1+iv)| ∈ X1 . X0 and |φ1 (1+iu)−φ |u−v| dφ1 Therefore, since dz z=θ = φ(θ)

|φ1 (iu)−φ1 (iv)| |u−v|

∈

= x, we get that x ∈ [X0 , X1 ]θ and

x[X0 ,X1 ]θ |φ1 (iu) − φ1 (iv)| |φ1 (1 + iu) − φ1 (1 + iv)| ≤ max sup , , sup |u − v| |u − v| u,v X0 u,v X1 which is smaller than 1 + ε. Since this holds for every ε > 0, we have proved that X01−θ X1θ → [X0 , X1 ]θ is continuous with norm smaller than or equal to one. In particular, we have an inclusion X0 ∩ X1

X01−θ X1θ

→ X0 ∩ X1

[X0 ,X1 ]θ

with norm smaller than one. Now, by [3], we have X0 ∩ X1

[X0 ,X1 ]θ

= X0 ∩ X1

[X0 ,X1 ]θ

with equality of norms. This proves the theorem.

,

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5. Factorization for operators which are both p-convex and q-concave In section 2, it was proved that every p-convex (resp. q-concave) operator factors in a nice way through a p-convex (resp. q-concave) Banach lattice. However, if the operator is both p-convex and q-concave, can this factorization be improved? It is well-known that if E is a q-concave Banach lattice and F a p-convex Banach lattice, then every operator T : E → F is both p-convex and q-concave. Moreover, if an operator T : E → F between Banach lattices, has a factorization of the following form E

T

ψ

φ

E1

/F O

R

/ F1

where φ and ψ are positive, E1 q-concave, and F1 p-convex, then T is both p-convex and q-concave [10]. Hence, the following question is natural: Can a p-convex and q-concave operator T : E → F factor always in this way? According to Theorems 1 and 3, this is true if the operator T : E → F can be written as T = T1 ◦ T2 , where T1 is p-convex, and T2 is q-concave. In fact, it turns out that the previous question is equivalent to the following one. If T : E → F is p-convex and q-concave, do there exist operators T1 and T2 , such that T = T1 ◦ T2 , where T1 is p-convex, and T2 is q-concave? In general, the answer to this question is negative, as the following examples show. Proposition 11. Let T : E → F be an operator from an ∞-convex Banach lattice (an AM -space) E to a q-concave Banach lattice F (q < ∞). If T can be factored as T = SR, with R q-concave and S ∞-convex, then T is compact. Proof. If T : E → F has such a factorization, then by Theorems 1 and 3 we must have T /F E O ϕ

φ

V

T1

/W

where V is q-concave, W an AM -space, and φ, ϕ lattice homomorphisms. Since φ and ϕ are positive and take values in q-concave Banach lattices, by [11, Prop. 1.d.9], they are q-concave operators. Moreover, since both operators are deﬁned on AM -spaces, by [11, Theorem 1.d.10], φ and ϕ are q-absolutely summing. Therefore, T = ϕ ◦ (T1 ◦ φ) is a product of two q-absolutely summing operators, hence it is compact, because every q-absolutely summing operator is weakly compact and Dunford-Pettis (cf. [2, Cor. 8.2.15]).

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Example 4. The formal inclusion T : C(0, 1) → Lq (0, 1) is q-concave and ∞-convex, but it does not factor as T = T1 ◦ T2 , with T1 ∞-convex, and T2 q-concave. Proof. Since T is positive, it is q-concave and ∞-convex by [11, Prop. 1.d.9]. However T is not compact since the closure in Lq (0, 1) of the unit ball of C(0, 1) contains the Rademacher functions. By duality, Proposition 11 immediately yields the following. Corollary 12. Let T : E → F be an operator from a p-convex Banach lattice E to a 1-concave Banach lattice (an AL-space) F . If T can be factored as T = SR, with R 1-concave and S p-convex, then T is compact. A diﬀerent argument can be used to see that the formal inclusion i : Lp (0, 1) → Lq (0, 1) with 1 < q < p < ∞ (which is clearly p-convex and q-concave) does not factor as i = T2 T1 with T1 q-concave and T2 p-convex. First we need the following lemma: Lemma 13. Let 1 < q < p < ∞. There is no disjointness preserving nonzero operator T : Lq (0, 1) → Lp (0, 1). Proof. Assume f := T h = 0 for some h ∈ Lq (0, 1) with hq = 1. If U : Lq (0, 1) → Lq (0, 1) is a linear isometry such that U χ[0,1] = h, then S := T U is also disjointness preserving. For each n ∈ N, let us consider the partition {0, n1 , n2 , . . . , 1}. Notice that for each n ∈ N there must exist kn ≤ n such that f p S(χ[ kn −1 , kn ] )p ≥ 1/p . n n n Otherwise, by the fact that S is disjointness preserving, we would have n n n

p1 f p p1 f = S(χ[ k−1 , k ] ) = S(χ[ k−1 , k ] )p < = f , n n n n n k=1

k=1

which is clearly a contradiction. Hence, since χ[ k−1 , k ] q = n

n

1 n1/q

k=1

for every k = 1, . . . , n, we have

f p f p = 1/p−1/q χ[ kn −1 , kn ] q . n n n1/p n Therefore, since q < p, for n large enough we get a contradiction with the fact that S is bounded. S(χ[ kn −1 , kn ] )p ≥ n

n

Recall that given a Banach lattice E and a Banach space X, an operator T : E → X is called AM-compact if T [−x, x] is relatively compact for every positive x ∈ E. Theorem 14. If a lattice homomorphism T : Lp (0, 1) → Lq (0, 1) (q < p) can be factored as T = T2 T1 with T1 q-concave and T2 p-convex, then T is AM-compact.

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Proof. Suppose that we have Lp (0, 1) GG GG GG T1 GGG #

T

X

/ Lq (0, 1) ; ww ww w ww T2 ww

with T1 q-concave and T2 p-convex. Hence, by Theorems 1 and 3 we have Lp (0, 1)

T

ϕ

φ

V

/ Lq (0, 1) O

S

/W

where V is q-concave, W p-convex, and φ, ϕ are lattice homomorphisms. Now, by Krivine’s Theorem ([11, Theorem 1.d.11]) we can factor φ /V Lp (0, 1) = II { { II { II { II {{ φ1 I$ {{ φ2 Lq (µ)

/ Lq (0, 1) WD DD u: DD uu u u D ϕ1 DD uu ϕ ! uu 2 Lp (ν) ϕ

where φi and ϕi are still lattice homomorphisms. Therefore, we can consider the closure of φ1 (Lp (0, 1)) in Lq (µ), which is lattice isomorphic to some µ), and the quotient Lp (ν)/ ker(ϕ2 ) which is lattice isomorphic to Lp ( ν) Lq ( for certain measures µ and ν. Thus, we can consider the following diagrams: / Lq (µ) Lp (0, 1) II v; II v v II v v II v 1 I$ φ , vvv i µ) Lq ( φ1

/ Lq (0, 1) Lp (ν) AA AA u: u AA uu u A π AA uu uu ϕ2 Lp ( ν) ϕ2

µ) → Lp ( ν ) be deﬁned by R = πϕ1 Sφ2 i. It follows that R Now, let R : Lq ( µ), we can consider (xn ) in is a lattice homomorphism. Indeed, given x ∈ Lq ( µ). Since T is a lattice homomorphism Lp (0, 1) such that φ1 (xn ) → x in Lq ( 2 is an injective lattice homomorT (|xn |) = |T xn | for every n, and since ϕ 1 (|xn |) = |Rφ 1 (xn )|, and by continuity and the fact phism we get that Rφ 1 is also a lattice homomorphism, we achieve R(|x|) = |R(x)|. that φ µ) and By considering the diﬀuse and atomic parts of the spaces Lq ( Lp ( ν ), we can decompose them as Lq (0, 1) ⊕ q and Lp (0, 1) ⊕ p (lattice isomorphically). Accordingly, every operator between them can be decomposed into four parts acting between each of the summands, that is R = R11 R12 with R21 R22 R11 : Lq (0, 1) → Lp (0, 1) R21 : Lq (0, 1) → p

R12 : q → Lp (0, 1) R22 : q → p .

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Clearly, if R is a lattice homomorphism, so are Rij , and since the intervals in p and q are compact, we have that R12 , R21 and R22 are AM-compact. Finally, by Lemma 13 we see that R11 has to be the zero operator. This ﬁnishes the proof. Example 5. For 1 < q < p < ∞, the formal inclusion i : Lp (0, 1) → Lq (0, 1) cannot be factored as i = T2 T1 with T1 q-concave and T2 p-convex. Proof. Since i is positive, by [11, Prop. 1.d.9], i is q-concave and p-convex. Moreover, since i : Lp (0, 1) → Lq (0, 1) is a lattice homomorphism and it is not AM-compact (consider for instance the Rademacher functions), by Theorem 14, we conclude that it cannot be factored as i = T2 T1 with T1 q-concave and T2 p-convex. Despite these facts, as an application of the results of section 4, we have the following factorization for operators which are both p-convex and q-concave. Theorem 15. Let E and F be Banach lattices, and let T : E → F be both p-convex and q-concave. For every θ ∈ (0, 1) we can factor T in the following way /F O

T

E

ϕθ

φθ

Eθ

/ Fθ

Rθ

q where φθ and ϕθ are interval preserving lattice homomorphisms, Eθ is ( 1−θ )p concave, and Fθ is ( θ+(1−θ)p )-convex.

Before the proof, we need some lemmas ﬁrst. Recall, that given a Banach space X, and 1 ≤ p < ∞, p (X) denotes the space of sequences (xn ) in X such that (xn X ) belongs to p . This is a Banach space with the norm ∞

p1 xn pX . (xn )p (X) = n=1

In order to keep a uniﬁed notation, for p = ∞, ∞ (X) will denote the space of sequences (xn ) of X such that (xn X ) belongs to c0 , equipped with the norm (xn )∞ (X) = sup xn X . Notice that this space is usually denoted c0 (X) in the literature. Analogously, given a Banach lattice E, and 1 ≤ p ≤ ∞, E(p ) denotes the completion of the space of eventually null sequences (xn ) of E under the norm ⎧

p1 ⎪ n ⎪ |xi |p E if 1 ≤ p < ∞, ⎨ sup n i=1 (xn )E(p ) = n ⎪ ⎪ if p = ∞. ⎩ sup |xi | n

i=1

E

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The following lemma consists of standard results. In the case of Banach lattices of measurable functions, this can be obtained from [5, Theorem 3], however, in general we cannot use the measurability tools and thus some functional calculus needs to be carried out. Lemma 16. Let (F, G) be a compatible pair of Banach lattices, let r, s ∈ θ [1, +∞] and θ ∈ (0, 1). For 1t = 1−θ r + s , we have: 1. r (F )1−θ s (G)θ = t (F 1−θ Gθ ), with equality of norms. r (F )1−θ s (G)θ

F 1−θ Gθ

2. r (F ) ∩ s (G) = t (F ∩ G ). 3. the inclusion F (nr )1−θ G(ns )θ → F 1−θ Gθ (nt ) is bounded of norm ≤ 1. 4. E(r )1−θ E(s )θ = E(t ), with equality of norms. We skip the proof of the lemma and proceed with the proof of the main result. Proof of Theorem 15. Since T is p-convex, it can be factored through a pconvex Banach lattice Z as in Theorem 3: E@ @@ @@ R @@

T

Z

/F ? ~ ~~ ~ ~ϕ ~~

where ϕ : Z → F is an injective interval preserving lattice homomorphism, and Rx = T x for all x ∈ E. Therefore, (Z, F ) can be considered as a compatible interpolation pair of Banach lattices, and we can interpolate T : E → F and R : E → Z by the complex method of interpolation (see [6]) with parameter θ, (thus, we complexify E and Z if they are not complex Banach lattices) and we get a Banach lattice Fθ = [(Z, F )]θ , and an operator Tθ : E → Fθ . Moreover, since ϕ is an inclusion, Fθ is also continuously included in F . Let us denote this inclusion by ϕθ : Fθ → F . p We claim that Fθ is pθ convex, with p1θ = pθ + 1−θ 1 , that is pθ = θ+(1−θ)p . Indeed, ﬁrst notice that if Z is p-convex then F 1−θ Z θ is pθ -convex. This is because for any positive operator S it holds that S(|x0 |1−θ |x1 |θ ) ≤ (S|x0 |)1−θ (S|x1 |)θ . This implies that for any positive operator S acting simultaneously from X0 into Y0 and from X1 into Y1 the interpolated operator S : X01−θ X1θ → Y01−θ Y1θ is bounded. In our particular case, (see the discussion following [11, 1.d.3]) for every n ∈ N, we have operators Iˆn : n1 (F ) (x1 , . . . , xn )

−→ F (n1 ) −→ (x1 , . . . , xn )

Iˆn : np (Z) −→ Z(np ) (x1 , . . . , xn ) −→ (x1 , . . . , xn )

which are bounded uniformly on n ∈ N. Since they are clearly positive, by the previous remark the following operators are also uniformly bounded In : n1 (F )1−θ np (Z)θ −→ F (n1 )1−θ Z(np )θ (x1 , . . . , xn ) −→ (x1 , . . . , xn )

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Using (1) and (3) of Lemma 16 we get that the operators pnθ (F 1−θ Gθ ) −→ n1 (F )1−θ np (Z)θ −→ F (n1 )1−θ Z(np )θ −→ F 1−θ Gθ (npθ ) −→ (x1 , . . . , xn )

(x1 , . . . , xn )

−→ (x1 , . . . , xn )

−→ (x1 , . . . , xn )

are also uniformly bounded on n. This means that F 1−θ Gθ is pθ -convex. Now, F 1−θ Z θ

by Theorem 9, Fθ = F ∩ Z , and since F ∩ Z is a sublattice of F 1−θ Z θ , Fθ is also pθ -convex. q )-concave. Indeed, since T : E → F is Now we claim that Tθ is ( 1−θ q-concave and R : E → Z is ∞-concave, the following maps are bounded: Tˇ : E(q ) (x1 , x2 , . . .)

ˇ : E(∞ ) R (x1 , x2 , . . .)

−→ q (F ) −→ (T x1 , T x2 , . . .)

−→ −→

∞ (Z) (Rx1 , Rx2 , . . .)

Therefore, the interpolated map Tˇθ : [(E(q ), E(∞ ))]θ → [(q (F ), ∞ (Z))]θ is also bounded (cf. [4] or [6, §4]). Note that by Theorem 9 and (4) of Lemma 16, we have [(E(q ), E(∞ ))]θ = E(q ) ∩ E(∞ )

E(q )1−θ E(∞ )θ

E(q )1−θ E(∞ )θ

= E(q ) where

1 qθ

=

θ ∞

+

1−θ q .

= E(qθ ),

And by Lemma 16, we have the identity q (F )1−θ ∞ (Z)θ

[(q (F ), ∞ (Z))]θ = q (F ) ∩ ∞ (Z) = qθ (F ∩ Z

F 1−θ Z θ

) = qθ (Fθ ),

with equality of norms. Therefore, the map Tˇθ : E(qθ ) → qθ (Fθ ) is bounded, q which means that Tθ is qθ -concave (qθ = 1−θ ). Hence, we can now apply Theorem 1 to Tθ : E → Fθ , and we get the factorization E@ @@ @@ @ φθ @@

Tθ

Eθ

/ Fθ > } }} } }} }} Rθ

through the qθ -concave Banach lattice Eθ . Therefore, T can be factorized as claimed. Remark 9. It is easy to see that in the case when the spaces E and F are real Banach lattices, after complexifying and making the previous argument, the obtained operators are all “complexiﬁed” operators, i.e. TC (x + iy) = T (x) + iT (y). Hence, by considering the restriction to the real part in each space, we obtain the same factorization result for real Banach lattices.

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6. Connections with Krivine’s theorem Recall the classical result proved in [10]: Given Banach spaces X, Y and a Banach lattice E, if T1 : X → E is p-convex and T2 : E → Y is p-concave, then T2 T1 factors through Lp (µ) for certain measure µ. We remark that the factorization Theorems 1 and 3 allow us to reduce Krivine’s theorem to the following purely lattice theoretical version: Lemma 17. If W , V are quasi-Banach lattices with W p-convex and V pconcave, then every lattice homomorphism h : W → V factors through some space Lp (µ), and the factors are lattice homomorphisms. Proof. We may assume by renorming that the p-convexity constant of W , resp. the q-concavity constant of V , are equal to one (see Rem. 1 and 5). Notice that by an elementary p-concaviﬁcation/ convexiﬁcation argument (see [11, pp. 53-54]), the proof of the Lemma reduces itself to the case p = 1 (this is because a lattice homomorphism h : W → V is bounded if and only if it is bounded between the p-convexiﬁcations h : W (p) → V (p) ). In this case, Krivine’s argument becomes transparent: indeed, let us consider F1 = {x ∈ W : xW < 1} and F2 = h−1 ({y ∈ V : y ≥ 0 and yV ≥ h}). Clearly, both sets are convex and satisfy F1 ∩ F2 = ∅. Hence, by HahnBanach’s Theorem we can ﬁnd a functional f ∈ W ∗ such that f (x) ≤ 1 for each x ∈ F1 and f (x) ≥ 1 for each x ∈ F2 . Thus, f is positive and for x ∈ W 1 we have h h(x)V ≤ f (|x|) ≤ xW . This allows us to deﬁne a seminorm on W by x → f (|x|) which induces a lattice norm norm on the vector lattice W/{x ∈ W : f (|x|) = 0}, the completion of which (for this new norm) is, by Kakutani’s theorem [11, Theorem 1.b.2], isomorphic as normed lattice to a space L1 (µ) for a certain measure µ. Moreover, if π denotes the map W → L1 (µ) induced by the quotient map W → W/{x ∈ W : f (|x|) = 0}, we have 1 h(x)V ≤ π(x) ≤ xW . h This means that we can factor /V WE y< EE y EE yy E yy π EE " yy h L1 (µ) h

where π and h are lattice homomorphisms and h is deﬁned so that h(π(x)) = h(x).

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Now, let T1 : X → E be a p-convex operator and T2 : E → Y be p-concave. Using Theorems 1 and 3 we have XB BB BB B R1 BB

T2 /E /Y >} @@ > ~ @@ ~ }} ~ } @ ~ ~ }} ϕ φ @@ }} ~~ S2 W V T1

where W is p-convex, V p-concave, and ϕ, φ are lattice homomorphisms. This diagram shows clearly how Krivine’s theorem can be obtained from the previous Lemma. Remark 10. The same argument plus a standard application of Maurey’s Theorem [2, Theorem 7.1.2] yields that if T1 : X → E is p-convex and T2 : E → Y is q-concave, with p > q, then T2 T1 can be factored through the canonical inclusion i : Lp (µ) → Lq (µ) for a certain measure µ (this was essentially proved in [17, Sec. 2, Corollary 7] when Y is reﬂexive). In a similar direction, as another application of Theorem 9, we have the following result (compare with [17, Sec. 3, Proposition 2]). Proposition 18. Let T : X → E be p-convex and S : E → Y q-concave. For every θ ∈ (0, 1) we can factor ST through a Banach lattice Uθ which is p q pθ -convex and qθ -concave (with as usual pθ = p(1−θ)+θ and qθ = 1−θ ). Proof. By Theorem 3, we can factor T in the following way, where W is a p-convex Banach lattice and i a positive operator: T /E XB BB }> } BB } BB }} B . }}} i T W

Moreover, since S ◦ i : W → Y is q-concave, by Theorem 1 we have the lattice seminorm ρS◦i which is continuous with respect to the norm in W (ρS◦i (x) ≤ Mq (S ◦ i)xW ), and such that W/ρ−1 S◦i (0) with the norm that ρS◦i induces becomes a q-concave Banach lattice, such that S ◦ i factors through it. But, since W is p-convex and ρ−1 S◦i (0) is a closed ideal, it follows that W/ρ−1 (0) with its quotient norm is also p-convex. S◦i (0) with its quotient norm, and X1 = We can consider X0 = W/ρ−1 S◦i −1 W/ρ (0) (the completion under ρ ) with the norm induced by ρ . Note S◦i

S◦i

S◦i

that, for all y with ρS◦i (y) = 0, we have that ρS◦i (x) = ρS◦i (x + y) ≤ M(q) (S ◦ i)x + y. Thus, xX1 = ρS◦i (x) ≤ M(q) (S ◦ i) inf{x + y : ρS◦i (y) = 0} = M(q) (S ◦ i)xX0 ,

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which means that the inclusion X0 → X1 is bounded of norm less than or equal to ≤ M(q) . Therefore, we can interpolate X0 and X1 . Since X0 is pconvex and X1 is q-concave, by [16] we get that Uθ = X01−θ X1θ is pθ -convex and qθ -concave. The following diagram illustrates the situation: /: E uu u uu uu , uuu i φ

T

XB BB BB B BB T W

X0

/ X 1−θ X θ 1 0

S

/ X1 O / X? 1

/Y }> } }} }} }} S1

Remark 11. A similar result to Proposition 18 was also given in [17, Sec. 3, Proposition 2] for interval preserving lattice homomorphisms from a p-convex to a q-concave Banach lattice with essentially the same proof. The idea of using interpolation to produce this kind of factorization has been initiated both by S. Reisner in [17] and, in parallel, by T. Figiel in [7] using the real method of interpolation. Corollary 19. If T : E → E is p-convex and q-concave, then T 2 factors through a pθ -convex and qθ -concave Banach lattice. In particular, it factors through a super reﬂexive Banach lattice. Acknowledgment We thank Prof. F. L. Hern´ andez for bringing the reference [17] to our attention after that a ﬁrst draft of the present paper was completed.

References [1] Y. A. Abramovich and C. D. Aliprantis, An invitation to operator theory. Graduate Studies in Mathematics, vol. 50, American Mathematical Society, Providence, RI, 2002. [2] F. Albiac, and N. J. Kalton, Topics in Banach space theory. Graduate Texts in Mathematics, 233, Springer, New York, 2006. [3] J. Bergh, On the relation between the two complex methods of interpolation. Indiana Univ. Math. J. 28 (1979), no. 5, 775–778. [4] J. Bergh, and J. L¨ ofstr¨ om, Interpolation spaces. An introduction. Grundlehren der Mathematischen Wissenschaften, no. 223, Springer-Verlag, Berlin-New York, 1976. [5] A. V. Bukhvalov, Interpolation of linear operators in spaces of vector-valued functions and with mixed norm. (Russian) Sibirsk. Mat. Zh. 28 (1987), no. 1, i, 37–51. English translation: Siberian Math. J. 28 (1987), no. 1, 24–36. [6] A. P. Calder´ on, Intermediate spaces and interpolation, the complex method. Studia Math. 24 (1964), 113–190. [7] T. Figiel, Uniformly convex norms on Banach lattices. Studia Math. 68 (1980), 215–247.

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[8] N. Ghoussoub, and W. B. Johnson, Factoring operators through Banach lattices not containing C(0, 1). Math. Z. 194 (1987), 153-171. [9] S. G. Kre˘ın, Yu. ¯I. Petun¯ın, and E. M. Sem¨enov, Interpolation of linear operators. Translations of Mathematical Monographs, 54, American Mathematical Society, 1982. [10] J. L. Krivine, Th´eor`emes de factorisation dans les espaces r´ eticul´es. (French) S´eminaire Maurey-Schwartz 1973–1974: Espaces Lp , applications radoniﬁantes ´ et g´eom´etrie des espaces de Banach, Exp. Nos. 22 et 23. Centre de Math., Ecole Polytech., Paris, 1974. [11] J. Lindenstrauss, and L. Tzafriri, Classical Banach Spaces II: Function Spaces. Springer-Verlag, Berlin-New York, 1979. ˇ 10 [12] G. Ya. Lozanovskii, Certain Banach lattices. (Russian) Sibirsk. Mat. Z. (1969), 584–599. ˇ 14 [13] G. Ya. Lozanovskii, Certain Banach lattices IV. (Russian) Sibirsk. Mat. Z. (1973), 140–155. [14] P. Meyer-Nieberg, Kegel p-absolutsummierende und p-beschr¨ ankende Operatoren. (German) Nederl. Akad. Wetensch. Indag. Math. 40 (1978), no. 4, 479– 490. [15] P. Meyer-Nieberg, Banach Lattices. Springer-Verlag, 1991. [16] G. Pisier, Some applications of the complex interpolation method to Banach lattices. J. Analyse Math. 35 (1979), 264–281. [17] S. Reisner, Operators which factor through convex Banach lattices. Canad. J. Math. 32 (1980), 1482–1500. [18] H. H. Schaefer, Banach lattices and positive operators. Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, 1974. ˇ [19] V. A. Sestakov, Complex interpolation in Banach spaces of measurable functions. (Russian) Vestnik Leningrad. Univ. No. 19 Mat. Meh. Astronom. Vyp. 4 (1974), 64–68, 171. Yves Raynaud Institut de Math´ematiques de Jussieu CNRS and UPMC-Univ. Paris-06 case 186 75005 Paris France e-mail: [email protected] Pedro Tradacete Departamento de An´ alisis Matem´ atico Universidad Complutense de Madrid 28040 Madrid Spain e-mail: [email protected] Submitted: January 20, 2009. Revised: August 25, 2009.

Integr. Equ. Oper. Theory 66 (2010), 113–140 DOI 10.1007/s00020-009-1735-5 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

Integral Equations, Dichotomy of Evolution Families on the Half-Line and Applications Adina Luminit¸a Sasu and Bogdan Sasu Abstract. The purpose of this paper is to provide a new, uniﬁed and complete study for uniform dichotomy and exponential dichotomy on the half-line. First we deduce conditions for the existence of uniform dichotomy, using classes of function spaces over R+ which are invariant under translations. After that, we obtain a classiﬁcation of the main classes of function spaces over R+ , in order to deduce necessary and suﬃcient conditions for the existence of exponential dichotomy, emphasizing on the main technical qualitative properties of the underlying spaces. We motivate our approach by illustrative examples and show that the main hypotheses cannot be dropped. We provide optimal methods regarding the input space in the study of dichotomy and deduce as particular cases some interesting situations as well as several dichotomy results published in the past few years. Mathematics Subject Classiﬁcation (2010). Primary 34D09; Secondary 34D05. Keywords. Uniform dichotomy; exponential dichotomy; evolution equation; integral equation; evolution family.

1. Introduction Exponential dichotomy is one of the most interesting asymptotic properties of evolution equations being in the front line of various research studies (see [1]–[4], [6]–[12], [14]–[16], [19]–[25]). For an evolution equation on a Banach space X, exponential dichotomy is described through the existence of a projections family {P (t)}, which determines a decomposition of the space X at every moment, into a direct sum of stable subspace and unstable subspace such that the norm of the projection onto the stable subspace of any orbit decays exponentially as t → ∞ and the norm of the projection onto the The work is supported by the Exploratory Research Grant PN II ID 1080 code 508/2009, director Professor Mihail Megan.

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unstable subspace of any orbit grows exponentially as t → ∞. This asymptotic property and its generalizations was at the center of intensive research studies, among which we mention the classical monographs of Massera and Sch¨ aﬀer [12], Daleckii and Krein [8] and Coppel [7], the notable contributions from [1]–[4] and also the recent results obtained in [6], [9], [10], [11], [14]–[17], [19]–[25]. Let X be a Banach space and let U = {U (t, s)}t≥s,t,s∈J be an evolution family on X (see e.g. [21], Deﬁnition 2.1 and the present paper Deﬁnition 3.1). If J = R then we say that U is deﬁned on the real line and if J = R+ then we say that U is deﬁned on the half-line. In the study of the exponential dichotomy, there are two crucial diﬀerences between the real line and the half-line. The ﬁrst one is that on the real line, if an evolution family is exponentially dichotomic with respect to a family of projections then the projections family is uniquely determined (see [6], [11], [21], [22], [25] and the references therein). In contrast, on the half-line an evolution family may be exponential dichotomic with respect to an inﬁnite class of projection families (see e.g. [16], Example 1.1.1). The second aspect is related to the characterization of the exponential dichotomy in terms of the admissibility with respect to an associated integral equation: on the real line the admissibility concept relies on the unique solvability of the integral equation, while on the half-line the uniqueness requirement may be removed. These arguments and also many others led to distinct studies concerning these two cases: the real line on the one side (see e.g. [6], [11], [21], [22], [25]) and the half-line on the other side (see e.g. [10], [14], [15], [19], [23], [24]). If X is a Banach space and U = {U (t, s)}t≥s≥0 is an evolution family on X, then a pair of function spaces (O(R+ , X), I(R+ , X)) is said to be admissible for U if for every v ∈ I(R+ , X) there exists a continuous function f ∈ O(R+ , X) such that t (EU ) f (t) = U (t, s)f (s) + U (t, τ )v(τ ) dτ, ∀t ≥ s ≥ 0. s

I(R+ , X) is called the input space and O(R+ , X) is called the output space. Taking into account that the input-output techniques rely on the veriﬁcation of the solvability of the equation (EU ) in the output space, for various input test functions, a natural concern is to obtain characterizations such that the input space is as small as possible, while the output space is as large as possible. For the case of evolution equations on the half-line, an interesting open problem is to classify the classes of function spaces where the input and the output space from the admissible pair would belong to, such that the admissibility provides the existence of dichotomy. For the case of evolution families deﬁned on the real line, this question was answered in some cases in papers [21] and [25]. Another open problem on the half-line is to determine the main diﬀerences between the case of uniform dichotomy and the case of exponential dichotomy and to identify the corresponding classes of input and output spaces

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in each case. These problems bring the subject on a new line of research and request an approach based on solid arguments from interpolation theory. In order to solve them, new methods must be provided at the interference of qualitative theory of evolution equations, dynamic control and interpolation theory. The main purpose of the present paper is to give complete answers to the above questions and to deduce new applications, emphasizing on the characteristic properties of the dichotomous families on the half-line. We propose a new, uniﬁed and complete study concerning the dichotomy of evolution families on the half-line using the input-output admissibility of pairs of function spaces. The paper is organized as follows: in the second section we present some basic properties of Banach function spaces arising from interpolation theory. We consider a general class Q(R+ ) of Banach function spaces which are invariant to translations, contain the continuous functions v with v(0) = 0 and compact support, and satisfy a simple integral property and we associate a subclass denoted V(R+ ), which contains the Banach function spaces in Q(R+ ) satisfying the ideal property. We also introduce three subclasses F (R+ ), L(R+ ), B(R+ ) (see Section 2) and discuss some properties that occur in the study of dichotomy on the half-line and establish the connections between the classes of function spaces considered throughout the paper. Next, our attention focuses on the property of uniform dichotomy using speciﬁc control tools. In several steps, we deduce the connections between the admissibility of the pair (O(R+ , X), I(R+ , X)) (see Deﬁnition 3.3), with I ∈ Q(R+ ) and O ∈ V(R+ ) and the uniform dichotomy of an evolution family U = {U (t, s)}t≥s≥0 . The main result of the third section shows that if the pair (O(R+ , X), I(R+ , X)) is admissible for U and the initial subspace Xs (0) := {x ∈ X : U (·, 0)x ∈ O(R+ , X)} is closed and complemented in X, then U is uniformly dichotomic. Naturally, the question arises under what kind of hypotheses the admissibility of the pair (O(R+ , X), I(R+ , X)), with I ∈ Q(R+ ) and O ∈ V(R+ ) implies the uniform exponential dichotomy of U = {U (t, s)}t≥s≥0 . The purpose of the fourth section is to answer this question. Using constructive methods, we analyze both cases O ∈ F(R+ ) and I ∈ L(R+ ), respectively. In these cases, we prove that if the pair (O(R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ O(R+ , X)} is closed and complemented in X, then U is uniformly exponentially dichotomic. Moreover, if I ⊂ O and one of the spaces I or O belongs to B(R+ ), then U is uniformly exponentially dichotomic if and only if the pair (O(R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) is closed and complemented in X. After that, by concrete examples we prove that the assumptions on the structure of the underlying function spaces are indeed necessary and we motivate our methods. We show that conditions “O ∈ F(R+ ) or I ∈ L(R+ )” cannot be dropped and that our context is the most general in this topic. The last section is also devoted to some optimization methods regarding the

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input space and to the connections between the main results of our paper and the previous contributions in this area. Finally, we present as particular cases many interesting situations among which we note several dichotomy results obtained in the past few years.

2. Banach function spaces: basic notations and preliminary results In this section, for the sake of clarity we recall some basic deﬁnitions and notations from the theory of Banach function spaces. Let M(R+ ) be the linear space of all Lebesgue measurable functions u : R+ → R, identifying the functions equal a.e. Deﬁnition 2.1. A linear subspace B of M(R+ ) is called a normed function space, if there is a mapping | · |B : B → R+ such that: (i) |u|B = 0 if and only if u = 0 a.e.; (ii) |αu|B = |α| |u|B , for all (α, u) ∈ R × B; (iii) |u + v|B ≤ |u|B + |v|B , for all u, v ∈ B; (iv) if u ∈ B, then |u| ∈ B; (v) if u, v ∈ B with |u(·)| ≤ |v(·)| a.e., then |u|B ≤ |v|B . If (B, | · |B ) is complete, then B is called a Banach function space. For A ⊂ R we denote by χA the characteristic function of the set A. Deﬁnition 2.2. A Banach function space (B, | · |B ) is said to be invariant to translations if for every u : R+ → R and every t > 0 we have that u ∈ B if and only if the function ut : R+ → R, ut (s) = u(s − t)χ[t,∞) (s) belongs to B and |ut |B = |u|B . Let Cc (R+ , R) be the linear space of all continuous functions v : R+ → R with compact support and let C0c (R+ , R) = {v ∈ Cc (R+ , R) : v(0) = 0}. Notation. We denote by Q(R+ ) the class of all Banach function spaces B, which are invariant to translations, C0c (R+ , R) ⊂ B and with the properties: (i) if q ∈ Cc (R, R) and u ∈ B then qu ∈ B; t (ii) for every t > 0 there is c(t) > 0 such that 0 |u(τ )| dτ ≤ c(t) |u|B , for all u ∈ B. Example. The space C00 (R+ , R) = {u : R+ → R : u continuous and lim u(t) = u(0) = 0} t→∞

with the norm |||u||| := sup ||u(t)||, C00 (R+ , R) is a Banach function space which belongs to Q(R+ ).

t≥0

Example. Let p ∈ [1, ∞). Then the space p L (R+ , R) = u ∈ M(R+ ) :

∞ 0

|u(τ )| dτ < ∞ p

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with respect to the norm ||u||p = space which belongs to Q(R+ ).

∞ 0

|u(τ )|p dτ

1/p

117

is a Banach function

Example. Let L∞ (R+ , R) be the linear space of all essentially bounded functions u ∈ M(R+ ). With respect to the norm ||u||∞ := ess supt≥0 |u(t)|, L∞ (R+ , R) is a Banach function space which belongs to Q(R+ ). Example. (Orlicz spaces) Let ϕ : R+ → [0, ∞] be a non-decreasing leftcontinuous function, which is not identically zero or ∞ on (0, ∞). The Young t function associated with ϕ is deﬁned by Yϕ (t) = 0 ϕ(s) ds. For every u ∈ ∞ M(R+ ) we deﬁne Mϕ (u) := 0 Yϕ (|u(s)|) ds. The set Oϕ of all u ∈ M(R+ ) with the property that there is k > 0 such that Mϕ (ku) < ∞ is easily checked to be a linear space. With respect to the norm |u|ϕ := inf{k > 0 : Mϕ (u/k) ≤ 1}, Oϕ is a Banach space, called the Orlicz space associated with ϕ. Proposition 2.3. If ϕ(1) < ∞, then Oϕ ∈ Q(R+ ). Proof. This follows using similar arguments with those in the proof of Proposition 3.4 in [21]. Remark 2.4. The Lp -spaces are particular cases of Orlicz spaces. Notation. In what follows, we denote by V(R+ ) the class of all Banach function spaces B ∈ Q(R+ ) with the properties: (i) Cc (R+ , R) ⊂ B; (ii) if |u(·)| ≤ |v(·)| a.e. and v ∈ B, then u ∈ B. Remark 2.5. If ϕ(1) < ∞, then Oϕ ∈ V(R+ ). Lemma 2.6. Let B ∈ V(R+ ). The following assertions hold: (i) χ[a,b) ∈ B, for all b > a ≥ 0; (ii) if un → u in B, then there is a subsequence (ukn ) such that ukn → u a.e. Proof. Since Cc (R+ , R) ⊂ B, the ﬁrst assertion is obvious. For the second we refer to [18]. Lemma 2.7. Let B ∈ V(R+ ) and ν > 0. Then the function eν : R+ → R, eν (t) = e−νt belongs to B. Proof. We observe that ∞ ∞ eν (t) = e−νt χ[n,n+1) (t) ≤ e−νn χ[n,n+1) (t), n=0

∀t ≥ 0.

n=0

This implies that eν ∈ B and |eν |B ≤ |χ[0,1) |B /(1 − e−ν ). Lemma 2.8. Let B ∈ V(R+ ), u ∈ B and t > 0. Then the function u ˜t : R+ → R, u˜t (s) = u(s + t) belongs to B and |˜ ut |B ≤ |u|B .

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Proof. Let v : R+ → R, v(s) = u(s)χ[t,∞) (s). Since B ∈ V(R+ ), we have that v ∈ B and |v|B ≤ |u|B . Using the invariance to translations of B we deduce that u ˜t ∈ B and |˜ ut | = |v|B , which concludes the proof. Lemma 2.9. Let B ∈ V(R+ ). Then, for every u ∈ B t+1 sup |u(τ )| dτ < ∞. t≥0

t

1 Proof. Let c(1) > 0 be such that 0 |v(τ )| dτ ≤ c(1) |v|B , for all v ∈ B. Let now u ∈ B and t ≥ 0. Deﬁne u˜t : R+ → X, u ˜t (s) = u(s+t). From Lemma 2.8 t+1 1 ut |B ≤ |u|B . Then t |u(τ )| dτ = 0 |˜ ut (s)| ds ≤ we have that u ˜t ∈ B and |˜ t+1 c(1) |˜ ut |B ≤ c(1) |u|B . It follows that supt≥0 t |u(τ )| dτ ≤ c(1) |u|B , for all u ∈ B. Deﬁnition 2.10. Let B ∈ V(R+ ). The function FB : (0, ∞) → R+ , FB (t) = |χ[0,t) |B is called the fundamental function of the space B. Remark 2.11. The fundamental function FB is non-decreasing. Notation. We consider F (R+ ) - the space of all Banach function spaces B ∈ V(R+ ) with the property that sup FB (t) = ∞. t>0

Remark 2.12. If ϕ(t) ∈ (0, ∞), for all t ∈ (0, ∞), then the Orlicz space Oϕ belongs to F (R+ ) (see e.g. [13], Proposition 2.1.). Notation. Let L(R+ ) denote the class of all Banach function spaces B ∈ Q(R+ ) with the property that B \ L1 (R+ , R) = ∅. Remark 2.13. If B ∈ L(R+ ), then there exists α : R+ → R+ such that α ∈ B \ L1 (R+ , R). Notation. In what follows we denote by B(R+ ) the class of all Banach function spaces B ∈ V(R+ ) with the property that for every u ∈ B the function t+1 λu : R+ → R+ , λu (t) = t u(s) ds belongs to B. Lemma 2.14. If ϕ(1) < ∞, then the Orlicz space Oϕ belongs to B(R+ ). Proof. The Orlicz spaces are rearrangement invariant (see e.g. [5]). So they are interpolation spaces between L1 (R+ , R) and L∞ (R+ , R) (see e.g. [5], Theo t+1 rem 2.2, p. 106). Let A : L∞ (R+ , R) → L∞ (R+ , R), (A(u))(t) = t u(s) ds. Then A is a bounded linear operator with the property that the restriction A| : L1 (R+ , R) → L1 (R+ , R) is correctly deﬁned and bounded. Then A(Oϕ ) ⊂ Oϕ , so Oϕ ∈ B(R+ ). Proposition 2.15. Let B ∈ B(R+ ) and ν > 0. Then for every u ∈ B the functions fu , gu : R+ → R given by t ∞ −ν(t−s) e u(s) ds and gu (t) = e−ν(s−t) u(s) ds fu (t) = 0

belong to B.

t

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Proof. Let u ∈ B. Then |u| ∈ B. Since B ∈ B(R+ ), the function t+1 ˜ u : R+ → R+ , λ ˜u (t) = λ |u(s)| ds t

belongs to B. 1 Let c > 0 be such that 0 |v(τ )| dτ ≤ c |v|B , for all v ∈ B. For t ∈ [0, 1), we have that t 1 |u(s)| ds ≤ |u(s)| ds ≤ c |u|B χ[0,1) (t). (2.1) |fu (t)| ≤ 0

0

For t ≥ 1 we observe that |fu (t)| ≤

[t]−1 t−j t−j−1

j=0 [t]−1

≤

e

−ν(t−s)

e−νj

t−j

|u(s)| ds +

{t}

0

|u(s)| ds + e−ν[t]

t−j−1

j=0

0

e−ν(t−s) |u(s)| ds {t}

|u(s)| ds

(2.2)

[t]−1

≤

˜u (t − j − 1) + (c eν |u|B )e−νt . e−νj λ

j=0

˜ u (t − j − 1)χ[j+1,∞) (t). Since B is For j ∈ N, let λj : R+ → R+ , λj (t) = λ ˜ u |B , ˜ invariant to translations and λu ∈ B, we have that λj ∈ B and |λj |B = |λ for all j ∈ N. From (2.1) and (2.2), setting m = c |u|B , we deduce that |fu (t)| ≤

∞

e−νj λj (t) + eν m e−νt + m χ[0,1) (t),

∀t ≥ 0.

j=0

Using Lemma 2.7, from the above inequality we deduce that fu ∈ B and ˜ u |B /(1 − e−ν )) + eν m |eν |B + m |χ[0,1) |B . In addition we have |fu |B ≤ (|λ that ∞ t+j+1 ∞ ˜ u (t + j), ∀t ≥ 0. e−ν(s−t) |u(s)| ds ≤ e−νj λ |gu (t)| ≤ j=0

t+j

j=0

˜ u |B /(1 − e−ν ). Using Lemma 2.8, we obtain that gu ∈ B and |gu |B ≤ |λ

Notation. Let (X, || · ||) be a real or complex Banach space. For every B ∈ Q(R+ ) we denote by B(R+ , X), the linear space of all Bochner measurable functions u : R+ → X with the property that the mapping Nu : R+ → R+ , Nu (t) = ||u(t)|| lies in B. Endowed with the norm ||u||B(R+ ,X) := |Nu |B , B(R+ , X) is a Banach space.

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3. Admissibility and uniform dichotomy of evolution families Let X be a real or complex Banach space. The norm on X and on B(X)-the Banach algebra of all bounded linear operators on X, will be denoted by ||·||. Denote by Id the identity operator on X. Deﬁnition 3.1. A family U = {U (t, s)}t≥s≥0 ⊂ B(X) is called an evolution family if the following properties hold: (i) U (t, t) = Id , for t ≥ 0 and U (t, s)U (s, t0 ) = U (t, t0 ), for all t ≥ s ≥ t0 ≥ 0; (ii) there are M ≥ 1 and ω > 0 such that ||U (t, s)|| ≤ M eω(t−s) , for all t ≥ s ≥ 0; (iii) for every t0 ≥ 0 and every x ∈ X the mapping t → U (t, t0 )x is continuous on [t0 , ∞) and the mapping s → U (t0 , s)x is continuous on [0, t0 ]. Deﬁnition 3.2. We say that the evolution family U = {U (t, s)}t≥s≥0 is uniformly dichotomic if there exist a constant K ≥ 1 and a family of projections {P (t)}t≥0 such that the following conditions are satisﬁed: (i) U (t, s)P (s) = P (t)U (t, s), for all t ≥ s ≥ 0; (ii) ||U (t, s)x|| ≤ K ||x||, for all x ∈ Im P (s) and all t ≥ s ≥ 0; 1 ||y||, for all y ∈ Ker P (s) and all t ≥ s ≥ 0; (iii) ||U (t, s)y|| ≥ K (iv) U (t, s)| : Ker P (s) → Ker P (t) is an isomorphism, for all t ≥ s ≥ 0. Let I, O be two Banach function spaces with I ∈ Q(R+ ) and O ∈ V(R+ ). Deﬁnition 3.3. The pair (O(R+ , X), I(R+ , X)) is admissible for the evolution family U = {U (t, s)}t≥s≥0 if for every v ∈ I(R+ , X) there exists a continuous function f ∈ O(R+ , X) such that the pair (f, v) satisﬁes the equation t f (t) = U (t, s)f (s) + U (t, τ )v(τ ) dτ, ∀t ≥ s ≥ 0. (EU ) s

In what follows we consider the initial stable subspace Xs (0) := {x ∈ X : U (·, 0)x ∈ O(R+ , X)} and we will work in the hypothesis that Xs (0) is closed and complemented in X. Let H be a closed complement of Xs (0), i.e. Xs (0) ⊕ H = X. For every t0 > 0 and every x ∈ X we consider the function U (t, t0 )x, t ≥ t0 , ϕxt0 : R+ → X, ϕxt0 (t) = 0, t ∈ [0, t0 ). We deﬁne the stable subspace at t0 > 0, by Xs (t0 ) := {x ∈ X : ϕxt0 ∈ O(R+ , X)}. Moreover, for every t0 ≥ 0 we deﬁne the unstable subspace at t0 by Xu (t0 ) = U (t0 , 0)H. Lemma 3.4. If the pair (O(R+ , X), I(R+ , X)) is admissible for the evolution family U = {U (t, s)}t≥s≥0 , then for every v ∈ I(R+ , X) there exists a unique continuous function fv ∈ O(R+ , X) such that fv (0) ∈ Xu (0) and the pair (fv , v) satisﬁes the equation (EU ).

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Proof. Let v ∈ I(R+ , X). According to our hypothesis, there is a continuous function f ∈ O(R+ , X) such that the pair (f, v) satisﬁes the equation (EU ). Let xs ∈ Xs (0), xu ∈ Xu (0) be such that f (0) = xs + xu . We deﬁne fv : R+ → X, fv (t) = f (t) − U (t, 0)xs . Then fv is a continuous function with fv ∈ O(R+ , X) and fv (0) = xu ∈ Xu (0). An easy computation shows that the pair (f, v) satisﬁes the equation (EU ). To prove the uniqueness, let f˜ be a continuous function with f˜ ∈ O(R+ , X), f˜(0) ∈ Xu (0) such that the pair (f˜, v) satisﬁes the equation (EU ). Setting ϕ = f − f˜ we have that ϕ is continuous, ϕ ∈ O(R+ , X), ϕ(0) ∈ Xu (0) and ϕ(t) = U (t, 0)ϕ(0), ∀t ≥ 0. (3.1) Since ϕ ∈ O(R+ , X), from (3.1) we deduce that ϕ(0) ∈ Xs (0). Then ϕ(0) ∈ Xs (0) ∩ Xu (0), so ϕ(0) = 0. Using (3.1), we obtain the uniqueness of f. Lemma 3.5. The linear space Θ(R+ , X) = {f ∈ O(R+ , X) : f|[0,1] is continuous and f (0) ∈ Xu (0)} endowed with the norm ||f ||Θ(R+ ,X) := ||f ||O(R+ ,X) + sup ||f (t)|| is a Banach t∈[0,1]

space.

Proof. Let (fn ) be a fundamental sequence in Θ(R+ , X). In particular, (fn ) is fundamental in O(R+ , X). Since O(R+ , X) is a Banach space, there is f ∈ O(R+ , X) such that fn → f in O(R+ , X). Setting ϕn = fn|[0,1] we have that (ϕn ) is fundamental in the sup-norm. This implies that there exists ϕ : [0, 1] → X such that ϕn → ϕ uniformly on [0, 1]. It follows that ϕ is continuous on [0, 1] and since ϕn (0) ∈ Xu (0), for every n ∈ N, and Xu (0) is closed, we deduce that ϕ(0) ∈ Xu (0). From fn → f in O(R+ , X) and Lemma 2.6 (ii) we deduce that there is a subsequence (fkn ) with fkn → f a.e. Then, we have that f|[0,1] = ϕ a.e. on [0, 1]. Since in O(R+ , X) one identiﬁes the functions equal a.e., we may consider that f|[0,1] = ϕ. In conclusion, we obtain that f ∈ Θ(R+ , X) and fn → f in Θ(R+ , X), so Θ(R+ , X) is a Banach space. Remark 3.6. According to Lemma 3.4, if the pair (O(R+ , X), I(R+ , X)) is admissible for U, then it makes sense to consider the operator Γ : I(R+ , X) → Θ(R+ , X),

Γ(v) = fv

where fv is continuous, belongs to Θ(R+ , X) and the pair (fv , v) satisﬁes the equation (EU ). We have that Γ is correctly deﬁned and it is easy to see that Γ is a linear operator. Theorem 3.7. Γ is a bounded linear operator. Proof. It is suﬃcient to prove that Γ is closed. Let (vn ) ⊂ I(R+ , X), v ∈ I(R+ , X) and f ∈ Θ(R+ X) be such that vn → v in I(R+ , X) and Γ(vn ) → f in Θ(R+ , X). We set fn = Γ(vn ), for all n ∈ N.

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Let M, ω > 0 be given by Deﬁnition 3.1. Let t > 0. Since I ∈ Q(R+ ) t there is c(t) > 0 such that 0 |u(τ )| dτ ≤ c(t) |u|I , for all u ∈ I. This implies that t t U (t, τ )vn (τ ) dτ − U (t, τ )v(τ ) dτ 0 0 t ≤ M eωt ||vn (τ ) − v(τ )|| dτ ≤ M eωt c(t) ||vn − v||I(R+ ,X) . (3.2) 0

From relation (3.2) we deduce that t t U (t, τ )vn (τ ) dτ → U (t, τ )v(τ ) dτ, 0

0

as n → ∞, ∀t > 0.

(3.3)

Since fn → f in Θ(R+ , X), in particular, it follows that fn (t) → f (t) as n → ∞, for every t ∈ [0, 1]. Then, using (3.3) we obtain that t f (t) = U (t, 0)f (0) + U (t, τ )v(τ ) dτ, ∀t ∈ [0, 1]. (3.4) 0

Moreover, from fn → f in O(R+ , X) and Lemma 2.6 (ii), it follows that there is a negligible set A ⊂ (1, ∞) and a subsequence (fkn ) ⊂ (fn ) such that fkn (t) → f (t), as n → ∞, for every t ∈ (1, ∞) \ A. Then using (3.3) we deduce that t f (t) = U (t, 0)f (0) + U (t, τ )v(τ ) dτ, ∀t ∈ (1, ∞) \ A. (3.5) 0

Since in O(R+ , X) we identify the functions equal a.e., according to relations (3.4) and (3.5), we may consider that t f (t) = U (t, 0)f (0) + U (t, τ )v(τ ) dτ, ∀t ≥ 0. 0

This implies that f is continuous and the pair (f, v) satisﬁes the equation (EU ), so f = Γ(v). It follows that Γ is closed and by the closed graph theorem we deduce that Γ is bounded. Theorem 3.8. If the pair (O(R+ , X), I(R+ , X)) is admissible for U, then there is K > 0 such that ||U (t, t0 )x|| ≤ K ||x||,

∀x ∈ Xs (t0 ), ∀t ≥ t0 ≥ 0.

Proof. Let α : R+ → [0, 2] be a continuous function with supp α ⊂ (0, 1) and 1 α(τ ) dτ = 1. Let M, ω > 0 be given by Deﬁnition 3.1. 0 Let t0 ≥ 0 and x ∈ Xs (t0 ). Since I is invariant to translations, the function αt0 : R+ → R, αt0 (t) = α(t − t0 )χ[t0 ,∞) (t) has the property that |αt0 |I = |α|I . We consider the functions v : R+ → X, v(t) = αt0 (t)U (t, t0 )x and t αt0 (τ ) dτ U (t, t0 )x. f : R+ → X, f (t) = 0

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Since v ∈ C0c (R+ , X) we have that v ∈ I(R+ , X). The function f is continuous and f (0) = 0 ∈ Xu (0). Let λ = supt∈[0,t0 +1] ||f (t)||. Observing that f (t) = U (t, t0 )x, for all t ≥ t0 + 1, it follows that ||f (t)|| ≤ λ χ[0,t0 +1) (t) + ||U (t, t0 )x||χ[t0 +1,∞) (t), If t0 = 0, let

ϕx0

: R+ → X,

ϕxt0 : R+ → X,

∀t ≥ 0.

(3.6)

ϕx0 (t)

= U (t, 0)x and if t0 > 0 let U (t, t0 )x, t ≥ t0 , x ϕt0 (t) = 0, t ∈ [0, t0 ).

Since x ∈ Xs (t0 ) we have that ϕxt0 ∈ O(R+ , X), for all t0 ≥ 0. Using relation (3.6) and Lemma 2.6 (i), we deduce that f ∈ O(R+ , X), so f ∈ Θ(R+ , X). It is easy to see that the pair (f, v) satisﬁes the equation (EU ), so f = Γ(v). It follows that ||f ||O(R+ ,X) ≤ ||f ||Θ(R+ ,X) ≤ ||Γ|| ||v||I(R+ ,X) .

(3.7)

From ||v(t)|| = αt0 (t) ||U (t, t0 )x|| ≤ M e ||x|| αt0 (t), for all t ≥ 0, we have that (3.8) ||v||I(R+ ,X) ≤ M eω ||x|| |α|I . Let t ≥ t0 + 2. Since f (τ ) = U (τ, t0 )x, for all τ ≥ t0 + 1, we deduce that ω

||U (t, t0 )x|| χ[t−1,t) (s) ≤ M eω ||U (s, t0 )x|| χ[t−1,t) (s) ≤ M eω ||f (s)||, ∀s ≥ 0. From the invariance to translations of the space O and from the above inequality, it follows that ||U (t, t0 )x|| FO (1) ≤ M eω ||f ||O(R+ ,X) .

(3.9)

From (3.7), (3.8) and (3.9) we obtain that ||U (t, t0 )x|| ≤

M 2 e2ω ||Γ|| |α|I ||x||, FO (1)

∀t ≥ t0 + 2.

Setting K = max{M e2ω , (M 2 e2ω ||Γ|| |α|I )/FO (1)} we have ||U (t, t0 )x|| ≤ K ||x||, for all t ≥ t0 . Taking into account that K does not depend on t0 or x, we conclude that ||U (t, t0 )x|| ≤ K ||x||, for all x ∈ Xs (t0 ) and t ≥ t0 ≥ 0. Theorem 3.9. If the pair (O(R+ , X), I(R+ , X)) is admissible for U, then there is K > 0 such that 1 ||x||, ∀x ∈ Xu (t0 ), ∀t ≥ t0 ≥ 0. ||U (t, t0 )x|| ≥ K Proof. Let α : R+ → [0, 2] be a continuous function with supp α ⊂ (0, 1) and 1 α(τ ) dτ = 1. Let M, ω > 0 be given by Deﬁnition 3.1. 0 Let t ≥ t0 ≥ 0 and x ∈ Xu (0). Since I is invariant to translations, the function αt : R+ → R, αt (s) = α(s−t)χ[t,∞) (s) belongs to I and |αt |I = |α|I . We consider the functions v : R+ → X, v(s) = −αt (s) U (s, 0)x and ∞ f : R+ → X, f (s) = αt (τ ) dτ U (s, 0)x. s

We have that f ∈ Cc (R+ , X) and v ∈ C0c (R+ , X), so in particular, f ∈ O(R+ , X) and v ∈ I(R+ , X). Moreover, f is continuous with

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f (0) = x ∈ Xu (0), so f ∈ Θ(R+ , X). An easy computation shows that the pair (f, v) satisﬁes the equation (EU ), so f = Γ(v). This implies that ||f ||Θ(R+ ,X) ≤ ||Γ|| ||v||I(R+ ,X) .

(3.10)

We observe that f (s) = U (s, 0)x, for all s ∈ [0, t]. If t0 ∈ [0, 1], since ||f ||Θ(R+ ,X) = ||f ||O(R+ ,X) + supt∈[0,1] ||f (t)|| from (3.10) it follows that ||U (t0 , 0)x|| = ||f (t0 )|| ≤ ||Γ|| ||v||I(R+ ,X) .

(3.11)

If t0 > 1 then ||U (t0 , 0)x|| χ[t0 −1,t0 ) (s) ≤ M eω ||U (s, 0)x|| χ[t0 −1,t0 ) (s) ≤ M eω ||f (s)||, for all s ≥ 0. Using the invariance to translations of the space O, from the above inequality we deduce that ||U (t0 , 0)x|| FO (1) ≤ M eω ||f ||O(R+ ,X) .

(3.12)

Then, from (3.10) and (3.12) we obtain that ||U (t0 , 0)x|| ≤

M eω ||Γ|| ||v||I(R+ ,X) . FO (1)

(3.13)

Setting γ = max{||Γ||, M eω ||Γ||/FO (1)}, from relations (3.11) and (3.13) we have that (3.14) ||U (t0 , 0)x|| ≤ γ ||v||I(R+ ,X) . Moreover, from ||v(s)|| = αt (s) ||U (s, 0)x|| ≤ αt (s) M eω ||U (t, 0)x||, for all s ≥ 0, it follows that ||v||I(R+ ,X) ≤ M eω |α|I ||U (t, 0)x||.

(3.15)

Setting K = γM e |α|I , from relations (3.14) and (3.15) we obtain that ||U (t, 0)x|| ≥ (1/K) ||U (t0 , 0)x||. Since K does not depend on x, t0 or t, we deduce that ||U (t, 0)x|| ≥ (1/K) ||U (t0 , 0)x||, for all x ∈ Xu (0) and all t ≥ t0 ≥ 0. Since Xu (t0 ) = U (t0 , 0)Xu (0), for all t0 ≥ 0, the proof is complete. ω

Theorem 3.10. If the pair (O(R+ , X), I(R+ , X)) is admissible for U, then for every t0 ≥ 0, the subspaces Xs (t0 ) and Xu (t0 ) are closed. Proof. Step 1. We prove that Xs (t0 ) is closed, for all t0 ≥ 0. Let α : R+ → [0, 2] 1 be a continuous function with supp α ⊂ (0, 1) and 0 α(τ ) dτ = 1. Let M, ω > 0 be given by Deﬁnition 3.1. Let t0 > 0 and αt0 : R+ → R, αt0 (t) = α(t − t0 )χ[t0 ,∞) (t). Then αt0 ∈ I and |αt0 |I = |α|I . Let (xn ) ⊂ Xs (t0 ) with xn → x. For every n ∈ N we consider the functions vn : R+ → X, vn (t) = αt0 (t)U (t, t0 )xn t fn : R+ → X, fn (t) = αt0 (τ ) dτ U (t, t0 )xn . 0

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Using similar arguments as in the proof of Theorem 3.8, we deduce that vn ∈ I(R+ , X), fn ∈ O(R+ , X) and that fn = Γ(vn ), for all n ∈ N. Let v : R+ → X,

v(t) = αt0 (t)U (t, t0 )x

and let f = Γ(v). Then we have that ||fn −f ||O(R+ ,X) ≤ ||fn −f ||Θ(R+ ,X) ≤ ||Γ|| ||vn −v||I(R+ ,X) ,

∀n ∈ N. (3.16)

Observing that ||vn (t)−v(t)|| ≤ αt0 (t) M e ||xn −x||, for all n ∈ N, we deduce that (3.17) ||vn − v||I(R+ ,X) ≤ |α|I M eω ||xn − x||, ∀n ∈ N. From (3.16) and (3.17) it follows that fn → f in O(R+ , X). From Lemma 2.6 (ii) we obtain that there is a subsequence (fkn ) ⊂ (fn ) such that fkn → f a.e. on R+ . In particular, there is h > t0 + 1 such that fkn (h) → f (h). We observe that fn (t) = U (t, t0 )xn , for all t ≥ t0 + 1 and all n ∈ N. Then, we deduce that f (h) = lim fkn (h) = lim U (h, t0 )xkn = U (h, t0 )x. ω

n→∞

n→∞

Since f = Γ(v), we have that f (t) = U (t, h)f (h) = U (t, t0 )x, for all t ≥ h. Because f ∈ O(R+ , X) and O ∈ V(R+ ) it follows that the function U (t, t0 )x, t ≥ h, ϕ : R+ → X, ϕ(t) = 0, t ∈ [0, h) belongs to O(R+ , X). Considering the function U (t, t0 )x, x x ϕt0 : R+ → X, ϕt0 (t) = 0,

t ≥ t0 t ∈ [0, t0 ),

we have that ||ϕxt0 (t)|| ≤ M eω(h−t0 ) ||x|| χ[t0 ,h) (t) + ||ϕ(t)||, for all t ≥ 0. This implies that ϕxt0 ∈ O(R+ , X), so x ∈ Xs (t0 ). In conclusion, the subspace Xs (t0 ) is closed. Step 2. We prove that Xu (t0 ) is closed, for all t0 ≥ 0. Let K > 0 be given by Theorem 3.9. Let t0 > 0 and (yn ) ⊂ Xu (t0 ) with yn → y. Since yn ∈ Xu (t0 ) = U (t0 , 0)Xu (0), there is zn ∈ Xu (0) such that yn = U (t0 , 0)zn , for all n ∈ N. From 1 ||yn − ym || = ||U (t0 , 0)(zn − zm )|| ≥ ||zn − zm ||, ∀n, m ∈ N K we deduce that the sequence (zn ) is fundamental in Xu (0). Since Xu (0) is closed, it follows that there is z ∈ Xu (0) such that zn → z. Then y = lim yn = lim U (t0 , 0)zn = U (t0 , 0)z. n→∞

n→∞

It follows that y ∈ U (t0 , 0)Xu (0) = Xu (t0 ), so Xu (t0 ) is closed.

Lemma 3.11. If the pair (O(R+ , X), I(R+ , X)) is admissible for U, then Xs (t0 ) ∩ Xu (t0 ) = {0},

∀t0 ≥ 0.

Proof. Let t0 > 0 and x ∈ Xs (t0 ) ∩ Xu (t0 ). Since x ∈ Xu (t0 ), there is z ∈ Xu (0) such that x = U (t0 , 0)z. From x ∈ Xs (t0 ) we have that the function U (t, t0 )x, t ≥ t0 , x x ϕt0 : R+ → X, ϕt0 (t) = 0, t ∈ [0, t0 )

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belongs to O(R+ , X). Let ϕ : R+ → X, ϕ(t) = U (t, 0)z. If M, ω > 0 are given by Deﬁnition 3.1, we obtain that ||ϕ(t)|| ≤ M eωt0 ||z|| χ[0,t0 ) (t) + ||ϕxt0 (t)||, for all t ≥ 0. Since O ∈ V(R+ ) from the above inequality we deduce that ϕ ∈ O(R+ , X). This implies that z ∈ Xs (0), so z ∈ Xs (0) ∩ Xu (0). It follows that z = 0, so x = 0, and the proof is complete. Lemma 3.12. If the pair (O(R+ , X), I(R+ , X)) is admissible for U, then Xs (t0 ) + Xu (t0 ) = X,

∀t0 ≥ 0.

Proof. Let t0 > 0 and x ∈ X. Let α : R+ → [0, 2] be a continuous function t +1 with supp α ⊂ (t0 , t0 + 1) and t00 α(τ ) dτ = 1. We consider the function v : R+ → X,

v(t) = α(t)U (t, t0 )x.

Since v ∈ C0c (R+ , X) we have that v ∈ I(R+ , X). Let f = Γ(v). Then for every t ≥ t0 + 1 we have that t U (t, τ )v(τ ) dτ = U (t, t0 )(f (t0 ) + x). (3.18) f (t) = U (t, t0 )f (t0 ) + t0

Let

ϕ : R+ → X,

ϕ(t) =

U (t, t0 )(f (t0 ) + x), 0,

t ≥ t0 , t ∈ [0, t0 ).

If M, ω > 0 are given by Deﬁnition 3.1, then using (3.18) we deduce that ||ϕ(t)|| ≤ M eω ||f (t0 )+x|| χ[t0 ,t0 +1) (t)+||f (t)||, for all t ≥ 0. Since O ∈ V(R+ ) the above inequality implies that ϕ ∈ O(R+ , X), so f (t0 ) + x ∈ Xs (t0 ). In addition, we observe that f (t0 ) = U (t0 , 0)f (0). From f = Γ(v), we have that f (0) ∈ Xu (0), so f (t0 ) ∈ Xu (t0 ). In conclusion, we have that x = (x + f (t0 )) − f (t0 ) ∈ Xs (t0 ) + Xu (t0 ) and the proof is complete.

The main result of this section is: Theorem 3.13. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X and let I, O be two Banach function spaces with I ∈ Q(R+ ) and O ∈ V(R+ ). If the pair (O(R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) := {x ∈ X : U (·, 0)x ∈ O(R+ , X)} is closed and complemented in X, then the evolution family U is uniformly dichotomic. Proof. Let Xu (0) be a closed linear subspace of X such that X = Xs (0) ⊕ Xu (0). For every t0 > 0 we consider Xs (t0 ) - as the linear subspace of all x ∈ X with the property that the function U (t, t0 )x, t ≥ t0 , x x ϕt0 : R+ → X, ϕt0 (t) = 0, t ∈ [0, t0 )

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belongs to O(R+ , X). We set Xu (t0 ) = U (t0 , 0)Xu (0), for all t0 > 0. Then, from Theorem 3.10, Lemma 3.11 and Lemma 3.12 we obtain that Xs (t0 ) ⊕ Xu (t0 ) = X, for all t0 ≥ 0. For every t0 ≥ 0 let P (t0 ) be the projection such that Im P (t0 ) = Xs (t0 ) and Ker P (t0 ) = Xu (t0 ). Then U (t, t0 )P (t0 ) = P (t)U (t, t0 ),

∀t ≥ t0 ≥ 0.

From the deﬁnition of the spaces Xu (t0 ) and from Theorem 3.9 we deduce that the restriction U (t, t0 )| : Ker P (t0 ) → Ker P (t) is an isomorphism, for all t ≥ t0 ≥ 0. Finally, from Theorem 3.8 and Theorem 3.9 we obtain that the evolution family U is uniformly dichotomic.

4. Uniform exponential dichotomy In this section we continue the investigation begun for the case of uniform dichotomy and we will give necessary and suﬃcient conditions for uniform exponential dichotomy of evolution families on the half-line in terms of the admissibility of pairs of function spaces. Let X be a real or complex Banach space. Deﬁnition 4.1. We say that an evolution family U = {U (t, s)}t≥s≥0 is uniformly exponentially dichotomic if there exist two constants K ≥ 1, ν > 0 and a family of projections {P (t)}t≥0 such that the following conditions are satisﬁed: (i) (ii) (iii) (iv)

U (t, s)P (s) = P (t)U (t, s), for all t ≥ s ≥ 0; ||U (t, s)x|| ≤ K e−ν(t−s) ||x||, for all x ∈ Im P (s) and all t ≥ s ≥ 0; 1 ||U (t, s)y|| ≥ K eν(t−s) ||y||, for all y ∈ Ker P (s) and all t ≥ s ≥ 0; U (t, s)| : Ker P (s) → Ker P (t) is an isomorphism, for all t ≥ s ≥ 0.

Let I, O be two Banach function spaces with I ∈ Q(R+ ) and O ∈ V(R+ ). Let U = {U (t, s)}t≥s≥0 be an evolution family on X. In what follows we suppose that the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ O(R+ , X)} is closed and complemented in X. Let Xu (0) be a closed linear subspace such that X = Xs (0) ⊕ Xu (0). As in the previous section, for every t0 > 0 we consider Xs (t0 ) - the stable subspace and Xu (t0 ) - the unstable subspace. In this context, according to the main result in Section 3, if the pair (O(R+ , X), I(R+ , X)) is admissible for U, then U is uniformly dichotomic with respect to a family of projections {P (t)}t≥0 such that Im P (t0 ) = Xs (t0 )

and

Ker P (t0 ) = Xu (t0 ),

∀t0 ≥ 0.

In what follows, we prove that if O ∈ F(R+ )

or

I ∈ L(R+ )

then the admissibility of the pair (O(R+ , X), I(R+ , X)) for U implies the uniform exponential dichotomy of U.

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Lemma 4.2. Suppose that the following assertions hold: (i) there is δ > 0 such that ||U (t, t0 )x|| ≤ δ ||x||, for all x ∈ Im P (t0 ) and all t ≥ t0 ≥ 0; (ii) there is r > 0 such that ||U (t0 + r, t0 )x|| ≤ 1e ||x||, for all x ∈ Im P (t0 ) and all t0 ≥ 0. Then ||U (t, t0 )x|| ≤ Ke−ν(t−t0 ) ||x||, for all x ∈ Im P (t0 ) and all t ≥ t0 ≥ 0, where K = δe and ν = 1/r. Proof. Let t ≥ t0 ≥ 0 and x ∈ Im P (t0 ). If k ∈ N and s ∈ [0, r) are such that t = t0 + kr + s, then ||U (t, t0 )x|| ≤ δ ||U (t0 + kr, t0 )x|| ≤ δe−k ||x|| ≤ Ke−ν(t−t0 ) ||x||.

Theorem 4.3. If O ∈ F(R+ ) or I ∈ L(R+ ) and the pair (O(R+ , X), I(R+ , X)) is admissible for U, then there are K, ν > 0 such that ||U (t, t0 )x|| ≤ Ke−ν(t−t0 ) ||x||,

∀x ∈ Im P (t0 ), ∀t ≥ t0 ≥ 0.

Proof. Let δ > 0 be such that ||U (t, t0 )x|| ≤ δ ||x||,

∀t ≥ t0 ≥ 0, ∀x ∈ Im P (t0 ).

(4.1)

Case 1: O ∈ F(R+ ). Then supt>0 FO (t) = ∞. Let α : R+ → [0, 2] be a 1 continuous function with supp α ⊂ (0, 1) and 0 α(τ ) dτ = 1. Let p > 0 be such that FO (p) ≥ e δ 2 ||Γ|| |α|I . (4.2) Let t0 ≥ 0 and let x ∈ Im P (t0 ). From I ∈ Q(R+ ) we have that the function αt0 : R+ → R, αt0 (t) = α(t − t0 )χ[t0 ,∞) (t) lies in I and |αt0 |I = |α|I . If U (t0 + 1, t0 )x = 0, we consider the functions v : R+ → X,

v(t) =

αt0 (t) U (t, t0 )x ||U (t, t0 )x||

t

αt0 (τ ) dτ U (t, t0 )x. ||U (τ, t0 )x|| 0 We have that v ∈ C0c (R+ , X), so v ∈ I(R+ , X). We observe that f is continuous with f (0) = 0. Moreover, setting t0 +1 αt0 (τ ) dτ λ := ||U (τ, t0 )x|| t0 f : R+ → X,

f (t) =

we have that f (t) = λ U (t, t0 )x, for all t ≥ t0 + 1. If t0 > 0 let U (t, t0 )x, t ≥ t0 , ϕxt0 : R+ → X, ϕxt0 (t) = 0, t ∈ [0, t0 ) and if t0 = 0 let ϕx0 : R+ → X, ϕx0 (t) = U (t, 0)x. Since x ∈ Im P (t0 ) = Xs (t0 ) we have that ϕxt0 ∈ O(R+ , X). Observing that ||f (t)|| ≤ λδ||x|| χ[t0 ,t0 +1) (t) + λ ||ϕxt0 (t)||, for all t ≥ 0 we deduce that

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f ∈ O(R+ , X). Thus f ∈ Θ(R+ , X). An easy computation shows that the pair (f, v) satisﬁes the equation (EU ), so f = Γ(v). This implies that ||f ||O(R+ ,X) ≤ ||Γ|| ||v||I(R+ ,X) .

(4.3)

From ||v(t)|| = αt0 (t), for all t ≥ 0, we have that ||v||I(R+ ,X) = |αt0 |I = |α|I . Then, from (4.3) we deduce that ||f ||O(R+ ,X) ≤ ||Γ|| |α|I .

(4.4)

It is easy to see that λ ≥ 1/(δ||x||). Then from ||U (t0 + p + 1, t0 )x|| χ[t0 +1,t0 +p+1) (s) ≤ δ ||U (s, t0 )x|| χ[t0 +1,t0 +p+1) (s) δ ||f (s)|| χ[t0 +1,t0 +p+1) (s) ≤ δ 2 ||x|| ||f (s)||, ∀s ≥ 0 λ using the invariance to translations of the space O and relation (4.4) we deduce that =

||U (t0 + p + 1, t0 )x|| FO (p) ≤ δ 2 ||Γ|| |α|I ||x||.

(4.5)

Setting r = p + 1 from relation (4.2) and (4.5) we obtain that ||U (t0 + r, t0 )x|| ≤ (1/e) ||x||. If U (t0 + 1, t0 )x = 0 then this inequality is obvious. Taking into account that r does not depend on t0 or x, it follows that 1 ||x||, ∀x ∈ Im P (t0 ), ∀t0 ≥ 0. e From relation (4.6), using Lemma 4.2 we obtain the conclusion. ||U (t0 + r, t0 )x|| ≤

Case 2: I ∈ L(R+ ). Let c > 0 be such that 1 |w(τ )| dτ ≤ c |w|O , 0

∀w ∈ O.

(4.6)

(4.7)

Since I ∈ L(R+ ) according to Remark 2.13, there is u : R+ → R+ with u ∈ I \ L1 (R+ , R). Let p > 0 be such that p u(s) ds ≥ e δ 2 c ||Γ|| |u|I . (4.8) 0

Let u0 = u and for t0 > 0, let ut0 : R+ → R, ut0 (t) = u(t − t0 )χ[t0 ,∞) (t). From the invariance to translations of I we have that |ut0 |I = |u|I , for all t0 ≥ 0. Let t0 ≥ 0 and x ∈ ImP (t0 ). Let ϕ : R+ → [0, 1] be a continuous function with compact support such that ϕ(t) = 1, for all t ∈ [0, t0 + p] and ϕ(t) = 0, for all t ≥ t0 + p + 1. We deﬁne the functions v : R+ → X, and

v(t) = ut0 (t) ϕ(t) U (t, t0 )x

ψ : R+ → R + ,

ψ(t) =

ϕ(t)||U (t, t0 )x||, ||x||,

t > t0 , t ∈ [0, t0 ].

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Then ||v(t)|| = ut0 (t)ψ(t), for all t ≥ 0. Since ψ ∈ Cc (R+ , R), ut0 ∈ I and I ∈ L(R+ ) we deduce that ut0 ψ ∈ I. It follows that v ∈ I(R+ , X). In addition ||v(t)|| ≤ ut0 (t)||U (t, t0 )x|| ≤ δ ||x|| ut0 (t), for all t ≥ 0. Hence we have that ||v||I(R+ ,X) ≤ δ ||x|| |ut0 |I = δ ||x|| |u|I . Let

f : R+ → X,

f (t) = 0

(4.9)

t

ut0 (τ ) ϕ(τ ) dτ U (t, t0 )x

t +p+1 ut0 (τ )ϕ(τ ) dτ. Then f is continuous, f (0) = 0 and f (t) = and λ := t00 λ U (t, t0 )x, for all t ≥ t0 + p + 1. Using similar arguments with those from Case 1, we deduce that f ∈ O(R+ , X), so f ∈ Θ(R+ , X). We observe that the pair (f, v) satisﬁes the equation (EU ), so f = Γ(v). Then from relation (4.9) we deduce that ||f ||O(R+ ,X) ≤ ||f ||Θ(R+ ,X) ≤ ||Γ|| ||v||I(R+ ,X) ≤ δ ||Γ|| |u|I ||x||.

(4.10)

We consider the function g : R+ → X, g(t) = f (t + t0 + p + 1). From Lemma 2.8 we have that g ∈ O(R+ , X) and ||g||O(R+ ,X) ≤ ||f ||O(R+ ,X) . Using relation (4.7) it follows that 1 ||g(τ )|| dτ ≤ c ||g||O(R+ ,X) ≤ c ||f ||O(R+ ,X) . 0

Taking into account that ||U (t0 + p + 2, t0 )x|| ≤ δ ||U (s, t0 )x||, for all s ∈ [t0 + p + 1, t0 + p + 2], we deduce that t0 +p+2 δ t0 +p+2 ||U (s, t0 )x|| ds = ||f (s)|| ds ||U (t0 + p + 2, t0 )x|| ≤ δ λ t0 +p+1 t0 +p+1 δ 1 δc ||f ||O(R+ ,X) . ||g(τ )|| dτ ≤ (4.11) = λ 0 λ We note that t0 +p ut0 (τ )ϕ(τ ) dτ = λ≥ t0

t0 +p

t0

ut0 (τ ) dτ =

p

u(τ ) dτ

(4.12)

0

and then from relations (4.8), (4.10), (4.11) and (4.12) it follows that ||U (t0 + p + 2, t0 )x|| ≤ (1/e) ||x||. We set r = p + 2 and since r does not depend on t0 or x, we obtain that ||U (t0 + r, t0 )x|| ≤ (1/e) ||x||, for all x ∈ Im P (t0 ) and all t0 ≥ 0. Finally, by applying Lemma 4.2 we deduce the conclusion. Lemma 4.4. Suppose that the following assertions hold: (i) there is δ > 0 such that ||U (t, t0 )x|| ≥ 1δ ||x||, for all x ∈ Ker P (t0 ) and all t ≥ t0 ≥ 0; (ii) there is r > 0 such that ||U (t0 + r, t0 )x|| ≥ e ||x||, for all x ∈ Ker P (t0 ) and all t0 ≥ 0. 1 Then ||U (t, t0 )x|| ≥ K eν(t−t0 ) ||x||, for all x ∈ Ker P (t0 ) and all t ≥ t0 ≥ 0, where K = δe and ν = 1/r.

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Proof. Let t ≥ t0 ≥ 0 and x ∈ Ker P (t0 ). Let k ∈ N and s ∈ [0, r) be such that t = t0 + kr + s. Then 1 1 1 ν(t−t0 ) e ||x||. ||U (t, t0 )x|| ≥ ||U (t0 + kr, t0 )x|| ≥ ek ||x|| ≥ δ δ K Theorem 4.5. If O ∈ F(R+ ) or I ∈ L(R+ ) and the pair (O(R+ , X), I(R+ , X)) is admissible for U, then there are K, ν > 0 such that ||U (t, t0 )x|| ≥

1 ν(t−t0 ) ||x||, e K

∀x ∈ Ker P (t0 ), ∀t ≥ t0 ≥ 0.

Proof. Let M, ω > 0 be given by Deﬁnition 3.1. Let δ > 0 be such that 1 ||x||, ∀x ∈ Ker P (t0 ), ∀t ≥ t0 ≥ 0. δ Case 1: O ∈ F(R+ ). Let α : R+ → [0, 2] be a continuous function with 1 supp α ⊂ (0, 1) and 0 α(τ ) dτ = 1. Let p > 0 be such that ||U (t, t0 )x|| ≥

FO (p) ≥ e δ 2 ||Γ|| |α|I .

(4.13)

Let t0 ≥ 0 and x ∈ Ker P (t0 ) \ {0}. Since I ∈ Q(R+ ) the function αt0 +p : R+ → R, αt0 +p (t) = α(t − t0 − p)χ[t0 +p,∞) (t) has the property that |αt0 +p |I = |α|I . We consider the functions v : R+ → X,

f : R+ → X,

f (t) =

v(t) = −

⎧ ∞ ⎪ ⎨ t

αt0 +p (t) U (t, t0 )x, ||U (t, t0 )x||

αt0 +p (τ ) ||U(τ,t0 )x||

dτ U (t, t0 )x, λ U (t, t0 )x, λ U (t0 , t)−1 | x,

⎪ ⎩

t ≥ t0 + p, t ∈ (t0 , t0 + p), t ∈ [0, t0 ],

denotes the inverse of the operator where U (t0 , t)−1 | U (t0 , t)| : Ker P (t) → Ker P (t0 ), for all t ∈ [0, t0 ] and

t0 +p+1

λ := t0 +p

αt0 +p (τ ) dτ. ||U (τ, t0 )x||

Since v ∈ C0c (R+ , X) we have that v ∈ I(R+ , X). From ||v(t)|| = αt0 +p (t), for all t ≥ 0, we deduce that ||v||I(R+ ,X) = |α|I . We observe that f ∈ Cc (R+ , X), so f ∈ O(R+ , X). In addition, we note that f (0) = λ U (t0 , 0)−1 | x ∈ Ker P (0) = Xu (0) and hence f ∈ Θ(R+ , X). An easy computation shows that the pair (f, v) satisﬁes the equation (EU ). Thus f = Γ(v), so ||f ||O(R+ ,X) ≤ ||Γ|| ||v||I(R+ ,X) = ||Γ|| |α|I . From ||U (t, t0 )x|| ≥ (1/δ)||x||, for all t ∈ [t0 , t0 + p) we have that ||x|| χ[t0 ,t0 +p) (t) ≤ δ ||U (t, t0 )x|| χ[t0 ,t0 +p) (t) ≤ (δ/λ) ||f (t)||,

(4.14)

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for all t ≥ 0. Now, using the invariance to translations of the space O we deduce that ||x|| FO (p) ≤ (δ/λ) ||f ||O(R+ ,X) . (4.15) From relations (4.13)-(4.15) it follows that e δ ||x|| ≤ 1/λ.

(4.16)

From ||U (t0 + p + 1, t0 )x|| ≥ (1/δ) ||U (τ, t0 )x||, for all τ ∈ [t0 + p, t0 + p + 1], we obtain that 1 . (4.17) λ≥ δ ||U (t0 + p + 1, t0 )x|| Then, from (4.16) and (4.17) it follows that ||U (t0 + p + 1, t0 )x|| ≥ e ||x||. We set r = p + 1 and taking into account that r does not depend on t0 or x we deduce that ||U (t0 + r, t0 )x|| ≥ e ||x||, for all x ∈ Ker P (t0 ) and all t0 ≥ 0. By applying Lemma 4.4 we obtain the conclusion. Case 2: I ∈ L(R+ ). Let c ≥ 1 be such that 1 |w(τ )| dτ ≤ c |w|O , 0

∀w ∈ O.

(4.18)

Since I ∈ L(R+ ) there is u : R+ → R+ with u ∈ I \ L1 (R+ , R). Let p > 0 be such that p u(s) ds ≥ e c M eω δ ||Γ|| |u|I . (4.19) 0

Let u0 = u and for t0 > 0 let ut0 : R+ → R, ut0 (t) = u(t − t0 )χ[t0 ,∞) (t). From the invariance to translations of the space I we have that |ut0 |I = |u|I , for all t0 ≥ 0. Let t0 ≥ 0 and x ∈ Ker P (t0 ). Let ϕ : R+ → [0, 1] be a continuous function with compact support such that ϕ(t) = 1, for t ∈ [0, t0 + p] and ϕ(t) = 0, for t ≥ t0 + p + 1. We consider the functions v : R+ → X, v(t) = −ut0 (t)ϕ(t)U (t, t0 )x ∞ t > t0 , t ut0 (τ )ϕ(τ ) dτ U (t, t0 )x, f : R+ → X, f (t) = λ U (t0 , t)−1 x, t ∈ [0, t0 ], | t0 +p+1 ut0 (τ )ϕ(τ ) dτ . Using similar arguments as in the proof where λ := t0 of Theorem 4.3, Case 2, we obtain that v ∈ I(R+ , X). From ||U (t0 + p + 1, t0 )x|| ≥ (1/δ) ||U (t, t0 )x||, for all t ∈ [t0 , t0 + p + 1], we deduce that ||v(t)|| = ut0 (t)ϕ(t)||U (t, t0 )x|| ≤ δ ut0 (t)||U (t0 + p + 1, t0 )x||, for all t ≥ 0. This implies that ||v||I(R+ ,X) ≤ δ |u|I ||U (t0 + p + 1, t0 )x||. Since f ∈ Cc (R+ , X) we have that f ∈ O(R+ , X). In addition, f (0) = λ U (t0 , 0)−1 | x ∈ Ker P (0) = Xu (0),

(4.20)

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so f ∈ Θ(R+ , X). An easy computation shows that the pair (f, v) satisﬁes the equation (EU ), so f = Γ(v). Using (4.20), it follows that ||f ||Θ(R+ ,X) ≤ ||Γ|| ||v||I(R+ ,X) ≤ δ ||Γ|| |u|I ||U (t0 + p + 1, t0 )x||.

(4.21)

If t0 ∈ [0, 1], then λ ||x|| = ||f (t0 )|| ≤ supt∈[0,1] ||f (t)|| ≤ ||f ||Θ(R+ ,X) . Hence using (4.21) we have that λ ||x|| ≤ δ ||Γ|| |u|I ||U (t0 + p + 1, t0 )x||.

(4.22)

If t0 > 1, then let g : R+ → X, g(t) = f (t + t0 − 1). Since f ∈ O(R+ , X), g ∈ O(R+ , X) and ||g||O(R+ ,X) ≤ ||f ||O(R+ ,X) (see Lemma 2.8). We observe that λx = U (t0 , t)f (t), for all t ∈ [t0 − 1, t0 ]. Thus, using relations (4.18) and (4.21) we successively deduce that t0 1 ω ω ||f (τ )|| dτ = M e ||g(τ )|| dτ ≤ c M eω ||g||O(R+ ,X) λ ||x|| ≤ M e 0

t0 −1

≤ c M eω ||f ||O(R+ ,X) ≤ c M eω δ ||Γ|| |u|I ||U (t0 + p + 1, t0 )x||. (4.23) p Observing that λ ≥ 0 u(τ ) dτ , from (4.22), (4.23) and (4.19) we obtain that ||U (t0 + p + 1, t0 )x|| ≥ e ||x||. We set r = p + 1 and since r does not depend on t0 or x we have that ||U (t0 + r, t0 )x|| ≥ e ||x||, for all t0 ≥ 0 and all x ∈ Ker P (t0 ). By applying Lemma 4.4 we obtain the conclusion. The main result of this section is: Theorem 4.6. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X and let I, O be two Banach function spaces with I ∈ Q(R+ ) and O ∈ V(R+ ). If O ∈ F(R+ ) or I ∈ L(R+ ), then the following assertions hold: (i) if the pair (O(R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ O(R+ , X)} is closed and complemented in X, then U is uniformly exponentially dichotomic; (ii) if I ⊂ O and one of the spaces I or O belongs to B(R+ ), then U is uniformly exponentially dichotomic if and only if the pair (O(R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) is closed and complemented in X. Proof. (i) This follows from Theorem 3.13, Theorem 4.3 and Theorem 4.5. (ii) Necessity. Suppose that U is uniformly exponentially dichotomic with respect to the family of projections {P (t)}t≥0 and the constants K, ν > 0. Then, we have that L := supt≥0 ||P (t)|| < ∞ (see e.g.[15]). Let v ∈ I(R+ , X). We consider the function t ∞ U (t, τ )P (τ )v(τ ) dτ − U (τ, t)−1 f : R+ → X, f (t) = | (I − P (τ ))v(τ ) dτ. 0

t

We have that f is continuous and for all t ≥ 0 t −ν(t−τ ) ||f (t)|| ≤ LK e ||v(τ )|| dτ + (L + 1)K 0

t

∞

e−ν(τ −t) ||v(τ )|| dτ. (4.24)

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If I ∈ B(R+ ), since v ∈ I(R+ , X) it follows that the function t+1 ||v(s)|| ds λv : R+ → R+ , λv (t) = t

lies in I. Using Proposition 2.15, relation (4.24) and the fact that I ⊂ O we deduce that f ∈ O(R+ , X). If O ∈ B(R+ ), then from I ⊂ O and v ∈ I(R+ , X) we have that v ∈ O(R+ , X). Then, from Proposition 2.15 and relation (4.24) we obtain that f ∈ O(R+ , X). It is easy to see that the pair (f, v) satisﬁes the equation (EU ). It follows that the pair (O(R+ , X), I(R+ , X)) is admissible for U. We prove that Xs (0) = Im P (0). Let x ∈ Im P (0). Then ||U (t, 0)x|| ≤ Ke−νt ||x||, for all t ≥ 0. Since O ∈ V(R+ ), using Lemma 2.7 it follows that U (·, 0)x ∈ O(R+ , X), so x ∈ Xs (0). This shows that Im P (0) ⊂ Xs (0). Conversely, let x ∈ Xs (0). Then, the function ϕ : R+ → X, ϕ(t) = U (t, 0)x belongs to O(R+ , X). Setting xs = P (0)x and xu = (I − P (0))x, we have that ||ϕ(t)|| ≥ ||U (t, 0)xu || − ||U (t, 0)xs || ≥ (1/K) eνt ||xu || − K||xs ||, for all t ≥ 0. It follows that t+1 1 νt e ||xu || − K ||xs ||, ∀t ≥ 0. ||ϕ(τ )|| dτ ≥ (4.25) K t t+1 From Lemma 2.9 we have that supt≥0 t ||ϕ(τ )|| dτ < ∞. Then using (4.25) we obtain that xu = 0, so x = P (0)x. This shows that x ∈ Im P (0), so Xs (0) = Im P (0). Thus, we conclude that Xs (0) is closed and complemented in X. Suﬃciency. This follows from (i).

5. Connections and applications In this section we present some applications of the main results of the paper and establish the connections with other dichotomy results published in the past few years. We will emphasize the generality of our conditions and provide some optimal methods in the study of dichotomy. The section is organized in two parts. In the ﬁrst part, we prove that the result obtained in Theorem 4.6 is the most general in this topic and, in the second part, we obtain as consequences several criteria with wide applicability area. Let Cb (R+ , X) denote the space of all continuous functions v : R+ → X with the property that supt≥0 ||v(t)|| < ∞ and let C0 (R+ , X) be the space

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of all continuous functions v : R+ → X with lim v(t) = 0. With respect to t→∞ the norm |||v||| := sup ||v(t)|| t≥0

the spaces Cb (R+ , X) and C0 (R+ , X) are Banach spaces. The ﬁrst natural question arises whether the hypothesis “O ∈ F(R+ ) or I ∈ L(R+ )” can be removed. Speciﬁcally, the main question is: if O ∈ F(R+ ) and I ∈ L(R+ ), then is the admissibility of the pair (O(R+ , X), I(R+ , X)) a suﬃcient condition for uniform exponential dichotomy? Generally, the answer is negative. To argue this fact we need a technical lemma. Lemma 5.1. Let O be a Banach function space with O ∈ V(R+ ) \ F (R+ ). Then C0 (R+ , R) ⊂ O. Proof. Let γ := supt>0 FO (t). Let u ∈ C0 (R+ , R). Then there is an unbounded increasing sequence (tn ) ⊂ (0, ∞) such that |u(t)| ≤ 1/(n + 1), for all t ≥ tn and all n ∈ N. Setting un = u χ[0,tn ) we have that |un+p − un |O ≤ 1/(n + 1) |χ[tn ,tn+p ) |O ≤ γ/(n + 1), for all p ∈ N∗ and all n ∈ N. The above inequality shows that the sequence (un ) is fundamental in the Banach space O, so there is v ∈ O such that un → v in O. From Lemma 2.6 (ii) there exists a subsequence (ukn ) such that ukn → v a.e. This implies that u = v a.e., so u = v in O. Thus u ∈ O and the proof is complete. In what follows we prove that the assumptions on the structure of the underlying function spaces are indeed necessary. Example. Let I, O be two Banach function spaces with I ∈ Q(R+ ) and O ∈ V(R+ ). We suppose that O ∈ F(R+ ) and I ∈ L(R+ ). This implies that I ⊂ L1 (R+ , R) and from Lemma 5.1 we have that C0 (R+ , R) ⊂ O. Let X = R2 with the norm ||(x1 , x2 )|| = |x1 | + |x2 |, for all (x1 , x2 ) ∈ X. For every t ≥ s ≥ 0 we consider the operator

s+1 t−s x1 , e x2 . U (t, s) : X → X, U (t, s)(x1 , x2 ) = t+1 Then U = {U (t, s)}t≥s≥0 is an evolution family on X. We prove that the pair (O(R+ , X), I(R+ , X)) is admissible for U. Let v = (v1 , v2 ) ∈ I(R+ , X). Then v1 , v2 ∈ L1 (R+ , R). We consider the function

t ∞ τ +1 v1 (τ ) dτ, − f : R+ → X, f (t) = e−(τ −t) v2 (τ ) dτ . 0 t+1 t We have that f is correctly deﬁned, continuous and an easy computation shows that the pair (f, v) satisﬁes the equation (EU ). Setting t ∞ τ +1 v1 (τ ) dτ, f2 (t) = − e−(τ −t) v2 (τ ) dτ, ∀t ≥ 0, f1 (t) = t + 1 0 t it is obvious that f2 ∈ C0 (R+ , R).

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We prove that f1 ∈ C0 (R+ , R).Let ε > 0. From v1 ∈ L1 (R+ , R) it ∞ follows that there is δ > 0 such that δ |v1 (τ )| dτ < ε/2. Then, for t > δ, we have that δ 1 ε (5.1) (τ + 1)|v1 (τ )| dτ + . |f1 (t)| ≤ t+1 0 2 Let δ˜ > δ be such that δ ε 1 ˜ (5.2) (τ + 1) |v1 (τ )| dτ < , ∀t ≥ δ. t+1 0 2 ˜ so Then from (5.1) and (5.2) it follows that |f1 (t)| < ε, for all t ≥ δ, f1 ∈ C0 (R+ , R). This shows that f ∈ C0 (R+ , X), so f ∈ O(R+ , X). It follows that the pair (O(R+ , X), I(R+ , X)) is admissible for U. Let Xs (0) := {x ∈ X : U (·, 0)x ∈ O(R+ , X)}. It is easy to see that R × {0} ⊂ Xs (0). Conversely, let x = (x1 , x2 ) ∈ Xs (0). Then 1 |x1 | + et |x2 | ≥ et |x2 |, ∀t ≥ 0. (5.3) ||U (t, 0)x|| = t+1 Since U (·, 0)x ∈ O(R+ , X) using similar arguments as in the proof of necessity of Theorem 4.6 (ii), from relation (5.3) it follows that x2 = 0, so x = (x1 , 0) ∈ R × {0}. Hence Xs (0) = R × {0}, so Xs (0) is closed and complemented in X. Suppose that U is uniformly exponentially dichotomic and let {P (t)}t≥0 be a family of projections such that U is uniformly exponentially dichotomic with respect to this family. Using similar arguments with those in the necessity of Theorem 4.6, we have that Im P (0) = Xs (0), so Im P (0) = R × {0}. Let K, ν > 0 be given by Deﬁnition 4.1. Then ||U (t, 0)x|| ≤ Ke−νt ||x||, for all t ≥ 0 and all x ∈ Im P (0). This implies that [1/(t + 1)] |x1 | ≤ Ke−νt |x1 |, for all t ≥ 0 and x1 ∈ R, which is absurd. It follows that U is not uniformly exponentially dichotomic. Remark 5.2. The above example shows that the hypotheses from Theorem 4.6 cannot be dropped and that the results given by Theorem 4.6 are the most general in this topic. It is important to point out that the input-output characterizations for exponential dichotomy relay on the veriﬁcation of the solvability of an integral equation for diverse input functions. The inputs are test functions and therefore, in order to improve the admissibility concept, the input set should be ”small”, while the output space should be a ”large” space. In this context, we note that all the classes of Banach function spaces, considered throughout this paper, are closed to ﬁnite intersections. This fact allows various optimizations of the input space. Remark 5.3. Let M 1 (R+ , R) be the linear space of all measurable functions t+1 u : R+ → R with supt≥0 t |u(s)| ds < ∞. With respect to the norm t+1 ||u||M 1 := sup |u(s)| ds < ∞ t≥0

t

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this is a Banach function space, which belongs to B(R+ ). Moreover, from Lemma 2.9 it follows that M 1 (R+ , R) is the largest space in V(R+ ). Theorem 5.4. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X and let I be a Banach function space in L(R+ ). The evolution family U is uniformly exponentially dichotomic if and only if the pair (M 1 (R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) := {x ∈ X : U (·, 0)x ∈ M 1 (R+ , X)} is closed and complemented in X. Remark 5.5. The above result allows the optimization of the input-output characterizations for uniform exponential dichotomy of evolution families on the half-line. Precisely, since the output space is the largest in V(R+ ), it suﬃces to consider smaller and smaller input spaces. Corollary 5.6. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X, let n ∈ N∗ , p1 , . . . , pn ∈ (1, ∞) and let I = Lp1 (R+ , R) ∩ . . . ∩ Lpn (R+ , R) ∩ C00 (R+ , R). Then U is uniformly exponentially dichotomic if and only if the pair (M 1 (R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ M 1 (R+ , X)} is closed and complemented in X. Theorem 5.7. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X. Let n ∈ N∗ and let Oϕ , Oψ1 , . . . , Oψn be Orlicz spaces with ϕ(t) ∈ (0, ∞), for all t > 0 and ψk (1) < ∞, for all k ∈ {1, . . . , n}. Let I = Oϕ ∩ Oψ1 ∩ . . . Oψn ∩ C00 (R+ , R). The family U is uniformly exponentially dichotomic if and only if the pair (Oϕ (R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ Oϕ (R+ , X)} is closed and complemented in X. Proof. From Remark 2.12 and Lemma 2.14 we have that Oϕ ∈ F(R+ )∩B(R+ ). By applying Theorem 4.6 the proof is complete. Corollary 5.8. Let p ∈ [1, ∞) and let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X. Let n ∈ N∗ , q1 , . . . , qn ∈ [1, ∞) and let I(R+ , X) = Lp (R+ , X) ∩ Lq1 (R+ , X) ∩ . . . ∩ Lqn (R+ , X) ∩ C00 (R+ , X). Then, the family U is uniformly exponentially dichotomic if and only if the pair (Lp (R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ Lp (R+ , X)} is closed and complemented in X. Proof. This immediately follows from Theorem 5.7.

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Remark 5.9. Using diﬀerent arguments, the above corollary was recently proved in [23] (see Theorem 3.3). Corollary 5.10. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X, let n ∈ N∗ , p1 , . . . , pn ∈ (1, ∞) and let I = Lp1 (R+ , R) ∩ . . . ∩ Lpn (R+ , R) ∩ C00 (R+ , R). Then U is uniformly exponentially dichotomic if and only if the pair (Cb (R+ , X), I(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ Cb (R+ , X)} is closed and complemented in X. Proof. We note that I ∈ L(R+ ). Hence, we apply Theorem 4.6 for O = L∞ (R+ , R), taking into account that a continuous function f lies in L∞ (R+ , R) if and only if f ∈ Cb (R+ , R). Remark 5.11. The above corollary was recently proved in [24], employing discrete-time arguments (see [24], Theorem 4.2). This also generalizes the main results of the papers [15] and [19] (see [15] Theorem 3.2 and [19] Theorem 4.3). Lemma 5.12. If V ∈ V(R+ ), then either V ∈ F(R+ ) or V ∈ L(R+ ). Proof. Indeed, suppose by contrary that there exists V ∈ V(R+ ) such V ∈ F(R+ ) and V ∈ L(R+ ). Then γ := supt>0 FV (t) < ∞ V ⊂ L1 (R+ , R). From V ⊂ L1 (R+ , R) it follows that there is c > 0 such ||v||1 ≤ c |v|V , for all v ∈ V . In particular, for v = χ[0,t) we deduce t ≤ c |χ[0,t) |V = c FV (t) ≤ cγ, for all t > 0, which is absurd.

that and that that

Theorem 5.13. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X and let E ∈ V(R+ ). The following assertions hold: (i) if the pair (E(R+ , X), E(R+ , X)) is admissible for U and the subspace Xs (0) := {x ∈ X : U (·, 0)x ∈ E(R+ , X)} is closed and complemented in X, then U is uniformly exponentially dichotomic; (ii) if E ∈ B(R+ ), then U is uniformly exponentially dichotomic if and only if the pair (E(R+ , X), E(R+ , X)) is admissible for U and the subspace Xs (0) is closed and complemented in X. Proof. This follows from Theorem 4.6 and Lemma 5.12.

Lemma 5.14. Let B ∈ V(R+ ) and let ν > 0. Then, for every u : R+ → R+ in B, the functions fu , gu : R+ → R+ given by t ∞ −ν(t−τ ) fu (t) = e u(τ ) dτ and gu (t) = e−ν(τ −t) u(τ ) dτ 0

t

belong to Cb (R+ , R+ ). Proof. This follows using similar arguments as in Proposition 2.15 and using t+1 that, according to Lemma 2.9, we have that sup t u(τ ) dτ < ∞. t≥0

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Corollary 5.15. Let U = {U (t, s)}t≥s≥0 be an evolution family on the Banach space X and let E ∈ B(R+ ). Then U is uniformly exponentially dichotomic if and only if the pair (Cb (R+ , X) ∩ E(R+ , X), E(R+ , X)) is admissible for U and the subspace Xs (0) = {x ∈ X : U (·, 0)x ∈ E(R+ , X)} is closed and complemented in X. Proof. Necessity follows from Theorem 4.6 (ii) and Lemma 5.14 and Suﬃciency follows from Theorem 5.13. Remark 5.16. Using diﬀerent arguments, the above corollary was proved by in [10] (see Theorem 4.2), under the hypothesis that the Banach function space E lies in a subclass of B(R+ ). We note that according to our proofs, conditions (1) and the Tτ− -invariance (from Deﬁnition 2.3 in [10]) may be removed. Acknowledgment The authors wish to express their special thanks to Professor Mihail Megan for very helpful discussions on this topic.

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[11] Y. Latushkin, T. Randolph, R. Schnaubelt, Exponential dichotomy and mild solutions of nonautonomous equations in Banach spaces, J. Dynam. Diﬀerential Equations 10 (1998), 489-509. [12] J. J. Massera, J. L. Sch¨ aﬀer, Linear Diﬀerential Equations and Function Spaces. Academic Press, New-York, 1966. [13] M. Megan, B. Sasu, A. L. Sasu, On uniform exponential stability of evolution families, Riv. Mat. Univ. Parma 4 (2001), 27-43. [14] M. Megan, A. L. Sasu, B. Sasu, On nonuniform exponential dichotomy of evolution operators in Banach spaces, Integral Equations Operator Theory 44 (2002), 71–78. [15] M. Megan, A. L. Sasu, B. Sasu, Discrete admissibility and exponential dichotomy for evolution families, Discrete Contin. Dynam. Systems 9 (2003), 383–397. [16] M. Megan, A. L. Sasu, B. Sasu, The Asymptotic Behavior of Evolution Families, Mirton Publishing House, 2003. [17] M. Megan, C. Stoica, On uniform exponential trichotomy of evolution operators in Banach spaces, Integral Equations Operator Theory 60 (2008), 499-506. [18] P. Meyer-Nieberg, Banach Lattices. Springer Verlag, Berlin, Heidelberg, New York, 1991. [19] N. Van Minh, F. R¨ abiger, R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the halfline, Integral Equations Operator Theory 32 (1998), 332–353. [20] K. J. Palmer, Exponential dichotomy and expansivity, Ann. Mat. Pura Appl. (4) 185 (2006), 171-185. [21] A. L. Sasu, B. Sasu, Exponential dichotomy on the real line and admissibility of function spaces, Integral Equations Operator Theory 54 (2006), 113-130. [22] B. Sasu, A. L. Sasu, Exponential trichotomy and p-admissibility for evolution families on the real line, Math. Z. 253 (2006), 515-536. [23] B. Sasu, A. L. Sasu, Exponential dichotomy and (p , q )-admissibility on the half-line, J. Math. Anal. Appl. 316 (2006), 397-408. [24] B. Sasu, Uniform dichotomy and exponential dichotomy of evolution families on the half-line, J. Math. Anal. Appl. 323 (2006), 1465-1478. [25] A. L. Sasu, Integral equations on function spaces and dichotomy on the real line, Integral Equations Operator Theory 58 (2007), 133-152. Adina Luminit¸a Sasu and Bogdan Sasu Department of Mathematics Faculty of Mathematics and Computer Science West University of Timi¸soara Pˆ arvan Blvd. No. 4 300223 Timi¸soara Romania e-mail: [email protected] [email protected] Submitted: April 2, 2009.

Integr. Equ. Oper. Theory 66 (2010), 141–152 DOI 10.1007/s00020-009-1732-8 Published online January 13, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

Quasi-Radial Quasi-Homogeneous Symbols and Commutative Banach Algebras of Toeplitz Operators Nikolai Vasilevski Abstract. We present here a quite unexpected result: Apart from already known commutative C ∗ -algebras generated by Toeplitz operators on the unit ball, there are many other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These last algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C ∗ -algebra, and for n = 1 all of them collapse to the algebra generated by Toeplitz operators with radial symbols. Mathematics Subject Classiﬁcation (2010). Primary 47B35; Secondary 47L80, 32A36. Keywords. Toeplitz operator, weighted Bergman space, commutative Banach algebra, quasi-radial quasi-homogeneous symbol.

1. Introduction As it often happens a multidimensional extension of the one-dimensional results brings more surprises that one might expect. The paper is a next illustration of that. Recall ﬁrst that the C ∗ -algebras generated by Toeplitz operators which are commutative on each weighted Bergman space over the unit disk were completely classiﬁed in [2]. Under some technical assumption on “richness” of a class of generating symbols the result was as follows. A C ∗ -algebra generated by Toeplitz operators is commutative on each weighted Bergman space if and only if the corresponding symbols of Toeplitz operators are constant on cycles of a pencil of hyperbolic geodesics on the unit disk, or if and only if the corresponding symbols of Toeplitz operators are invariant under the action a maximal commutative subgroup of the M¨ obius transformations of the unit This work was partially supported by CONACYT Project 80503, M´exico.

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disk. We note that the commutativity on each weighted Bergman space was crucial in the part “only if” of the above result. Generalizing this result to Toeplitz operators on the unit ball, it was proved in [6, 7] that, given a maximal commutative subgroup of biholomorphisms of the unit ball, the C ∗ -algebra generated by Toeplitz operators, whose symbols are invariant under the action of this subgroup, is commutative on each weighted Bergman space. The geometric description of corresponding symbols in terms of so-called Lagrangian foliations (which generalize the notion of a pencil of hyperbolic geodesics to multidimensional case) was also given. It turned out that for the unit ball of dimension n there are n + 2 essentially diﬀerent “model” commutative C ∗ -algebras, all others are conjugated with one of them via biholomorphisms of the unit ball. It was ﬁrmly expected that the above algebras exhaust all possible algebras of Toeplitz operators on the unit ball which are commutative on each weighted Bergman space. Inspired by [8], we present here a quite unexpected result. There exist other Banach algebras generated by Toeplitz operators which are commutative on each weighted Bergman space. These algebras are non conjugated via biholomorphisms of the unit ball, non of them is a C ∗ -algebra, and for n = 1 all of them collapse to the C ∗ -algebra, which is generated by Toeplitz operators with radial symbols.

2. Preliminaries Let Bn be the unit ball in Cn , that is, Bn = {z = (z1 , . . . , zn ) ∈ Cn : |z|2 = |z1 |2 + . . . + |zn |2 < 1}, and let Sn be the corresponding unit sphere, the boundary of the unit ball Bn . In what follows we will use the notation τ (Bm ) for the base of the unit ball Bm , considered as a Reinhard domain, i.e., 2 ∈ [0, 1)}. τ (Bm ) = {(r1 , . . . , rm ) = (|z1 |, . . . , |zm |) : r2 = r12 + . . . + rm

Given a multi-index α = (α1 , α2 , . . . , αn ) ∈ Zn+ we will use the standard notation, |α| = α1 + α2 + . . . + αn , α! = α1 ! α2 ! · · · αn !, z α = z1α1 z2α2 · · · znαn . Two multi-indices α and β are called orthogonal, α ⊥ β, if α · β = α1 β1 + α2 β2 + . . . + αn βn = 0.

(2.1)

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Denote by dV = dx1 dy1 . . . dxn dyn , where zl = xl + iyl , l = 1, 2, . . . , n, the standard Lebesgue measure in Cn ; and let dS be the corresponding surface measure on Sn . We introduce the one-parameter family of weighted measures, Γ(n + λ + 1) (1 − |z|2 )λ dV (z), λ > −1, dvλ (z) = n π Γ(λ + 1) which are probability ones in Bn ; and recall two known equalities (see, for example, [9, Section 1.3]) 2π n α! β , (2.2) ξ α ξ dS(ξ) = δα,β (n − 1 + |α|)! n S α! Γ(n + λ + 1) . (2.3) z α z β dvλ (z) = δα,β Γ(n + |α| + λ + 1) n B We introduce the weighted space L2 (Bn , dvλ ) and its subspace, the weighted Bergman space A2λ = A2λ (Bn ), which consists of all functions analytic in Bn . The (orthogonal) Bergman projection Bλ of L2 (Bn , dvλ ) onto A2λ (Bn ) is known to have the following integral form ϕ(ζ) dvλ (ζ) . (Bλ ϕ)(z) = n+λ+1 n B (1 − z · ζ) Finally, given a function a(z) ∈ L∞ (Bn ), the Toeplitz operator Ta with symbol a acts on A2λ (Bn ) as follows Ta : ϕ ∈ A2λ (Bn ) −→ Bλ (aϕ) ∈ A2λ (Bn ).

3. Quasi-radial symbols Let k = (k1 , . . . , km ) be a tuple of positive integers whose sum is equal to n: k1 + . . . + km = n. The length of such a tuple may obviously vary from 1, for k = (n), to n, for k = (1, . . . , 1). Given a tuple k = (k1 , . . . , km ), we rearrange the n coordinates of z ∈ Bn in m groups, each one of which has kj , j = 1, . . . , m, entries and introduce the notation z(1) = (z1,1 , . . . , z1,k1 ), z(2) = (z2,1 , . . . , z2,k2 ), . . . , z(m) = (zm,1 , . . . , zm,km ). We represent then each z(j) = (zj,1 , . . . , zj,kj ) ∈ Bkj in the form z(j) = rj ξ(j) , where rj = |zj,1 |2 + . . . + |zj,kj |2 and ξ(j) ∈ Skj . Given a tuple k = (k1 , . . . , km ), a bounded measurable function a = a(z), z ∈ Bn , will be called k-quasi-radial if it depends only on r1 , . . . , rm . Varying k we have a collection of the partially ordered by inclusion sets Rk of k-quasi-radial functions. The minimal among these sets is the set R(n) of radial functions and the maximal one is the set R(1,...,1) of separately radial functions. There is some ambiguity in the above deﬁnition. Indeed given a tuple k there are many corresponding sets Rk which diﬀer by perturbation

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of coordinates. At the same time each perturbation of coordinates of z is a biholomorphism, say κ, of the unit ball Bn , which generates the unitary equivalence of the Toeplitz operators Ta and Ta◦κ . Thus it is suﬃcient, in fact, to consider only one of these perturbation diﬀerent sets. To avoid all possible repetitions and ambiguities in what follows we will always assume ﬁrst, that k1 ≤ k2 ≤ . . . ≤ km , and second, that z1,1 = z1 , z1,2 = z2 , . . . , z1,k1 = zk1 , z2,1 = zk1 +1 , . . . , z2,k2 = zk1 +k2 , . . . , zm,km = zn .

(3.1)

Given k = (k1 , . . . , km ) and any n-tuple α = (α1 , . . . , αn ), we deﬁne α(1) = (α1 , . . . , αk1 ), α(2) = (αk1 +1 , . . . , αk1 +k2 ), . . . , α(m) = (αn−km +1 , . . . , αn ). As each set Rk is a subset of the set R(1,...,1) of separately radial functions, the Toeplitz operator Ta with symbol a ∈ Rk , by [6], is diagonal with respect to the standard monomial basis in A2λ (Bn ). The exact form of the corresponding spectral sequence gives the next lemma. Lemma 3.1. Given a k-quasi-radial function a = a(r1 , . . . , rm ), we have Ta z α = γa,k,λ (α) z α ,

α ∈ Zn+ ,

where γa,k,λ (α) = γa,k,λ (|α(1) |, . . . , |α(m) |) =

2m Γ(n + |α| + λ + 1) Γ(λ + 1) m j=1 (kj − 1 + |α(j) |)! ×

m

2|α(j) |+2kj −1

rj

τ (Bm )

a(r1 , . . . , rm )(1 − |r|2 )λ

drj .

j=1

Proof. We calculate Ta z α , z α = az α , z α =

Γ(n + λ + 1) π n Γ(λ + 1)

Bn

a(r1 , . . . , rm )z α z α (1 − |z|2 )λ dV (z).

Changing the variables z(j) = rj ξ(j) , where rj ∈ [0, 1] and ξ(j) ∈ Skj , j = 1, . . . , m, we have m Γ(n + λ + 1) 2|α |+2kj −1 az α , z α = n a(r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj π Γ(λ + 1) τ (Bm ) j=1 m α(j) α × ξ(j)(j) ξ (j) dS(ξ(j) ) j=1

Skj

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2m Γ(n + λ + 1) α! m Γ(λ + 1) j=1 (kj − 1 + |α(j) |)! m 2|α |+2kj −1 a(r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj . × τ (Bm )

j=1

Then the result follows by (2.3).

Remark 3.2. For extreme cases of radial symbols (k = (n)) and separately radial symbols (k = (1, . . . , 1))) the above formula for γa,k,λ reduces to the corresponding formula from [1] and [6], respectively. Given k = (k1 , . . . , km ), we use the representations z(j) = rj ξ(j) , j = 1, . . . , m, to deﬁne the vector ξ = (ξ(1) , ξ(2) , . . . , ξ(m) ) ∈ Sk1 × Sk2 × . . . × Skm . We introduce now an extension of k-quasi-radial functions, which may be called following [3, 5, 8] the quasi-homogeneous functions. A function ϕ(z) is called quasi-homogeneous (or k-quasi-homogeneous) function if it has the form ϕ(z) = ϕ(z(1) , z(2) , . . . , z(m) ) s

s

s

(1) (2) (m) ξ(2) . . . ξ(m) , = a(r1 , r2 , . . . , rm ) ξ s = a(r1 , r2 , . . . , rm ) ξ(1)

where a(r1 , r2 , . . . , rm ) ∈ Rk and s ∈ Zn . After separating positive and negative entries in s, it admits the unique representation s = p − q, where p, q ∈ Zn+ and p ⊥ q. Then ξ s , for s ∈ Zn , is always understood as q

ξs = ξp ξ , where s = p − q, with p, q ∈ Zn+ and p ⊥ q. We will call the pair (p, q) the quasi-homogeneous degree of the k-quasi-homogeneous function q a(r1 , r2 , . . . , rm ) ξ p ξ . Lemma 3.3. The Toeplitz operator Ta ξp ξq with k-quasi-homogeneous symbol q

a ξ p ξ acts on monomials z α , α ∈ Zn+ as follows 0, if ∃ l such that αl < ql − pl α , Ta ξ p ξ q z = γ a,k,p,q,λ (α) z α+p−q , if ∀ l αl ≥ ql − pl where γ a,k,p,q,λ (α) =

2m Γ(n + |α + p − q| + λ + 1) (α + p)! m (3.2) Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! (α + p − q)! m |2α +p −q |+2kj −1 × a(r1 , . . . , rm )(1 − |r|2 )λ rj (j) (j) (j) drj . τ (Bm )

j=1

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Proof. For each two multi-indices α, β ∈ Zn+ , we calculate q

Ta ξp ξq z α , z β = a ξ p ξ z α , z β Γ(n + λ + 1) q = n a(r1 , . . . , rm ) ξ p ξ z α z β (1 − |z|2 )λ dV (z). π Γ(λ + 1) Bn Changing the variables z(j) = rj ξ(j) , where rj ∈ [0, 1) and ξ(j) ∈ Skj , j = 1, . . . , m, we have Γ(n + λ + 1) p q α β a ξ ξ z , z = n a(r1 , . . . , rm )(1 − |r|2 )λ π Γ(λ + 1) τ (Bm ) × ×

m j=1 m j=1

|α(j) +β(j) |+2kj −1

rj

= δα+p,β+q ×

α

Skj

τ (Bm )

ξ(j)(j)

drj

+p(j) β(j) +q(j) ξ (j)

dS(ξ(j) )

2m Γ(n + λ + 1) (α + p)! m Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! a(r1 , . . . , rm )(1 − |r|2 )λ

m

|λ(j) +β(j) |+2kj −1

rj

drj .

j=1

Now the right-hand side is non zero (for a generic a ∈ Rk ) if and only if α + p = β + q and αl + pl − ql ≥ 0, for each l = 1, 2, . . . , n. Then, for β = α + p − g with αl + pl − ql ≥ 0 for each l = 1, 2, . . . , n, we have by (2.3), z β , z β = z α+p−g , z α+p−g = and the result follows.

(α + p − q)! Γ(n + λ + 1) , Γ(n + |α + p − q| + λ + 1)

4. Commutativity results A particular case of the next theorem when k = (n) and λ = 0 was proved in [8]. Theorem 4.1. Let k = (k1 , k2 , . . . , km ) and p, q be a pair of orthogonal multiindices. Then for each pair of non identically zero k-quasi-radial functions a1 and a2 , the Toeplitz operators Ta1 and Ta2 ξp ξ q commute on each weighted Bergman space A2λ (Bn ) if and only if |p(j) | = |q(j) | for each j = 1, 2, . . . , m. Proof. For those multi-indices α with αl + pl − ql ≥ 0, for each l = 1, 2, . . . , n,

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by Lemmas 3.1 and 3.3 we have Ta2 ξp ξ q Ta1 z α =

2m Γ(n + |α + p − q| + λ + 1) (α + p)! m Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! (α + p − q)! m |2α +p −q |+2kj −1 × a2 (r1 , . . . , rm )(1 − |r|2 )λ rj (j) (j) (j) drj τ (Bm )

j=1

2m Γ(n + |α| + λ + 1) × Γ(λ + 1) m j=1 (kj − 1 + |α(j) |)! m 2|α |+2kj −1 × a1 (r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj z α+p−q τ (Bm )

j=1

and Ta1 Ta2 ξp ξq z α =

2m Γ(n + |α + p − q| + λ + 1) m Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) − q(j) |)! m 2|α +p −q |+2kj −1 a1 (r1 , . . . , rm )(1 − |r|2 )λ rj (j) (j) (j) drj × τ (Bm )

j=1

2m Γ(n + |α + p − q| + λ + 1) (α + p)! m × Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! (α + p − q)! m |2α +p −q |+2kj −1 a2 (r1 , . . . , rm )(1 − |r|2 )λ rj (j) (j) (j) drj × τ (Bm )

×z

j=1

α+p−q

.

α

That is Ta2 ξp ξq Ta1 z = Ta1 Ta2 ξp ξq z α (for each pair of k-quasi-radial functions a1 and a2 ) if and only if |p(j) | = |q(j) | for each j = 1, 2, . . . , m Remark 4.2. For those j with kj = 1 both p(j) and q(j) are of length one, and the condition |p(j) | = |q(j) | is equivalent to p(j) = q(j) = 0. We note that under the condition |p(j) | = |q(j) |, for each j = 1, 2, . . . , m, formula (3.2) reads as γ a,k,p,q,λ (α) =

2m Γ(n + |α| + λ + 1) (α + p)! m Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! (α + p − q)! m 2|α |+2kj −1 2 λ × a(r1 , . . . , rm )(1 − |r| ) rj (j) drj τ (Bm )

m

= m

j=1 (kj

j=1 (kj

=

m j=1

(4.1)

j=1

− 1 + |α(j) |)! (α + p)!

− 1 + |α(j) + p(j) |)! (α + p − q)!

γa,k,λ (α)

(α(j) + p(j) )! (kj − 1 + |α(j) |)! γa,k,λ (α). (kj − 1 + |α(j) + p(j) |)! (α(j) + p(j) − q(j) )!

As surprising corollaries we have:

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Corollary 4.3. Given k = (k1 , k2 , . . . , km ), for each pair of orthogonal multiindices p and q with |p(j) | = |q(j) |, for all j = 1, 2, . . . , m, and each a(r1 , r2 , . . . , rm ) ∈ Rk , we have Ta Tξp ξq = Tξp ξq Ta = Taξp ξq . Given k = (k1 , k2 , . . . , km ), and a pair of orthogonal multi-indices p and q with |p(j) | = |q(j) |, for all j = 1, 2, . . . , m, let p(j) = (0, . . . , 0, p(j) , 0, . . . , 0)

and

q(j) = (0, . . . , 0, q(j) , 0, . . . , 0).

Then, of course, p = p(1) + p(2) + . . . + p(m) and q = q(1) + q(2) + . . . + q(m) . For each j = 1, 2, . . . , m, we introduce the Toeplitz operator Tj = Tξp(j) ξ q(j) . Corollary 4.4. The operators Tj , j = 1, 2, . . . , m, mutually commute. Given an h-tuple of indices (j1 , j2 , . . . , jh ), where 2 ≤ h ≤ m, let ph = p(j1 ) + p(j2 ) + . . . + p(jh )

and

qh = q(j1 ) + q(j2 ) + . . . + q(jh ) .

Then h g=1

Tjg = Tξph ξqh .

In particular, m

Tj = Tξ p ξ q .

j=1

Given k = (k1 , k2 , . . . , km ), we consider two bounded measurable kq v quasi-homogeneous symbols a(r1 , r2 , . . . , rm )ξ p ξ and b(r1 , r2 , . . . , rm )ξ u ξ , which satisfy the conditions of Theorem 4.1, i.e., a(r1 , r2 , . . . , rm ) and b(r1 , r2 , . . . , rm ) are arbitrary k-quasi-radial functions, p ⊥ q, u ⊥ v, and |p(j) | = |q(j) |

and |u(j) | = |v(j) |, q

for all j = 1, 2, . . . , m. v

Theorem 4.5. Let a(r1 , r2 , . . . , rm )ξ p ξ and b(r1 , r2 , . . . , rm )ξ u ξ be as above. Then the Toeplitz operators Taξp ξq and Tbξu ξv commute on each weighted Bergman space A2λ (Bn ) if and only if for each l = 1, 2, . . . , n one of the next conditions is fulﬁlled 1. 2. 3. 4.

pl = ql = 0; ul = vl = 0; pl = ul = 0; ql = vl = 0.

Proof. We calculate and compare ﬁrst Taξp ξq Tbξu ξv z α and Tbξu ξ v Taξp ξq z α for those multi-indices α when both these expressions are non zero.

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By (4.1) we have Taξp ξq Tbξu ξv z α =

2m Γ(n + |α| + λ + 1) (α + u − v + p)! m Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! (α + u − v + p − q)! m 2|α |+2kj −1 a(r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj × τ (Bm )

j=1 m

2 Γ(n + |α| + λ + 1) (α + u)! m Γ(λ + 1) j=1 (kj − 1 + |α(j) + u(j) |)! (α + u − v)! m 2|α |+2kj −1 b(r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj × ×

τ (Bm )

×z

α+u−v+p−q

j=1

,

and Tbξu ξ v Taξp ξq z α =

2m Γ(n + |α| + λ + 1) (α + p − q + u)! m Γ(λ + 1) j=1 (kj − 1 + |α(j) + u(j) |)! (α + p − q + u − v)! m 2|α |+2kj −1 b(r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj × τ (Bm )

j=1

2m Γ(n + |α| + λ + 1) (α + p)! m × Γ(λ + 1) j=1 (kj − 1 + |α(j) + p(j) |)! (α + p − q)! m 2|α |+2kj −1 a(r1 , . . . , rm )(1 − |r|2 )λ rj (j) drj × τ (Bm )

×z

α+p−q+u−v

j=1

.

α

That is, Taξp ξq Tbξu ξv z = Tbξu ξ v Taξp ξq z α (for each pair of k-quasi-radial functions a and b) if and only if (α + p − q + u)! (α + p)! (α + u − v + p)! (α + u)! = . (α + u − v)! (α + p − q)! Varying α it is easy to see that the last equality holds if and only if for each l = 1, 2, . . . , n one of the next conditions is fulﬁlled 1. pl = ql = 0; 2. ul = vl = 0; 3. pl = ul = 0; 4. ql = vl = 0. To ﬁnish the proof we mention that under either of the above conditions both quantities Taξp ξq Tbξu ξv z α and Tbξu ξv Taξp ξq z α are zero or non zero simultaneously only. Example. Let n = 7 and k = (2, 5). Then by Theorem 4.1 the Toeplitz q operators with symbols a(r1 , r2 ) ∈ Rk and b ξ p ξ , where b(r1 , r2 ) ∈ Rk , p = (1, 0, 0, 3, 0, 1, 0), q = (0, 1, 1, 0, 1, 0, 2), commute. We mention that here p(1) = (1, 0), p(2) = (0, 3, 0, 1, 0) and q(1) = (0, 1), q(2) = (1, 0, 1, 0, 2).

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As easy to see, all pairs (u, v) of orthogonal multi-indices such that (by Theorem 4.5) the Toeplitz operators with k-quasi-homogeneous symbols having that quasi-homogeneous degrees mutually commute, and commute with both Ta and Tbξp ξ q are of the form u = (u1 , 0, 0, u4 , 0, u6 , 0),

v = (0, v2 , v3 , 0, v5 , 0, v7 ),

(4.2)

where u1 , u4 , u6 ∈ Z+ , v2 , v3 , v5 , v7 ∈ Z+ , and u1 = v2 ,

u4 + u6 = v3 + v5 + v7 .

(4.3)

That is, the Banach algebra generated by all Toeplitz operators Ta ξu ξv , where a(r1 , r2 ) ∈ Rk , and the orthogonal multi-indices u and v of the form (4.2) satisfy the condition (4.3), is commutative. We formalize the above example as follows. First, to avoid the repetition of the unitary equivalent algebras and to simplify the classiﬁcation of the (non unitary equivalent) algebras, in addition to (3.1), we can rearrange the variables zl and correspondingly the components of multi-indices in p and q so that (i) for each j with kj > 1, we have p(j) = (pj,1 , . . . , pj,hj , 0, . . . , 0) and q(j) = (0, . . . , 0, qj,hj +1 , . . . , qj,kj ); (4.4) (ii) if kj = kj with j < j , then hj ≤ hj . Now, given k = (k1 , . . . , km ), we start with m-tuple h = (h1 , . . . , hm ), where hj = 0 if kj = 1 and 1 ≤ hj ≤ kj − 1 if kj ≥ 1; in the last case, if kj = kj with j < j , then hj ≤ hj . We denote by Rk (h) the linear space generated by all k-quasi-homogeneous functions q a(r1 , r2 , . . . , rm ) ξ p ξ , where a(r1 , r2 , . . . , rm ) ∈ Rk , and the components p(j) and q(j) , j = 1, 2, . . ., m, of multi-indices p and q are of the form (4.4) with pj,1 + . . . + pj,hj = qj,hj +1 + . . . + qj,kj , pj,1 , . . . , pj,hj , qj,hj +1 , . . . , qj,kj ∈ Z+ . We note that Rk ⊂ Rk (h) and that the identity function e(z) ≡ 1 belongs to Rk (h). The main informatory result of the paper gives the next corollary. Corollary 4.6. The Banach algebra generated by Toeplitz operators with symbols from Rk (h) is commutative. We would like to emphasize the following features of such algebras: – for diﬀerent k and h these algebras are not conjugated via biholomorphisms of the unit ball; – these algebras are just Banach and not C ∗ -algebras; extending them to C ∗ -algebras they become non commutative;

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– given k = (1, 1, . . . , 1), there is a ﬁnite number of diﬀerent m-tuples h and thus a ﬁnite number of diﬀerent corresponding commutative algebras; – these algebras remain commutative for each weighted Bergman space A2λ (Bn ), with λ > −1, – for n = 1 all of them collapse to the single C ∗ -algebra generated by Toeplitz operators with radial symbols. We ﬁnish the paper presenting another application of Theorems 4.1 and 4.5. Studying commutativity properties of Toeplitz operators on the Bergman space on the unit disk I. Louhichi and N. V. Rao [4] conjectured that if two Topelitz operators commute with a third one, none of them being the identity, then they commute with each other. As next example shows, this conjecture is wrong when formulated for Toeplitz operators on the unit ball Bn , with n > 1. Example. Given n > 1, let k = (2, 1, . . . , 1). Consider the following three symbols a0 = a(r1 , r2 , . . . , rn−1 ), (1,0) (0,1)

a1 = b(r1 , r2 , . . . , rn−1 )ξ(1) ξ (1) , (0,1) (1,0)

a2 = c(r1 , r2 , . . . , rn−1 )ξ(1) ξ (1) , where a, b, c ∈ Rk . Then by Theorem 4.1 Ta0 commutes with both Ta1 and Ta2 , while by Theorem 4.5 the operators Ta1 and Ta2 do not commute.

References [1] S. Grudsky, A. Karapetyants, and N. Vasilevski. Toeplitz operators on the unit ball in Cn with radial symbols. J. Operator Theory, 49:325–346, 2003. [2] S. Grudsky, R. Quiroga-Barranco, and N. Vasilevski. Commutative C ∗ -algebras of Toeplitz operators and quantization on the unit disk. J. Funct. Anal., 234(1):1–44, 2006. [3] I. Louhichi and N.V. Rao. On Toeplitz operators with quasihomogeneous symbols. Atch. Math., 851:248–257, 2005. [4] I. Louhichi and N.V. Rao. Bicommutants of Toeplitz operators. Atch. Math., 91:256–264, 2008. [5] Issam Louhichi, Elizabeth Strouse, and Lova Zakariasy. Products of Toeplitz operators on the Bergman space. Integral Equations Operator Theory, 54(4):525– 539, 2006. [6] Raul Quiroga-Barranco and Nikolai Vasilevski. Commutative C ∗ -algebras of Toeplitz operators on the unit ball, I. Bargmann-type transforms and spectral representations of Toeplitz operators. Integral Equations Operator Theory, 59(3):379–419, 2007.

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[7] Raul Quiroga-Barranco and Nikolai Vasilevski. Commutative C ∗ -algebras of Toeplitz operators on the unit ball, II. Geometry of the level sets of symbols. Integral Equations Operator Theory, 59(1):89–132, 2008. [8] Ze-Hua Zhou and Xing-Tang Dong. Algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball. Integral Equations and Operator Theory (to appear), 18 p. [9] Kehe Zhu. Spaces of Holomorphic Functions in the Unit Ball. Springer Verlag, 2005. Nikolai Vasilevski Departamento de Matem´ aticas CINVESTAV Apartado Postal 14-740 07000, M´exico, D.F. M´exico e-mail: [email protected] Submitted: March 4, 2009. Revised: July 10, 2009.

Integr. Equ. Oper. Theory 66 (2010), 153–181 DOI 10.1007/s00020-010-1742-6 Published online January 29, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

Quasianalytic Wave Front Sets for Solutions of Linear Partial Diﬀerential Operators A. A. Albanese, D. Jornet and A. Oliaro Abstract. In the present paper, we introduce and study Beurling and Roumieu quasianalytic (and nonquasianalytic) wave front sets, W F∗ , of classical distributions. In particular, we have the following inclusion W F∗ (u) ⊂ W F∗ (P u) ∪ Σ,

u ∈ D (Ω),

where Ω is an open subset of Rn , P is a linear partial diﬀerential operator with coeﬃcients in a suitable ultradiﬀerentiable class, and Σ is the characteristic set of P . Some applications are also investigated. Mathematics Subject Classiﬁcation (2010). Primary 46F05; Secondary 35A18, 35A21. Keywords. Quasianalytic weight function, wave front set, propagation of singularities.

1. Introduction Classes of ultradiﬀerentiable functions have been investigated intensively since the 20ies of the last century. According to the theorem of Denjoy– Carleman they split in two groups: the quasianalytic and the nonquasianalytic classes. Several authors have introduced the classes in diﬀerent ways (e.g., Beurling [1]; Komatsu [16]; H¨ ormander [14]; Braun, Meise, and Taylor [9]). In general, the classes deﬁned in one way cannot be deﬁned in another way (see [8]). On the other hand, independently of the deﬁnition there are several known methods to construct functions with prescribed properties in nonquasianalytic classes, while this is not the case for quasianalytic classes, for which other techniques are needed, for example, complex variable methods. The research of the authors was partially supported by MEC and FEDER, Project MTM2007-62643, and MEC, Project MTM2007-30904-E, and Conselleria d’Educaci´ o de la GVA, Ajuda complementaria ACOMP/2009/253.

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The notion of wave front set was introduced by H¨ ormander in 1970 to simplify the study of the propagation of singularities of distributional solutions of linear partial diﬀerential operators. In 1971, H¨ ormander [14] established the following microlocal regularity result: W FL (u) ⊂ Σ ∪ W FL (P u),

u ∈ D (Ω),

(1)

where P is a linear partial diﬀerential operator with analytic coeﬃcients, and Σ is the characteristic set of P . This type of result represents a fundamental tool in the study of propagation of singularities. This inclusion has been also applied, modiﬁed and adapted several times since the seventies of the last century to study the problem of iterates from a microlocal point of view (we refer to [2, 3, 5, 24], among others; see also [4] for a survey on this topic). In the inclusion (1) the wave front set W FL (u) is deﬁned with respect to the ultradiﬀerentiable class C L of Roumieu type and u is always a classical distribution on an open set Ω of Rn . The function spaces C L were introduced in [14] and cover the classical spaces of Gevrey functions and of analytic functions. In the case of the analytic wave front set, W FA , such an inclusion was proved by Kato [15] for hyperfunctions, but for a very diﬀerent deﬁnition of W FA . We also mention that a version of inclusion (1) for wave front sets with respect to the C ∞ -class and an operator P with C ∞ -coeﬃcients can be found in [13, Chapter VIII]. In this paper we present a version of inclusion (1) for wave front sets deﬁned with respect to the ultradiﬀerentiable classes as introduced by Braun, Meise, and Taylor [9] and classical distributions. By modifying the arguments in [14] in a suitable way, we cover both quasianalytic and non-quasianalytic cases at the same time. These classes have the advantage that can be studied using the decay properties of their Fourier transform and the decay behaviour of their derivatives. We give two versions of the result. First, we prove the Beurling case in Theorem 4.1. In this case, we require the assumption ω(t) = o(t), as t tends to inﬁnity, on the weight function. The Roumieu version is obtained as a consequence of the Beurling case and the description of Roumieu wave front set of a classical distribution as the closure of the union of all the Beurling wave front sets contained in it. Such a description is proved for arbitrary weight functions, quasianalytic or not, in Proposition 4.5 (see [11] for a version for non-quasianalytic weight functions). To show our inclusion results, Theorem 4.1 and Theorem 4.8, we need to assume that the weight functions are equivalent to subadditive weight functions (see [18, 19]). In particular, our results apply to the most relevant cases considered by Komatsu [16] (see [8]). Finally, our results are applied to study the propagation of singularities of the operator P = ∂xn in Rn .

2. Notation and preliminaries In this preliminary section we ﬁx the notation and provide a number of basic results that will be used in the subsequent sections. Throughout this article

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| · | denotes the Euclidian norm on Rn or Cn and Br (a) denotes the open ball of radius r and center a. Deﬁnition 2.1. A weight function is a continuous increasing function ω : [0, ∞[→ [0, ∞[ with the following properties: (α) (β) (γ) (δ)

there exists L ≥ 0 such that ω(2t) ≤ L(ω(t) + 1) for all t ≥ 0, ω(t) = O(t), as t tends to ∞, log(t) = o(ω(t)) as t tends to ∞, ϕ : t → ω(et ) is convex. A weight function ω is called quasianalytic if ∞ ω(t) dt = ∞. t2 1

If this integral is ﬁnite, then ω is called nonquasianalytic. A weight function ω is equivalent to a sub-additive weight if, and only if, it has the property (α0 ) ∃D > 0 ∃t0 > 0 ∀λ ≥ 1 ∀t ≥ t0 : ω(λt) ≤ λDω(t). The condition above should be compared with [19, p.19] and [18, Lemma 1]. The Young conjugate ϕ∗ : [0, ∞[→ R of ϕ is given by ϕ∗ (s) := sup{st − ϕ(t), t ≥ 0}. There is no loss of generality to assume that ω vanishes on [0, 1]. Then ϕ∗ has only non-negative values, it is convex and ϕ∗ (t)/t is increasing and tends to ∞ as t → ∞ and ϕ∗∗ = ϕ. For more details on properties of ω and ϕ∗ we refer to [9, 12]. Example 2.2. The following are examples of weight functions (eventually after a change on the interval [0, δ] for a suitable δ > 0): (1) (2) (3) (4)

ω(t) = tα , 0 < α < 1; β ω(t) = (log(1 + t)) , β > 1; −β ω(t) = t (log(e + t)) , β > 0; ω(t) = t.

The weight function in (3) is quasianalytic for β ∈]0, 1] and nonquasianalytic for β > 1. The weight function in (4) is also quasianalytic. Moreover, all the weight functions above satisfy property (α0 ). For further examples of quasianalytic weight functions we refer to [8]. Deﬁnition 2.3. Let ω be a weight function. For an open set Ω ⊂ Rn we set E(ω) (Ω) :={f ∈ C ∞ (Ω) : f K,λ < ∞, for every K ⊂⊂ Ω and every λ > 0}, and E{ω} (Ω) := {f ∈ C ∞ (Ω) : for every K ⊂⊂ Ω, there exists λ > 0 such that f K,λ < ∞},

156

where

Albanese, Jornet and Oliaro

IEOT

|α|

f K,λ := sup sup |f (α) (x)| exp − λϕ∗ . λ x∈K α∈NN 0

The spaces E(ω) (Ω) and E{ω} (Ω) are endowed with their natural topologies. Then E(ω) (Ω) is a nuclear Fr´echet space, while E{ω} (Ω) is a countable projective limit of (DFN)-spaces, which is reﬂexive and complete. If ω is nonquasianalytic, the space E{ω} (Ω) is ultrabornological, see [9, Proposition 4.9]. If ω is quasianalytic, the space E{ω} (Ω) is surely ultrabornological under the assumption of convexity of Ω and, this follows from [21, Satz 3.25], together with [22, Theorem 3.4], and [23, Theorem 3.5]. The elements in the space E(ω) (Ω) (respectively, in the space E{ω} (Ω)) are called ω-ultradifferentiable functions of Beurling type (respectively, of (Ω) and E{ω} (Ω) we denote the duals of E(ω) (Ω) Roumieu type) in Ω. By E(ω) and E{ω} (Ω). When ω is quasianalytic the elements of E(ω) (Ω) (respectively, E{ω} (Ω)) are called quasianalytic functionals of Beurling (respectively, Roumieu) type. We observe that in the case ω(t) = tα , 0 < α ≤ 1, the corresponding Roumieu class is the Gevrey class with exponent s = 1/α. In particular, E{ω} (Ω) coincides with the space A(Ω) of all real analytic functions on Ω. We will write ∗ to denote (ω) or {ω} when it is not necessary to distinguish between both cases. If ω is quasianalytic, the elements with compact support in E{ω} (Ω) or in E(ω) (Ω) are trivial. While, if ω is nonquasianalytic, the space D∗ (K) := E∗ (Ω) ∩ D(K) = {0}, being K ⊂ Ω a compact set. Then D∗ (Ω) := indn D∗ (Kn ), where (Kn ) is any compact exhaustion of Ω. The elements of (Ω) (respectively, D{ω} (Ω)) are called ω-ultradistributions of Beurling D(ω) (respectively, Roumieu) type. The ∗-singular support of a classical distribution u ∈ D (Ω), denoted by sing∗ supp u, is the complementary in Ω of the biggest open set U ⊂ Ω satisfying u|U ∈ E∗ (U ). Remark 2.4. We also recall the following properties (see, for example, [12, Remark 2.8]): (a) If σ(t) = o(ω(t)) as t tends to inﬁnity, then E{ω} (Ω) ⊂ E(σ) (Ω) with continuous inclusion. (b) If ω(t) = o(t) as t tends to inﬁnity, then for each constant l ∈ N, there is a constant Cl > 0 such that y + Cl , y > 0. y log y ≤ y + lϕ∗ l

3. Quasianalytic wave front sets and properties In this section, we introduce and study Beurling and Roumieu quasianalytic and non quasianalytic wave front sets of a classical distribution. To provide

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this, we begin with two lemmas which clarify the behaviour of the Fourier transform with respect to the weight function in Roumieu and Beurling settings. The ﬁrst one treats the Roumieu case. Lemma 3.1. Let f be a continuous function deﬁned in a cone Γ ⊂ Rn \ {0} taking values in [0, +∞[. The following statements are equivalent: (i) There exist constants C, ε > 0 such that f (ξ) ≤ C exp(−εω(ξ)),

ξ ∈ Γ,

(ii) There exists a constant C > 0 such that f (ξ) ≤ C

N +1

N!

1 ω(ξ)

N N = 0, 1, 2, . . . , ξ ∈ Γ,

,

(iii) There exists a constant C > 0 such that f (ξ) ≤ C

N

CN ω(ξ)

N = 0, 1, 2, . . . , ξ ∈ Γ,

,

(iv) There exist constants C > 0 and k ∈ N satisfying 1

|ξ|N f (ξ) ≤ Ce k ϕ

∗

(N k)

N = 0, 1, 2, . . . , ξ ∈ Γ.

,

(v) There exist constants C > 0 and k ∈ N such that 1

|ξ|N f (ξ) ≤ C N +1 e k ϕ

∗

(N k)

N = 0, 1, 2, . . . , ξ ∈ Γ.

,

Proof. (i) ⇐⇒ (ii) ⇐⇒ (iii) can be proved as in [20, Lemma 1.6.2]. We ﬁrst show (i) ⇐⇒ (iv). To prove this, it is suﬃcient to check, for t > 1, that: 1

1

e− k ω(t) ≤ inf t−N e k ϕ

∗

(N k)

N ∈N0

1

≤ e− k ω(t)+log t .

(2)

Indeed, since log(t) = o(ω(t)) as t → ∞, for 0 < ε < k1 there is t0 > 0 so that − k1 ω(t) + log t ≤ −εω(t) for all t ≥ t0 . The ﬁrst inequality in (2) follows by observing that (ϕ∗ )∗ = ϕ and hence, we have ω(t) = sup{s log t − ϕ∗ (s)} s>0

≥ sup {N k log t − ϕ∗ (N k)} N ∈N0

= k sup {N log t − N ∈N0

1 ∗ ϕ (N k)}. k

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The second inequality in (2) follows by observing that ϕ∗ is increasing and hence, we have ω(t) = sup{s log t − ϕ∗ (s)} s>0 sup

= sup N ∈N0

N k≤s≤(N +1)k

{s log t − ϕ∗ (s)}

≤ sup {(N + 1)k log t − ϕ∗ (N k)} N ∈N0

= k log t + sup {N k log t − ϕ∗ (N k)} N ∈N0 1 = k log t + sup {N log t − ϕ∗ (N k)} . k N ∈N0 It is obvious that (iv) implies (v). To ﬁnish the proof, it remains to show that (v) implies (iv). We proceed as in [10, p. 404]. We take s ∈ N to be the smallest natural number such that C ≤ es , being C the constant that appears in (iv). Let m the smallest natural number bigger than kLs , where k is the constant of (iv) and L > 1 the one that appears in property (α) of the weight function ω. We have s 1 1 j 1 ∗ ϕ (N k) + N s ≤ ϕ∗ (N m) + L , k m m j=1 and hence, 1

C N +1 e k ϕ

∗

(N k)

1

≤ CeN s+ k ϕ

∗

(N k)

1

≤ Ce m

s

j=1

Lj

1

emϕ

∗

(N m)

.

The following lemma treats the Beurling case. Lemma 3.2. Let f be a continuous function deﬁned in a cone Γ ⊂ Rn \ {0} taking values in [0, +∞[. The following statements are equivalent: (i) For every k ∈ N there exists a constant Ck > 0 such that f (ξ) ≤ Ck exp(−kω(ξ)),

ξ ∈ Γ,

(ii) For every k ∈ N there exists a constant Ck > 0 such that N 1 f (ξ) ≤ Ck N ! , N = 0, 1, 2, . . . , ξ ∈ Γ, kω(ξ) (iii) For every k ∈ N there exists a constant Ck > 0 such that N N f (ξ) ≤ Ck , N = 0, 1, 2, . . . , ξ ∈ Γ, kω(ξ) (iv) For every k ∈ N there exists a constant Ck > 0 such that |ξ|N f (ξ) ≤ Ck ekϕ

∗

(N/k)

N = 0, 1, 2, . . . , ξ ∈ Γ.

,

(v) There exists a constant C > 0 such that for every k ∈ N there exists a constant Ck > 0 for which |ξ|N f (ξ) ≤ Ck C N ekϕ

∗

(N/k)

,

N = 0, 1, 2, . . . , ξ ∈ Γ.

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Proof. It follows easily that (i) implies (ii) by using the Taylor expansion of the exponential function. It is clear that (ii) implies (iii). On the other hand, (iii) implies (ii) since N N ≤ eN N ! and therefore, if f satisﬁes (iii) for some sequence {Ck }, then for every k ∈ N, −N

f (ξ) ≤ C3k N ! [k ω(ξ)]

,

N = 0, 1, 2, . . . , ξ ∈ Γ.

(ii) implies that, for each k ∈ N, f (ξ)k N ω(ξ)N /N ! ≤ C2k (1/2)N for all N = 0, 1, 2, . . . . So, using the expansion of the exponential function, we obtain f (ξ) exp(k ω(ξ)) ≤ 2C2k , and hence, (i) is satisﬁed. (i) ⇐⇒ (iv) is proved in a similar way to Lemma 3.1. It remains to prove that (v) implies (iv). As in the Roumieu setting, we take s ∈ N such that C ≤ es , being C the constant that appears in (v). By (v) for each k ∈ N there is Ck > 0 such that, for Ak = k(Ls + · · · + L), f (ξ) ≤ Ck C N ekL

s

ϕ∗ (N/(kLs ))

≤ Ck eAk ekϕ

∗

(N/k)

.

The following proposition describes the ∗-singular support of a classical distribution. The proof follows the lines of [14] (see also [12, Lemma 4.7]). Proposition 3.3. Let Ω ⊂ Rn be an open set, u ∈ D (Ω) and x0 ∈ Ω. (a) Then u is a E{ω} -function in some neighborhood of x0 if and only if for some neighborhood U of x0 there exists a bounded sequence uN ∈ E (Ω) which is equal to u in U and satisﬁes, for some C > 0 and k ∈ N, the estimates 1

uN (ξ)| ≤ Ce k ϕ |ξ|N |

∗

(N k)

,

N = 1, 2, . . . ,

ξ ∈ Rn .

(3)

(b) Then u is a E(ω) -function in some neighborhood of x0 if and only if for some neighborhood U of x0 there exists a bounded sequence uN ∈ E (Ω) which is equal to u in U and such that for every k ∈ N there exists a constant Ck > 0 satisfying uN (ξ)| ≤ Ck ekϕ |ξ|N |

∗

(N/k)

,

N = 1, 2, . . . ,

ξ ∈ Rn .

(4)

Proof. (a) Necessity. Let u ∈ E{ω} (U ) with U = B3r (x0 ) and choose χN ∈ D(Ω) so that χN = 1 in Br (x0 ) and χN = 0 on (B2r (x0 ))c in such a way that |Dα χN | ≤ (C1 N )|α| ,

|α| ≤ N,

(5)

where C1 does not depend on N (we refer to [14] for a proof of the existence of sequences {χN } satisfying (5)). Now, we put uN = χN u. Then, for |α| ≤ N ,

α ∗ 1 (6) (C1 N )|β| Dr e m ϕ (m|α−β|) , |Dα (χN u)| ≤ β β≤α

being Dr the value of the seminorm u B2r (x0 ),1/m . Since ϕ∗ is a convex 1 ∗ 1 ∗ 1 ∗ function, we have m ϕ (m|α − β|) + m ϕ (m|β|) ≤ m ϕ (m|α|) and therefore,

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the right hand side of (6) is less than or equal to

α ∗ 1 1 ϕ∗ (m|α|) m e|β| log(C1 N )− m ϕ (m|β|) Dr e β β≤α

N

≤ 2 Dr e

∗ 1 m ϕ (mN )

1

sup e|β| log(C1 N )− m ϕ

∗

(m|β|)

|β|≤N 1

∗

(mN )

e m sup|β|≤N {m|β| log(C1 N )−ϕ

1

∗

(mN )

emϕ

≤ 2N Dr e m ϕ ≤ 2N Dr e m ϕ

1

1

1

∗∗

1

∗

≤ Dr eN + m ω(C1 N )+ m ϕ

∗

(m|β|)}

(log(C1 N ))

(mN )

,

where we have used the fact that ϕ∗∗ = ϕ, the deﬁnition of ϕ and |α| ≤ N . Then, we have, for |α| = N ,

∗ 1 1 α −ix,ξ α

N (ξ)| = e D uN (x)dx

≤ CDr eN + m ω(C1 N )+ m ϕ (mN ) , |ξ u where the constant C depends on the Lebesgue measure of B2r (x0 ). Now, we select i = 1, . . . , n such that |ξi | = max1≤j≤n |ξj | and set α = N ei , where ei is the i−th vector of the canonical basis of Rn . Then, |ξ|N ≤ nN/2 max |ξj |N = nN/2 |ξi |N = nN/2 |ξ α |. 1≤j≤n

Consequently, uN (ξ)| ≤ nN/2 |ξ α u N (ξ)| |ξ|N | 1

1

≤ CDr nN/2 eN + m ω(C1 N )+ m ϕ N

= CDr e 2

∗

(N m)

1 1 log n+N + m ω(C1 N )+ m ϕ∗ (N m)

.

Since ω(t) = O(t) as t tends to inﬁnity, we can ﬁnd a positive constant C2 such that ω(t) ≤ C2 t + C2 for all t > 0. Now, proceeding as in the proof Lemma 3.2, (v) ⇒ (iv), we take s ∈ N greater than 1 + log2 n + C1mC2 . Then, if k is the smallest natural number bigger than mLs , where L > 1 is the constant that appears in property (α) of the weight function ω, we have s 1 1 j 1 ∗ ϕ (N m) + N s ≤ ϕ∗ (N k) + L . m k k j=1

Hence, we obtain |ξ|N | uN (ξ)| ≤ CDr eC2 +N (

log n ∗ 1 1 2 +1+ m C2 C1 )+ m ϕ (N m) 1

≤ CDr eC2 eN s+ m ϕ 1

s

∗

(N m) j

1

≤ Cm e k ϕ

∗

(N k)

,

where the constant Cm = CDr eC2 + k j=1 L depends on m, on the weight ω, on r, and the selection of χN . Suﬃciency. For x ∈ U , we have Dα u(x) = (2π)−n ξ α u N (ξ)eix,ξ dξ,

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when N ≥ |α| + n + 1, for ξ α u N (ξ) is then integrable since (3) is satisﬁed by hypothesis. Now, as {uN } is a bounded sequence in E (Ω), an application of the Banach-Steinhaus Theorem gives, for all N ∈ N, | uN (ξ)| ≤ C1 (1 + |ξ|)M , for some constants C1 > 0 and M ∈ N that do not depend on N . Hence, for x ∈ U , |Dα u(x)| ≤ (2π)−n |ξ α || uN (ξ)|dξ = I1 + I2 , 1 where I1 denotes the integral when |ξ| ≤ exp( kN ϕ∗ (kN )), and I2 denotes 1 ϕ∗ (kN )). Since |ξ α | ≤ |ξ||α| , the integral when |ξ| ≥ exp( kN M |α| ∗ ∗ 1 , I1 ≤ mn C1 e kN ϕ (N k) 1 + e kN ϕ (kN ) 1 being mn the Lebesgue measure of the set {ξ : |ξ| ≤ exp( kN ϕ∗ (kN ))} n in R , that is less than or equal to the Lebesgue measure of the hypercube 1 ϕ∗ (kN ))},

· ∞ denotes the maximum norm in {ξ : ξ ∞ ≤ exp( kN n where n R , which is equal to 2n exp kN ϕ∗ (kN ) . Summing up, we have 1 ∗ n+|α|+M 1 ∗ ≤ 2n+M C1 e k ϕ (k(n+|α|+M )) , I1 ≤ 2n+M C1 e kN ϕ (kN )

when N = n + |α| + M . On the other hand, by (3), I2 ≤ |ξ|N −n−1 | uN (ξ)|dξ 1 ∗ |ξ|≥e N k ϕ (N k) 1 ∗ 1 ≤ Ce k ϕ (N k) dξ 1 ϕ∗ (kN ) n+1 |ξ| kN |ξ|≥e 1 ∗ 1 ≤ Ce k ϕ (N k) dξ. n+1 |ξ| |ξ|≥1 Therefore, we deduce that there is a constant C > 0, that depends only on n and M , such that 1

|Dα u| ≤ C e k ϕ

∗

(k(|α|+n+M ))

,

on U . Now, from the convexity of ϕ∗ we obtain 1 ∗ 1 ∗ 1 ∗ ϕ (k(|α| + n + M )) ≤ ϕ (2k|α|) + ϕ (2k(n + M )), k 2k 2k 1

∗

1

∗

and hence, C e k ϕ (k(|α|+n+M )) ≤ Ck e m ϕ (m|α|) , with m = 2k, which concludes the proof of (a). (b) The proof is similar to the one of (a), and we only indicate the main changes. Necessity. As in (a), we put uN = χN u. Then, for |α| ≤ N and k ∈ N there is Ck > 0 such that

α ∗ α |D (χN u)| ≤ Ck (7) (C1 N )|β| ekϕ (|α−β|/k) . β β≤α

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Proceeding as in the Roumieu case, the right hand side of (7) is less than or equal to Ck eN +kω(C1 N )+kϕ

∗

(N/k)

,

and therefore, N

|ξ|N | uN (ξ)| ≤ DCk e 2

log n+N +kω(C1 N )+kϕ∗ (N/k)

,

where the constant D depends on the Lebesgue measure of B2r (x0 ). Now, since ω(t) = o(t) as t tends to inﬁnity, we can ﬁnd, for each k ∈ N, a positive constant Ak such that kω(t) ≤ t + Ak for t > 0. Hence, if we put log n C k = DCk eAk and B = e 2 +1+C1 , we obtain |ξ|N | uN (ξ)| ≤ C k B N ekϕ

∗

(N/k)

.

Proceeding as in Lemma 3.2, (v) ⇒ (iv), we obtain an estimate like (4). Suﬃciency. As in (a), for x ∈ U , and a ﬁxed k ∈ N, |Dα u(x)| ≤ (2π)−n |ξ α || uN (ξ)|dξ = I1 + I2 , ∗ N , and I2 denotes where I1 denotes the last integral when |ξ| ≤ exp 2k Nϕ 2k 2k ∗ N the integral when |ξ| ≥ exp N ϕ 2k . Since |ξ α | ≤ |ξ||α| , M 2k|α| ∗ N 2k ∗ N , I1 ≤ mn C1 e N ϕ ( 2k ) 1 + e N ϕ ( 2k ) ∗ N } in Rn . being mn the Lebesgue measure of the set {ξ : |ξ| ≤ exp 2k Nϕ 2k We obtain, as in (a), 2k ∗ N n+|α|+M ∗ n+|α|+M ≤ 2n+M C1 e2kϕ ( 2k ) , I1 ≤ 2n+M C1 e N ϕ ( 2k ) when N = n + |α| + M , and by (4), I2 ≤ |ξ|N −n−1 | uN (ξ)|dξ 2k ϕ∗ N ( 2k ) |ξ|≥e N ∗ N 1 dξ. ≤ Ce2kϕ ( 2k ) n+1 |ξ|≥1 |ξ| Therefore, we deduce that there is a constant C > 0, that depends only on n and M , such that ∗ |α|+n+M |Dα u| ≤ C e2kϕ ( 2k ) , on U . Now, from the convexity of ϕ∗ we obtain |α| + n + M ∗ 2kϕ ≤ kϕ∗ (|α|/k) + kϕ∗ ((n + M )/k). 2k

We can now give the deﬁnition of quasianalitic (and non quasianlytic) wave front set of a classical distribution in Roumieu and Beurling settings.

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Deﬁnition 3.4. Let Ω ⊂ Rn be an open set and u ∈ D (Ω). Let ω be a weight function. The {ω}−wave (respectively (ω)−wave) front set W F{ω} (u) (respectively W F(ω) (u)) of u is the complement in Ω × (Rn \ 0) of the set of points (x0 , ξ0 ) such that there exist an open neighborhood U of x0 in Ω, a conic neighborhood Γ of ξ0 and a bounded sequence uN ∈ E (Ω) equal to u in U satisfying (3) (respectively satisfying (4)) in Γ. Lemma 3.5. Let u ∈ D (Ω) and let K be a compact subset of Ω with nonempty interior, F a closed cone in Rn . Let ω be a weight function. Suppose that {χN } ⊂ D(K) and, for every α ∈ Nn0 and N ∈ N, that |β|

|Dα+β χN | ≤ Cα (Cα N ) Then {χN u} is a bounded sequence in E hood of K. Moreover:

M

,

|β| ≤ N.

(8)

if u is of order M in a neighbor-

(a) If W F{ω} (u) ∩ (K × F ) = ∅, there exist a constant C > 0 and k ∈ N such that 1

N +1 k ϕ e |ξ|N |χ N u(ξ)| ≤ C

∗

(kN )

,

N = 1, 2, . . . ,

ξ ∈ F.

(9)

(b) If W F(ω) (u) ∩ (K × F ) = ∅ and ω(t) = o(t) as t tends to inﬁnity, there is a constant C > 0 such that for every k ∈ N there is Ck > 0 for which N kϕ |ξ|N |χ N u(ξ)| ≤ Ck C e

∗ N (k

)

,

N = 1, 2, . . . ,

ξ ∈ F.

(10)

Proof. The condition (8) with β = 0 implies that the sequence χN is bounded M in D(K) and hence χN u is bounded in E if u is of order M in a neighborhood of K. Let x0 ∈ K, ξ0 ∈ F \ {0} and choose U , Γ and uN according to Deﬁnition 3.4. If the support of χN is in U , we have χN u = χN uN . We ﬁrst prove (a). By hypothesis uN satisﬁes in Γ 1

|ξ|N | uN (ξ)| ≤ D1 e k ϕ

∗

(N k)

,

N = 1, 2, . . . ,

(11)

for some constant D1 > 0 that does not depend on N . On the other hand, | uN (ξ)| ≤ D2 (1 + |ξ|)M for some constants D2 , M > 0 and all ξ ∈ Rn , as the sequence uN is bounded in E M . Since the function ϕ∗ (x)/x is increasing and k ≥ 1, it follows from (8) that |β| ∗ 1 |Dα+β χN | ≤ Cα Cα e kN ϕ (N k) , |β| ≤ N, where Cα denotes again a suitable positive constant depending only on α. Therefore, we obtain 1 ∗ N exp kN ϕ (N k) N +1 −n−1−M , η ∈ Rn . (12) | χN (η)| ≤ C 1 N (1 + |η|) |η| + exp kN ϕ∗ (N k) The properties of the Fourier transform give −n uN (ξ − η)dη. χ N (η) χ N u(ξ) = (2π)

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Let 0 < c < 1. We split the integral into two parts: χ N u(ξ) −n

= (2π)

χ N (η) uN (ξ − η)dη +

χ N (η) uN (ξ − η)dη .

|η|≤c|ξ|

(13)

|η|≥c|ξ| −1

In the second integral we have |ξ−η| ≤ (1+c )|η|, and from the boundedness of {uN }, it can be estimated by −1 M | χN (η)|(1 + |η|)M dη. (14) D2 (1 + c ) |η|≥c|ξ|

Now, we estimate the ﬁrst integral in (13) by uN (ξ − η)|.

χN L1 sup |

(15)

|η|≤c|ξ|

On the other hand, for a conic closed neighborhood Γ1 of ξ0 contained in Γ \ {0} we can choose the constant c so that η ∈ Γ if ξ ∈ Γ1 and |ξ − η| ≤ c|ξ|. We observe that |η| ≥ (1−c)|ξ| and that the last supremum can be written as uN (η)|. Therefore, we conclude, from formulas (11), (12), (14) sup|ξ−η|≤c|ξ| | and (15), and for N ≥ 0, −N sup |ξ|N |χ

χN L1 sup | uN (η)| · |η|N N u(ξ)| ≤ (1 − c) Γ1 Γ −1 N +M M N + D2 (1 + c ) χN (η)|dη (1 + |η|) |η| | 1

≤ (1 − c)−N C1 C N D1 e k ϕ

∗

(N k)

1

+ D2 (1 + c−1 )N +M C2 e k ϕ

∗

(N k)

,

where the constants C1 , C2 , D1 , D2 do not depend on N . Now, F can be covered by a ﬁnite number of neighborhoods like Γ1 so that (9) is valid if supp χN ⊂ U for a suﬃciently small neighborhood of x0 . We can cover K by such neighborhoods Uj , j = 1, . . . , J, and choose χN,j ∈ D(Uj ) so that χN,j = 1 in K and χN,j satisﬁes (8) for j = 1, . . . , J. But, if χN ∈ D(K) satisﬁes (8), the same is valid also for the product χN,j χN with some other constants. Hence (9) is valid with χN replaced by χN,j χN . The proof of (a) is complete since χN,j χN = χN . We now prove (b). By hypothesis, uN satisﬁes in Γ |ξ|N | uN (ξ)| ≤ Ck ekϕ

∗

(N/k)

,

N = 1, 2, . . . ,

for some sequence {Ck } of positive constants that does not depend on N . On the other hand, | uN (ξ)| ≤ D(1 + |ξ|)M for some constants D, M > 0 and all n ξ ∈ R , as the sequence uN is bounded in E M . It follows from (8) that | χN (η)| ≤ C N +1

NN N

(|η| + N )

(1 + |η|)−n−1−M ,

η ∈ Rn .

Now we can proceed as in the Roumieu case (a) taking into account that N N ≤ eN N ! and the following property: as ω(t) = o(t) as t tends to inﬁnity,

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from Remark 2.4(b) and Lemma 3.2, it follows that for every B > 0 and λ > 0 there is a constant CB,λ > 0 such that ∗ n B n n! ≤ CB,λ eλϕ ( λ ) ,

n ∈ N.

At this point, we can give some properties and consequences. Theorem 3.6. Let Ω ⊂ Rn be an open set and ω be a weight function. Then the projection of W F∗ (u) in Ω is equal to sing∗ supp u if u ∈ D (Ω). Proof. We give the proof only for the Roumieu case. The Beurling case is similar. If u is a E{ω} -function in a neighborhood of x0 it follows from Proposition 3.3 that (x0 , ξ0 ) ∈ / W F{ω} (u) for each ξ0 ∈ Rn \ {0}. Assume that (x0 , ξ0 ) ∈ / W F{ω} (u) for every ξ0 ∈ Rn \ {0}. Then we can choose a compact neighborhood K of x0 so that W F{ω} (u)∩(K ×Rn ) = ∅. By Lemma 3.5 there is a sequence χN ∈ D(K) which is equal to 1 in a neighborhood of x0 such / sing{ω} supp u that χN u is bounded in E and satisﬁes (3). Therefore, x0 ∈ by Proposition 3.3. The condition (8) is satisﬁed by any ﬁxed function in E∗ (Ω) with support in K, where ∗ = {ω} or (ω). Therefore, if ω is a non-quasianalytic weight function, an equivalent deﬁnition of wave front set is given by the following proposition. Proposition 3.7. Let Ω ⊂ Rn be an open set and u ∈ D (Ω). Let ω be a nonquasianalytic weight function. Then (x0 , ξ0 ) ∈ Ω × (Rn \ 0) is not in the wave front set W F{ω} (u) (resp. W F(ω) (u)) of u if and only if there is a neighborhood U ⊂ Ω of x0 , a conic neighborhood Γ of ξ0 and v ∈ E{ω} (Ω) (resp. v ∈ E(ω) (Ω)) which is equal to u in U and has a Fourier transform satisfying (3) (resp. (4)) in Γ. Combining Proposition 3.7 with Proposition 3.3 and Lemmas 3.1 and 3.2, we recover the deﬁnition of wave front set in the Gevrey setting, [20, p. 36], and in the nonquasianalytic Beurling setting, [11]. The properties of the Young conjugate ϕ∗ω of ϕω for a given weight function ω lead to the following property: Proposition 3.8. Let Ω ⊂ Rn be an open set. If ω and σ are two weight functions such that ω = O(σ), then W F{ω} (u) ⊂ W F{σ} (u), and W F(ω) (u) ⊂ W F(σ) (u), for each u ∈ D (Ω). Since log t = o(ω), and ω = O(ω1 ), where ω1 (t) = max(t−1, 0) for every weight function ω, we observe that W F (u) ⊂ W F{ω} (u) ⊂ W FA (u),

u ∈ D (Ω),

being W F (u) the classical wave front set, and W FA (u) the Roumieu wave front set with respect to ω1 . Moreover, if ω(t) = o(t) as t tends to inﬁnity, we have W F (u) ⊂ W F{ω} (u) ⊂ W F(ω) (u) ⊂ W FA (u),

u ∈ D (Ω).

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Proposition 3.9. If Ω ⊂ Rn is an open set and S is a closed cone in Ω×(Rn \0), then there is u ∈ D (Ω) with W F (u) = W F{ω} (u) = S

(respectively, W F (u) = W F(ω) (u) = S)

for every weight function ω (respectively, for every weight function ω satisfying ω(t) = o(t) as t tends to inﬁnity). Proof. By [13, Theorem 8.4.14], there is u ∈ D (Ω) such that W F (u) = W FA (u) = S. Since W F (u) ⊂ W F{ω} (u) ⊂ W FA (u) (respectively, W F (u) ⊂ W F(ω) (u) ⊂ W FA (u)) for every weight function ω (respectively, for every weight function ω satisfying ω(t) = o(t) as t tends to inﬁnity), the result follows. The conditions in (8) remain valid if we multiply all χN by the same function in E∗ (Ω). Therefore, we obtain Theorem 3.10. W F∗ (au) ⊂ W F∗ (u) if a ∈ E∗ (Ω) and u ∈ D (Ω). On the other hand, it is clear that W F{ω} (∂u/∂xj ) ⊂ W F{ω} (u). In fact,

∂u

N +1 ∗ 1

N +1 (ξ) = |ξj u N +1 (ξ)| ≤ C|ξ| e (N +1)k ϕ ((N +1)k) /|ξ|

∂xj

1

= Ce k ϕ

∗

((N +1)k)

1

/|ξ|N ≤ C|ξ|−N e 2k ϕ

∗

(2N k)

1

e 2k ϕ

∗

(2k)

.

If we combine this property with Theorem 3.10, we obtain Theorem 3.11. Let Ω ⊂ Rn be an open set and P (x, D) = |α|≤m aα (x)Dα be a linear partial diﬀerential operator with coeﬃcients in E∗ (Ω). Then W F{ω} (P (x, D)u) ⊂ W F{ω} (u),

u ∈ D (Ω),

W F(ω) (P (x, D)u) ⊂ W F(ω) (u),

u ∈ D (Ω))

(respectively,

for every weight function ω (respectively, for every weight function ω satisfying ω(t) = o(t) as t tends to inﬁnity).

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4. Propagation of singularities In this section we prove a converse of Theorem 3.11 related to the propagation of singularities of solutions of linear partial diﬀerential operators. We ﬁrst study the Beurling case. The corresponding Roumieu version is then obtained as a consequence of a description of Roumieu wave front sets via a suitable union of Beurling wave front sets. Theorem 4.1. Let Ω ⊂ Rn be an open set. Let σ be a weight function satisfying property (α0 ) and ω be a weight function such that ω(t) = o(σ(t)) as t tends to inﬁnity. If P := P (x, D) = |α|≤m aα (x)Dα is a linear partial diﬀerential operator with coeﬃcients in the Roumieu class E{σ} (Ω), then (16) W F(ω) (u) ⊂ W F(ω) (P u) ∪ Σ, u ∈ D (Ω), α where Pm (x, ξ) = |α|=m aα (x)ξ is the principal symbol of P and Σ the characteristic set of P which is deﬁned by Σ = {(x, ξ) ∈ Ω × (Rn \ 0) : Pm (x, ξ) = 0} . Proof. We must prove that if (x0 , ξ0 ) does not belong to the right hand side of / W F(ω) (u). If we assume this hypothesis, we (16) and ξ0 = 0, then (x0 , ξ0 ) ∈ can choose a compact neighborhood K of x0 and a closed conic neighborhood Γ of ξ0 in Rn \ 0 such that Pm (x, ξ) = 0 in K × Γ, (K × Γ) ∩ W F(ω) (P u) = ∅.

(17) (18)

Now, we take a sequence χN ∈ D(K) equal to 1 in a ﬁxed neighborhood U of x0 satisfying property (8) for every α. Then, the sequence uN = χ2N u is bounded in E and equal to u in U . The theorem will be proved if we show that (4) is valid for the weight ω when ξ ∈ Γ and |ξ| ≥ N , since (4) is true for |ξ| ≤ N. In fact, when |ξ| ≤ N we can argue in the following way. As M ≤ C1 (1 + N )M ≤ C N for some in Proposition 3.3, |u N (ξ)| ≤ C1 (1 + |ξ|) positive constants C1 , C and a natural number M ∈ N. On the other hand, since ω(t) = o(t) as t tends to inﬁnity, we have that, for each k ∈ N, there ∗ exists Ck > 0 satisfying N N ≤ Ck ekϕω (N/k) . Then, we obtain ∗

N N |ξ|N |u ≤ Ck C N ekϕω (N/k) . N (ξ)| ≤ C N

To estimate u N (ξ) in Γ we will solve in an approximate way the equation P t v(x) = χ2N (x)e−ix,ξ .

(19)

−ix,ξ

w/Pm (x, ξ) and observe that the principal Following [14], we put v = e symbol of P t is Pm (x, −ξ). Hence, we obtain instead of (19) an equation of the form (20) w − Rw = χ2N , R = R1 + R2 + · · · + Rm where Rj |ξ|j is a diﬀerential operator of order less than or equal to j with E{σ} -coeﬃcients which is homogeneous of degree 0 with respect to ξ for ξ ∈ Γ and x ∈ K.

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We now write

wN =

IEOT

Rj1 · · · Rjk χ2N .

(21)

j1 +···+jk ≤N −m

From this formula, we obtain

wN − RwN = χ2N −

Rj1 · · · Rjk χ2N

(22)

j1 +···+jk >N −m≥j2 +···+jk

= χ2N − eN , which means that P t (x, D)(e−ix,ξ wN (x, ξ)/Pm (x, ξ)) = e−ix,ξ (χ2N (x) − eN (x, ξ)). With integrals denoting action of distributions we obtain, with f = P (x, D)u, u(x)χ2N (x)e−ix,ξ dx = u(x)eN (x, ξ)e−ix,ξ dx (23) + f (x)e−ix,ξ wN (x, ξ)/Pm (x, ξ)dx. To estimate the right-hand side of (23) we need the following lemma: Lemma 4.2. There exists a constant C > 0 such that for j = j1 + · · · + jk and j + |β| ≤ 2N , 1 ∗ j+|β| |Dβ Rj1 · · · Rjk χ2N | ≤ C N +1 e 2hN ϕσ (2hN ) |ξ|−j , ξ ∈ Γ. (24) Proof of Lemma 4.2. By homogeneity it suﬃces to prove the lemma when |ξ| = 1 (x ∈ K). But then, this lemma is a consequence of the next one. Lemma 4.3. Let K ⊂ Ω be a compact set and χN ∈ D(K) be a sequence satisfying property (8). If a1 , . . . , aj−1 are functions in E{σ} (Ω) such that, for some constant C > 0 and some h ∈ N, that 1

∗

sup |Dα as | ≤ Ce h ϕσ (h|α|) K

(25)

for every α ∈ Nn0 and s = 1, . . . , j − 1, we have, for some constant C1 > 0 that depends only on C and the sequence {χN }, and j ≤ N , 1 ∗ j (26) sup |Di1 a1 Di2 · · · aj−1 Dij χN | ≤ C1j+1 e hN ϕσ (hN ) K

Proof of Lemma 4.3. Since σ(t) = O(t) as t tends to inﬁnity, N N ≤ ∗ C N eϕσ (N ) for a suﬃciently large constant C > 0. Therefore, from (8), for α = 0, we obtain |β| ∗ 1 , |β| ≤ N. (27) |Dβ χN | ≤ C0 C0 e N ϕσ (N ) It is clear that Di1 a1 Di2 · · · aj−1 Dij χN is a sum of terms of the form (Dα1 a1 ) · · · (Dαj−1 aj−1 )Dαj χj with |α1 | + · · · + |αj | = j.

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If there are Ck1 ,...,kj terms with |α1 | = k1 ,. . . ,|αj | = kj , we have |Di1 a1 Di2 · · · aj−1 Dij χN | 1 ∗ kj

1 ∗ 1 ∗ k ≤ Ck1 ,...,kj C j−1 e h ϕσ (hk1 ) · · · e h ϕσ (hkj−1 ) C0 C0 j e N ϕσ (N )

e ≤ Ck1 ,...,kj k1 ! · · · kj−1 !C j−1

1 ∗ h ϕσ (hk1 )

k1 !

···

e

1 ∗ h ϕσ (hkj−1 )

kj−1 !

(28)

1 ∗ kj k C0 C0 j e N ϕσ (N ) .

As in [10], since σ satisﬁes property (α0 ), we can suppose that σ is equivalent to a sub-additive weight and then, we have 1

∗

1

∗

1

∗

e h ϕσ (hkj−1 ) e h ϕσ (h(j−kj )) e h ϕσ (hk1 ) ··· ≤ . k1 ! kj−1 ! (j − kj )! We also observe that k1 ! · · · kj−1 ! k1 ! · · · kj !j! k1 ! · · · kj ! = ≤ 2j (j − kj )! (j − kj )!kj !j! j! and that Ck1 ,...,kj k1 ! · · · kj ! = (2j − 1)!!. We can assume that C, C0 > 1 k and hence, if we put C1 = CC0 , we have C j−1 C0 C0 j ≤ C1j+1 . Since ϕ∗σ (x)/x is increasing, from (28), we obtain |Di1 a1 Di2 · · · aj−1 Dij χN | 1 ∗ kj

1 ∗ ≤C1j+1 2j /j! Ck1 ,...,kj k1 ! · · · kj !e h ϕσ (h(j−kj )) e hN ϕσ (hN ) 1 ∗ kj

1 ϕ∗ (h(j−kj )) j−kj ≤(2C1 )j+1 /j! Ck1 ,...,kj k1 ! · · · kj ! e h(j−kj ) σ e hN ϕσ (hN ) 1 ∗ j (2j − 1)!! ≤(4C1 )j+1 e hN ϕσ (hN ) , j!2j which concludes the proof of the lemma since

(2j−1)!! j!2j

≤ 1.

We now ﬁnish the proof of Theorem 4.1. If M is the order of u in a neighborhood of K, we can estimate the ﬁrst term on the right-hand side of (23) for large N and |ξ| > N, with ξ ∈ Γ, by

(1 + |ξ|)M −|α| sup |Dα eN (x, ξ)|. C |α|≤M

x

The number of terms in eN cannot exceed 2N , and each term can be estimated by means of (24), where j1 + . . . + jk = j > N − m by (22), which gives the bound 1 ∗ N +|α| |Dxα eN (x, ξ)| ≤ C N +1 2N e 2hN ϕσ (2hN ) |ξ|−N +m . Therefore, the ﬁrst term on the right-hand side of (23) can be estimated by 1 ∗ N +M |ξ|M −N +m . (29) C N +1 2N +M e 2hN ϕσ (2hN )

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Now, as ϕ∗σ is convex and ϕ∗σ (x)/x is increasing, we have N +M N +M 1 ∗ ∗ 1 e 2hN ϕσ (2hN ) ≤ e 2h(N +M ) ϕσ (2h(N +M )) ≤e

∗ 1 2h ϕσ (2h(N +M ))

= Ch,M e

∗ 1 4h ϕσ (4hN )

≤e

∗ 1 4h ϕσ (4hM )

(30) e

∗ 1 4h ϕσ (4hN )

.

Since ω(t) = o(σ(t)) as t tends to ∞, if N is replaced by N + m + M , the bound (29) and (30) imply an estimate of the form (11) for the ﬁrst integral on the right of (23). Indeed, from ω(t) = o(σ(t)) as t tends to ∞ we deduce that for every k ∈ N there exists dk > 0 such that N 1 ∗ ∗ ϕσ (4hN ) ≤ dk + kϕω , N ∈ N. (31) 4h k Now, combining (29) togheter (31) and (30) with N replaced by N + m + M , we easily obtain an estimate of the form (11) for the ﬁrst integral on the right of (23). To estimate the last term in (23) we observe that (24) gives 1 ∗ |β| , |β| ≤ N, ξ ∈ Γ, |ξ| > N. (32) |Dβ wN | ≤ C1N +1 e 2hN ϕσ (2hN ) We have a similar bound for wN |ξ|m /Pm (x, ξ). The proof is completed by the following lemma: Lemma 4.4. Let f ∈ D (Ω). Let K ⊂ Ω be a compact set and Γ a closed cone ⊂ Rn \ 0 such that W F(ω) (f ) ∩ (K × Γ) = ∅. If wN ∈ D(K) and (32) is fulﬁlled, then there exists C2 > 0 such that for every k ∈ N there exists Ck > 0 such that N −M −n ∗ N −M −n k N |w (33) e N −M −n ϕω ( k ) /|ξ| N f (ξ)| ≤ Ck C2 if ξ ∈ Γ, |ξ| > N , and N > M +n. Here M is the order of f in a neighborhood of K. Proof. By Lemma 3.5 we can ﬁnd a sequence fN which is bounded in E M and equal to f in a neighborhood of K so that there exists D > 0 such that for every k ∈ N there exists Dk > 0 for which k ∗ N |f N (η)| ≤ Dk DN e N ϕω (N/k) /|η| , N = 0, 1, 2, . . . , η ∈ Γ , where Γ is a conic neighborhood of Γ. Then wN f = wN fN , N = N −M −n. On the other hand, using (32) and proceeding as in Lemma 3.5 to show (12), we obtain that there exists E > 0 such that 1

∗

1

∗

|w N (η)| ≤ E N +1 (e 2hN ϕσ (2hN ) /(e 2hN ϕσ (2hN ) + |η|))N ,

η ∈ Rn .

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Proceeding again as in the proof of Lemma 3.5, it follows (the integrals that appear there are convergent from the selection of N ), that there exists D > 0 such that for every k ∈ N there exists Dk > 0 for which ∗ ∗ 1 N sup |ξ|N |w ekϕω (N /k) + e 2h ϕσ (2hN ) , ξ ∈ F, |ξ| > N. N f (ξ)| ≤ Dk D ξ∈Γ

We now observe that the convexity of ϕ∗σ implies that 1 ∗ 1 ϕσ (2hN ) ≤ [ϕ∗ (4h(N − M − n)) + ϕ∗σ (4h(M + n))] , 2h 4h σ

(34)

N > M + n.

(35) On the other hand, ω(t) = o(σ(t)) as t tends to ∞ and hence, for every k ∈ N there exists Dk > 0 such that N 1 ∗ ∗ ϕσ (4hN ) ≤ Dk + kϕω , N ∈ N. (36) 4h k Combining (34) with (35) and (36), (33) follows. Thus the lemma is proved. The last lemma gives the key to estimate the second integral on the right-hand side of (23), and ﬁnishes the proof of Theorem 4.1. An examination of the proofs above shows that we can repeat the same arguments to prove the analogous result of Theorem 4.1 in the Roumieu setting, even assuming the coeﬃcients of P (x, D) in E{ω} (Ω) with ω a weight function satisfying property (α0 ). We point out that it is necessary only to state and prove Lemma 4.4 appropriately. But, we also present another proof of the Roumieu version based on an application of Theorem 4.1 and of the following proposition, which is an extension to the quasianalytic case of [11, Proposition 2] and could be of independent interest. Given two weight functions σ0 and ω such that σ0 (t) = o(ω(t)) as t tends to inﬁnity, we set S := {σ weight function : σ0 ≤ σ = o(ω)}. Proposition 4.5. Let ω be weight function. If σ0 and σ are as above, we have W F(σ) (u), u ∈ D (Ω). W F{ω} (u) = σ∈S

Proof. The inclusion ∪σ∈S W F(σ) (u) ⊂ W F{ω} (u) follows easily from the deﬁnition of S, and the facts that E{ω} (Ω) ⊂ E(σ) (Ω) if σ = o(ω) (see Remark 2.4) and that the wave front set is closed. Conversely, suppose that (x0 , ξ0 ) ∈ Γ := ∪σ∈S W F(σ) (u). We can then ﬁnd a compact neighborhood K of x0 and a closed conic neighborhood F of ξ0 in Rn \ 0 such that (K × F ) ∩ ∪σ∈S W F(σ) (u) = ∅.

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For each N ∈ N let χN ∈ D(K) be equal to 1 in a ﬁxed neighborhood U of x0 such that, for every α ∈ Nn0 , |Dα+β χN | ≤ Cα (Cα N )|β| ,

|β| ≤ N.

Then, by Lemma 3.5. (b), χN u is a bounded sequence in E M if M is the order of u in a neighborhood of K and, for every σ ∈ S and k ∈ N, there is Ckσ > 0 so that ∗

σ kϕσ (N/k) , |ξ|N |χ N u(ξ)| ≤ Ck e

ξ ∈ F, N = 0, 1, 2, . . . .

(37)

We will deduce from this fact that (x0 , ξ0 ) ∈ W F{ω} (u), after showing that there exist C > 0 and h ∈ N for which 1

∗

N +1 h ϕω (hN ) e , |ξ|N |χ N u(ξ)| ≤ C

ξ ∈ F, N = 0, 1, 2, . . . .

(38)

In order to prove such an inequality, we will proceed as follows. By (37) we obtain that, for every σ ∈ S, k ∈ N, and r > 0, gN (r) := rN and hence

sup |ξ|=r,ξ∈F

∗

σ kϕσ (N/k) |χ , N u(ξ)| ≤ Ck e

∗

sup gN (r) ≤ Ckσ ekϕσ (N/k) , r>0

N = 0, 1, 2, . . .

N = 0, 1, 2, . . . .

This implies that, for every σ ∈ S and k ∈ N, N aN := log sup gN (r) ≤ kϕ∗σ + log Ckσ , k r>0

N = 0, 1, 2, . . . .

(39)

We claim that (39) implies that there exist h ∈ N and C > 0 so that 1 (40) aN ≤ ϕ∗ω (N h) + C, N = 0, 1, 2, . . . . h Proceeding by contradiction we can construct an increasing sequence (N (h))h of positive integers (N (1) := 0) such that 1 aN (h) > ϕ∗ω (N (h)h) + Ch , (41) h for every h ∈ N (Ch := h). We will show that inequalities (41) are in contradiction with inequalities (39) by constructing a weight function σ ∈ S for which the inequality 1 ∗ ϕ (N (h)h) + Ch < ϕ∗σ (N (h)) + log C1σ (42) h ω does not hold for inﬁnitely many indices h ∈ N. Without loss of generality, we can suppose that ω|[R,+∞[ is a C 1 function for some R ≥ 0 (see [9, Lemma 1.7]). Then the function ϕ := ϕω is a C 1 function too on [R, +∞[ and ϕ is a nondecreasing continuous function on [R, +∞[. In particular, limx→+∞ ϕ (x) = +∞ because log(1 + t) = o(ω(t)) as t tends to inﬁnity. Then, ϕ ([0, +∞[) ⊃ [ϕ (R), +∞[. Consequently, we can ﬁnd an increasing sequence (xh )h ⊂ [0, +∞[ (x1 := 0) satisfying xh → ∞ and ϕ (xh ) = N (h)h for all h ∈ N. Set x1 = y1 = z1 = 0. As xh → ∞, we can inductively deﬁne an increasing

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sequence (h(n))n of positive integers (h(0) = h(1) = 1) and the sequences (yn )n and (zn )n with xh(2) > R, y2 = z2 = xh(2) and for all n ≥ 3 N (h(n)) >

N (h(n − 1))h(n − 1) , h(n − 3)

xh(n) > yn−1 + n, σ0 (ex ) ≤

(44)

ϕ(x) for all x ≥ xh(n) , h2 (n − 1)

ϕ(xh(n) ) ≥ h(n − 1)

n−1

(43)

(45)

ϕ(zi ),

(46)

h(n − 1) ϕ (xh(n) ), h(n − 2)

(47)

i=1

ϕ (yn ) = and [h(n − 1) − h(n − 2)]ϕ(zn )

= h(n − 1)ϕ(xh(n) ) − h(n − 2)ϕ(yn ) + h(n − 1)(yn − xh(n) )ϕ (xh(n) ).

(48)

We have that xh(n) ≤ zn ≤ yn

(49)

for all n ∈ N. From (47) we get that yn ≥ xh(n) as ϕ is a nondecreasing function. Hence, by (48) we get ϕ(zn ) − ϕ(xh(n) ) yn − xh(n) 1 = h(n − 1) − h(n − 2) h(n − 1)ϕ(xh(n−1) ) − h(n − 2)ϕ(yn ) − (h(n − 1) − h(n − 2))ϕ(xh(n) ) × yn − xh(n) h(n − 1) ϕ (xh(n) ) h(n − 1) − h(n − 2) ϕ(xh(n) ) − ϕ(yn ) h(n − 1) h(n − 2) ϕ (xh(n) ) + = h(n − 1) − h(n − 2) yn − xh(n) h(n − 1) − h(n − 2) ϕ(yn ) − ϕ(xh(n) ) h(n − 2) ≥ ≥ 0, ϕ (xh(n) ) − h(n − 1) − h(n − 2) yn − xh(n) +

because ϕ is a nondecreasing function. On the other hand, (48) also implies that [h(n − 1) − h(n − 2)]ϕ(zn ) = h(n − 1)[ϕ(xh(n) ) + (yn − xh(n) )ϕ (xh(n) )] − h(n − 2)ϕ(yn ) ≤ [h(n − 1) − h(n − 2)]ϕ(yn ) because ϕ is a convex function.

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We deﬁne a function ψ on [0, +∞[ by setting ⎧ n−2 1 ⎪ ϕ(xh(n) ) + i=1 h(i)−h(i−1) ⎪ h(i−1)h(i) ϕ(zi+1 ) ⎪ ⎨h(n−2) if xh(n) ≤ x < yn , x−xh(n) ψ(x) = + h(n−2) ϕ (xh(n) ) (50) ⎪ ⎪ ⎪ ⎩ 1 ϕ(x) + n−1 h(i)−h(i−1) ϕ(z ) if yn ≤ x ≤ xh(n+1) . i+1 i=1 h(i−1)h(i) h(n−1) From (47) and (48) it follows that ψ is a C 1 function. Moreover, it is convex because it consists of linear parts and of dilated and shifted parts of ϕ. We deﬁne σ by σ(t) = ψ(max(log t, 0)), which is again a C 1 function. Proceeding as in the proof of [9, Lemma 1.7], it is easily seen that 1 ϕ(x) for all x ∈ [xh(n) , xh(n+1) ], n ≥ 3; ψ(x) ≥ h(n − 1) hence by (45) we deduce 1 1 ϕ(x) ≤ ψ(x) σ0 (t) ≤ 2 h (n − 1) h(n − 1)

for all x ∈ [xh(n) , xh(n+1) ], n ≥ 3.

Therefore σ0 (t) = o(σ(t)) as t → ∞. We also have ψ(x) 2 ≤ ϕ(x) h(n − 2)

for all x ∈ [xh(n) , xh(n+1) ], n ≥ 2,

and hence, σ(t) = o(ω(t)) as t → ∞. Now, we have, for every n ∈ N, that ϕ∗ (N (h(n))h(n)) = N (h(n))h(n)xh(n) − ϕ(xh(n) ).

(51)

Indeed,

ϕ (xh(n) ) = N (h(n))h(n), and, since ϕ is nondecreasing, N (h(n))h(n) − ϕ (s) ≥ 0 for all 0 ≤ s ≤ xh(n) . Moreover, N (h(n))h(n)s − ϕ(s) → −∞ as s → +∞ and (N (h(n))h(n)s − ϕ(s))(0) = 0. On the other hand, by (50) and proceeding as above we deduce, for every n ∈ N, that ψ ∗ (N (h(n))) = N (h(n))ζn − ψ(ζn ),

(52)

where ψ (ζn ) = N (h(n)) (indeed, the function δn (s) = N (h(n))s − ψ(s) is C 1 with derivative δn (s) = N (h(n)) − ψ (s) and hence δn (s) ≥ 0 if and only if ψ (s) ≤ N (h(n))). In particular, ζn ∈ [yn−1 , xh(n) ]. In fact, if x ∈ [yn−1 , xh(n) ] then, by (50), ϕ (xh(n) ) ϕ (yn−1 ) ≤ ψ (x) ≤ . h(n − 2) h(n − 2) Moreover, by (43) and (47), ϕ (yn−1 ) 1 N (h(n − 1))h(n − 1) = ϕ (xh(n−1) ) = < N (h(n)), h(n − 2) h(n − 3) h(n − 3) while

ϕ (xh(n) ) N (h(n))h(n) = > N (h(n)). h(n − 2) h(n − 2)

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Consequently, we have N (h(n)) ∈ ψ ([yn−1 , xh(n) ]) and so we can conclude from the fact that ψ is continuous. Therefore, we have constructed a C 1 –function σ : [0, +∞[→ [0, +∞[ such that σ0 (t) = o(σ(t)) and σ(t) = o(ω(t)) as t → +∞. Then, by [9, Lemma 1.7] we can ﬁnd a weight function v such that σ(t) = o(v(t)) and v(t) = o(ω(t)) as t → +∞ (hence, v ∈ S), and ψ(t) ≤ ϕv (t)

for all t ≥ 0.

ϕ∗v (t) ≤ ψ ∗ (t)

for all t ≥ 0.

This implies that (53)

As v ∈ S, by (39) and (41) we then obtain 1 ϕ∗ (N (h(n))h(n)) + h(n) < ϕ∗v (N (h(n))) + log C1v , h(n) for all n ∈ N. Thus, by (53) we get 1 ϕ∗ (N (h(n))h(n)) + h(n) < ψ ∗ (N (h(n)) + log C1v , h(n) for all n ∈ N. By (51) and (52) we also have N (h(n))xh(n) −

ϕ(xh(n) ) + h(n) < N (h(n))ζn − ψ(ζn ) + log C1v . h(n)

Then, by (50), since ϕ is a nondecreasing function and since ϕ (xh(n) ) = N (h(n))h(n), we obtain N (h(n))(xh(n) − ζn ) + h(n) n−2

<

h(i) − h(i − 1) ϕ(xh(n) ) ϕ(ζn ) − − ϕ(zi+1 ) + log C1v h(n) h(n − 2) i=1 h(i − 1)h(i)

ϕ(xh(n) ) ϕ(ζn ) ϕ(ζn ) ϕ(ζn ) − + − + log C1v h(n) h(n) h(n) h(n − 2) xh(n) ϕ (s) ds + log C1v ≤ h(n) ζn xh(n) ϕ (xh(n) ) ds + log C1v ≤ h(n) ζn

<

= N (h(n))(xh(n) − ζn ) + log C1v ,

which is a contradiction. Clearly, (41) implies (38) and the proof is complete.

An immediate consequence of Proposition 4.5 is the following result. Corollary 4.6. Let ω be a weight function and Ω be an open set in Rn . Then E(σ) (Ω). E{ω} (Ω) = σ∈S

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Proof. Let g ∈ D (Ω). By Theorem 3.6, we have that g ∈ E{ω} (Ω) if, and only if, W F{ω} (g) is empty. Then, W F(σ) (g) is empty for all σ ∈ S, and the conclusion follows. There exists an unpublished version of Corollary 4.6, with a diﬀerent proof, for open convex sets, due to J. Bonet and R. Meise, and is an extension for quasianalytic classes of [6, Proposition 3.5]. Another consequence of Proposition 4.5 is the following: Corollary 4.7. Let ω be a weight function and Ω be an open set in Rn . Then sing(ω) supp (u), u ∈ D (Ω). sing{ω} supp (u) = σ∈S

Proof. Fix u ∈ D (Ω). Then, combining Proposition 4.5 and Theorem 3.6, we easily obtain sing(ω) supp (u). sing{ω} supp (u) ⊂ σ∈S

The opposite inclusion follows from Remark 2.4(a) and the fact that the singular support is always a closed set. We now state and prove the Roumieu version of Theorem 4.1. We do not need any change of weight function, but the weight function must satisfy property (α0 ), as in the Beurling version (Theorem 4.1). Theorem 4.8. Let ω be a weight function satisfying property (α0 ) and Ω ⊂ Rn be an open set. If P (x, D) is a linear partial diﬀerential operator whose coeﬃcients belong to E{ω} (Ω), then W F{ω} (u) ⊂ W F{ω} (P u) ∪ Σ,

u ∈ D (Ω),

where Σ is the characteristic set of P . Proof. Let S be the set of weight functions σ such that σ = o(ω). Then, for each σ ∈ S and u ∈ D (Ω), by Theorem 4.1 we have W F(σ) u ⊂ W F(σ) (P u) ∪ Σ,

u ∈ D (Ω),

as the coeﬃcients of P (x, D) belong to E{ω} (Ω). Now, we apply Proposition 4.5 to conclude. Finally, we point out that from Theorems 4.1 and 4.8 we immediately get (see, for example, [20, p.105]): Corollary 4.9. Let P = P (x, D) be an elliptic linear partial diﬀerential operator deﬁned on an open set Ω of Rn (elliptic means that Σ = ∅). Then: (a) If ω is a weight function satisfying property (α0 ) and the coeﬃcients of P belong to E{ω} (Ω), we have W F{ω} (u) = W F{ω} (P u),

u ∈ D (Ω),

and hence, sing{ω} supp (u) = sing{ω} supp (P u),

u ∈ D (Ω).

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(b) If ω and σ are two weight functions such that ω satisﬁes property (α0 ) and ω = o(σ), and the coeﬃcients of P belong to E{σ} (Ω), then W F(ω) (u) = W F(ω) (P u),

u ∈ D (Ω),

and hence, sing(ω) supp (u) = sing(ω) supp (P u),

u ∈ D (Ω).

We conclude the paper by studying the wave front set of the solutions of the following (non hypoelliptic) partial diﬀerential operator of principal type in Rn : ∂ P = . ∂xn Observe that the characteristic set of P is Σ = {(x, ξ) ∈ R2n : ξn = 0, ξ = 0}. Moreover, we point out that u ∈ D (Rn ) is a solution of P u = 0 if, and only if, u = v ⊗ 1 for some v ∈ D (Rn−1 ), being 1 the function identically 1 in the xn -variable. Indeed, if u is of the form v ⊗ 1, then ∂x∂n (v ⊗ 1) = 0. On the other hand, if P u = 0, then u satisﬁes τh u = u for every h = (0, . . . , 0, hn ), where τh u denotes the h-translation of the distribution u (see, for example, [20]). From this fact, by an approximation procedure, it is easy to conclude that u must be of the form v ⊗ 1 for some distribution v ∈ D (Rn−1 ). We can now state the following result: Proposition 4.10. Let ω be a weight function satisfying property (α0 ), and write, as usual, ∗ for {ω} or (ω). Let u ∈ D (Rn ) be a solution of the equation P u = 0. If (x0 , ξ0 ) ∈ W F∗ (u), then (x0 , ξ0 ) ∈ Σ, and splitting Rn x = (x , xn ) = (x1 , . . . , xn−1 , xn ), we have that the straight line L = {(x0 , xn , ξ0 ), xn ∈ R} is contained in W F∗ (u). Moreover, if ω is non-quasianalytic, for every (x0 , ξ0 ) ∈ Σ there exists a solution u ∈ D∗ (Rn ) of P u = 0 whose ∗-wave front set is given by the set {(x0 , xn , λξ0 ), xn ∈ R, λ > 0}. Proof. Since u ∈ D (Rn ) is a solution of P u = 0, by Theorems 4.1 and 4.8 we have W F∗ (u) ⊂ Σ and u = v ⊗ 1 for some suitable v ∈ D (Rn−1 ). We claim that W F∗ (u) = {(x, ξ) ∈ Σ : (x , ξ ) ∈ W F∗ v}.

(54)

First, we show property (54) in Beurling case proceeding as follows. Let / W F(ω) (v) then, by Deﬁnition 3.4, there exist an open (x, ξ) ∈ Σ. If (x , ξ ) ∈ neighborhood U of x , a conic neighborhood Γ of ξ and a bounded sequence vN ∈ E (Rn−1 ), vN = v in U , such that for every k ∈ N there exists a positive constant Ck satisfying |ξ |N | vN (ξ )| ≤ Ck ekϕ

∗

(N/k)

(55)

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in Γ , for N = 1, 2, . . . . Let χ ∈ D(R), be a function equal to 1 in a neighborhood I of xn . Then, we have that uN := vN ⊗ χ is a bounded sequence in E (Rn ), uN = u in U := U × I. Let Γ be a conic neighborhood of (ξ , 0) (since (x, ξ) ∈ Σ we have ξ n = 0) satisfying Γ ∩ {ξn = 0} ⊂ Γ . Then there exists a positive constant c1 such that |ξn | ≤ c1 |ξ | for ξ = (ξ , ξn ) ∈ Γ.

(56)

We also observe that u ˆN (ξ) = v N (ξ ) χ(ξn ). N ⊗ χ(ξ) = v Now, from (55) and (56), it follows that for every k ∈ N, N uN (ξ)| ≤ |ξ | + |ξn | | vN (ξ )| | χ(ξn )| |ξ|N | vN (ξ )| | χ(ξn )| ≤ (1 + c1 )N |ξ |N | ≤ c2 Ck (1 + c1 )N ekϕ

∗

(N/k)

,

for each N ∈ N and ξ ∈ Γ. In view of Deﬁnition 3.4 and Lemma 3.2, this inequality implies that (x, ξ) ∈ / W F(ω) (u). Then, we deduce that W F(ω) (u) ⊂ {(x, ξ) ∈ Σ : (x , ξ ) ∈ W F(ω) (v)}. / W F(ω) (u). By Deﬁnition 3.4 Conversely, let (x, ξ) ∈ Σ such that (x, ξ) ∈ there exist an open neighborhood U of x, a conic neighborhood Γ of ξ and a bounded sequence uN ∈ E (Rn ), uN = u in U , such that (4) is satisﬁed in Γ. Since u = v ⊗1, without loss of generality we can assume that uN = vN ⊗χN , eventually multiplying uN by a tensor product test function equal to 1 in a neighborhood V of x with V ⊂ U (see Lemmas 3.5 and 3.2 and remarks before Proposition 3.7). Therefore, there exists a sequence Ck of positive constants such that |ξ|N | uN (ξ)| = |ξ|N | vN (ξ )| | χN (ξn )| ≤ Ck ekϕ

∗

(N/k)

(57)

for N ∈ N and ξ ∈ Γ. We may also suppose that χN =: χ is independent of N and satisﬁes χ (0) = 0. So, from (57) it follows that |ξ |N | vN (ξ )| ≤

Ck kϕ∗ (N/k) e | χ(0)|

for N = 1, 2, . . . and ξ ∈ Γ = Γ ∩ {ξn = 0}. Then (x , ξ ) ∈ / W F(ω) (v), and so we have shown that {(x, ξ) ∈ Σ : (x , ξ ) ∈ W F(ω) (v)} ⊂ W F(ω) (u). This concludes the proof of (54). In the Roumieu case the proof is similar with minor changes. Now, it follows immediately that if (x, ξ) ∈ W F∗ (u), being u a solution of P u = 0, then every point of the kind (x0 , xn , ξ0 ), with xn ∈ R, belongs to W F∗ (u). Thereby, the straight line L is contained in W F∗ (u).

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In the following, let ω be a non-quasianalytic weight function. Now, we show that there exists an ∗-ultradistribution u ∈ D∗ (Rn ) solution of P u = 0 with the prescribed wave front set W F∗ (u) = {(x, ξ) ∈ Rn × Rn \ {0} : x = (x0 , xn ), ξ = λξ0 , xn ∈ R, λ > 0}. (58) Fix (x0 , ξ0 ) ∈ Σ. Proceeding in a similar way as in Example 1 in [11], we can construct v˜ ∈ E∗ (Rn−1 ) satisfying v) W F∗ (˜ = {(x , ξ ) ∈ Rn−1 × Rn−1 \ {0} : x = 0, ξ = (0, . . . , 0, ξn−1 ), ξn−1 > 0}. By a linear change of variable and a translation, we then ﬁnd v satisfying W F∗ (v) = {(x , ξ ) ∈ Rn−1 × Rn−1 \ {0} : x = x0 , ξ = λξ0 , λ > 0}. Set u = v ⊗ 1 ∈ D∗ (Rn ). Then, we have that P u = 0 and that, by (54), equality (58) is satisﬁed. Observe that an analogous result holds for the equation P u = f , with f ∈ C ∞ (Rn ). Indeed, every solution u ∈ D (Rn ) of P u = f can be written as xn u(x) = u0 (x) + f (x , t) dt, x = (x , xn ) ∈ Rn , 0

/ W F∗ (f ), then where u0 is a solution of P u = 0. If (x0 , ξ0 ) ∈ Σ and (x0 , ξ0 ) ∈ (x0 , ξ0 ) ∈ W F∗ (u) implies that (x0 , xn , ξ0 ) ∈ W F∗ (u) for xn in a suitable / W F∗ (f ) then there exists a interval I containing xn,0 . In fact, if (x0 , ξ0 ) ∈ neighborhood U of (x0 , ξ0 ) with empty intersection with W F∗ (f ). So, in a neighborhood of x0 the wave front set of u coincides with the wave front set of u0 . Acknowledgement The authors wish to thank professor Luigi Rodino for proposing the problem. The second author is indebted to professor Rodino for his hospitality and his helpful suggestions on mathematics during his diﬀerent visits in Torino. He thanks also the members of the Department of Mathematics of the University of Lecce for their hospitality, where part of the present paper was written.

References [1] A. Beurling, Quasi-analiticity and general distributions, Lecture 4 and 5, AMS Summer Institute, Stanford, 1961. [2] P. Bolley, J. Camus, R´egularit´e Gevrey et it´er´es pour une classe d’op´erateurs hypoelliptiques, Comm. Partial Diﬀerential Equations 6 (1981), no. 10, 1057– 1110. [3] P. Bolley, J. Camus, C. Mattera, Analyticit´e microlocale et it´er´es d’operateurs, S´eminaire Goulaouic-Schwartz, 1978–79. [4] P. Bolley, J. Camus, L. Rodino, Hypoellipticit´e analytique-Gevrey et it´er´es d’op´erateurs, Rend. Sem. Mat. Univ. Politec. Torino 45 (1987), no. 3, 1–61 (1989).

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[5] C. Bouzar, L. Chaili, A Gevrey microlocal analysis of multi-anisotropic diﬀerential operators, Rend. Semin. Mat. Univ. Politec. Torino 64 (2006), no. 3, 305–317. [6] J. Bonet, C. Fern´ andez, R. Meise, Characterization of the ω-hypoelliptic convolution operators on ultradistributions, Ann. Acad. Sci. Fenn. Math. 25 (2000), no. 2, 261–284. [7] J. Bonet, A. Galbis, and S. Momm, Nonradial H¨ ormander algebras of several variables and convolution operators, Trans. Amer. Math. Soc. 353 (2001), no. 6, 2275–2291. [8] J. Bonet, R. Meise, S.N. Melikhov, A comparison of two diﬀerent ways to deﬁne classes of ultradiﬀerentiable functions, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 3, 425–444. [9] R.W. Braun, R. Meise, B.A. Taylor, Ultradiﬀerentiable functions and Fourier analysis, Results Math. 17 (1990) 206–237. [10] C. Fern´ andez, A. Galbis, Superposition in classes of ultradiﬀerentiable functions, Publ. RIMS, Kyoto Univ. 42 (2006), 399–419. [11] C. Fern´ andez, A. Galbis, D. Jornet, Pseudodiﬀerential operators of Beurling type and the wave front set, J. Math. Anal. Appl. 340 (2008) 1153–1170. [12] T. Heinrich, R. Meise, A support theorem for quasianalytic functionals, Math. Nachr. 280, No. 4, 364–387 (2007). [13] L. H¨ ormander, The Analysis of Linear Partial Diﬀerential Operators I, Springer, Berlin 1983. [14] L. H¨ ormander, Uniqueness theorems and wave front sets for solutions of linear diﬀerential equations with analytic coeﬃcients, Comm. Pure Appl. Math. 24 (1971), 671–704. [15] M. Kato, Some results on potential scattering, Proc. Internat. Conf. on Functional Analysis and Related topics (Tokyo, 1969), 91–94, Univ. of Tokyo Press, Tokyo, 1970. [16] H. Komatsu, Ultradistributions I. Structure theorems and a characterization, J. Fac. Sci. Tokyo Sec. IA 20 (1973), 25–105. [17] R. Meise and D. Vogt, Introduction to Functional Analysis, Oxford Univ. Press, Oxford, 1997. [18] J. Peetre, Concave majorants of positive functions, Acta Math. Acad. Sci. Hungaricae 21 (1970), 327–333. [19] H.J. Petzsche, D. Vogt, Almost analytic extension of ultradiﬀerentiable functions and the boundary values of holomorphic functions, Math. Ann. 267 (1984), 17–35. [20] L. Rodino, Linear Partial Diﬀerential Operators in Gevrey Spaces, World Scientiﬁc Pub. (1993). [21] T. R¨ osner, Surjectivit¨ at partieller Diﬀerentialoperatoren auf quasianalytischen Roumieu-Klassen, Dissertation, D¨ usseldorf 1997. [22] D. Vogt, Topics in projective spectra of (LB)-spaces, in “Advances in the theory of Fr´echet spaces” (ed. T. Terzioglu), NATO Advanced Sciencie Institute, Series C, 287, 11–27. [23] J. Wengenroth, Acyclic inductive spectra of Fr´echet spaces, Studia Math., 120 (1996), 247–258.

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[24] L. Zanghirati, Iterati di operatori e regolarit` a Gevrey microlocale anisotropa, Rend. Sem. Mat. Univ. Padova, 67 (1982). A. A. Albanese Dipartimento di Matematica “E. De Giorgi” Universit` a del Salento-Lecce Via Per Arnesano, P.O. Box 193 I-73100 Lecce Italy e-mail: [email protected] D. Jornet Instituto Universitario de Matem´ atica Pura y Aplicada IUMPA-UPV Universidad Polit´ecnica de Valencia C/Camino de Vera, s/n E-46022 Valencia Spain e-mail: [email protected] A. Oliaro Dipartimento di Matematica Universit` a di Torino Via Carlo Alberto, 10 I-10123 Torino Italy e-mail: [email protected] Submitted: April 8, 2009. Revised: October 1, 2009.

Integr. Equ. Oper. Theory 66 (2010), 183–195 DOI 10.1007/s00020-010-1740-8 Published online January 26, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

A Levi-Civit´a Equation on Compact Groups and Nonabelian Fourier Analysis Jinpeng An and Dilian Yang Abstract. In this paper, we study the following Levi-Civit´ a equation m w(xy) + w(yx) = fi (x)gi (y) (LC) i=1

on a compact group G, where w, fi ’s, and gi ’s are continuous complex-valued functions to determine. Our main ingredient is (nonabelian) Fourier analysis on compact groups. We apply the Fourier transform to Eq. (LC) on the product group G × G so that we obtain its several equivalent operator equations. Using those equivalent equations, we derive some crucial properties of solutions to Eq. (LC). Consequently, Eq. (LC) with m ≤ 2 is completely solved. In particular, a Wilson type equation arising from and playing a central role in [4] is solved on compact groups. Mathematics Subject Classiﬁcation (2010). Primary 39B52; Secondary 22C05. Keywords. Functional equation, nonabelian Fourier analysis.

1. Introduction Let G be a group, and let w and f be complex-valued functions on G. The equation w(xy) + w(yx) = 2f (x)w(y) + 2w(x)f (y) (1.1) came up recently in [4], when Davison studied a pre-d’Alembert equation, a generalization of the d’Alembert functional equation. (For the d’Alembert and related equations, refer to [1, 2, 7, 10].) It turns out that Eq. (1.1) plays a signiﬁcant role in [4]. In particular, Davison [4] used Eq. (1.1) to study the Wilson functional equation g(xy) + g(xy −1 ) = 2g(x)f (y)

(1.2)

J. An is partially supported by the NSFC grant 10901005 and the FANEDD grant 200915. D. Yang is partially supported by an NSERC Discovery grant and a Start-up grant.

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on groups. Surprisingly, he proved that, if f satisﬁes the pre-d’Alembert equation with some other properties, the dimension of the space consisting of the solutions g of Eq. (1.2) is either 2 or 4. We should mention that [3, Corollary 6.3] implies that this holds true on compact groups without any restriction on f except that f = 0. According to our best knowledge, the above result has not been well-understood yet in general. We remark that Eq. (1.1) has not been solved although some nice properties of its solutions were given in [4]. In this paper, we study a much more general functional equation, which is called a (nonclassical ) Levi-Civit´ a equation, i.e., w(xy) + w(yx) =

m

fi (x)gi (y)

(LC)

i=1

on a compact group G, where w, f1 , ..., fm , g1 , ..., gm are the unknown complex-valued functions on G, and m ≥ 1 is an integer. When G is abelian, Eq. (LC) is nothing but the classical Levi-Civit´ a equation, which includes many important and well-known equations, such as the Cauchy equation, the Pexider equation, and trigonometric equations, as special cases. Refer to [1, 12] for more details. In [1, 12], the classical Levi-Civit´ a equation was studied on abelian groups. The classical one was solved on any locally compact group in [8], where only the role of w in the solution was emphasized. The argument there is slick, but unfortunately heavily depends on the form of the left hand side on the classical Levi-Civit´ a equation being w(xy), rather than w(xy) + w(yx). It seems that the argument there can not be adapted to the above nonclassical case. For more information about the classical Levi-Civit´a equation, refer to the recent work [11] and the references therein. Our main tool to solve Eq. (LC) is (nonabelian) Fourier analysis on compact groups. In our recent paper [3], Fourier analysis on a compact group G was used to solve a class of functional equations on G. For Eq. (LC), we ﬁnd that it is more convenient (especially for the proof of Theorem 5.1(ii) below) to use Fourier analysis on the product group G × G. By regarding the functions on both sides of Eq. (LC) as functions on G × G and taking the Fourier transform on G × G, we obtain several equivalent operator equations which have very nice properties and are easy to analyze. After exploring these operator equations, we are able to obtain some crucial properties of solutions to Eq. (LC). Making use of these properties, we completely solve the LeviCivit´ a equation (LC) in the case of m ≤ 2. In particular, Eq. (1.1) is solved. Furthermore, we shall see the assumption that f satisﬁes the pre-d’Alembert equation given in [4] is redundant on compact groups. This paper is organized as follows. In Section 2, we construct two types of solutions of Eq. (LC). It will be shown later that those solutions are generic in certain cases. With the aid of the Fourier transform on G × G, in Section 3 Eq. (LC) is converted to its several equivalent operator equations. In Section 4 some useful lemmas are given, which are crucial for obtaining the main results in Section 5. The last section gives our main theorems, from which,

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particularly, Eq. (1.1) in [4] is easily solved. Furthermore, it turns out that the solutions constructed in Section 2 indeed exhaust all of the cases of m ≤ 2. We should mention that Eq. (LC) is also meaningful even when G is a monoid, which is actually the context used in [4]. However, since, as mentioned above, we will use the Fourier transform, G needs to be a group, not just a monoid. We end the introduction with some notation. Throughout the paper, we suppose that G is a compact group, unless otherwise speciﬁed. Let dx be the normalized Haar measure on G, and L2 (G) the Hilbert space of all square integrable functions on G with respect to dx. We always assume that representations of G and solutions of an equation on G are continuous. Let Mm,n (C) denote the space of all m × n complex matrices. If m = n, then we simply write it as Mn (C). By diag(a1 , ..., an ), we mean the diagonal matrix with (i, i)-th entry ai . The notation Rn stands for the (row) space Cn whose vectors are regarded as row vectors.

2. Constructing solutions Consider the Levi-Civit´ a equation (LC) on a compact group G. We denote its solution by a row vector (w, f1 , . . . , fm , g1 , . . . , gm ) of continuous complexvalued functions. It is natural to focus on the case where both {fi }m i=1 and are linearly independent (over C). Such solutions are said to be non{gi }m i=1 degenerate. The aim of this section is to construct two types of solutions of Eq. (LC). As we shall see later, these solutions actually exhaust all solutions in certain cases. 2.1. Construction 1. Let π1 , . . . , πm be m 1-dimensional representations of G. We want to ﬁnd solutions of Eq. (LC) which are linear combinations of π1 , . . . , πm . That is, we wish to ﬁnd a matrix Q ∈ Mm,2m+1 (C) such that (w, f1 , . . . , fm , g1 , . . . , gm ) = (π1 , . . . , πm )Q

(2.1) t

is a solution of Eq. (LC). We write Q = (ε A B), where ε = (ε1 , . . . , εm ) ∈ Rm , and A, B ∈ Mm (C). Then Eq. (LC) becomes 2(π1 (x), . . . , πm (x)) diag(ε1 , . . . , εm )(π1 (y), . . . , πm (y))t = (π1 (x), . . . , πm (x))AB t (π1 (y), . . . , πm (y))t . Thus if AB t = 2 diag(ε1 , . . . , εm ), then the row vector of functions determined by (2.1) is a solution of Eq. (LC). Moreover, such a solution is nondegenerate iﬀ π1 , . . . , πm are distinct and all components εi ’s of ε are nonzero. Indeed, it suﬃces to notice that π1 , ..., πm distinct is the same as π1 , ..., πm linearly independent. This is a direct consequence of Schur’s orthogonality relations ([5]).

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2.2. Construction 2. This construction applies only to the case of m = 2. We ﬁrst prove a simple lemma. We denote sl(2, C) = {A ∈ M2 (C) : Tr A = 0}. Lemma 2.1. Let W ∈ sl(2, C). Then for any X, Y ∈ M2 (C) we have Tr(W XY + W Y X) = Tr(W X) Tr Y + Tr(W Y ) Tr X. Proof. It is easy to check that AB + BA = Tr(AB)I

for all A, B ∈ sl(2, C).

Substituting A and B with W and X − Tr(X)I/2, respectively, we get W X + XW = Tr(W X)I + Tr(X)W. In order to obtain the desired formula, it suﬃces to multiply Y from the right at both sides and then take the trace. Now let π : G → U (2) be a representation, and χπ the character of π: for all x ∈ G. 0 Let W ∈ sl(2, C), and C, D ∈ M2 (C) with CDt = 1 the functions on G deﬁned by χπ (x) = Tr π(x)

w(x) = Tr(W π(x)),

1 . We claim that 0

(f1 , f2 , g1 , g2 ) = (w, χπ )(C D)

give a solution of Eq. (LC) with m = 2. Indeed, by Lemma 2.1, we have f1 (x)g1 (y) + f2 (x)g2 (y) = (f1 (x), f2 (x))(g1 (y), g2 (y))t = (w(x), χπ (x))CDt (w(y), χπ (y))t = Tr(W π(x)) Tr π(y) + Tr(W π(y)) Tr π(x) = Tr(W π(xy) + W π(yx)) = w(xy) + w(yx). Furthermore, one can check that if π is irreducible, the solution (w, f1 , f2 , g1 , g2 ) constructed above is nondegenerate iﬀ W = 0. In fact, this is a straight-forward consequence of Schur’s orthogonality relations, and the fact that both {f1 , f2 } and {g1 , g2 } are linearly independent iﬀ so is {w, χπ }.

3. Operator Equations via the Fourier transform 3.1. The Fourier transform In this subsection, we brieﬂy review some background in Fourier analysis on compact groups, which will be used later. For more information, refer to [5, Chapter 5] and [6, Section 27]. ˆ a complete set of mutually Let G be a compact group. We denote by G ˆ let inequivalent unitary irreducible matrix representations of G. For π ∈ G,

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dπ stand for the dimension of the representation space of π. According to ˆ [6, Theorem 27.43], we may choose G × G = {π ⊗ ρ : π, ρ ∈ G}. For f ∈ L2 (G), the Fourier transform of f is deﬁned by ˆ f (x)π(x)−1 dx for all π ∈ G. fˆ(π) = dπ G

So fˆ(π) ∈ Mdπ (C). Note that, for convenience, our deﬁnition is diﬀerent from the one in [5] by a factor dπ . Then the Fourier inversion formula is given by Tr fˆ(π)π(x) for all x ∈ G. f (x) = ˆ π∈G

Note that if f is continuous, the above series converges pointwise. We denote ˆ : fˆ(π) = 0} and supp{fˆ1 , ..., fˆm } = m supp(fˆi ). supp(fˆ) = {π ∈ G i=1 3.2. Operator equations equivalent to Eq. (LC) In this subsection, with the aid of the Fourier transform, we convert Eq. (LC) to some equivalent forms, which will be used later to explore some crucial properties of solutions to Eq. (LC). Let n Eij ⊗ Eji ∈ Mn (C) ⊗ Mn (C), Pn = i,j=1

where Eij , as usual, is the matrix unit whose (i, j)-entry is 1 and others 0. It is easy to verify that Pn has the following properties: Pn2 = In ⊗ In ,

Pn (A ⊗ B)Pn = B ⊗ A,

where A, B ∈ Mn (C). The importance of Pn for us lies in the following lemma. ˆ we have Lemma 3.1. For π, ρ ∈ G, 0, −1 dπ π(x) ⊗ ρ(x) dx = Pdπ , G

π= ρ; π = ρ.

Proof. By Schur’s orthogonality relations, we have dρ dπ −1 dπ π(x) ⊗ ρ(x) dx = dπ πij (x)ρlk (x)Eij ⊗ Ekl dx G

i,j=1 k,l=1

G

⎧ ⎪ π = ρ; ⎪ ⎨0, dπ dπ = ⎪ δil δjk Eij ⊗ Ekl = Pdπ , π = ρ. ⎪ ⎩ i,j=1 k,l=1

This proves the lemma. Proposition 3.2. Eq. (LC) is equivalent to m Pdπ (w(π) ˆ ⊗ Idπ ) + (w(π) ˆ ⊗ Idπ ) Pdπ , ˆ fi (π) ⊗ gˆi (ρ) = 0, i=1

π = ρ; π = ρ.

(3.1)

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Proof. The idea of the proof is simple: Regard both sides of Eq. (LC) as functions on G × G, and then take their Fourier transforms on G × G. The value of the Fourier transform of the function (x, y) → w(xy) + w(yx) at π ⊗ ρ is given by (w(xy) + w(yx))π(x)−1 ⊗ ρ(y)−1 dxdy dπ dρ G×G = d π dρ w(x)(π(y)π(x)−1 ) ⊗ ρ(y)−1 dxdy G×G + dπ dρ w(x)(π(x)−1 π(y)) ⊗ ρ(y)−1 dxdy G×G = dπ dρ π(y) ⊗ ρ(y)−1 dy w(x)π(x)−1 ⊗ Idρ dx G G −1 w(x)π(x) ⊗ Idρ dx π(y) ⊗ ρ(y)−1 dy + dπ dρ G G 0, π = ρ, = Pdπ (w(π) ˆ ⊗ Idπ ) + (w(π) ˆ ⊗ Idπ )Pdπ , π = ρ, where Lemma 3.1 was used to get the last equality. The value of the Fourier transform of the function fi ⊗ gi : (x, y) → fi (x)gi (y) at π ⊗ ρ is fi (x)gi (y)π(x)−1 ⊗ ρ(y)−1 dxdy fi ⊗ gi (π ⊗ ρ) =dπ dρ G×G = d π dρ fi (x)π(x)−1 dx ⊗ gi (y)ρ(y)−1 dy G

G

= fˆi (π) ⊗ gˆi (ρ). The proposition now immediately follows from the above calculations and the Fourier inversion formula. In order to obtain our main results, we need to convert the operator equation (3.1) of Eq. (LC) to another more convenient one. To this end, we need to introduce more notation ﬁrst. For any n ∈ N, deﬁne a linear isomorphism R : Mn (C) → Rn2 by R(A) = (v1 · · · vn ), where vi (i = 1, ..., n) is the i-th row vector of A ∈ Mn (C). Then for any m, n, the linear mapping Mat : Mm (C) ⊗ Mn (C) → Mm2 ,n2 (C) determined by Mat(A ⊗ B) = R(A)t R(B), is also a linear isomorphism, where A ∈ Mm (C), B ∈ Mn (C). Finally, we deﬁne for any n a linear mapping Φ : Mn (C) → Mn2 (C) by Φ(W ) = Mat (Pn (W ⊗ In ) + (W ⊗ In )Pn ) . Keeping the notation as above and by Proposition 3.2, we have

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Corollary 3.3. Eq. (LC) is equivalent to

Φ(w(π)), ˆ R(fˆi (π)) R(ˆ gi (ρ)) = 0, i=1

m

t

π = ρ; π = ρ.

(3.2)

ˆ = {π1 , π2 , . . .}, then (3.2) can be rewritten as If G m

(R(fˆi (π1 )) · · · )t (R(ˆ gi (π1 )) · · · ) = diag(Φ(w(π ˆ 1 )), . . .),

(3.3)

i=1

which is an equation on matrices of inﬁnite size.

4. Some auxiliary lemmas As we shall see later, the mapping Φ deﬁned in Section 3 plays an important role in analyzing properties of solutions to Eq. (LC). We take a closer look at it in this section. Lemma 4.1. For any 0 = W ∈ Mn (C), we have rank Φ(W ) ≥ n. In particular, Φ is injective. Proof. Set W = (Wij ). Direct computations give Pn (W ⊗ In ) + (W ⊗ In )Pn = =

=

n

n

r,s=1 q=1 n

n

(Esr W + W Esr ) ⊗ Ers

r,s=1

Wrq Esq +

n

Wps Epr

⊗ Ers

p=1

(δps Wrq + δrq Wps )Epq ⊗ Ers .

p,q,r,s=1

But Mat(Epq ⊗ Ers ) = R(Epq )t R(Ers ) = E(p−1)n+q,(r−1)n+s . So the ((p − 1)n + q, (r − 1)n + s)-th entry of Φ(W ) is Φ(W )(p−1)n+q,(r−1)n+s = δps Wrq + δrq Wps .

(4.1)

If W is not diagonal, then there are p, s ∈ Nn = {1, ..., n} with p = s s, from (4.1) we have such that Wps = 0. We ﬁx such a pair p, s. As p = Φ(W )(p−1)n+q,(r−1)n+s = δrq Wps . Let I = {(p − 1)n + q : q ∈ Nn }

and J = {(r − 1)n + s : r ∈ Nn }.

Then the submatrix {(Φ(W ))i,j : i ∈ I, j ∈ J } of Φ(W ) has exactly one nonzero entry in each row and each column, and so it has rank n. This implies that rank Φ(W ) ≥ n. If W is diagonal, then it follows from (4.1) that Φ(W )(p−1)n+q,(r−1)n+s = δps δqr (Wpp + Wqq ).

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This implies that rank Φ(W ) = |{(p, q) : Wpp + Wqq = 0}|. Let J = {(p, p) : Wpp = 0} ∪ {(p, q) : Wpp = 0, Wqq = 0}. Then (p, q) ∈ J implies that Wpp + Wqq = 0. If k(≥ 1) is the number of nonzero diagonal entries in W , then |J| = k(n − k + 1) ≥ n. Hence we also have rank Φ(W ) ≥ |J| ≥ n. The proof of the following lemma is an easy exercise of linear algebra, and left to the reader. m Lemma 4.2. Let α1 , . . . , αm , β1 , . . . , βm ∈ RN . Then rank ( i=1 αit βi ) ≤ m. m The equality holds iﬀ both {αi }m i=1 and {βi }i=1 are linearly independent. In this case, for u1 , . . . , um , v1 , . . . , vm ∈ RN , m i=1

uti vi

=

m

αit βi

⇐⇒

ui =

i=1

m

cji αj , vi =

j=1

m

dji βj

j=1

for some matrices C = (cij ), D = (dij ) ∈ Mm (C) with CDt = Im . For n = 2, the following lemma gives some very nice characterizations of rank Φ(W ) = 2, which is the key to completely solving Eq. (LC) in the case of m = 2. Lemma 4.3. Let 0 = W ∈ M2 (C). Then rank Φ(W ) = 2 ⇐⇒ W ∈ sl(2, C) ⇐⇒ Φ(W ) = R(W )t R(I) + R(I)t R(W ). Proof. From (4.1) we see that ⎛ 2W11 ⎜ W12 Φ(W ) = ⎜ ⎝ W21 0

W12 0 Tr W W12

W21 Tr W 0 W21

⎞ 0 W12 ⎟ ⎟. W21 ⎠ 2W22

Suppose that rank Φ(W ) = 2. Let vi be the i-th row of Φ(W ). If W ∈ sl(2, C), then v2 and v3 are linearly independent. Hence v1 − v4 = (2W11 , 0, 0, −2W22 ) must be a linear combination of v2 and v3 . This implies W11 = W22 = 0, a contradiction. So we have W ∈ sl(2, C). Conversely, if W ∈ sl(2, C), it is straightforward to verify that Φ(W ) = R(W )t R(I) + R(I)t R(W ), which, by Lemmas 4.1 and 4.2, is a matrix of rank 2.

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5. Main theorems 5.1. The general case The ﬁrst main theorem in this section provides some properties of solutions to the Levi-Civit´ a equation (LC). Theorem 5.1. Suppose that (w, f1 , . . . , fm , g1 , . . . , gm ) is a nondegenerate solution of Eq. (LC) on a compact group G. Then we have (i) supp w ˆ = supp{fˆ1 , . . . , fˆm } = supp{ˆ g1 , . . . , gˆm }; d ≤ m. In particular, supp w, ˆ supp fˆi and supp gˆi (i = (ii) π π∈supp w ˆ 1, ..., m) all 2are ﬁnite; (iii) π∈supp w ˆ dπ ≥ m. ˆ If π ∈ supp w, Proof. (i) Let π ∈ G. ˆ then from (3.2) and the injectivity of Φ, ˆ ˆ ˆ ⊆ supp{fˆ1 , . . . , fˆm }. we see that π ∈ supp{f1 , . . . , fm }, so supp w ˆ we have If π ∈ / supp w, ˆ then, by (3.2), for all 1 ≤ k ≤ d2π and ρ ∈ G, m

R(fˆi (π))k gˆi (ρ) = 0,

i=1

where R(fˆi (π))k is the k-th component of R(fˆi (π)). By the Fourier inversion m m ˆ formula, we get i=1 R(fi (π))k gi = 0. Since {gi }i=1 is linearly indepenˆ ˆ dent, we have R(fi (π))k = 0. Thus fi (π) = 0 for all 1 ≤ i ≤ m, and so ˆ = supp{fˆ1 , . . . , fˆm }. Similarly, π ∈ / supp{fˆ1 , . . . , fˆm }. This proves supp w we have supp w ˆ = supp{ˆ g1 , . . . , gˆm }. This proves (i). ˆ from (3.3) we have (ii) For any distinct π1 , . . . , πk ∈ supp w, m

(R(fˆi (π1 )) · · · R(fˆi (πk )))t (R(ˆ gi (π1 )) · · · R(ˆ gi (πk )))

i=1

ˆ k ))). = diag(Φ(w(π ˆ 1 )), . . . , Φ(w(π By Lemmas 4.1 and 4.2, the matrix on the right hand side of the above k equation has rank at least j=1 dπj , while the one on the left hand side has k rank at most m. Hence j=1 dπj ≤ m. Now (ii) follows from the arbitrariness of πj . The rest of (ii) is obvious. (iii) Let ˆ 1 ≤ i, j ≤ dπ }. H = span{πij : π ∈ supp w, By (i) and the Fourier inversion formula, we have {f1 , ..., fm } ⊆ H.

2 Then by Schur’s orthogonality relations, we have dim H = π∈supp w ˆ dπ m (which is ﬁnite from (ii)). Now the linear independence of {fi }i=1 yields m ≤ π∈supp wˆ d2π . It immediately follows from Theorem 5.1 that one can strengthen the above results in the two extremal cases, as shown below.

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Corollary 5.2. Suppose that (w, f1 , . . . , fm , g1 , . . . , gm ) is a nondegenerate solution of Eq. (LC) on a compact group G. Then we have (i) dπ = m for some π ∈ supp w ˆ =⇒ supp w ˆ = {π}; (ii) dπ = 1 for all π ∈ supp w ˆ ⇐⇒ | supp w| ˆ = m. Corollary 5.3. If | supp w| ˆ = m, then any nondegenerate solution of Eq. (LC) is given by Construction 1 in Section 2. More precisely, there are distinct representations π1 , ..., πm : G → U (1) and Q ∈ Mm,2m+1 (C) such that (w, f1 , . . . , fm , g1 , . . . , gm ) = (π1 , . . . , πm )Q,

(5.1)

where Q = (ε A B) for some ε = (ε1 , . . . , εm ) ∈ Rm , A, B ∈ Mm (C) with AB t = 2 diag(ε1 , . . . , εm ). t

Proof. By Theorem 5.1 (i) and Corollary 5.2 (ii), we have that dπ = 1 for all π ∈ supp w ˆ and that supp w ˆ = supp{fˆ1 , ..., fˆm } = supp{gˆ1 , ..., gˆm } = {π1 , ..., πm }. Let ˆ i ), A = (fˆj (πi )), B = (ˆ gj (πi )). εi = w(π Then the Fourier inversion formula implies that (5.1) holds true. From (3.3) we see that ε, A, B satisfy all required conditions. Remark 5.4. Although, by Theorem 5.1, we have nice estimates on supp w ˆ and the dimensions of representation spaces in supp w, ˆ it is far from completely solving Eq. (LC). It is still unsolved even for the extremal case (i) of Corollary 5.2, because here we are not able to obtain a result analogous to Lemma 4.3. However, we can completely solve Eq. (LC) for the cases of m ≤ 2. We are devoted to considering these cases in the rest of the paper. 5.2. The case m = 1 For this case, the general solution is of the form obtained from Construction 1 in Section 2. Corollary 5.5. Let (w, f, g) be a nondegenerate solution of the equation w(xy) + w(yx) = f (x)g(y)

(5.2)

on a compact group G. Then there are a representation π : G → U (1) and nonzero constants a, b ∈ C such that ab π, aπ, bπ . (5.3) (w, f, g) = 2 Proof. By Theorem 5.1 (i), (ii), and Corollary 5.2 (i), we see that supp w ˆ= ˆ ˆ supp f = supp gˆ = {π} for some π ∈ G with dπ = 1. Then the theorem follows from the Fourier inversion formula. Actually, the general solution (without any regularity) of Eq. (5.2) on any (not necessarily compact) group G is in fact of the form in Corollary 5.5. It is probably known in the literature. But we are lack of a reference; we give a short proof here for completeness.

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Lemma 5.6. Any nondegenerate solution (without any regularity) of Eq. (5.2) on an arbitrary group G is of the form (5.3) for some homomorphism π : G → C∗ and nonzero complex numbers a, b. Proof. Letting x = e (resp. y = e) in Eq. (5.2) gives w(y) = f (e) 2 g(y) (resp. w(x) = g(e) f (x)). Since w ≡ 0, we have f (e) = 0 and g(e) = 0. Thus 2 g(e) f (x) g(x) = f (e) f (x). Put π(x) = f (e) . It is now easy to check that π(xy) + π(yx) = 2π(x)π(y). From [9], π is indeed a homomorphism from G to C∗ .

5.3. The case m = 2 We are now in a position to obtain the general solution to Eq. (LC) with m = 2. It turns out that every solution has to be one of the forms constructed in Section 2. Theorem 5.7. Suppose that (w, f1 , f2 , g1 , g2 ) is a nondegenerate solution of the equation w(xy) + w(yx) = f1 (x)g1 (y) + f2 (x)g2 (y)

(5.4)

on a compact group G. Then either (i) there are two distinct homomorphisms π1 , π2 : G → U (1) and Q = (ε A B) ∈ M2,5 (C) with AB t = 2 diag(ε1 , ε2 ) such that (w, f1 , f2 , g1 , g2 ) = (π1 , π2 )Q, where ε = (ε1 , ε2 ) , and A, B ∈ M2 (C); or (ii) there exist an irreducible representation π : G → U (2), 0 = W 0 1 ∈ sl(2, C), C, D ∈ M2 (C) with CDt = such that 1 0 t

w(x) = Tr(W π(x)),

(f1 , f2 , g1 , g2 ) = (w χπ )(C D).

Proof. Applying Theorem 5.1 to the case of m = 2, we see that either supp w ˆ = supp{fˆ1 , fˆ2 } = supp{gˆ1 , gˆ2 } = {π1 , π2 } ˆ with dπ = dπ = 1; or for some distinct π1 , π2 ∈ G 1 2 supp w ˆ = supp{fˆ1 , fˆ2 } = supp{gˆ1 , gˆ2 } = {π} ˆ with dπ = 2. for some π ∈ G For the former case, the conclusion follows directly from Corollary 5.3. For the latter case, if we denote W = w(π), ˆ it follows from Corollary 3.3 that 2 R(fˆi (π))t R(ˆ gi (π)) = Φ(W ). i=1

By Lemmas 4.1 and 4.2, both sides of the above equation have rank 2. Then Lemma 4.3 implies 0 = W ∈ sl(2, C)

and

Φ(W ) = R(W )t R(I) + R(I)t R(W ).

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It now follows from the last part of Lemma 4.2 that there are C = (cij ), ˜ = (dij ) ∈ M2 (C) with C D ˜ t = I such that D R(fˆi (π)) = c1i R(W ) + c2i R(I),

R(gˆi (π)) = d1i R(I) + d2i R(W ).

Hence we have fˆi (π) = c1i W + c2i I, gˆi (π) = d1i I + d2i W 0 1 ˜ D. Then from the Fourier inversion foras R is injective. Let D = 1 0 mula, we see that the solution is as in case (ii) of the theorem. As a consequence of Theorem 5.7, we can easily solve Eq. (1.1), which plays an important role in [4]. However, in [4], Eq. (1.1) is only solved in the context where f is given to be a solution of the pre-d’Alembert equation. In Theorem 5.8 below, we derive the solution formulas for Eq. (1.1) without assuming that f satisﬁes the pre-d’Alembert equation, but in contrast to [4] we require that the group G is compact. In the following theorem, we discard the trivial solutions: w = 0, and f arbitrary. Theorem 5.8. Any nontrivial solution (w, f ) of the equation w(xy) + w(yx) = 2f (x)w(y) + 2w(x)f (y), on a compact group G has one of the following forms: (i) There are a representation π : G → U (1) and 0 = a ∈ C such that 1 w(x) = aπ(x) and f (x) = π(x). 2 (ii) There are two distinct representations π, π : G → U (1) and 0 = a ∈ C such that 1 w(x) = a(π(x) − π (x)) and f (x) = (π(x) + π (x)). 2 (iii) There are an irreducible representation π : G → U (2) and 0 = W ∈ sl(2, C) such that 1 w(x) = Tr(W π(x)) and f (x) = Tr(π(x)). 2 Proof. If w and f are linearly dependent, a straightforward calculation using Corollary 5.5 shows that (w, f ) is of the form in (i). Otherwise, (w,2f,w,w,2f ) is a nondegenerate solution of Eq. (5.4). Applying Theorem 5.7 to the case where f1 = g2 = 2f and f2 = g1 = w, we obtain (ii) and (iii). By Theorem 5.8, one can easily see that f satisﬁes the pre-d’Alembert equation deﬁned in [4] automatically. It should be mentioned that f in [4] is normalized, i.e., f (e) = 1. So only f ’s given in (ii) or (iii) of Theorem 5.8 have such a property. Acknowledgment We are very grateful to the anonymous referee for his/her careful reading and many valuable comments.

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References [1] J. Acz´el, Lectures on functional equations and their applications, Academic Press, New York-London, 1966. [2] J. Acz´el and J. Dhombres, Functional equations in several variables, Cambridge University Press, Cambridge, 1989. [3] J. An and D. Yang, Nonabelian harmonic analysis and functional equations on compact groups, preprint, 2008. [4] T. M. K. Davison, D’Alembert’s functional equation on topological monoids, preprint, 2008. [5] G. Folland, A course in abstract harmonic analysis, CRC Press, Boca Raton, FL, 1995. [6] E. Hewitt and K. A. Ross, Abstract harmonic analysis, Vol. II, Die Grundlehren der mathematischen Wissenschaften. Springer-Verlag, New York-Berlin, 1970. [7] Pl. Kannappan, Functional equations and inequalities with applications, Springer Monographs in Mathematics. Springer, New York, 2009. [8] E. V. Shulman, Group representations and stability of functional equations, J. London Math. Soc. 54 (1996), 111–120. [9] H. Stetkær, On multiplicative maps, Semigroup Forum 63 (2001), 466–468. [10] H. Stetkær, Functional equations on groups—recent results, presented in an invited talk at the 42nd International Symposium on Functional Equations, Opava, Czech Republic, 2004. [11] H. Stetkær, Functional equations and matrix-valued spherical functions, Aequationes Math. 69 (2005), 271–292. [12] L. Sz´ekelyhidi, Convolution type functional equations on topological abelian groups, World Scientiﬁc Publishing Co., Inc., Teaneck, NJ, 1991. Jinpeng An LMAM, School of Mathematical Sciences, Peking University, Beijing, 100871, P. R. China e-mail: [email protected] Dilian Yang Department of Mathematics & Statistics, University of Windsor, Windsor, ON, N9B 3P4, Canada e-mail: [email protected] Submitted: October 1, 2009. Revised: November 6, 2009.

Integr. Equ. Oper. Theory 66 (2010), 197–214 DOI 10.1007/s00020-010-1741-7 Published online February 5, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

Summability Properties for Multiplication Operators on Banach Function Spaces O. Delgado and E. A. S´anchez P´erez Abstract. Consider a couple of Banach function spaces X and Y over the same measure space and the space X Y of multiplication operators from X into Y . In this paper we develop the setting for characterizing certain summability properties satisﬁed by the elements of X Y . At this end, using the “generalized K¨ othe duality” for Banach function spaces, we introduce a new class of norms for spaces consisting of inﬁnite sums of products of the type xy with x ∈ X and y ∈ Y . Mathematics Subject Classiﬁcation (2010). Primary 46E30; Secondary 47B38. Keywords. Banach function spaces, K¨ othe dual and generalized dual spaces, product spaces, multiplication operators, summability properties.

1. Introduction Let (Ω, Σ, µ) be a ﬁxed σ-ﬁnite measure space and consider a couple of Banach function spaces X and Y related to µ. In this paper we introduce a technique based on topological products of Banach function spaces for analyzing the summability properties of the multiplication operators from X into Y . For the deﬁnition of such topologies we use the so called generalized duality for Banach function spaces, which was originally studied by Maligranda and Persson in [9]. The Y -dual space of X, denoted by X Y , is the space of measurable functions g deﬁning a multiplication operator (also denoted by g) from X into Y , that is, g, x = gx ∈ Y for all x ∈ X. This notion includes 1 the classical K¨othe dual (or associate) space X = X L . The ﬁrst author thanks the support by UPV (PAID-06-08 Ref. 3093), MEC (TSGD-08 and D.G.I. #MTM2006-13000-C03-01) (Spain) and FEDER. The second author thanks the support by UPV (PAID-06-08 Ref. 3093), MEC (D.G.I. #MTM2006-11690-C02-01) (Spain) and FEDER.

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Given 1 ≤ p < ∞ and Z another Banach function space related to µ, our goal is to characterize when a multiplication operator g ∈ X Y is what we call (p, Z)-summing, that is, when there exists K > 0 such that for every x1 , ..., xn ∈ X, n n 1/p 1/p gxi pY ≤ K sup f xi pZ . (1.1) i=1

f ∈BX Z

i=1

An operator satisfying this kind of inequality is interesting as it transforms sequences which are summable in a certain weak sense into strongly summable sequences. Some relevant well known geometric and topological properties involving vector norm inequalities for operators can be written as particular examples of this general class of inequalities when Z is chosen adequately. For instance, in the case when X is order continuous, the positive p-summing multiplication operators coincide with the (p, L1 )-summing ones, or in the case when X is order semi-continuous and p-convex with constant 1, the pconcave multiplication operators coincide with the (p, Lp )-summing ones (see Section 4). Inspired in part by the representation theory of operator ideals as dual spaces of topological tensor products (see for instance [5]), we show that the subspace of X Y of all (p, Z)-summing multiplication operators can be described as the K¨ othe dual of a product space with a particular normed topology given by a certain dp,Z -norm. Actually, there is an abuse of the notation as the “dp,Z -product space” of X and Y consists of inﬁnite sums of products of the type xy with x ∈ X and y ∈ Y . As a consequence of the above description, some factorization theorems for multiplication operators which play a central role in the theory of the Banach function spaces (Reisner and Maurey-Rosenthal’s theorems) provide suﬃcient conditions for (p, Z)summability type properties to hold. The paper is organized as follows. Section 2 contains the deﬁnitions and some results concerning product spaces which will be necessary for our work. In Section 3 we introduce the dp,Z -product spaces which will allow us to characterize in Section 4 the (p, Z)-summing multiplication operators. Moreover, in Section 4 we show conditions on X, Y and Z guaranteeing that every multiplication operator from X into Y is (p, Z)-summing. Examples in which these conditions hold are provided in Section 5 by using the already quoted factorization theorems.

2. Preliminaries and ﬁrst results Let (Ω, Σ, µ) be a ﬁxed σ-ﬁnite measure space and denote by L0 the space of all (a.e. classes of) real measurable functions deﬁned on Ω. A Banach function space is a Banach space X ⊂ L0 with norm · X , satisfying that if f ∈ L0 , g ∈ X and |f | ≤ |g| a.e. then f ∈ X and f X ≤ gX . Note that in this case, X is a Banach lattice for the pointwise a.e. order. A Banach function

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space is order continuous if every order bounded increasing sequence is norm convergent. A Banach function space X has the Fatou property if for every sequence (fn ) ⊂ X such that 0 ≤ fn ↑ f a.e. and supn fn X < ∞, it follows that f ∈ X and fn X ↑ f X . A Banach function space X is order semicontinuous if f, fn ∈ X with 0 ≤ fn ↑ f a.e. implies fn X ↑ f X . Of course, if a Banach function space X is order continuous or has the Fatou property, then X is order semi-continuous. For issues related to Banach function spaces, see [12, Ch. 15] considering the function norm ρ deﬁned as ρ(f ) = f X if f ∈ X and ρ(f ) = ∞ in other case. Given two Banach function spaces X and Y , the Y -dual space of X is deﬁned by X Y = {h ∈ L0 : hf ∈ Y for all f ∈ X}, that is, the space of functions in L0 deﬁning a continuous linear operator from X into Y . The continuity follows from the fact that every positive linear operator between Banach lattices is continuous, see [6, p. 2]. The space X Y is a Banach function space with norm hX Y = sup hf Y , f ∈BX

for h ∈ X Y ,

if and only if X is saturated, that is, there is no A ∈ Σ with µ(A) > 0 such that f χA = 0 a.e. for all f ∈ X, see [9, Proposition 2] and [2, p. 3]. The saturation property is equivalent to the following one: for all A ∈ Σ with µ(A) > 0 there exists B ∈ Σ such that B ⊂ A, µ(B) > 0 and χB ∈ X. This is also equivalent to X having a weak unit, i.e. a function g ∈ X such that g > 0 a.e. As we have already noted, the classical K¨othe dual space X coincide with 1 X L , the L1 -dual space of X. In this case, X is saturated whenever X is so. However, the generalized dual X Y of X may be non saturated even if X is saturated. For these and other comments about saturation involving the spaces X Y see [2]. Let us introduce now the product spaces which will be the basic setting for deﬁning the dp,Z -product spaces in Section 3. 0 Deﬁnition 2.1. The π–product space XπY is the space of functions z ∈ L such that |z| ≤ i≥1 |xi yi | a.e. for some sequences (xi ) ⊂ X and (yi ) ⊂ Y satisfying i≥1 xi X yi Y < ∞. For z ∈ XπY , consider π(z) = inf xi X yi Y , i≥1

where the inﬁmum is taken over all sequences (xi ) ⊂ X and (yi ) ⊂ Y such that |z| ≤ i≥1 |xi yi | a.e. and i≥1 xi X yi Y < ∞. The space XπY is clearly an ideal of L0 and π(v) ≤ π(z) whenever |v| ≤ |z| a.e. It can be routinely checked that π is a seminorm. However, there are cases in which π is not a norm. Example. Let [0, 1], B([0, 1]), m be the ﬁxed measure space, where B([0, 1]) is the σ–algebra of all Borel sets of [0, 1] and m is the Lebesgue measure on

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i 1 [0, 1], and consider the product space L πL2 . For the intervals Ani = [ i−1 n , n] n with i = 1, ..., n, we have that χ[0,1] ≤ i=1 χAni and

π(χ[0,1] ) ≤

n

χAni L1 χAni L2 =

i=1

n 1 1 12 1 =√ . n n n i=1

Then, taking limit as n → ∞ we have that π(χ[0,1] ) = 0, while χ[0,1] > 0. So, π is not a norm. Saturation conditions will be crucial for π to be a norm under which XπY is a Banach function space. We write “X →c Y ” (“X →i Y ”) if X is continuously contained in Y with xY ≤ c xX ( xY = xX ) for all x ∈ X. If X = Y with equal norms, we write X ≡ Y . Proposition 2.2. The following conditions are equivalent: (a) XπY is a saturated Banach function space. (b) X, Y and X Y are saturated. Moreover, if (a)-(b) holds, we have that

(i) XπY →1 (X Y ) , (ii) (XπY ) ≡ X Y ≡ Y X . Proof. (a) ⇒ (b) Let us see that X is saturated. If this is not the case, there exists A ∈ Σ with µ(A) > 0 such that xχA = 0 a.e. for all x ∈ X. Since XπY is saturated we can take B ∈ Σ such that B ⊂ A, µ(B) > 0 and χB ∈ XπY . Let (xi ) ⊂ X and (yi ) ⊂ Y be such that i≥1 xi X yi Y < ∞ and χB ≤ i≥1 |xi yi | a.e. Then, χB = χB · χA ≤ i≥1 |xi χA yi | = 0 a.e. and so µ(B) = 0, which is a contradiction. Similarly, Y is saturated. Then, the spaces Y and X Y are Banach function spaces. Let us prove that X Y ≡ (XπY ) and so we will have that X Y is saturated as (XπY ) is so. Let h ∈ X Y . Given z ∈ XπY and (xi ) ⊂ X, (yi ) ⊂ Y with i≥1 xi X yi Y < ∞ such that |z| ≤ i≥1 |xi yi | a.e., using the monotone convergence theorem, we have hxi Y yi Y ≤ hX Y xi X yi Y . |hxi yi |dµ ≤ |hz|dµ ≤

i≥1

i≥1

i≥1

Then, |hz|dµ ≤ hX Y · π(z). So, h ∈ (XπY ) and h(XπY ) ≤ hX Y . Consider now h ∈ (XπY ) . For every x ∈ X and y ∈ Y , we have that xy ∈ XπY and so hxy ∈ L1 (µ). Then, hx ∈ Y for every x ∈ X, that is, h ∈ X Y . Moreover, since π(xy) ≤ xX yY for x ∈ X and y ∈ Y , it follows hX Y = sup sup |hxy|dµ ≤ sup |hz|dµ = h(XπY ) . x∈BX y∈BY

z∈BXπY

(b) ⇒ (a). Note that from the hypothesis (X Y ) is a Banach function space. Let us see that XπY →1 (X Y ) . Given z ∈ XπY , (xi ) ⊂ X, (yi ) ⊂ Y

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such that i≥1 xi X yi Y < ∞ and |z| ≤ i≥1 |xi yi | a.e., for every h ∈ XY , hxi Y yi Y ≤ hX Y xi X yi Y |zh|dµ ≤ |hxi yi |dµ ≤

i≥1

i≥1

i≥1

and so |zh|dµ ≤ hX Y · π(z). Then, z ∈ (X Y ) and z(X Y ) ≤ π(z). Hence, π(z) = 0 implies z = 0 a.e. That is, π is a norm on XπY . Let (zn )n≥1 ⊂ XπY be such that zn ≥ 0 and n≥1 π(zn ) < ∞. Let us prove that n≥1 zn ∈ XπY (i.e. XπY has the Riesz-Fischer property) and so XπY will be complete, see [12, Ch.15, §64, Theorem 2]. Given ε > 0 there exist (xnj )j ⊂ X and (yjn )j ⊂ Y such that zn ≤ j≥1 |xnj yjn | a.e. and ε n n n n j≥1 xj X yj Y ≤ π(zn ) + 2n . So, n≥1 zn ≤ n≥1 j≥1 |xj yj | a.e. and xnj X yjn Y ≤ π(zn ) + ε < ∞. n≥1 j≥1

Note that z := we have that

n≥1 zn

n≥1

< ∞ a.e., since taking h ∈ X Y such that h > 0 a.e.,

|hz|dµ ≤ hX Y

xnj X yjn Y < ∞

n≥1 j≥1

which implies that |hz| < ∞ a.e. Then, z ∈ XπY and π(z) ≤ n≥1 π(zn ). Therefore, it follows that XπY is a Banach function space. Moreover, given A ∈ Σ with µ(A) > 0, since X is saturated, there exists B ∈ Σ such that B ⊂ A, µ(B) > 0 and χB ∈ X. Since Y is also saturated, there exists C ∈ Σ such that C ⊂ B, µ(C) > 0 and χC ∈ Y . Then, χC = χB · χC ∈ XπY and so XπY is saturated. Suppose (a)-(b) holds. Claim (i) has been proved in (b) ⇒ (a). The ﬁrst equivalence in (ii) has been obtained in (a) ⇒ (b). For the second equivalence, just note that XπY ≡ Y πX. The proof of the completeness of XπY in the previous proposition can also be obtained as a consequence of the fact that XπY is saturated. This can be found in [8], where these notions are developed in the general frame work of the function norms. The space deﬁned in a similar way as XπY by taking ﬁnite sums has been independently studied in [11] obtaining similar results, although for the completeness the pointwise product BX · BY of the unit balls of X and Y is required to be convex. Remark 2.3. Suppose (a)-(b) in Proposition 2.2 holds.Then, if (xi ) ⊂ X and (yi ) ⊂ Y with i≥1 xi X yi Y < ∞, we have that i≥1 |xi yi | ∈ XπY . In deed, taking h ∈ X Y such that h > 0 a.e., it follows that h i≥1 |xi yi |dµ < ∞ and so i≥1 |xi yi | < ∞ a.e. As a consequence, every z ∈ XπY is actually an inﬁnite sum of products of the type xy with x ∈ X and y ∈ Y . Indeed, if (xi ) ⊂ X and (yi ) ⊂ Y with i≥1 xi X yi Y < ∞ and |z| ≤ ˜i = vz χsupp (v) |xi | ∈ X (as i≥1 |xi yi | a.e., taking v = i≥1 |xi yi | and x

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∈ BL∞ ), we have that z = i≥1 x ˜i |yi | a.e. This fact may fail if X is not saturated. The series can be even divergent a.e. For instance, taking xni = χAni ∈ L1 and yin = n1 χAni ∈ L2 for i = 1, ..., n in Example 2, we have that n n 1 1 1 1 12 · xni L1 · yin L2 = = 3 < ∞, n n n n2 n≥1 i=1 n≥1 i=1 n≥1 n while n≥1 i=1 |xni yin | = n≥1 n1 = ∞ a.e. z v χsupp (v) Y

Example. Let us show some particular cases of π-product spaces. (i) XπL∞ ≡ X even if X is not saturated. This is direct from the deﬁnition of π-product space. (ii) If X is saturated, from a classical Lozanovskii’s result ([7, Theorem 6]) it follows that XπX ≡ L1 . (iii) Let 1 ≤ p < ∞. The p-power of a saturated Banach function space X is the Banach function space given by X p = {x ∈ L0 : |x|p ∈ X} 1/p with norm xX p = |x|p X for x ∈ X p , see [9, Proposition 1]. If 1 ≤ r, q < ∞ satisfy 1/r = 1/p + 1/q, from [9, Lemma 1], it follows that X p πX q ≡ X r . Moreover, if Y is another Banach function space and 0 < θ < 1 we obtain the Calder´ on-Lozanovskii interpolation space X θ Y 1−θ 1/θ as the π-product space X πY 1/(1−θ) (see [3] and [11, Section 2]). We end this section with a result which will be useful along the paper. Lemma 2.4. Assume that X, Y and Z are saturated Banach function spaces such that X Z and Z Y are saturated. Then, X Z πZ Y is a saturated Banach function space and satisﬁes X Z πZ Y →1 X Y . Y

Proof. Let us see that (X Z )(Z ) is saturated and so, by Proposition 2.2, we will have that X Z πZ Y is a saturated Banach function space. Take x ∈ X such that x > 0 a.e. and y ∈ Y such that y > 0 a.e. Then xy > 0 a.e. and for every f ∈ X Z and g ∈ Z Y , as Z Y →1 Z Y ≡ (Y )Z (see for instance [2, §2(3) and Lemma 3.1(a)] and Proposition 2.2(ii)), it follows |xy f g| dµ ≤ xf Z y gZ < ∞ Y

and so xy ∈ (X Z )(Z ) . Given z ∈X Z πZ Y , consider sequences (fi )⊂ X Z and (gi ) ⊂ Z Y satisfying that i≥1 fi X Z gi Z Y < ∞ and |z| ≤ i≥1 |fi gi | a.e. For every x ∈ X, we have that |zx| ≤ i≥1 |fi gi x| a.e. with fi gi xY ≤ fi xZ gi Z Y ≤ xX fi X Z gi Z Y i≥1

i≥1

i≥1

and so zx ∈ Y with zxY ≤ xX · π(z). Hence, z ∈ X Y and satisﬁes zX Y ≤ π(z).

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Note that the hypothesis of Lemma 2.4 are satisﬁed for instance if X is saturated and X ⊂ Z ⊂ Y since in this case L∞ is contained in both X Z and Z Y .

3. The dp,Z -product spaces Throughout this section, X, Y and Z will be saturated Banach function spaces such that X Z and Z Y are saturated. Then, by Lemma 2.4, we can consider the saturated Banach function space X Z πZ Y which is contained in Y Y X . In particular, X is saturated and so, by Proposition 2.2, we also can consider the saturated Banach function space XπY . Let 1 ≤ p ≤ ∞. For any Banach space E and (ei ) ⊂ E, we will denote (ei )E,p =

ei pE

1/p

i≥1

if p < ∞ and for the case p = ∞, (ei )E,∞ = sup ei E . i≥1

Deﬁnition 3.1. Thedp,Z -product space Xdp,Z Y is the space of functions h ∈ L0 such that |h| ≤ i≥1 |xi yi | a.e. for some (xi ) ⊂ X and (yi ) ⊂ Y satisfying (yi )Y,p · sup (f xi )Z,p < ∞,

(3.1)

f ∈BX Z

where 1 ≤ p ≤ ∞ is such that 1/p + 1/p = 1. For h ∈ Xdp,Z Y , we denote dp,Z (h) = inf (yi )Y,p · sup (f xi )Z,p , f ∈BX Z

where the inﬁmum is taken over all (xi ) ⊂ X, (yi ) ⊂ Y satisfying (3.1) such that |h| ≤ i≥1 |xi yi | a.e. Proposition 3.2. The space Xdp,Z Y is a Banach function space with norm dp,Z . Moreover,

Xdp,Z Y →1 (X Z πZ Y ) . Proof. Let h ∈ Xdp,Z Y and take (xi ) ⊂ X, (yi ) ⊂ Y satisfying (3.1) such that |h| ≤ i≥1 |xi yi | a.e. Consider a function ξ ∈ X Z πZ Y and (fj ) ⊂ X Z , (gj ) ⊂ Z Y ≡ Y Z with |ξ| ≤ j≥1 |fj gj | a.e. and j≥1 fj X Z gj Y Z < ∞.

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Then |hξ| dµ ≤

|xi yi fj gj | dµ ≤

j≥1 i≥1

≤

gj Y Z

j≥1

≤

xi fj Z yi gj Z

j≥1 i≥1

xi fj Z yi Y

i≥1

gj Y Z (fj xi )i Z,p (yi )Y,p

j≥1

≤

IEOT

fj X Z gj Y Z · (yi )Y,p · sup (f xi )Z,p . f ∈BX Z

j≥1

So, h ∈ (X Z πZ Y ) and h(X Z πZ Y ) ≤ dp,Z (h). In particular, dp,Z (h) = 0 implies h = 0 a.e. Note that if h ∈ Xdp,Z Y and (xi ) ⊂ X, (yi ) ⊂ Y are such that |h| ≤ (f xi )Z,p < ∞, then i≥1 |xi yi | a.e. and satisfy 0 < (yi )Y,p · supf ∈BX Z there exists (˜ xi ) ⊂ X and (˜ yi ) ⊂ Y such that |h| ≤ i≥1 |˜ xi y˜i | a.e., 1/p , (˜ yi )Y,p = (yi )Y,p · sup (f xi )Z,p f ∈BX Z

and 1/p sup (f x ˜i )Z,p = (yi )Y,p · sup (f xi )Z,p .

f ∈BX Z

f ∈BX Z

Indeed, the vectors 1/p

x ˜i = (yi )Y,p ·

−1/p

y˜i = (yi )Y,p ·

sup (f xi )Z,p

f ∈BX Z

−1/p

· xi ,

1/p sup (f xi )Z,p ) · yi

f ∈BX Z

work. Let (hn ) ∈ Xdp,Z Y such that n≥1 dp,Z (hn ) < ∞. Let us prove that h = n≥1 hn ∈ Xdp,Z Y with dp,Z (h) ≤ n≥1 dp,Z (hn ) and so we will have that dp,Z satisﬁes the triangular inequality and Xdp,Z Y has the RieszFischer property. Given ε > 0, we can take (xni )i ⊂ X, (yin )i ⊂ Y satisfying that hn ≤ i≥1 |xni yin | a.e. and (yin )i Y,p · sup (f xni )i Z,p ≤ dp,Z (hn ) + f ∈BX Z

ε . 2n

Note that h ∈ L0 , since taking ξ ∈ X Z πZ Y such that ξ > 0 a.e. and (fj )j ∈ X Z , (gj )j ∈ Z Y ≡ Y Z with j≥1 fj X Z gj Y Z < ∞ and ξ ≤ j≥1 |fj gj |

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˜= a.e., denoting h n≥1 |hn | we have that ˜ |xni yin fj gj | dµ ≤ xni fj Z yin gj Z h ξ dµ ≤ j≥1 n≥1 i≥1

≤

gj Y Z

j≥1

≤

gj Y Z

j≥1

≤

j≥1 n≥1 i≥1

xni fj Z yin Y

n≥1 i≥1

j≥1

≤

(yin )i Y,p · (fj xni )i Z,p

n≥1

(yin )i Y,p · sup (f xni )i Z,p gj Y Z fj X Z · f ∈BX Z

n≥1

gj Y Z fj X Z · ε + dp,Z (hn ) < ∞

j≥1

n≥1

n ˜ < ∞ a.e. We can assume that (y n )i Y,p ·sup and so h f ∈BX Z (f xi )i Z,p > 0 i as in other case dp,Z (hn ) = xni )i and (˜ yin )i as 0 and nsonhn = 0 a.e. Consider (˜ xi y˜i | a.e. and it can be checked that above. Then, |h| ≤ n≥1 i≥1 |˜ (yin )i Y,p · sup (f xni )i Z,p . (˜ yin )n,i Y,p · sup (f x ˜ni )n,i Z,p ≤ f ∈BX Z

n≥1

f ∈BX Z

Thus, h ∈ Xdp,Z Y and dp,Z (h) ≤ n≥1 dp,Z (hn ). The remaining conditions for Xdp,Z Y to be a Banach function space are clear and we have already shown in the beginning of the proof that Xdp,Z Y →1 (X Z πZ Y ) . The norm π of XπY can be described as follows. For every z ∈ XπY , π(z) = inf (xi )X,p · (yi )Y,p (3.2) where the inﬁmum is taken over all sequences (xi ) ⊂ X and (yi ) ⊂ Y such that |z| ≤ i≥1 |xi yi | a.e. and (xi )X,p · (yi )Y,p < ∞. The proof of this fact is a routine computation after noting that if (xi ) ⊂ X and (yi ) ⊂ Y are such that |z| ≤ i≥1 |xi yi | a.e., then for x ˜i = (xi X yi Y )1/p xxi i X ∈ X

and y˜i = (xi X yi Y )1/p yyi iY ∈ Y we have that x ˜i y˜i = xi yi and (˜ xi )X,p · X ∞ (˜ yi )Y,p = i≥1 xi X yi Y . Since X ≡ L (see [9, Theorem 1]) and sup (f xi )X,p = (xi )X,p ,

f ∈BL∞

from (3.2), it follows that XπY ≡ Xdp,X Y . Hence, the π-product spaces are particular cases of the dp,Z -product spaces. Proposition 3.3. The space Xdp,Z Y is saturated and satisﬁes XπY →1 Xdp,Z Y.

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Proof.Let h ∈ XπY . By (3.2) there exist (xi ) ⊂ X, (yi ) ⊂ Y such that |h| ≤ i≥1 |xi yi | a.e. and (xi )X,p · (yi )Y,p < ∞. Since sup (f xi )Z,p ≤ (xi )X,p ,

f ∈BX Z

we have that h ∈ Xdp,z Y and dp,Z (h) ≤ π(z). Then, XπY →1 Xdp,Z Y and in particular, Xdp,Z Y is saturated.

4. (p,Z)-summing multiplication operators Let us recall the deﬁnition given in (1.1). Given X, Y , Z saturated Banach function spaces, a multiplication operator g : X → Y is (p, Z)-summing if there exists a constant K > 0 such that for every x1 , ..., xn ∈ X, n

gxi pY

1/p

f ∈BX Z

i=1

if 1 ≤ p < ∞ and

n

≤ K sup

sup gxi Y ≤ K sup

1/p

i=1

f ∈BX Z

i=1,...,n

f xi pZ

sup f xi Z

i=1,...,n

if p = ∞.

Some relevant classes of multiplication operators between Banach function spaces can be obtained as particular (p, Z)-summing operators. Let us show some examples. (I) p-concave multiplication operators. Let 1 ≤ p < ∞. Recall that a linear operator T : E → F , from a Banach lattice E into a Banach space F , is p-concave if there exists C > 0 such that for every x1 , ..., xn ∈ E, n

T (xi )pF

1/p

n 1/p ≤ C |xi |p .

i=1

E

i=1

Every (p, Lp )-summing multiplication operator g : X → Y is p-concave. p p Indeed, noting that X →1 X L L (see for instance [2, §2(3)]), we have that there exists K > 0 such that for every x1 , ..., xn ∈ X, n

gxi pY

1/p

≤K

i=1

sup f ∈BX Lp

=K

sup f ∈BX Lp

n

f xi pLp

i=1

n 1/p |xi |p f

Lp

i=1

n 1/p =K |xi |p i=1 n

≤K

i=1

1/p

p Lp

XL

|xi |p

1/p . X

(4.1)

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Recall that a Banach lattice E is p-convex with constant K if for all x1 , ..., xn ∈ E, n n 1/p 1/p |xi |p xi pE . ≤K i=1

E

i=1

If the contrary inequality holds, then E is called p-concave. In the case when X is p-convex with constant 1 and order semi-continuous, the inequality (4.1) is an equality (see [2, Proposition 5.3(ii)]) and so the class of the p-concave multiplication operators from X into Y coincides with the class of the (p, Lp )-summing ones. (II) Positive p-summing multiplication operators. Let 1 ≤ p < ∞. An operator T : E → F , with E a Banach lattice and F a Banach space, is positive p-summing if there exists K > 0 such that for every x1 , ..., xn ∈ E, n n 1/p 1/p T (|xi |)p ≤ K sup |x∗ , |xi ||p , i=1

x∗ ∈BE ∗

i=1

∗

where E is the topological dual of E, see [1]. Every (p, L1 )-summing multiplication operator g : X → Y is positive p-summing. Indeed, noting that each f ∈ X can be identiﬁed with an element of the dual space X ∗ via f, x = f x dµ for all x ∈ X with f X = f X ∗ , we have that there exists K > 0 such that for every x1 , ..., xn ∈ X, n n 1/p 1/p p g|xi | Y = gxi pY i=1

i=1

≤ K

n

sup f ∈B

1 XL

= K sup ≤K

sup

x∗ ∈BX ∗

1/p

i=1

n

f ∈BX

f xi pL1

|f |, |xi | p

1/p

i=1

n

|x∗ , |xi ||p

1/p

.

(4.2)

i=1

In the case when X is order continuous, the K¨ othe dual X can be iden∗ tiﬁed with the whole space X (see for instance [6, p. 29]) and then the inequality (4.2) is just an equality, so the class of the positive p-summing multiplication operators from X into Y coincides with the class of the (p, L1 )-summing ones. The analogous result holds for p = ∞. From now and on X, Y and Z will be saturated Banach function spaces such that X Z and Z Y are saturated. In this case, by Lemma 2.4, we have that X Z πZ Y is a saturated Banach function space. Moreover, since Z Y is Y also saturated (as it contains Z ), we can consider the space Xdp,Z Y . Let us show now our main result which gives a characterization of the space of all (p, Z)-summing multiplication operators from X into Y , whenever Y is order semi-continuous. Note that, adopting the notation given in Section 3,

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a function g ∈ X Y is (p, Z)-summing if there exists a constant K > 0 such that for every x1 , ..., xn ∈ X, (gxi )Y,p ≤ K sup (f xi )Z,p . f ∈BX Z

In this case, the inequality also holds for inﬁnite sequences. Theorem 4.1. Assume that Y is order semi-continuous and let g : X → Y be a multiplication operator. Then, g is (p, Z)-summing if and only if g ∈ Xdp,Z Y . Proof. Suppose that g is (p, Z)-summing. Given h ∈ Xdp,Z Y and(xi ) ⊂ X, (yi ) ⊂ Y satisfying (3.1) for Y instead of Y and such that |h| ≤ i≥1 |xi yi | a.e., we have that |gh| dµ ≤

|gxi yi | dµ

i≥1

≤

gxi Y yi Y

i≥1

≤ (gxi )Y,p · (yi )Y ,p ≤ K (yi )Y ,p · sup (f xi )Z,p < ∞ f ∈BX Z

and so g ∈ (Xdp,Z Y ) . Let us prove the converse. Suppose that g ∈ (Xdp,Z Y ) and let x1 , ..., xn ∈ X. Suppose ﬁrst that 1 ≤ p < ∞. Given ε > 0, since yY = yY for all y ∈ Y as Y is order semi-continuous (see

for instance [2, p. 4,5]), there exists yi ∈ BY such that gxi Y ≤ εc + |gxi yi | dµ, where n p 1/p n p−1 c= . Then, denoting y˜i = gxi p−1 / Y yi , i=1 gxi Y i=1 gxi Y we have that n

gxi pY =

i=1

n i=1

≤

n

gxi p−1 Y gxi Y gxi Yp−1 εc + |gxi yi | dµ

i=1

=ε

n

gxi pY

i=1

≤ε

n i=1

gxi pY

1/p 1/p

+

|g|

n

|xi y˜i | dµ

i=1

+ g(Xdp,Z Y ) dp,Z

n i=1

|xi y˜i | .

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Noting that n n n 1/p 1/p dp,Z |xi y˜i | ≤ ˜ yi pY · sup f xi pZ i=1

≤

i=1 n

f ∈BX Z

(p−1)p gxi Y

1/p

i=1

· sup

n

f ∈BX Z

i=1

f xi pZ

1/p

,

i=1

we obtain

n n n 1/p 1/p p p p gxi Y ≤ gxi Y · ε + g(Xdp,Z Y ) sup f xi Z i=1

f ∈BX Z

i=1

i=1

and so n

gxi pY

1/p

n

≤ ε + g(Xdp,Z Y ) sup

f ∈BX Z

i=1

f xi pZ

1/p

.

i=1

Since ε is arbitrary, g is (p, Z)-summing. For the case p = ∞, given ε > 0, there exists yi ∈ BY such that gxi Y ≤ ε + |gxi yi | dµ, and then sup gxi Y ≤ ε + sup |gxi yi | dµ i=1,...,n

i=1,...,n

≤ ε + g(Xd∞,Z Y ) sup d∞,Z (xi yi ) i=1,...n

≤ ε + g(Xd∞,Z Y ) sup

i=1,...n

≤ ε + g(Xd∞,Z Y ) sup

f ∈BX Z

yi Y · sup f xi Z

f ∈BX Z

sup f xi Z ,

i=1,...n

so g is (∞, Z)-summing.

Note that under conditions of Theorem 4.1, from the proof it follows that g(Xdp,Z Y ) is the smallest constant K satisfying the inequality of the deﬁnition of (p, Z)-summing. By Proposition 3.2 and Proposition 3.3, XπY →1 Xdp,Z Y →1 X Z πZ Y . Then, X Z πZ Y →1 Xdp,Z Y →1 XπY (see for instance [2, and Lemma 3.1(b)]) and since X Z πZ Y →1 X Z πZ Y →1 X Z πZ Y (XπY ) ≡ X Y (see Proposition 2.2(ii)), we have that X Z πZ Y →1 Xdp,Z Y →1 X Y . (4.3) On the other hand, from Lemma 2.4, it follows that

X Z πZ Y →1 X Y →1 X Y . Rewriting Theorem 4.1, whenever Y is order semi-continuous, we get g ∈ X Y : g is (p, Z)-summing = X Y ∩ Xdp,Z Y .

(4.4)

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In the case when Y has the Fatou property (i.e. Y ≡ Y ), from (4.3), it follows that the space of (p, Z)-summing multiplication operators from X into Y is just Xdp,Z Y , in particular, it has the Fatou property as it coincides with the K¨ othe dual of a Banach function space. From (4.3), (4.4) and Theorem 4.1, we obtain that under the assumption of order semi-continuity for Y , if g ∈ X Z πZ Y then g is a (p, Z)-summing multiplication operator from X into Y . A direct computation proves that this holds also without any assumption on Y . In particular, if a multiplication operator g : X → Y factorizes through Z via two multiplication operators (i.e. g = f h for some f ∈ X Z and h ∈ Z Y ), then g is (p, Z)-summing. Let us show a useful consequence of (4.3), (4.4) and Theorem 4.1. Corollary 4.2. Let Y a Banach function space with the Fatou property and suppose that X Z πZ Y = X Y . Then g ∈ X Y : g is (p, Z)-summing = (Xdp,Z Y ) = X Y . The Fatou property for Y is necessary in the result above. An easy counterexample can be given if it is not satisﬁed. Take X = ∞ , Y = c0 and Z = ∞ . Then X Y = c0 and X Z πZ Y = ∞ πc0 = c0 = X Y . However, 1 = ∞ π 1 = ∞ dp,∞ 1 (see the comments before Proposition 3.3), and then (Xdp,Z Y ) = ( ∞ dp,∞ 1 ) = ∞ . Corollary 4.2 provides conditions guaranteeing that the space consisting of all (p, Z)-summing multiplication operators from X into Y coincides with the whole space X Y . This is not a general fact, as the following example shows. Example. Consider 1 < p < q < r < ∞ and the spaces X = p , Y = q and Z = r . Note that, since X Z ≡ X Y ≡ ∞ and Z Y ≡ s for 1/s = 1/q−1/r (see [9, Theorem 2 and Proposition 3]), in this case X Z πZ Y ≡ ∞ π s ≡ s X Y . The space of (t, r )-summing multiplication operators from p into q is just the space of sequences g ∈ ∞ satisfying (gxi )q ,t ≤ K sup (f xi )r ,t = K (xi )r ,t f ∈B∞

for some constant K > 0 and for every x1 , ..., xn ∈ p . Clearly, there exist elements of ∞ which do not satisfy the above inequality (e.g. g = (1, 1, 1, ...)). Note that, since X Z πZ Y ≡ s , every g ∈ s is (t, r )-summing for all 1 ≤ t ≤ ∞. Other conditions diﬀerent from those in Corollary 4.2 under which every multiplication operators from X into Y is (p, Z)-summing are presented in the following result. Proposition 4.3. Assume Y is p-concave, Z is p-convex and X →i X ZZ . Then, every g ∈ X Y is (p, Z)-summing.

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Proof. Let g ∈ X Y . Given x1 , ..., xn ∈ X, we have that n n 1/p 1/p gxi pY ≤ K1 |gxi |p i=1

Y

i=1

n 1/p ≤ K1 gX Y |xi |p i=1

= K1 gX Y

X

n

sup

f ∈BX Z

≤ K1 K2 gX Y

|f xi |p

1/p

i=1

sup f ∈BX Z

n

f xi pZ

Z

1/p

,

i=1

where K1 is the p-concavity constant of Y , K2 is the p-convexity constant of Z and for the equality we have used that xX = xX ZZ for all x ∈ X. Actually in Proposition 4.3 X and X Z being saturated are enough instead of the saturation conditions required before Theorem 4.1. Conditions under which X →i X ZZ are studied in [2]. Finally, note that every multiplication operator from X into Y is (p, X)-summing and (p, Y )-summing.

5. Applications Let us ﬁnish the paper by applying several important factorization theorems for multiplication operators to the results obtained for the dp,Z -product spaces. Summability properties of these operators are obtained in a straightforward way from Corollary 4.2. 5.1. Reisner’s theorem 1 ∞ Let Z be a Banach function space satisfying that L∞ F ⊂ Z ⊂ Lloc , where LF ∞ denotes the space of functions in L with support having ﬁnite measure and L1loc denotes the space of locally integrable functions. Given 1 ≤ p < q ≤ ∞, consider r deﬁned by 1/r = 1/p − 1/q. If Z is p-convex with constant K1 and p q-concave with constant K2 , then for every ε > 0 and g ∈ (Lq )L ≡ Lr (see [9, q p Proposition 3]), the multiplication operator g : L → L has a factorization as Lq HH j H f H

g

Z

- Lp * h

p

(i.e. g = f h) where f ∈ (Lq )Z and h ∈ Z L are such that f (Lq )Z · hZ Lp ≤ (1 + ε)K1 K2 g(Lq )Lp p

(see [10, Theorem 1]) and so g ∈ (Lq )Z πZ L with π(g) ≤ K1 K2 g(Lq )Lp . p p p Hence, (Lq )L →K1 K2 (Lq )Z πZ L . Note that Z L and (Lq )Z are saturated, see the comments in [2] after Proposition 5.3 and Theorem 5.4. Then, by

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p

p

p

p

Lemma 2.4, we have (Lq )Z πZ L →1 (Lq )L . Hence, (Lq )Z πZ L = (Lq )L (with equal norms if K1 K2 ≤ 1). Therefore, from Corollary 4.2, we obtain the following result. Proposition 5.1. Let 1 ≤ p < q ≤ ∞ and r such that 1/r = 1/p − 1/q. If Z is 1 a p-convex and q-concave Banach function space such that L∞ F ⊂ Z ⊂ Lloc , q then the space of (s, Z)-summing multiplication operators from L into Lp (for any 1 ≤ s ≤ ∞) coincides with the whole space Lr , which also coincides with (Lq ds,Z Lp ) . p

Note that under conditions of Proposition 5.1, the norm π of (Lq )Y πY L p is equivalent to π ˜ , deﬁned on g ∈ (Lq )Y πY L as p π ˜ (g) = inf f (Lq )Y hY Lp : g = f h with f ∈ (Lq )Y and h ∈ Y L . (5.1) The norms π and π ˜ coincide whenever K1 K2 ≤ 1. 5.2. Maurey-Rosenthal’s theorem Let 1 ≤ p < ∞. Recall that a linear operator T : E → F , from a Banach space E into a Banach lattice F , is p-convex if there exists C > 0 such that for every x1 , ..., xn ∈ E, n n 1/p 1/p |T (xi )|p xi pE . ≤C F

i=1

i=1

Consider a saturated Banach function space Y being order semi-continuous and p-concave with constant K1 . If T : E → Y is p-convex with constant K2 , then there exists a function 0 ≤ h ∈ (Lp )Y and an operator R : E → Lp such that T factorizes as E

T

H R

H H j H

Lp

- Y * h

and h(Lp )Y RE→Lp ≤ K1 K2 (see [4, Corollary 2]). If a saturated Banach function space X is p-convex with constant K, it is direct to check that every multiplication operator g : X → Y is p-convex with constant KgX Y . Then, there exists 0 ≤ h ∈ (Lp )Y and an operator R : X → Lp such that gx = hR(x) for all x ∈ X and h(Lp )Y RX→Lp ≤ K1 KgX Y . Taking f := hg χ[h>0] , p we have that f ∈ X L (as |f x| ≤ |R(x)| for all x ∈ X) and g = f h (as gx = hR(x) with 0 < x ∈ X implies g(ω) = 0 whenever h(ω) = 0). Then, p p g ∈ X L π(Lp )Y with π(g) ≤ K1 KgX Y . Hence, X Y →K1 K X L π(Lp )Y . Suppose that (Lp )Y is saturated. For instance, this is the case when L∞ F ⊂ p Y ⊂ L1loc , as Y is p-concave. Since X L is saturated (as X is p-convex), p p by Lemma 2.4 we have that X L π(Lp )Y →1 X Y . So, X L π(Lp )Y = X Y (with equal norms if K1 K ≤ 1). Therefore, from Corollary 4.2, we obtain the following result.

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Proposition 5.2. Let 1 ≤ p < ∞. Given two saturated Banach function spaces X and Y such that X is p-convex and Y is p-concave, has the Fatou property and satisﬁes that (Lp )Y is saturated, then the space of (s, Lp )-summing multiplication operators from X into Y (for any 1 ≤ s ≤ ∞) coincides with the whole space X Y , which also coincides with (Xds,Lp Y ) . p

Note that under conditions of Proposition 5.2, the norm π of X L π(Lp )Y p is equivalent to π ˜ deﬁned on X L π(Lp )Y in a similar way as (5.1). If K1 K ≤ 1, the norms π and π ˜ coincide. Acknowledgment The authors thank the referee for providing the reference to the paper by Luxemburg and Zaanen quoted after the proof of Proposition 2.2.

References [1] O. Blasco, Positive p-summing operators on Lp -spaces, Proc. Amer. Math. Soc. 100 (1987), 275–280. [2] J. M. Calabuig, O. Delgado and E. A. S´ anchez P´erez, Generalized perfect spaces, Indag. Math. 19 (2008), 359–378. [3] A. P. Calder´ on, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 113–190. [4] A. Defant, Variants of the Maurey-Rosenthal Theorem for quasi-K¨ othe function spaces, Positivity 5 (2001), 153–175. [5] A. Defant and K. Floret, Tensor norms and operator ideals, North Holland Math. Studies, Amsterdam, 1993. [6] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Springer-Verlag, Berlin, 1979. [7] G. Ya. Lozanovskii, On some Banach lattices, Siberian Math. J. 10 (1969), 419–430. [8] W. A. J. Luxemburg and A. C. Zaanen, Notes on Banach function spaces, Note III, Nederl. Akad. Wet., Proc., Ser. A 66 (1963), 239-250. [9] L. Maligranda and L. E. Persson, Generalized duality of some Banach function spaces, Indag. Math. 51 (1989), 323–338. [10] S. Reisner, A factorization theorem in Banach lattices and its application to Lorentz spaces, Annales de l´institut Fourier 31 (1981), 239–255. [11] A. R. Schep, Products and factors of Banach function spaces, preprint. [12] A. C. Zaanen, Integration, 2nd rev. ed. North Holland, Amsterdam; Interscience, New York, 1967. O. Delgado and E. A. S´ anchez P´erez Instituto Universitario de Matem´ atica Pura y Aplicada Universidad Polit´ecnica de Valencia 46071 Valencia Spain e-mail: [email protected] [email protected]

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Submitted: February 24, 2009. Revised: October 14, 2009.

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Integr. Equ. Oper. Theory 66 (2010), 215–229 DOI 10.1007/s00020-010-1747-1 Published online February 5, 2010 © The Author(s). This article is published with open access at Springerlink.com 2010

Integral Equations and Operator Theory

The Non-symmetric Discrete Algebraic Riccati Equation and Canonical Factorization of Rational Matrix Functions on the Unit Circle A.E. Frazho, M.A. Kaashoek and A.C.M. Ran Abstract. Canonical factorization of a rational matrix function on the unit circle is described explicitly in terms of a stabilizing solution of a discrete algebraic Riccati equation using a special state space representation of the symbol. The corresponding Riccati diﬀerence equation is also discussed. Mathematics Subject Classiﬁcation (2010). Primary 47A68, 15A24; Secondary 47B35, 42A58, 39A99. Keywords. Toeplitz operators, rational matrix functions, canonical factorization, discrete algebraic Riccati equation, Riccati diﬀerence equation, Schur complements, ﬁnite section method.

1. Introduction Throughout R is a rational m × m matrix function with no poles on the unit ∞ j circle T, and we write R(z) = j=−∞ z Rj for the Laurent expansion of R on T. The corresponding (block) Toeplitz operator on 2+ (Cm ) is denoted by T , that is, ⎡ ⎤ R0 R−1 R−2 · · · ⎢R1 R0 R−1 · · ·⎥ ⎢ ⎥ (1.1) T = ⎢R2 R1 on 2+ (Cm ). R0 · · ·⎥ ⎣ ⎦ .. .. .. .. . . . . Recall (see, e.g., [9], Section XXIV.3) that R is said to admit a right canonical factorization (with respect to T) if R can be factored as R(z) = Ψ(z)Θ(z),

z ∈ T,

(1.2)

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where Θ and Ψ are regular m × m matrix rational functions such that Θ(z) and Θ(z)−1 have no poles in {z ∈ C : |z| ≤ 1}, while Ψ(z) and Ψ(z)−1 have no poles in {z ∈ C : |z| ≥ 1} inﬁnity included. It is well-known that R admits such a factorization if and only if the Toeplitz operator T is invertible. Moreover, in that case, T −1 = TΘ−1 TΨ−1 , where TΘ−1 and TΨ−1 are the Toeplitz operators deﬁned by Θ(z)−1 and Ψ(z)−1 , respectively. In this paper we analyze canonical factorization of R using a special state space representation, namely R(z) = R0 + zC(I − zA)−1 B + γ (zI − α)

−1

β.

(1.3)

Here A and α are square matrices of sizes n × n and ν × ν, say, which are assumed to be stable, that is, the eigenvalues of A and α are contained in the open unit disc. The I’s in (1.3) stand for identity matrices of appropriate sizes, and B, C, β, γ in (1.3) are matrices, again of appropriate sizes. We shall refer to (1.3) as a stable representation of R. With the representation (1.3) we associate the algebraic Riccati equation Q = αQA + (β − αQB)(R0 − γQB)−1 (C − γQA).

(1.4)

We say that Q is a stabilizing solution to this Riccati equation if the matrix R0 − γQB is invertible, Q is a solution to (1.4), and the matrices A◦ = A − B(R0 − γQB)−1 (C − γQA) −1

α◦ = α − (β − αQB)(R0 − γQB)

γ

(1.5) (1.6)

are both stable. We are now ready to state the main result of this note. Theorem 1.1. Let R be an m × m rational matrix function with no poles on the circle, and let (1.3) be a stable representation for R. Then R admits a right canonical factorization with respect to the unit circle if and only if the algebraic Riccati equation (1.4) has a stabilizing solution Q, and in that case a canonical factorization R(z) = Ψ(z)Θ(z) is obtained by taking Θ(z) = D + zC◦ (I − zA)−1 B,

Ψ(z) = δ + γ(zI − α)−1 β◦ .

(1.7)

Here

(1.8) C◦ = δ −1 (C − γQA), β◦ = (β − αQB)D−1 , and δ and D are any invertible matrices satisfying δD = R0 −γQB. Moreover, the inverses of the factors are given by Θ(z)−1 = D−1 − zD−1 C◦ (I − zA◦ )−1 BD−1 Ψ(z)−1 = δ −1 − δ −1 γ(zI − α◦ )−1 β◦ δ −1 ,

(1.9) (1.10)

where A◦ and α◦ are deﬁned by (1.5) and (1.6), respectively. Finally, if (1.4) has a stabilizing solution, then this solution is unique and given by ⎡ ⎤ C ⎢ CA ⎥ ⎢ ⎥ Q = β αβ α2 β · · · T −1 ⎢CA2 ⎥ , (1.11) ⎣ ⎦ .. .

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where T is the block Toeplitz operator (1.1) deﬁned by R. In Section 2 below we prove Theorem 1.1 and specify this theorem for the case when the Toeplitz operator T is tri-diagonal and for the case when R has no poles on the closed unit disc. Theorem 1.1 is of particular interest for the case when the values of R on the unit circle are hermitian matrices. In that case, one takes α = A∗ , β = C ∗ , and γ = B ∗ . The corresponding Riccati equation is then given by Q = A∗ QA + (C ∗ − A∗ QB)(R0 − B ∗ QB)−1 (C − B ∗ QA),

(1.12)

and the rational matrix function R(z) plays the role of the so-called Popov function or spectral density. This symmetric algebraic Riccati equation originates from stochastic realization theory (see the books [3], [5], [6], [12]), and for this symmetric case Theorem 1.1 is basically known. See, for instance, [15] where R(z) is positive deﬁnite on the unit circle and the resulting canonical factorization is a spectral factorization (cf., Appendix A1 in [7]). When R(z) is just hermitian on the unit circle symmetric canonical factorization is usually referred to as a J-spectral factorization, and for this type of factorization the relation with the stabilizing solution of (1.12) is covered by Lemma 12.4.1 (iv) in [11]. The existence of a (unique) stabilizing solution of the symmetric algebraic Riccati equation and its connection with the invertibility of the Toeplitz operator can be found in Section 4.7 of [13]. For further references on the symmetric algebraic Riccati equation and a more detailed description of the history of the subject, see the books [3], [11], [13], and [14]. For the non-symmetric case Theorem 1.1 seems to be new. Our proof of Theorem 1.1, which is self-contained, is based on a Schur complement argument as in [6]. This proof and formula (1.11) for the stabilizing solution also may be of interest for the symmetric case. We see Theorem 1.1 as an addition to [10], where canonical factorization of R and invertibility of T are described explicitly in terms of a diﬀerent state space representation, namely R(z) = I + C(zG − A)−1 B, where zG − A is a square matrix pencil which is invertible for z on the unit circle. In [10] canonical factorization is obtained by matching of spectral subspaces of the pencils zG − A and zG − (A − BC) (see also Chapter 2 of [1] for the special case when G = I). In Section 3 below we reconsider the example discussed in Section 10 of [10] (cf., Section XXIV.10 in [9]), and we use this example to illustrate Theorem 1.1. Solving (1.4) by iteration leads to the Riccati diﬀerence equation QN +1 = αQN A + (β − αQN B)(R0 − γQN B)−1 (C − γQN A). As one may expect from formula (1.11) for the stabilizing solution, solving this equation is closely related to inverting Toeplitz operators by the ﬁnite section method. This connection is the main topic of Section 4, the ﬁnal section of the paper. Other solutions of (1.4), not just the stabilizing ones, are also of interest. For instance, if Q is an arbitrary solution of (1.4), then the rational matrix functions Θ and Ψ deﬁned by (1.7) and (1.8) are analytic on the closed unit

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disc and R(z) = Ψ(z)Θ(z). Moreover, this factorization is a so-called pseudocanonical factorization (see [16] or Section 9.2 in [2]) of R if, in addition, the matrices A◦ and α◦ deﬁned by (1.5) and (1.6) have eigenvalues only in the closed unit disc. Note that in this case (1.11) does not hold, and to prove that conversely any pseudo-canonical factorization of R is obtained in this way one has to require additional minimality conditions on the representation (1.3). We will come back to this in a future publication. Finally, we mention that canonical factorization of rational matrix functions with respect to the real line or the imaginary axis and its connection with continuous time algebraic Riccati equations is well understood; see, e.g., Chapter 5 in [2], and Chapter 19 in [14], and the references in these books.

2. Proof of Theorem 1.1 Let T be the (block) Toeplitz operator on 2+ (Cm ) deﬁned in (1.1). The Toeplitz structure of T allows us to partition T as a 2 × 2 operator matrix

R0 Γ Cm T = (2.1) on 2 m . + (C ) Ξ T Here Γ is the row operator and Ξ is the column operator deﬁned by ⎡ ⎤ R1 2 m ⎢R2 ⎥ m Γ = R−1 R−2 · · · : + (C ) → C , Ξ = ⎣ ⎦ : Cm → 2+ (Cm ). .. . If T is an invertible operator on 2+ (Cm ), then (see, e.g., pages 28, 29 in [2]) the Schur complement ∆ = R0 − ΓT −1 Ξ is a well-deﬁned invertible operator on Cm . Moreover, the inverse of T admits the block matrix representation

∆−1 −∆−1 ΓT −1 Cm −1 on 2 m . (2.2) T = + (C ) −T −1Ξ∆−1 T −1 + T −1 Ξ∆−1 ΓT −1 This yields the following useful result. Lemma 2.1. Assume that T is an invertible Toeplitz operator on 2+ (Cm ), and let ∆ = R0 − ΓT −1 Ξ be the corresponding Schur complement. Then the following identities hold: T −1 = ST −1 S ∗ + (E − ST −1 Ξ)(R0 − ΓT −1 Ξ)−1 (E ∗ − ΓT −1 S ∗ ), (2.3) S ∗ T −1 = T −1 S ∗ − T −1 Ξ∆−1 (E ∗ − ΓT −1 S ∗ ), T

−1

S = ST

−1

− (E − ST

−1

−1

Ξ)∆

ΓT

−1

.

(2.4) (2.5)

Here E denotes the canonical embedding of Cm onto the ﬁrst coordinate space of 2+ (Cν ), and S is the block forward shift on 2+ (Cm ). Note that the identity in (2.3) can be viewed as an algebraic Riccati equation with T −1 as the solution. We shall see below (in Part 1 of the proof of Theorem 1.1) that equation (1.4) follows from (2.3) in a straightforward way.

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Proof. A simple computation shows that

0 0 −1 ∗ ST S = . 0 T −1 This identity, together with the identity (2.2), yields T

−1

− ST

−1

∆−1 −∆−1 ΓT −1 S = −1 −1 −1 −T Ξ∆ T Ξ∆−1 ΓT −1

I = ∆−1 I −ΓT −1 . −T −1Ξ ∗

Next observe that

I 0 I = = E − ST −1 Ξ, − 0 T −1 Ξ −T −1 Ξ I −ΓT −1 = I 0 − 0 ΓT −1 = E ∗ − ΓT −1 S ∗ .

(2.6)

(2.7) (2.8)

Using the latter identities in (2.6), we obtain T −1 = ST −1 S ∗ + (E − ST −1 Ξ)∆−1 (E ∗ − ΓT −1 S ∗ ).

(2.9)

This is precisely equation (2.5). Multiplying (2.3) by S ∗ on the left yields (2.4). Likewise multiplying (2.3) by S on the right gives (2.5). Proof of Theorem 1.1. The proof is broken up into four parts. In the ﬁrst two parts we assume that R admits a right canonical factorization, or equivalently, that T is invertible, and we show that the matrix Q deﬁned by (1.11) is a stabilizing solution to the algebraic Riccati equation (1.4). In the third part we start from a stabilizing solution to (1.4) and derive the desired canonical factorization. The ﬁnal part deals with uniqueness statement. Part 1. Assume that T is invertible, and let Q be the matrix deﬁned by (1.11). In this part we show that Q is a solution to (1.4). Notice that Q can be written Q = ωT −1W , where W is the observability and ω is the controllability operator deﬁned by ⎡ ⎤ C ⎢ CA ⎥ ⎢ ⎥ W = ⎢CA2 ⎥ : Cn → 2+ (Cm ), ⎣ ⎦ .. . ω = β αβ α2 β · · · 2+ (Cm ) → Cν . By comparing (1.1) with (2.1), and using the representation (1.3), we obtain Ξ = W B, Γ = γ,

C = E∗W β = ωE

and S ∗ W = W A,

and ωS = α.

(2.10) (2.11)

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Using the ﬁrst identities in (2.10) and (2.11) together with Q = ωT −1W , we see that ∆ = R0 − ΓT −1 Ξ = R0 − γωT −1W B = R0 − γQB.

(2.12)

Furthermore, the identities in (2.10) and (2.11) yield (E ∗ − ΓT −1 S ∗ )W = C − γωT −1 W A = C − γQA, ω(E − ST

−1

Ξ) = β − αωT

−1

W B = β − αQB.

(2.13) (2.14)

Now multiplying equation (2.3) on the left by ω and on the right by W , we obtain that Q = ωT −1 W is a solution to the algebraic Riccati equation (1.4); here we used (2.12), (2.13) and (2.14). Part 2. As in the previous part Q = ωT −1W . In this part we show that for this choice of Q the matrices A◦ and α◦ deﬁned by (1.5) and (1.6) are stable matrices. In fact, we shall only prove that A◦ is stable. The proof of the stability of α◦ can be obtained in the same way using a duality argument. First we note that S ∗ T −1 W = T −1 W A◦ . This identity follows from (2.4) together with (2.10) and (2.13). Indeed, we have S ∗ T −1 W = T −1 S ∗ − Ξ∆−1 (E ∗ − ΓT −1 S ∗ ) W = T −1 W A − W B∆−1 (C − γQA = T −1 W A◦ . Next, we decompose Cn as Cn = X1 ⊕ X2 , where X2 = Ker W and X1 = C Ker W . Notice that X2 is an invariant subspace for A, and C|X2 = 0. We claim that QA|X2 = 0. This follows from the fact that n

QAX2 ⊆ QX2 = ωT −1 W X2 = {0}. By using C|X2 = 0 and QA|X2 = 0 in (1.5), we see that A◦ |X2 = A|X2 and X2 is also an invariant subspace for A◦ . In other words, A◦ admits a matrix representation of the form

A11 0 X1 A◦ = on (2.15) A21 A22 X2 where A22 = A|X2 on X2 . Since X2 is an invariant subspace for A and A is stable, A22 is also stable. Let E1 be the natural embedding of X1 into Cn = X1 ⊕ X2 . Let W1 be the one to one operator deﬁned by W1 = W E1 mapping X1 into 2+ (Cm ). Using S ∗ T −1 W = T −1 W A◦ with A11 = E1∗ A◦ E1 , we obtain S ∗ T −1 W1 = S ∗ T −1 W E1 = T −1 W A◦ E1 = T −1 W1 E1 A◦ E1 = T −1 W1 A11 . In other words, S ∗ T −1 W1 = T −1 W1 A11 . Because W1 is one to one, T −1 W1 is also one to one. Notice that S ∗n T −1 W1 = T −1 W1 An11 for all integers n ≥ 0. Since S ∗n converges to zero in the strong operator topology and A11 acts on a ﬁnite dimensional space, An11 converges to zero. Therefore A11 is stable. Recall that A22 = A|X2 is stable. By consulting the matrix representation for A◦ in (2.15), we see that A◦ is stable.

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Part 3. In this part Q is a stabilizing solution of (1.4), and we derive the desired canonical factorization. Let Θ(z) and Ψ(z) be the rational m × m matrix functions deﬁned by (1.7) and (1.8). First we prove that R(z) = Ψ(z)Θ(z). Note that C − γQA = δC◦ ,

β − αQB = β◦ D,

R0 − γQB = δD.

(2.16)

Using these identities we see that the Riccati equation (1.4) can be rewritten as a Stein equation (a discrete Lyapunov equation), namely Q = αQA + β◦ C◦ .

(2.17)

From the identity (2.17) we see that zβ◦ C◦ = z(Q − αQA) = (zI − α)Q − (zI − α)Q(I − zA) + zQ(I − zA). It follows that γ(zI − α)−1 (zβ◦ C◦ )(I − zA)−1 B = γQ(I − zA)−1 B − γQB + zγ(zI − α)−1 QB = zγQA(I − zA)−1 B + γQB + γ(zI − α)−1 αQB, and hence

Ψ(z)Θ(z) = δ + γ(zI − α)−1 β◦ D + zC◦ (I − zA)−1 B = δD + γ(zI − α)−1 β◦ D + zδC◦ (I − zA)−1 B + γ(zI − α)−1 (zβ◦ C◦ )(I − zA)−1 B = (δD + γQB) + γ(zI − α)−1 (β◦ D + αQB)+ + z(γQA + δC◦ )(I − zA)−1 B.

From the third identity in (2.16) we see that δD + γQB = R0 , the second identity in (2.16) yields β◦ D + αQB = β, and the ﬁrst identity in (2.16) shows that γQA + δC◦ = C. But then we see that Ψ(z)Θ(z) is equal to the right hand side of (1.3), that is, R(z) = Ψ(z)Θ(z). It remains to show that this factorization is a right canonical one, i.e., we have to show (see, e.g., [9], Section XXIV.3) that Θ(z) and Θ(z)−1 have no poles in {z ∈ C : |z| ≤ 1}, while Ψ(z) and Ψ(z)−1 have no poles in {z ∈ C : |z| ≥ 1} inﬁnity included. But these properties follow directly from the stability of the matrices A, α, A◦ , and α◦ . Thus R(z) = Ψ(z)Θ(z) is a right canonical factorization of R relative to the unit circle. Part 4. In this part we prove the uniqueness of the stabilizing solution. In fact, we show that the stabilizing solution to (1.4) is given by (1.11). So let Q be a stabilizing solution to (1.4). By the result of the previous part, R admits a right canonical factorization R(z) = Ψ(z)Θ(z), where Θ and Ψ are given by (1.7). It follows that the block Toeplitz operator T deﬁned by R is invertible. Moreover, its inverse can be expressed explicitly in terms of the × × −1 at zero and the Taylor coeﬃTaylor coeﬃcients Θ× 0 , Θ1 , Θ2 , . . . of Θ(z) × × × −1 cients Ψ0 , Ψ1 , Ψ2 , . . . of Ψ(z) at inﬁnity. In fact (see Theorem XXIV.4.1

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in [9]), we have T −1 = TΘ−1 TΨ−1 , where ⎤ ⎡ × ⎡ × 0 0 ··· Θ0 Ψ0 × ⎥ ⎢Θ× ⎢ Θ 0 · · · 0 ⎥ ⎢ 1 ⎢ 0 TΘ−1 = ⎢Θ× Θ× Θ× ⎥ , TΨ−1 = ⎢ 0 1 0 ⎦ ⎣ 2 ⎣ .. .. .. .. . . . .

Ψ× 1 Ψ× 0 0 .. .

Ψ× 2 Ψ× 1 Ψ× 0

⎤ ··· · · ·⎥ ⎥ ⎥ . (2.18) ⎦ .. .

As Θ(z)−1 and Ψ(z)−1 are given by (1.9) and (1.10), we see that −1 , Θ× 0 = D

Ψ× 0

=δ

−1

,

−1 −1 Θ× C◦ Aj−1 ◦ BD j = −D

Ψ× j

= −δ

−1

−1 γαj−1 ◦ β◦ δ

(j = 1, 2, . . .),

(j = 1, 2, . . .).

Using these identities and (2.18) we have ωTΘ−1 = β αβ α2 β · · · , where ⎞ ⎛ ∞ β = βD−1 − α ⎝ αj βD−1 C◦ Aj◦ ⎠ BD−1 ,

(2.19)

j=0

and

⎡ ⎤ C ⎢ CA ⎥ ⎢ ⎥ TΨ−1 W = ⎢CA 2⎥ , ⎣ ⎦ .. .

where

⎞ ⎛ ∞ = δ −1 C − δ −1 γ ⎝ C αj◦ β◦ δ −1 CAj ⎠ A.

(2.20)

j=0

= C◦ . To do this set ∆ = R0 − γQB. We shall prove that β = β◦ and C Because Q is a solution to the algebraic Riccati equation (1.4) and δD = ∆, we have Q = αQA + (β − αQB)(R0 − γQB)−1 (C − γQA) = αQ(A − B∆−1 (C − γQA)) + β∆−1 (C − γQA) = αQA◦ + βD−1 C◦ . In other Q = αQA◦ + βD−1 C◦ . Since α and A◦ are both stable, ∞ words, j −1 Q = j=0 α βD C◦ An◦ . So according to (2.19) and the second identity in (1.8), we see that β = βD−1 −αQBD−1 = (β−αQB)D−1 = β◦ . Analogously, Q = αQA + (β − αQB)∆−1 (C − γQA) = α − (β − αQB)∆−1 γ QA + (β − αQB)∆−1 C = α◦ QA + β◦ δ −1 C. In other Q = α◦ QA + β◦ δ −1 C. Since α◦ and A are both stable, ∞words, j −1 Q = CAj . So according to (2.20) and the ﬁrst identity in j=0 α◦ β◦ δ (1.8), we see that C◦ = C.

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To complete the proof, recall that Q also satisﬁes the Lyapunov equation ∞

Q = αQA + β◦ C◦ .

Hence Q = j=0 αj β◦ C◦ Aj . By consulting (2.19) and (2.20) with β = β◦ = C◦ , we obtain and C ωT −1 W = ωTΘ−1 TΨ−1 W =

∞

αj β◦ C◦ Aj = Q.

j=0

Therefore Q = ωT −1W and the stabilizing solution Q is unique.

Next we specify Theorem 1.1 for the simple case when R has no poles on the closed unit disc D. In this case the block Toeplitz operator T on 2+ (Cm ) deﬁned by R is block lower triangular, and R admits a representation of the form R(z) = R0 + zC(I − zA)−1 B

with A a stable n × n matrix.

(2.21)

2+ (Cm )

Notice that T deﬁnes an invertible operator on if and only if det R(z) had no zeros in D, or equivalently, R0 is invertible and A − BR0−1 C is stable. By choosing α = 0 on the zero space {0}, and β = 0 from Cm into {0}, and γ = 0 from {0} into Cm , we see that (2.21) is of the form (1.3). In this case the corresponding Riccati equation (1.4) is just the equation Q = 0, where Q maps Cn into {0}. Hence Q = 0 is the only solution to (1.4). Moreover, α◦ = 0 and A◦ = A − BR0−1 C. So Q is a stabilizing solution if and only if A − BR0−1 C is stable, or equivalently, T is invertible. By choosing D = R0 and δ = I, we see that R(z) = Ψ(z)Θ(z) is a right canonical factorization with Θ = R and Ψ = I. From the simple case considered in the previous paragraphs it already follows that the algebraic Riccati equation (1.4) may not have any stabilizing solution. For example, take R as in (2.21) above, with A = 0, B = −2, C = 1 and R0 = 1. Then Q = 0 is the only solution to the corresponding Riccati equation (1.4) and A◦ = 2 is unstable. The next proposition is a corollary of Theorem 1.1 for the case when the Toeplitz operator is tri-diagonal. Proposition 2.2. Assume R(z) = R0 + zR1 + z −1 R−1 . Then R admits a right canonical factorization if and only if the equation Q = R−1 (R0 − Q)−1 R1

(2.22)

has a solution Q with R0 − Q invertible and with A◦ = −(R0 − Q)−1 R1

and

α◦ = −R−1 (R0 − Q)−1

(2.23)

being stable matrices. In this case, take any δ and D invertible so that δD = R0 − Q. Then the corresponding canonical factorization is given by R(z) = Ψ(z)Θ(z), where the factors Ψ and Θ and their inverses are determined by

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Θ(z) = D + zδ −1 R1 ,

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Ψ(z) = δ + z −1 R−1 D−1 ,

Θ(z)−1 = D−1 + zA◦ (I + zA◦ )−1 D−1 , −1

Ψ(z)

=δ

−1

+δ

−1

−1

(z − α◦ )

(2.24)

α◦ .

Proof. To see this we simply set A = α = 0 on Cm , and B = γ = I on Cm . Moreover, take C = R1 and β = R−1 . Then this proposition follows from Theorem 1.1. Put Y = R0 − Q. Then (2.22) can be rewritten as R0 = Y + R−1 Y −1 R1 .

(2.25)

So we can also reformulate the above proposition in terms of Y rather than in terms of Q. The stabilizing solution in this reformulation is the one for which Y is invertible and A◦ = −Y −1 R1 and α◦ = −R−1 Y −1 are stable matrices. With R0 positive deﬁnite, and R−1 = R1∗ , equation (2.25) is studied in [4]. The case when R is a trigonometric polynomial, R(z) = pj=−t Rj , is also of special interest. In this case we obtain a representation (1.3) by taking ⎤ ⎡ ⎤ ⎡ R1 0 I ⎥ ⎢R2 ⎥ ⎢ . . . . ⎥ ⎢ ⎥ ⎢ . . A=⎢ ⎥ , B = ⎢ .. ⎥ , C = I 0 · · · 0 , ⎣ . ⎦ ⎣ 0 I⎦ 0

⎡

0 ⎢I ⎢ α=⎢ ⎣

0 .. .

.. I

.

⎤

⎥ ⎥ ⎥, ⎦ 0

Rp ⎡ ⎤ I ⎢0 ⎥ ⎢ ⎥ β = ⎢.⎥ , ⎣ .. ⎦ 0

γ = R−1

R−2 · · ·

R−t .

Using (1.11) one computes that for this special representation of R the solution Q of the correponding algebraic Riccati equation is just equal to the p × t block matrix in the top left corner of the operator T −1 .

3. An example To illustrate how one can use Theorem 1.1, we reconsider the example analyzed in Section 10 of [10] (cf., Section XXIV.10 in [9]). Consider the rational matrix function

1 − z −1 12 z −1 . R(z) = −3z 1+z As in [10] we seek a canonical factorization of R with respect to the unit circle. For this R we have R0 = I and a stable representation is obtained by taking

0 1 A = 0, B = −3 1 , C = , α = 0, β = −1 12 , γ = . 1 0

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In this setting, Q = q is a scalar,

1 + 3q −q (1 + 3q)−1 R0 − γqB = , (R0 − γqB)−1 = 0 1 0

225

q(1 + 3q)−1 . 1

The corresponding Riccati equation is now determined by the following scalar equation:

(1 + 3q)−1 q(1 + 3q)−1 0 −1 1 q = β(R0 − γqB) C = −1 2 0 1 1 1 q 1+q = − = . 2 1 + 3q 2 + 6q Rewriting this leads to the quadratic equation 6q 2 + q − 1 = 0. The zeros of this equation are −1/2 and 1/3. Since A and α are zero, equations (1.5) and (1.6) yield A◦ = −B(R0 − γqB)−1 C

and α◦ = −β(R0 − γqB)−1 γ.

A simple calculation shows that A◦ and α◦ are both equal to −1/(1 + 3q). So the stabilizing solution is obtained with q = 1/3, and A◦ = α◦ = −1/2. We now take δ = I and D = R0 − γqB, that is

1 1 2 − 13 −1 D= and D = 2 6 . 0 1 0 1 Then we compute the factors Θ(z) and Ψ(z) in (1.7):

0 0 2 − 13 2 − 13 +z = , Θ(z) = 0 1 −3 1 −3z 1 + z and Ψ(z) = I +

1 1 −1 z 0

1 2

D−1 = I +

1 1 1 −2 z 0

1 3

=

1 1 − 2z 0

1 3z

1

.

This is exactly the canonical factorization of R derived in Section 10 of [10].

4. Riccati iteration and ﬁnite sections Throughout this section R is a rational m × m matrix function given by the stable representation (1.3), and (1.4) is the corresponding Riccati equation. Solving (1.4) by iteration leads to a Riccati diﬀerence equation (cf., Appendix A2 in [7]): QN +1 = αQN A + (β − αQN B)(R0 − γQN B)−1 (C − γQN A).

(4.1)

Let us assume that starting from an initial condition at N = k, at each step of the iteration the matrix R0 − γQN B is invertible. In this way we obtain from (4.1) a sequence QN , QN +1 , QN +2 , . . . of ν × n matrices. Moreover, assume that this sequence converges with limit Q and the matrix R0 − γQB is invertible. Then Q is a solution to the Riccati equation (1.4) and this solution will be the stabilizing solution of (1.4) provided the matrices A◦ and α◦ deﬁned by (1.5) and (1.6) are both stable. In that case the Toeplitz operator

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deﬁned by R is invertible and a right canonical factorization for R is given by R(z) = Ψ(z)Θ(z) where Ψ and Θ are deﬁned by (1.7). Formula (1.11) for the stabilizing solution suggests to deﬁne the iterates QN in terms of the ﬁnite sections of T . By deﬁnition the N -th section of T is the N × N block matrix TN given by ⎤ ⎡ R−1 · · · R1−N R0 ⎢ R1 R0 · · · R2−N ⎥ ⎥ ⎢ TN = ⎢ . .. ⎥ . . .. .. ⎣ .. . . ⎦ RN −1 RN −2 · · · R0 In what follows we assume that the Toeplitz operator T deﬁned by R is invertible and that the same holds true for its block transpose T # = [Tk−j ]∞ j,k=0 . # −1 Note that the latter is equivalent to requiring that R (z) = R(z ) admits a right canonical factorization relative to the circle. Thus in the scalar case the invertibility of T # follows from the invertibility of T and conversely. This is also true in the symmetric case (when R(z) is Hermitian for each z on the unit circle). Since R is a continuous matrix symbol, we know (see Section VIII.5 in [8]) that invertibility of T and T # implies (in fact, is equivalent to the statement) that the ﬁnite section method for T converges. In particular, in this case, there exists a positive integer k such that TN is invertible for N ≥ k and lim ωn TN−1 WN = ωT −1 W. n→∞

Here

⎡

ωj = β

αβ

α2 β

···

αj−1 β ,

C CA CA2 .. .

⎤

⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Wj = ⎢ ⎥, ⎥ ⎢ ⎦ ⎣ CAj−1

j ≥ 1.

In checking the invertibility of the ﬁnite section the following lemma is useful. Lemma 4.1. Assume the N -the section TN of T is invertible, and put QN = ωN TN−1 WN . Then TN +1 is invertible if and only if R0 − γQN B is invertible, and in that case the matrix QN +1 = ωN +1 TN−1+1 WN +1 is given by QN +1 = αQN A + (β − αQN B)(R0 − γQN B)−1 (C − γQN A). Note that the matrix R0 − γQB is a square matrix of order m while TN +1 is of order m(N + 1). Hence, in general, checking the invertibility of R0 − γQB will be a much easier task than checking the invertibility TN +1 . Proof. Note that TN +1 admits the 2 × 2 block matrix representation:

R0 γωN TN +1 = . (4.2) WN B TN

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Moreover, the Schur complement ∆N of TN corresponding 2 × 2 block matrix in (4.2) is given by ∆N = R0 − γωN Tn−1 WN B = R0 − γQN B.

(4.3)

Hence TN +1 will be invertible if and only if R0 − γQN B is invertible. The fact that QN +1 = ωN +1 TN−1+1 WN +1 is then given by the right hand side of (4.3) follows by proving the analogue of Lemma 2.1 with T being replaced by TN +1 and using the same type of arguments as in Part 1 of the proof of Theorem 1.1. We summarize the preceding discussion with the following proposition which is a partial converse to the result stated in the second paragraph of this section. Proposition 4.2. Let R be given by the stable representation (1.3), and consider the Riccati diﬀerence equation in (4.1). Assume the Toeplitz operator T deﬁned by R and its block transpose T # are invertible. Then there exists a positive integer k such that the following holds (i) Tk is invertible; (ii) R0 − γQN B is invertible for all N ≥ k where QN is the solution to (4.1) subject to the initial condition Qk = ωk Tk−1 Wk ; (iii) QN converges to Q and R0 − γQB is invertible; (iv) the matrices α◦ and A◦ are stable. In this case, Q is the unique stabilizing solution to the Riccati equation (1.4). It can happen that Qn converges to Q and R0 − γQB is invertible and α◦ or A◦ may not be stable. In fact, this follows from the example considered in the ﬁnal paragraphs of the previous section. Indeed, take R(z) = 1 − 2z, and represent R(z) as in (1.3) with A, B, C matrices of size 1 × 1, A = 0, B = −2, C = 1, and with α = 0 on {0}, β = 0 from C into {0}, and γ = 0 from {0} into C. Then (4.1) and (1.4) reduce to QN = 0 and Q = 0. Thus limN →∞ QN = Q, but Q is not a stabilizing solution. In this case, R does not admit a right canonical factorization. In conclusion we note that for the example considered in Section 3 the Riccati diﬀerence equation is given by

(1 + 3qN )−1 qN (1 + 3qN )−1 0 1 + qN 1 = . qN +1 = −1 2 1 2 + 6qN 0 1 Starting with the initial condition q0 = 0, we see that the sequence qN converges to 1/3. In fact, qN = 1/3 for all N ≥ 11, and the ﬁrst eleven values for qN are given by 1 3 13 51 205 819 3277 13107 52429 209715 1 , , , , , , , , , , . 2 10 38 154 614 2458 9830 39322 157286 629146 3

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References [1] H. Bart, I. Gohberg, and M.A. Kaashoek, Wiener-Hopf integral equations, Toeplitz matrices and linear systems, in: Toeplitz Centennial (ed. I. Gohberg), OT 4, Birkh¨ auser Verlag, 1982; pp. 85-135. [2] H. Bart, I. Gohberg, M.A. Kaashoek, and A.C.M. Ran, Factorization of matrix and operator functions: the state space method, OT 178, Birkh¨ auser Verlag, Basel, 2008. [3] P.E. Caines, Linear Stochastic Systems, Wiley, New York, 1988. [4] J.C. Engwerda, A.C.M. Ran and A. Rijkeboer, Necessary and suﬃcient conditions for the existence of a positive solution of the matrix equation X + A∗ X −1 A = I, Lin. Alg. Appl. 184 (1993), 255-275. [5] P. Faurre, M. Clerget and F. Germain, Op´erateurs rationels positifs, Dunod, Paris, 1979. [6] P.L. Faurre, Stochastic realization algorithms, in: System Identiﬁcation: Advances and case studies, (R.K. Mehra and D.F. Lainiotis, Eds), Academic Press, New York, 1976. [7] C. Foias, A.E. Frazho, I. Gohberg, M. A. Kaashoek, Metric constrained interpolation, commutant lifting and systems, OT 100, Birkh¨ auser-Verlag, Basel, 1998. [8] I.C. Gohberg and I.A. Fel’dman, Convolution equations and projection methods for their solution, Transl. Math. Monographs, vol 41, Amer. Math. Soc., Providence, R.I., 1974. [9] I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of Linear Operators, Volume I, OT 63, Birkh¨ auser Verlag, Basel, 1993. [10] I. Gohberg and M.A. Kaashoek, Block Toeplitz operators with rational symbols, in: Contributions to Operator Theory and its Applications (Eds. I. Gohberg, J.W. Helton and L. Rodman), OT 35, Birkh¨ auser Verlag, Basel, 1988; pp. 385-440. [11] B. Hassibi, A.H. Sayed and T. Kailath. Indeﬁnite-Quadratic Estimation and Control, A Uniﬁed Approach to H 2 and H ∞ Theories., SIAM, Philadelphia, 1999. [12] C. Heij, A.C.M. Ran and F. van Schagen, Introduction to Mathematical Systems Theory, Birkh¨ auser Verlag, Basel, 2007. [13] V. Ionescu, C. Oara and M. Weiss. Generalized Riccati Theory and robust control. A Popov function approach, John Wiley, Chichester, 1999. [14] P. Lancaster and L. Rodman, Algebraic Riccati equations, Clarendon Press, Oxford, 1995. [15] B.P. Molinari, The stabilizing solution of the discrete algebraic Riccati equation, IEEE Trans. Automat. Control 20 (1975), 396-399. [16] L. Roozemond, Canonical pseudo-spectral factorization and Wiener-Hopf integral equations, in: Constructive methods of Wiener-Hopf factorization, OT 21, Birkh¨ auser Verlag, Basel, 1986, pp. 127–156.

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A.E. Frazho Department of Aeronautics and Astronautics Purdue University West Lafayette, IN 47907 USA e-mail: [email protected] M.A. Kaashoek and A.C.M. Ran Afdeling Wiskunde Faculteit der Exacte Wetenschappen Vrije Universiteit De Boelelaan 1081a 1081 HV Amsterdam The Netherlands e-mail: [email protected] [email protected] Submitted: July 11, 2009. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Integr. Equ. Oper. Theory 66 (2010), 231–251 DOI 10.1007/s00020-010-1739-1 Published online January 29, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

Semiseparable Integral Operators and Explicit Solution of an Inverse Problem for a Skew-Self-Adjoint Dirac-Type System B. Fritzsche, B. Kirstein and A. L. Sakhnovich Abstract. Inverse problem to recover the skew-self-adjoint Dirac-type system from the generalized Weyl matrix function is treated in the paper. Suﬃcient conditions under which the unique solution of the inverse problem exists, are formulated in terms of the Weyl function and a procedure to solve the inverse problem is given. The case of the generalized Weyl functions of the form φ(λ) exp{−2iλD}, where φ is a strictly proper rational matrix function and D = D∗ ≥ 0 is a diagonal matrix, is treated in greater detail. Explicit formulas for the inversion of the corresponding semiseparable integral operators and recovery of the Dirac-type system are obtained for this case. Mathematics Subject Classiﬁcation (2010). Primary 34A55; Secondary 34B20; 47G10; 34A05. Keywords. Skew-self-adjoint Dirac system, Weyl function, inverse problem, semiseparable operator, operator identity, explicit solution.

1. Introduction A skew-self-adjoint Dirac-type system has the form d u(x, λ) = iλj + jV (x) u(x, λ), x ≥ 0, (1.1) dx where Ip 0 v 0 j= , (1.2) , V = v∗ 0 0 −Ip Ip is the p × p identity matrix, and v is a p × p matrix function. Such systems have been actively studied in analysis and soliton theory (see, for instance, [1, 12] and the references therein). The system (1.1) is an auxiliary linear The work of A. L. Sakhnovich was supported by the Austrian Science Fund (FWF) under Grant no. Y330.

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system for the matrix nonlinear Schr¨ odinger equation, Sine-Gordon and other important integrable equations. In this paper we consider an inverse problem for the system (1.1), namely, the problem to recover the system (1.1) from its Weyl function. We provide an explicit procedure to recover the potential v when the Weyl function is of the form ϕ(λ) = φ(λ) exp{−2iλD}R,

D ≥ 0,

(1.3)

where φ is a strictly proper rational p×p matrix function, D is a p×p diagonal matrix, and R is a p×p unitary matrix. First, we derive the new Theorem 2.3, which allows us to apply to a wider class of functions, including functions of the form (1.3), the general (non-explicit) scheme for solving inverse problems presented in [29, 33]. Next, we modify the expression for the solution of the inverse problem so that this solution is presented in a more traditional and convenient form. Finally, we use the result mentioned above and the results on the inversion of semiseparable operators from [18] to construct the explicit solution of the inverse problem. The solution of the inverse problem given in this paper has a simple form for the case D = 0, see Corollary 4.10. To present the solution for this case, we ﬁrst recall that all strictly proper rational matrix functions admit representations (realizations in the terminology of system theory) of the form ϕ(λ) = iθ1∗ (λIn − β)−1 θ2 ,

(1.4)

where β is an n × n (n > 0) matrix and θm (m = 1, 2) are n × p matrices. Using this realization, it is shown in Section 4 that the potential v is given by −1 θ2 , v(l) = 2θ1∗ ρ− 11 (l) − where ρ− mj (l) (m, j = 1, 2) are n × n blocks of ρ (l), iβ θ2 θ2∗ , ρ− (l) := e−2lζ , ζ := θ1 θ1∗ iβ ∗

and det ρ− 11 (l) = 0. In another way the inverse problem for this case was solved in [21]. Let us consider connections with the self-adjoint case in some detail and give some references. The inverse problem to recover a self-adjoint Dirac-type system from its Weyl or spectral function is closely related to the inversion of integral operators with diﬀerence kernels, see [9, 26, 32, 36, 37] and various references therein. For the discrete analogues of Dirac systems, Toeplitz matrices appear instead of the operators with diﬀerence kernels [7, 10, 15, 38]. (Various results on Toeplitz matrices and related j-theory one can ﬁnd, for instance, in [5, 8, 13, 14].) When the Weyl functions of the self-adjoint Dirac-type system are rational, one can solve the inverse problem explicitly. One of the approaches to solve the inverse problem explicitly is connected with a version of the B¨acklund-Darboux transformation and some notions from system theory [20, 22]. (See also [15, 16, 24, 27] for this approach, and see [39] and the

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references therein for explicit formulas for the radial Dirac equation.) Another method is to apply the general theory. It proves [2] that for the case of rational Weyl functions the corresponding operators with diﬀerence kernels can be inverted explicitly by formulas from [4]. The case of the skew-self-adjoint Dirac-type system with the rational Weyl function was treated in [21] . It was shown that any strictly proper rational p × p matrix function is the Weyl function of a skew-self-adjoint Dirac type system on semi-axis and the solution of the inverse problem was constructed explicitly similar to the self-adjoint case treated in [20]. The analogues of the operators with diﬀerence kernel for the skew-selfadjoint system (1.1) are bounded operators Sl in L2p (0, l) (0 < l < ∞), which have the form [30, 33] 1 l x+t z + x − t z + t − x ∗ k dzf (t)dt, (1.5) Sl f = Sf = f (x) + k 2 0 |x−t| 2 2 where sup0<x 0. The class of bounded linear operators acting from H1 into H2 is denoted by {H1 , H2 }, identity operators are denoted by I, and the spectrum of an operator A is denoted by σ(A).

2. Inverse problem. Preliminaries First, normalize the fundamental solution u(x, λ) of system (1.1) by the initial condition (2.1) u(0, λ) = I2p ,

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and assume that the real and imaginary parts vij and vij of the entries vij of the potential v are measurable functions. If sup v(x) ≤ M,

(2.2)

0<x<∞

the unique p×p Weyl matrix function ϕ(λ) of the skew-self-adjoint Dirac type system (1.1) on the semi-axis [0, ∞) can be deﬁned [29] (see also [6, 21, 33]) by the inequality ∞ ϕ(λ) ϕ(λ)∗ Ip u(x, λ)∗ u(x, λ) dx < ∞, (2.3) Ip 0 which holds for all λ in the halfplane λ < −M < 0. Under condition (2.2) such a Weyl function always exists. An example of bounded potentials is given by the class of so called pseudo-exponential potentials [21], which are denoted by the acronym PE. A potential v ∈ PE is determined by three parameter matrices, that is, by the n × n matrix α (n > 0) and two p × n matrices θ1 and θ2 , which satisfy the identity (2.4) α − α∗ = i(θ1 θ1∗ + θ2 θ2∗ ). The pseudo-exponential potential has the form ∗

v(x) = 2θ1∗ eixα Σ(x)−1 eixα θ2 , where

Σ(x) = In +

x

Λ(t)jΛ(t)∗ dt,

Λ(x) =

0

e−ixα θ1

(2.5)

eixα θ2

.

(2.6)

By Proposition 1.4 in [21] the pseudoexponential potential v, i.e., the potential given by (2.5) is bounded on the semi-axis. The Weyl function of the system (1.1) with v ∈ PE is a rational matrix function, which is also expressed in terms of the parameter matrices [21]: ϕ(λ) = iθ1∗ (λIn − β)−1 θ2 ,

β = α − iθ2 θ2∗ .

(2.7)

In spite of the requirement β − β ∗ = i(θ1 θ1∗ − θ2 θ2∗ ),

(2.8)

α − iθ2 θ2∗ ,

which is implied by the equalities (2.4) and β = any strictly proper rational matrix function can be presented in the form (2.7). The inverse problem to recover v from the strictly proper rational matrix function ϕ is solved explicitly in [21], using a minimal realization of ϕ and formula (2.5). When (2.2) is true, inequality (2.3) implies other inequalities: ixλ ϕ(λ) < ∞ for all 0 < l < ∞, (2.9) sup e u(x, λ) Ip x≤l, λ<−M which can be treated as a more general deﬁnition of the Weyl function. Deﬁnition 2.1. Let the system (1.1) be given on the semi-axis [0, ∞). Then a p × p matrix function ϕ(λ) analytic in some halfplane λ < −M < 0 is called a Weyl function of this system, if inequalities (2.9) hold.

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If sup v(x) < ∞

0<x
for all 0 < l < ∞,

(2.10)

then there is at most one Weyl function. Deﬁnition 2.2. The inverse spectral problem (ISP) for system (1.1) on the semi-axis is the problem to recover v(x) satisfying (2.9) and (2.10) from the Weyl function ϕ. For an analytic matrix function ϕ satisfying the condition

sup λ2 ϕ(λ) − α/λ < ∞,

(2.11)

λ<−M

where α is some p × p matrix, the solution of the inverse problem always exists (see Lemma 1 [30] for the scalar version of this result and the matrix case can be proved in a quite similar way). The general (non-explicit) procedure to solve ISP is described in [28, 29, 30, 33]. Fix a positive value l (0 < l < ∞). The ﬁrst step to solve ISP is to recover a p × p matrix function s(x) with the entries from L2 (0, l) (l < ∞), i.e., s(x) ∈ L2p×p (0, l) via the Fourier transform. That is, we put a i −ηx e l.i.m.a→∞ eiξx λ−1 ϕ(λ/2)dξ (λ = ξ + iη, η <−2M ), (2.12) s(x) = 2π −a the limit l.i.m. being the limit in L2p×p (0, l). As (2.12) has sense for any l < ∞ the matrix function s(x) is deﬁned on the non-negative real semi-axis x ≥ 0. Moreover, it is easily checked that s is absolutely continuous, it does not depend on the choice of η < −2M , s is bounded on any ﬁnite interval, and s(0) = 0. To deﬁne the operator Sl we substitute k(x) = s (x) into (1.5). Next, denote the p × 2p block rows of u by ω1 and ω2 : ω1 (x) = [Ip

0]u(x, 0),

ω2 (x) = [0 Ip ]u(x, 0).

(2.13)

It follows from (1.1) that u(x, 0)∗ u(x, 0) = I2p . Hence, by (1.1) and (2.13) we have (2.14) v(x) = ω1 (x)ω2 (x)∗ , and ω1 , ω2 satisfy the equalities ω1 (0) = [Ip

0],

ω1 ω1∗ ≡ Ip ,

ω1 ω1∗ ≡ 0,

ω1 ω2∗ ≡ 0.

It is immediate that ω1 is uniquely recovered from ω2 using (2.15). Finally, we obtain ω2 via the formula l ∗ ω2 (l) = [0 Ip ] − Sl−1 s (x) [Ip s(x)]dx (0 < l < ∞),

(2.15)

(2.16)

0

where Sl−1 is applied to s columnwise. From the considerations in [29, 30] (see also similar constructions in [31], where the Weyl theory for the linear system auxiliary to the nonlinear optics equation is treated) it follows that one can solve ISP under requirements weaker than (2.11). Namely, we assume sup ϕ(λ) < ∞,

λ<−M

(2.17)

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ϕ(λ) ∈ L2p×p (−∞, ∞), and s(0) = 0,

λ = ξ + iη (−∞ < ξ < ∞) for all η < −M, (2.18)

sup k(x) < ∞ for all 0 < l < ∞,

0<x
IEOT

∞

0

k(x) := s (x),

e−cx k(x)dx < ∞

(2.19) (2.20)

for s given by (2.12) and for some c > 0. Theorem 2.3. Let the matrix function ϕ be analytic in the halfplane λ < −M and satisfy the relations (2.17) and (2.18). Assume also that the function s(x), deﬁned via ϕ by formula (2.12), is absolutely continuous and satisﬁes (2.19) and (2.20). Then ISP has a unique solution, which is given by formulas (2.14)– (2.16), where Sl ≥ I has the form (1.5) with k = s .

3. Factorization of S and operator identity Consider again the operator S = Sl . It is easy to see that functions, which are bounded on the interval, can be approximated in the L1 -norm by the continuous functions. As k = s is bounded on the ﬁnite intervals, one can see that the kernel K of S, which is given by (1.6), is continuous with respect to x and t. Hence, the kernel of Sl−1 is continuous with respect to x, t, and l

([23], p. 185). Therefore, Sl−1 k has the form Sl−1 k (x) = k(x)+k1 (x), where k1 is continuous, and the matrix function Sl−1 k (l) is well-deﬁned:

Sl−1 k (l) = k(l)+k1 (l) = k(l)+ lim Sl−1 k (x)−k(x) (0 < l < ∞). (3.1) x→l−0 It is useful to express v in terms of Sl−1 k (l). Theorem 3.1. Under the conditions of Theorem 2.3 we have

v(l) = Sl−1 s (l).

(3.2)

Proof. STEP 1. According to [34] there are triangular operators Vl ∈ {L2p (0, l), L2p (0, l)}, such that x x −1 V− (x, t)f (t)dt, Vl AVl = iω1 (x) ω1 (t)∗ · dt, (3.3) (Vl f )(x) = f (x)+ 0

0

where A = Al = i

0

x

· dt,

A ∈ {L2p (0, l), L2p (0, l)},

V− (x, t) does not depend on l, and the operators Vl and Vl−1 map functions with bounded derivatives into functions with bounded derivatives. Moreover, as bounded functions on an interval can be approximated in the L1 -norm by the continuous functions, it follows from the construction in [34] that V− (x, t) (x ≥ t) is continuous with respect to x and t.

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Next, introduce the operator x V − (x − t)f (t)dt, (V l f )(x) = f (x) + where ω11

d −1 Vl ω11 (x), (3.4) V − (x) := dx 0 is the ﬁrst p × p block of ω1 , and put x V− (x, t) · dt. (3.5) Vl := Vl V l = I + 0

It is easy to see that V l A = AV l , and so the second equality in (3.3) yields x −1 ω1 (t)∗ · dt. (3.6) Vl AVl = iω1 (x) 0

By (2.13) we see that ω11 (0) = Ip . Hence, using deﬁnition (3.4) one gets x

d −1 Vl ω11 (x − t)dt Vl Ip = Ip + (3.7) dx 0 x

d −1 Vl ω11 (t)dt = Vl−1 ω11 (x). = Ip + dx 0 Formula (3.7) implies Vl−1 ω11 = Ip . Moreover, from [29, 33] it follows that under the conditions of Theorem 2.3 the equalities −1

Vl ω1 (x) = [Ip s(x)] (3.8) and

Sl−1 = Vl∗ Vl

(3.9) are also true. STEP 2. Using (3.9) and changing variables l and x into x and t, correspondingly, we rewrite (2.16) in the form x (3.10) ω2 (x) = [0 Ip ] − Vx s (t)∗ Vx [Ip s(t)]dt. 0

As V− does not depend on l we have Vx s (t) = Vl s (t) for t ≤ x ≤ l. Thus, according to (3.8) and (3.10), we get

(3.11) ω2 (x) = − Vl s (x)∗ ω1 (x). Multiplying both sides of (3.11) by ω1∗ from the right and taking into account (2.14) and (2.15), one derives −v(x)∗ = − Vl s (x)∗ , i.e., the equality

v(x) = Vl s (x) (3.12) is true. is continuous, taking into account (3.4) and (3.5) we see

V− (x, t) As that Vl s (x) − s (x) is continuous. It is also immediate from (3.5) that l ∗ V− (t, x)∗ f (t)dt. (3.13) (Vl f )(x) = f (x) + x

Hence, according to (3.1), (3.9), and (3.13) we have Sl−1 s (l) = Vl s (l). Finally, formula (3.2) follows from (3.12) and (3.14).

(3.14)

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By (3.9) the equality

∗ −1 ∗ Vl AS − SA∗ = Vl−1 Vl AVl−1 − Vl AVl−1 is valid for S = Sl . Therefore, taking into account (3.6) and (3.8) one can see that S satisﬁes the operator identity AS − SA∗ = iΠΠ∗ ,

Π = [Φ1

Φ2 ],

Φ1 g ≡ g,

Φ2 g = s(x)g.

(3.15)

Here Φm ∈ {Cp , L2p (0, l)} (m = 1, 2) and C denotes the complex plane. This identity diﬀers from the identity AS − SA∗ = i(Φ1 Φ∗2 + Φ2 Φ∗1 ) [35, 36] for an operator with diﬀerence kernel. Matrices satisfying a discrete analogue of (3.15) were treated in [17]. The operator identity (3.15) for the case, when k in (1.5) is a vector, was studied in [25]. It could be useful also to prove (3.15) directly. In fact, we prove below a somewhat more general identity. Proposition 3.2. Let the operator S in L2p (0, l) (0 < l < ∞) be deﬁned by 1 l x+t z + x − t z + t − x Sf = f (x) + k k dzf (t)dt, (3.16) 2 0 |x−t| 2 2

k(x) < ∞. Then S satisﬁes the operator identity where sup0<x
x 0

k(t)dt, ψ(x) =

l

0

x 0

Ip + ψ(x)ψ(t) · dt,

(3.17)

k(t)dt.

Proof. Taking into account (3.16) and changing the order of integration we have l γ1 (x, t)f (t)dt, (3.18) ASf = Af + i 0 1 x y+t z + y − t z + t − y k γ1 (x, t) := k dzdy. (3.19) 2 0 |y−t| 2 2 Denote the kernel of S by K(x, t). Using the equality (A∗ f, g)l =(f, Ag)l for the scalar product (·, ·)l in L2p (0, l), we see that 0

l

K(x, t)(A∗ f )(t)dt =

0

l

AK(x, ·)∗ (t)∗ f (t)dt = −i

l 0

0

t

K(x, y)dyf (t)dt.

Hence, we rewrite SA∗ in the form l SA∗ f = A∗ f − i γ2 (x, t)f (t)dt, (3.20) 0 1 t x+y z + x − y z + y − x γ2 (x, t) := k k dzdy. (3.21) 2 0 |x−y| 2 2

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First, consider the case t ≥ x. From (3.19), after changes of variables ξ = (z + y − t)/2 and η = t − y + ξ, we get 1 x y+t z + y − t z + t − y k dzdy (3.22) γ1 (x, t) = k 2 0 t−y 2 2 x y x x = k(ξ) k(t − y + ξ)dξdy = k(ξ) k(t − y + ξ)dydξ 0

0

x

0

t

k(ξ) k(η)dηdξ.

= 0

ξ

t−x+ξ

Next, calculate γ2 (x, t) (t ≥ x). From (3.21) it follows that γ2 (x, t) = γ21 (x, t) + γ22 (x, t), where

(3.23)

z + x − y z + y − x k dzdy, (3.24) 2 2 0 x−y 1 t x+y z + x − y z + y − x k k dzdy. (3.25) γ22 (x, t) := 2 x y−x 2 2

γ21 (x, t) :=

1 2

x

x+y

k

Replace the variable z by η = (z + y − x)/2 in (3.24) , then change the order of integration, and after that put ξ = x − y + η and change the order of integration again to obtain x ξ k(ξ) k(η)dηdξ. (3.26) γ21 (x, t) = 0

0

In (3.25), replace z by ξ = (z + x − y)/2, change the order of integration and put η = y − x + ξ. We get x t−x+ξ γ22 (x, t) = k(ξ) k(η)dηdξ. (3.27) 0

ξ

By (3.22), (3.23), (3.26), and (3.27) the equality x t k(ξ) k(η)dηdξ = ψ(x)ψ(t) γ1 (x, t) + γ2 (x, t) = 0

(3.28)

0

is true for t ≥ x. Using similar calculations one can show that (3.28) holds also for x ≥ t, i.e., (3.28) is true for all 0 ≤ x, t ≤ l. Finally, formulas (3.18), (3.20), and (3.28) yield (3.17).

4. ISP and semiseparable operators Sl In this section we consider matrix functions of the form ϕ(λ) = iθ1∗ (λIn − β)−1 θ2 e−2iλD R, D = diag{d1 , . . . , dp }, dk1 ≥ dk2 ≥ 0

(4.1) for k1 > k2 ,

(4.2)

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where θj (j = 1, 2) is an n × p matrix with the m-th column denoted by θj,m , β is an n × n matrix, R is a p × p matrix, and D is a p × p diagonal matrix. We do not suppose here that θj and β satisfy the identity (2.8). Proposition 4.1. Let the matrix function ϕ be given by (4.1). Then, the matrix function s, which is deﬁned via ϕ by (2.12), has the form s = CR, where C = c1 c2 . . . cp , the columns cm (p ≥ m ≥ 1) being given by the formulas for 0 ≤ x ≤ dm , x−dm cm (x) = 2θ1∗ exp{2itβ}dtθ2,m

cm (x) = 0

0

(4.3) for

x ≥ dm ,

(4.4)

and the function ϕ is the Weyl function of system (1.1) with potential v satisfying (2.10). Proof. First, choose M > 0 such that σ(β + iM In ) ⊂ C+ , where σ means spectrum and C+ is the open upper halfplane. According to (4.1) ϕ(λ) is analytic and the function λϕ(λ) is bounded in the halfplane λ < −M . So, the conditions (2.17) and (2.18) on ϕ are fulﬁlled. The fact that s is absolutely continuous and satisﬁes conditions (2.19) and (2.20) is immediate from (4.3) and (4.4). Therefore, after we have proved (4.3) and (4.4) , it will follow from Theorem 2.3 that ϕ is the Weyl function of system (1.1) with potential v satisfying (2.10). Now, let us prove (4.3) and (4.4). As λϕ(λ) is bounded, one can rewrite (2.12) as a pointwise limit: ∞ 1 eiλ(x−dm ) (4.5) s = c1 c2 . . . cp R, cm (x) = − θ1∗ π −∞ ×λ−1 (λIn − 2β)−1 dξθ2,m

(λ = ξ + iη,

η < −2M ).

Introduce the counterclockwise oriented contours, where ξ may take complex values: {ξ : |ξ| = a, ξ > 0}, Γ− {ξ : |ξ| = a, ξ < 0}. Γ+ a = [−a, a] a = [−a, a] For λ = ξ + iη and for the ﬁxed values of η < −2M , it follows from (4.5) that 1 eiλ(x−dm ) λ−1 (λIn − 2β)−1 dξθ2,m (4.6) cm (x) = − θ1∗ lim π a→∞ Γ+ a in the case x ≥ dm , and 1 eiλ(x−dm ) λ−1 (λIn − 2β)−1 dξθ2,m cm (x) = θ1∗ lim π a→∞ Γ− a

(4.7)

in the case x ≤ dm . As eiλ(x−dm ) λ−1 (λIn − 2β)−1 is analytic with respect to ξ inside Γ− a and on the contour itself, equality (4.3) is immediate from (4.7). Next, consider the case x ≥ dm . For suﬃciently large a all the poles of (λIn − 2β)−1 (and the pole ξ = −iη of λ−1 ) are contained inside Γ+ a and taking into account (4.6) we have 1 ∗ eiλ(x−dm ) λ−1 (λIn − 2β)−1 dξθ2,m . (4.8) cm (x) = − θ1 π Γ+ a

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Let us approximate β by matrices βε such that β − βε < ε and det βε = 0 (if det β = 0 we put β = βε ). It is easy to see that

λ−1 (λIn − 2βε )−1 = (2βε )−1 (λIn − 2βε )−1 − λ−1 In . (4.9) For suﬃciently small ε all the poles of (λIn − 2βε )−1 are contained inside Γ+ a and we have 1 eiξx (λIn − 2βε )−1 dξ = eηx exp(2ixβε ) (x ≥ 0). (4.10) 2πi Γ+ a Finally, using (4.8)-(4.10) we get cm (x) = lim

ε→0

2θ1∗

x−dm

0

exp{2itβε }dtθ2,m .

Hence, formula (4.4) is immediate.

Remark 4.2. Note that the matrix functions ϕ of the form (4.1) in general position do not satisfy (2.11) and so they do not satisfy in a scalar case conditions of Lemma 1 [30], but the conditions of Theorem 2.3 are fulﬁlled. By Proposition 4.1 the matrix function k in the expression (1.6) for the kernel of the operator Sl , generated by the Weyl function ϕ of the form (4.1), is given by the formula k(x) = s (x) = 2θ1∗ e2ixβ νχ(x)R,

ν := {exp(−2idm β)θ2,m }pm=1 , (4.11) 0, 0 ≤ x < dm , χ(x) = diag{χ1 (x), χ2 (x), . . . , χp (x)}, χm (x) := 1, x > dm .

According to (1.6) and (4.11) we have K(x, t) ∗ = 2θ1

(4.12) exp i(z + x − t)β Q(z, x, t) exp − i(z + t − x)β ∗ dzθ1 ,

x+t

|x−t|

where

z + t − x z + x − t (4.13) Q(z, x, t) = νχ RR∗ χ ν∗. 2 2 The matrix function Q(z, x, t) is piecewise constant with respect to z and without loss of generality we assume Q(0, x, t) = 0. It is easy to see that Q(z, x, t) has only a ﬁnite number of jumps {Qj }. Moreover, if σ(β)∩σ(β ∗ ) = ∅, the matrix identity i(βXj − Xj β ∗ ) = Qj always has the solution Xj . Therefore we have ∗ ∗ d izβ (4.14) e Xj e−izβ . eizβ Qj e−izβ = dz Hence, according to (4.12) and (4.14) we can express the kernel K(x, t) of S explicitly in terms of matrix exponents and {Xj }. It follows also from (1.6) that K(x, t) = K(t, x)∗ , (4.15) and so we need to simplify (4.12) only for x > t.

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Another approach to the presentation of K in terms of matrix exponents is given in the following lemma. Lemma 4.3. Put gj (z) := [0

izβ

e

zEj

]e

Then we have

Ip 0

Ej :=

,

−iβ ∗ Qj

0 −iβ

.

(4.16)

d ∗ gj (z) = eizβ Qj e−izβ . dz

Proof. By (4.16) we have d gj = iβgj + [0 eizβ ]Ej ezEj dz

Ip 0

(4.17)

∗

= iβgj + eizβ Qj e−izβ − iβgj ,

and (4.17) is immediate.

Recall [18] that the operator S is called semiseparable, when K admits representation K(x, t) = F1 (x)G1 (t)

for x > t,

K(x, t) = F2 (x)G2 (t)

for x < t, (4.18)

where F1 and F2 are p × p matrix functions and G1 and G2 are p × p matrix functions. For the operator S to be semiseparable, assume RR∗ = Ip .

(4.19)

Then the matrix function Q has the form z + t − x Q(z, x, t) = νχ ν∗ 2 Rewrite (4.2) as D = diag{d 1 Ip1 , . . . , d r Ipr },

for x > t.

p1 + . . . + pr = p,

(4.20)

d k1 > d k2 ≥ 0 for k1 > k2 ,

(4.21)

Qj = νPj ν ∗ ,

(4.22)

and put Pj = diag{0, . . . , 0, Ipj , 0, . . . , 0}.

Remark 4.4. Some notations. Further we consider K(x, t) (x > t) on the intervals d m < t < min(x, d m+1 ), where we choose such m that the inequalities d m < x hold. If d 1 > 0 we put d 0 = 0 and include the interval d 0 < t < min(x, d 1 ) into consideration. If x > d r , we include the interval d r < t < d r+1 (d r+1 = x). Some matrix functions, like B(t) and C(t), will be considered on the intervals as above, but with x = l. In the following, in mall such cases we simply write dm < t < dm+1 . We also assume that j=1 . . . = 0, when m = 0. Proposition 4.5. Let the matrix function ϕ be given by (4.1), where D satisﬁes (4.21) and R is unitary. Assume also that the matrix identities i(βXj − Xj β ∗ ) = Qj ,

(4.23)

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where Qj are given by (4.22), have solutions Xj . Then the operator S, which is deﬁned via ϕ by formulas (1.5), k = s and (2.12), is semiseparable, and its kernel K(x, t) (0 < x, t < l) is given by relation (4.15) and by the equalities ∗ m θ1 (d m < t < d m+1 ) (4.24) K(x, t) = 2θ1∗ e2ixβ Zm e−2itβ − e2i(x−t)β Z for t < x < l. Here m Xj , Zm = j=1

m = Z

m

exp 2id j β Xj exp − 2id j β ∗ .

(4.25)

j=1

Moreover, there are self-adjoint solutions of (4.23) and we suppose Xj = Xj∗ in (4.23) and (4.25). Proof. First, note that we can choose Xj = Xj∗ because the adjoint of each solution of (4.23) also satisﬁes (4.23). Next, using (4.12), (4.20), and (4.22) we get equalities m x+t

exp i(z + x − t)β Qj K(x, t) = 2θ1∗ j=1

x−t+2d j

× exp − i(z + t − x)β ∗ dz θ1

(4.26)

for t < x and d m < t < d m+1 . From (4.14) and (4.26) it follows that (4.24) holds. Formula (4.15) was derived earlier. Remark 4.6. By (2.14)–(2.16) and (4.11) the equality v(x) = 0 is valid for 0 < x < d1 in the case d1 > 0. Therefore inequality (2.9) for l = d1 takes the form ixλ ixλj φ(λ)e−2iλD e e < ∞, sup (4.27) Ip x≤d1 , λ<−M and can be easily checked directly.

l When the operator S = I + 0 K(x, t) · dt is semiseparable and its kernel K is given by (4.18), the kernel of the operator T = S −1 is expressed in terms of the 2 p × 2 p solution U of the diﬀerential equation d U (x) = H(x)U (x), x ≥ 0, U (0) = I2 p , (4.28) dx where H(x) := B(x)C(x), (4.29) −G1 (x) , C(x) = F1 (x) F2 (x) . B(x) = G2 (x) Namely, we have (see, for instance, [18]) l T (x, t) · dt, T = S −1 = I + T (x, t) =

(4.30)

0

C(x)U (x) I2 p − P × U (t)−1 B(t), x > t, −C(x)U (x)P × U (t)−1 B(t), x < t.

(4.31)

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Here P × is given in terms of the p × p blocks U21 (l) and U22 (l) of U (l): 0 0 × P = , (4.32) U22 (l)−1 U21 (l) Ip and the invertibility of U22 (l) is a necessary and suﬃcient condition for the invertibility of S. If K admits the representation

CexA I2 p − P e−tA B, x > t, (4.33) K(x, t) = −CexA P e−tA B, x < t, where A, B, and C are constant matrices, then U is calculated explicitly [19]. In our case a representation

Cm exA I2 p − Pm e−tA Bm , t < x < l, d m < t < d m+1 , (4.34) K(x, t) = −Cm exA Pm e−tA Bm , x < t < l, d m < x < d m+1 , where p = n and

A = 2i

β 0

0

β∗

,

(4.35)

easily follows from (4.15) and (4.24). However, (4.34) is insuﬃcient for the explicit construction of U and we shall construct U and T explicitly, using more general formulas (4.28)-(4.32). For this purpose we introduce B(x) and C(x) (0 < x < l) by the equalities −2ixβ √ m − Zm e−2ixβ ∗ e Z (4.36) θ1 (d m < x < d m+1 ), B(x) = 2 ∗ e−2ixβ √ m e2ixβ ∗ C(x) = 2θ1∗ e2ixβ e2ixβ Zm − Z (d m < x < d m+1 ), (4.37) ∗ ∗ m = Z m where Zm = Zm and Z are deﬁned in (4.25).

Proposition 4.7. Let the conditions of Proposition 4.5 be fulﬁlled and let S be deﬁned via ϕ by formulas (1.5), k = s and (2.12). Then the operator T = S −1 is given by formulas (4.30)-(4.32), (4.36), (4.37), and U (x) = Ωm e−xA exAm Ξ−1 m U (dm )

(d m ≤ x ≤ d m+1 ),

×

where A is deﬁned by (4.35) and A× m := A + 2Ym , Ωm :=

In 0

Ym := −Zm In

,

m Z In

θ1 θ1∗

m −Z

In

U (x) JU (x) = J,

−1

U (x)

,

(4.39)

×

Ξm := Ωm e−dm A edm Am .

Moreover, we have ∗

U (0) = I2n , (4.38)

∗

∗

= JU (x) J ,

J :=

0 In

−In 0

(4.40) .

(4.41)

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Proof. Recall that B and C are recovered from K by formulas (4.18) and (4.29). In view of (4.15) and (4.24) we have √ F2 (x) = G1 (x)∗ , (4.42) F1 (x) = 2θ1∗ e2ixβ , √

∗ m θ1 (d m < x < d m+1 ), G1 (x) = 2 Zm e−2ixβ − e−2ixβ Z (4.43) G2 (x) = F1 (x)∗ .

(4.44)

Therefore, formulas (4.29) and (4.42)-(4.44) imply that B and C corresponding to S are given by (4.36) and (4.37). It follows from (4.29) and (4.35)-(4.37) that H(x) = 2Ωm e−xA Ym exA Ω−1 (d m < x < d m+1 ), (4.45) m where Ym is given in (4.39), Ωm is given in (4.40), and In Zm −1 Ωm = . (4.46) 0 In According to (4.38), (4.39), and (4.45) we get d xA× m ) = H(x)U (x) m Ξ−1 U (d U (x) = Ωm e−xA A× m−A e m dx for d m < x < d m+1 , and so U of the form (4.38) satisﬁes (4.28). In other words, formulas (4.36)-(4.38) deﬁne explicitly B, C and U , which are used in the expressions (4.31) and (4.32) to construct the kernel of T = S −1 . It remains to prove (4.41). Note that JA∗ J ∗ = −A, ∗

JΩ∗m J ∗ = Ω−1 m ,

JYm∗ J ∗ = −Ym .

(4.47)

∗

Hence, we have JH J = −H, i.e.,

d U (x)∗ JU (x) ≡ 0. dx Formula (4.41) follows from (4.48) and from U (0) = I2n . Taking into account (4.36)-(4.38) we get √ m ), m ]exA× m Ξ−1 U (d F (x) := C(x)U (x) = 2θ1∗ [In − Z m √ m Z := U (t)−1 B(t) = 2U (d m )−1 Ξm e−tA× m G(t) θ1 , In for d m < x < d m+1 , d m < t < d m+1 ,

(4.48)

(4.49) (4.50)

and F (d m+1 ) is given by the substitution of x = d m+1 into the right-hand side of (4.49). In particular, we deﬁne in this way F (l) for l = d r+1 . Corollary 4.8. Let the conditions of Proposition 4.5 be fulﬁlled. Then the kernel T (x, t) of the operator T = Sl−1 has the form

x > t, F (x) I2n − P × G(t), (4.51) T (x, t) = × −F (x)P G(t), x < t, are given by (4.49) and (4.50). where F and G

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By (4.11), (4.39), and (4.50) for d m < t < d m+1 we get m √ × In Pj R G(t)k(t) = 2U (d m )−1 Ξm e−tAm (2Ym )etA ν 0 j=1 m √ d −tA× In m etA = − 2U (d m )−1 Ξm ν Pj R. e 0 dt

(4.52)

j=1

From Theorem 3.1 and formulas (4.30), (4.51), and (4.52) the explicit solution of the inverse problem is immediate. Theorem 4.9. Let the Weyl matrix function ϕ be given by (4.1), where D satisﬁes (4.21) and R is unitary. Assume also that the matrix identities (4.23), where Qj are given by (4.22), have solutions Xj = Xj∗ . Then the ISP solution v is given by the formula N √

v(l) = k(l) + F (l) I2n − P × 2U (dm )−1 Ξm

m=1 −dm A× m dm A

× e

e

−dm+1 A× m dm+1 A

−e

e

(4.53)

m I n Pj R, ν 0 j=1

×

where k is given by (4.11), U is given by (4.38), P is given by (4.32), and Ξm is given by (4.40). The number N in the sum is chosen in the following way: if l < d 1 then N = 0; if d j < l < d j+1 then N = j; if l > d r then N = r. We put dm = d m for m ≤ N and dN +1 = l. Corollary 4.10. Let the Weyl matrix function ϕ be given by (4.1), where D = 0 and R = Ip . Then we have s (x) = k(x) = 2θ1∗ e2ixβ θ2 , the kernel K of the operator Sl is given by the formula t ∗ e2iyβ θ2 θ2∗ e−2iyβ dy θ1 (x > t), K(x, t) = 4θ1∗ e2i(x−t)β

(4.54)

(4.55)

0

and −1 v(l) = 2θ1∗ ρ− θ2 , 11 (l)

where

ρ− mj (l)

(4.56)

−

(m, j = 1, 2) are n × n blocks of ρ (l), iβ θ2 θ2∗ − −2lζ ρ (l) := e , ζ := , θ1 θ1∗ iβ ∗

(4.57)

and det ρ− 11 (l) = 0. Proof. By (4.11) the equality (4.54) is true. As D = 0 we get (see Remark 4.4): r = 1, d 1 = 0, d 2 = l. (4.58)

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In view of (4.58) it follows from (4.26) after substitution 2y = z + t − x that K(x, t) = F1 (x)G1 (t) for x > t, where t ∗ ∗ 2ixβ −2itβ , G1 (t) = 2e e2iyβ θ2 θ2∗ e−2iyβ dy θ1 , (4.59) F1 (x) = 2θ1 e 0

and (4.55) is immediate. Next assume that there is X1 such that i(βX1 − X1 β ∗ ) = Q1 = θ2 θ2∗ ,

X1 = X1∗ .

(4.60)

According to (4.25) the equalities 1 = X1 Z1 = Z

(4.61)

hold. Taking into account (4.58) and (4.61) we derive from (4.40) that In −X1 In X1 = Ω1 = Ξ1 = , Ω−1 . (4.62) 1 0 In 0 In Moreover, in view of (4.49), (4.58), (4.61), and (4.62) we have √

F (l) = 2θ1∗ [In 0] exp lΩ1 A× Ω−1 . 1 Formulas (4.35), (4.60), and (4.62) imply iβ θ2 θ2∗ Ω1 AΩ−1 = 2 . 1 0 iβ ∗ Hence, using (4.39), (4.57), (4.62), and (4.64) we get X1 ∗ −X I Ω1 A× Ω−1 = Ω θ A + 2 θ Ω−1 n 1 1 1 1 1 1 = 2ζ. In Finally, by (4.63) and (4.65) we have √ F (l) = 2θ1∗ [In

0]e2lζ .

(4.63)

(4.64)

(4.65)

(4.66)

From (4.32), (4.53), (4.58), and (4.66) it follows that (4.67) v(l) = k(l) + 2θ1∗ [In 0]e2lζ × 0 In In Ip Ξ1 − e−lA e2ilβ θ2 . × 0 0 −U22 (l)−1 U21 (l) 0 According to (4.62) and (4.65) one can rewrite (4.67) as (4.68) v(l) = k(l) + 2θ1∗ [In 0]e2lζ In 0 In Ip − e−2lζ e2ilβ θ2 . × 0 0 −U22 (l)−1 U21 (l) 0 By (4.38), (4.62), and (4.65) we get ∗

U2,m (l) = e−2ilβ ρ+ 2m (l)

(m = 1, 2),

ρ+ (l) := e2lζ .

(4.69)

Taking into account (4.54) and (4.69) rewrite (4.68) in the form v(l) = 2θ1∗ e2ilβ θ2

+ + −1 + 2ilβ θ2 . + 2θ1∗ ρ+ ρ21 (l) In − ρ− 11 (l) − ρ12 (l)ρ22 (l) 11 (l)e

(4.70)

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As ρ− (l) = ρ+ (l)−1 , one can see that + + −1 + −1 ρ21 (l) = ρ− . ρ+ 11 (l) − ρ12 (l)ρ22 (l) 11 (l)

Hence, formula (4.70) implies (4.56). Now consider an arbitrary strictly proper rational matrix function ϕ(λ) = iθ1∗ (λIn − β)−1 θ2 without requirement (4.60). Notice that for each ε > 0 there is a matrix βε , which satisﬁes condition (4.60) and inequality β − βε < ε. For matrices and functions corresponding to ϕε (λ) = iθ1∗ (λIn − βε )−1 θ2 we shall use the index ε. It is easily checked that lim vε (l) = v(l),

ε→0

− lim ρ− ε,11 (l) = ρ11 (l).

ε→0

(4.71)

+ Moreover, we have limε→0 Uε,22 (l) = U22 (l), limε→0 ρ+ ε,22 (l) = ρ22 (l), and −2ilβε∗ + −2ilβ ∗ + so Uε,22 (l) = e ρε,22 (l) implies U22 (l) = e ρ22 (l). As Sl is invertible and semiseparable, so det U22 (l) = 0, that is, det ρ+ 22 (l) = 0. The last inequality yields (4.72) det ρ− 11 (l) = 0. −1 (l) θ . So, by (4.71) and (4.72) It was shown above that vε (l) = 2θ1∗ ρ− 2 ε,11 formula (4.56) is valid without requirement (4.60).

Another way to prove Corollary 4.10 would be to use the procedure from [19] for explicitly inverting semiseparable operators. Acknowledgment The authors are grateful to the referee for very useful remarks.

References [1] M. J. Ablowitz, B. Prinari, and A. D. Trubatch, Discrete and continuous nonlinear Schr¨ odinger systems, London Math. Soc. Lect. Note Ser. 302, Cambridge University Press, 2004 [2] D. Alpay, I. Gohberg, L. Lerer, M. A. Kaashoek, and A. L. Sakhnovich, Krein systems, OT: Adv. Appl. 191 (2009), Birkh¨ auser, 19–36. [3] D. Alpay, I. Gohberg, M. A. Kaashoek, and A. L. Sakhnovich, Direct and inverse scattering problem for canonical systems with a strictly pseudoexponential potential, Math. Nachr. 215 (2000), 5–31. [4] H. Bart, I. Gohberg, and M. A. Kaashoek, Convolution equations and linear systems, IEOT 5 (1982), 283–340. [5] A. B¨ ottcher and B. Silbermann, Analysis of Toeplitz operators, Springer Monographs in Mathematics, Berlin: Springer, 2006. [6] S. Clark and F. Gesztesy, On Self-adjoint and J-self-adjoint Dirac-type Operators: A Case Study, Contemp. Math. 412 (2006), 103–140. [7] Ph. Delsarte, Y. Genin, and Y. Kamp, Schur parametrization of positive deﬁnite block-Toeplitz systems, SIAM J. Appl. Math. 36:1 (1979), February, 34–46. [8] V. K. Dubovoj, B. Fritzsche, and B. Kirstein, Matricial version of the classical Schur problem, in: Teubner-Texte zur Mathematik [Teubner Texts in Mathematics] 129, B. G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1992.

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[9] H. Dym and A. Iacob, Positive deﬁnite extensions, canonical equations and inverse problems, OT: Adv. Appl. 12 (1984), Birkh¨ auser, 141–240. [10] H. Dym, Hermitian block Toeplitz matrices, orthogonal polynomials, reproducing kernel Pontryagin spaces, interpolation and extension, OT: Adv. Appl. 34 (1988), Birkh¨ auser, 79–135. [11] Y. Eidelman and I. Gohberg, Algorithms for inversion of diagonal plus semiseparable operator matrices, IEOT 44 (2002), 172–211. [12] L. D. Faddeev and L. A. Takhtajan, Hamiltonian methods in the theory of solitons, Berlin: Springer, 1986. [13] B. Fritzsche, B. Kirstein, An extension problem for non-negative Hermitian block Toeplitz matrices, Math. Nachr., Part I: 130 (1987), 121–135; Part II: 131 (1987), 287–297; Part III: 135 (1988), 319–341; Part IV: 143 (1989), 329– 354; Part V: 144 (1989), 283–308. [14] B. Fritzsche, B. Kirstein, On the Weyl matrix balls associated with nondegenerate matrix-valued Caratheodory functions, Z. Anal. Anwendungen 12 (1993), 239–261. [15] B. Fritzsche, B. Kirstein, I. Roitberg, and A. L. Sakhnovich, Weyl matrix functions and inverse problems for discrete Dirac-type self-adjoint systems: explicit and general solutions, Operators and Matrices 2 (2008), 201-231. [16] B. Fritzsche, B. Kirstein, and A. L. Sakhnovich, Completion problems and scattering problems for Dirac type diﬀerential equations with singularities, J. Math. Anal. Appl. 317 (2006), 510–525. [17] B. Fritzsche, B. Kirstein, and A. L. Sakhnovich, On a new class of structured matrices related to the discrete skew-self-adjoint Dirac systems, ELA 17 (2008), 473-486. [18] I. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of Linear Operators vol. 1, OT: Adv. Appl. 49, Birkh¨ auser, 1990. [19] I. Gohberg and M. A. Kaashoek, Time varying linear systems with boundary conditions and integral operators I. The transfer operator and its properties, IEOT 7 (1984), 325–391. [20] I. Gohberg, M. A. Kaashoek, and A. L. Sakhnovich, Canonical systems with rational spectral densities: explicit formulas and applications, Math. Nachr. 194 (1998), 93–125. [21] I. Gohberg, M. A. Kaashoek, and A. L. Sakhnovich, Pseudocanonical systems with rational Weyl functions: explicit formulas and applications, J. Diﬀ. Eqs. 146 (1998), 375–398. [22] I. Gohberg, M. A. Kaashoek, and A. L. Sakhnovich, Scattering problems for a canonical system with a pseudo-exponential potential, Asymptotic Analysis 29 (2002), 1–38. [23] I. Gohberg and M.G. Krein, Theory and applications of Volterra operators in Hilbert space, Transl. of math. monographs 24, Providence, RI, 1970. [24] M. A. Kaashoek and A. L. Sakhnovich, Discrete skew self-adjoint canonical system and the isotropic Heisenberg magnet model, J. Functional Anal. 228 (2005), 207–233. [25] I. Koltracht, B. Kon, and L. Lerer, Inversion of structured operators, IEOT 20 (1994), 410–480.

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[26] M. G. Krein, Continuous analogues of propositions on polynomials orthogonal on the unit circle (Russian), Dokl. Akad. Nauk SSSR 105 (1955), 637–640. [27] R. Mennicken, A. L. Sakhnovich, and C. Tretter, Direct and inverse spectral problem for a system of diﬀerential equations depending rationally on the spectral parameter, Duke Math. J. 109:3 (2001), 413–449. [28] A. Sakhnovich, The mixed problem for nonlinear Shr¨ odinger equation and the inverse spectral problem, Preprint, Branch of Hydroacoustics, Inst. Hydromech. Acad. Sci. Ukrainian SSR, Odessa, 1989, Manuscript No. 3255-B89 deposited at VINITI AN SSSR, 1989. [29] A. L. Sakhnovich, Nonlinear Schr¨ odinger equation on a semi-axis and an inverse problem associated with it, Ukr. Math. J. 42:3 (1990), 316–323. [30] A. L. Sakhnovich, The Goursat problem for the sine-Gordon equation and the inverse spectral problem, Russ. Math. Iz. VUZ 36:11 (1992), 42–52. [31] A. L. Sakhnovich, Inverse spectral problem related to the N -wave equation, OT: Adv. Appl. 117 (2000), Birkh¨ auser, M.G. Krein volume, 323–338. [32] A. L. Sakhnovich, Dirac type and canonical systems: spectral and WeylTitchmarsh functions, direct and inverse problems. Inverse Problems 18 (2002), 331–348. [33] A. L. Sakhnovich, Skew-self-adjoint discrete and continuous Dirac-type systems: inverse problems and Borg-Marchenko theorems, Inverse Problems 22 (2006), 2083–2101. [34] L. A. Sakhnovich, Spectral analysis of Volterra’s operators deﬁned in the space of vector-functions L2m (0, l), Ukr. Mat. Zh. 16 (1964), 259-268. [35] L. A. Sakhnovich, An integral equation with a kernel dependent on the diﬀerence of the arguments, Mat. Issled. 8 (1973), 138–146. [36] L. A. Sakhnovich, Factorisation problems and operator identities, Uspekhi Mat. Nauk 41 :1 (1986), 3–55; English transl. in Russian Math. Surveys 41 (1986), 1–64. [37] L. A. Sakhnovich, Spectral theory of canonical diﬀerential systems, method of operator identities, OT: Adv. Appl. 107, Birkh¨ auser, 1999. [38] B. Simon, Orthogonal polynomials on the unit circle, Parts 1,2, Colloquium Publications, American Mathematical Society 51, 54, Providence, RI, 2005. [39] G. Teschl, Deforming the point spectra of one-dimensional Dirac operators, Proc. Amer. Math. Soc. 126:10 (1998), 2873–2881. [40] R. Vandebril, M. Van Barel, G. Golub, and N. Mastronardi, A bibliography on semiseparable matrices, Calcolo 42 (2005), 249–270. B. Fritzsche and B. Kirstein Fakult¨ at f¨ ur Mathematik und Informatik Mathematisches Institut Universit¨ at Leipzig Johannisgasse 26 D-04103 Leipzig Germany e-mail: [email protected] [email protected]

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Semiseparable Operators and an Inverse Problem

A. L. Sakhnovich Fakult¨ at f¨ ur Mathematik Universit¨ at Wien Nordbergstrasse 15 A-1090 Wien Austria e-mail: al [email protected] Submitted: April 20, 2009. Revised: September 18, 2009.

251

Integr. Equ. Oper. Theory 66 (2010), 253–264 DOI 10.1007/s00020-010-1746-2 Published online January 26, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

Translation-Invariant Bilinear Operators with Positive Kernels Loukas Grafakos and Javier Soria Abstract. We study Lr (or Lr,∞ ) boundedness for bilinear translationinvariant operators with nonnegative kernels acting on functions on Rn . We prove that if such operators are bounded on some products of Lebesgue spaces, then their kernels must necessarily be integrable functions on R2n , while via a counterexample we show that the converse statement is not valid. We provide certain necessary and some suﬃcient conditions on nonnegative kernels yielding boundedness for the corresponding operators on products of Lebesgue spaces. We also prove that, unlike the linear case where boundedness from L1 to L1 and from L1 to L1,∞ are equivalent properties, boundedness from L1 × L1 to L1/2 and from L1 × L1 to L1/2,∞ may not be equivalent properties for bilinear translation-invariant operators with nonnegative kernels. Mathematics Subject Classiﬁcation (2010). Primary 42A85; Secondary 47A07. Keywords. Bilinear operators, convolution, positive kernels.

1. Introduction For a nonnegative regular Borel measure µ on Rn × Rn , we deﬁne the bilinear convolution operator: Tµ (f, g)(x) = f (x − y)g(x − z) dµ(y, z), (1) Rn ×Rn

n

where x ∈ R and f, g are nonnegative functions on Rn . If dµ(y, z) = K(y, z) dydz, for some nonnegative function K, then we denote TK (f, g)(x) = f (x − y)g(x − z) K(y, z) dydz, Rn

Rn

The ﬁrst author was supported by the NSF under grant DMS 0900946. The second author was partially supported by grants MTM2007-60500 and 2005SGR00556.

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assuming no confusion occurs in the notation. We are interested in studying boundedness properties of these operators on diﬀerent products of Lp (Rn ) spaces and on more general rearrangement-invariant quasi-Banach function spaces. We discuss necessary conditions for boundedness in terms of the range of the Lebesgue indices and of the kernels of such operators. A suﬃcient condition for boundedness is obtained in a particular case, see Theorem 3.2. Theorem 4.3 provides a characterization, in terms of the Lorentz space L1/2n,1/2 (R+ ), of the boundedness of TK from L1 × L1 to L1/2 , if K(y, z) = ϕ(|y| + |z|) and ϕ is decreasing. The study of bilinear operators within the context of harmonic analysis was initiated by Coifman and Meyer [2, 3] in the late seventies but recent attention in the subject was rekindled by the breakthrough work of Lacey and Thiele [9, 10] on the bilinear Hilbert transform. The behavior of this operator is still not understood on spaces near L1 × L1 . Although the results obtained in this paper are not applicable to the bilinear Hilbert transform, they suggest that bilinear translation-invariant operators may exhibit behavior at the endpoint L1 × L1 diﬀerent from that of their linear counterparts on L1 (see Theorems 3.4 and 4.1). An interesting example of an operator of type (1) is given by the measure µ = δ0 (y + z)χ|y|≤1 on Rn × Rn , where δ0 denotes the Dirac delta mass on the diagonal in Rn . This operator (which appeared in the study of bilinear fractional integrals) can be written as f (x − y)g(x + y)χ|y|≤1 dy , B(f, g)(x) = Rn

and maps L1 (Rn ) × L1 (Rn ) to L1/2 (Rn ), as proved independently by Kenig and Stein [8] and Grafakos and Kalton [5]. The bilinear fractional integrals are also operators of the form (1) associated with the singular measures µα = δ0 (y + z)|y|−n+α on Rn × Rn , where 0 < α < n, and they map Lp × Lq → Lr , when 1/p + 1/q = α/n + 1/r.

2. Necessary conditions We begin by exhibiting a general restriction on a set of indices p, q, r for which an operator Tµ of the form (1) is bounded. The next result is analogous to H¨ormander’s [7] in the linear case; see also [6]. Proposition 2.1. Let µ be a nonnegative regular Borel measure on Rn × Rn . Suppose that the bilinear operator Tµ maps Lp (Rn ) × Lq (Rn ) to Lr (Rn ) for some 0 < p, q, r ≤ ∞. Then one has 1/p + 1/q ≥ 1/r. In particular, if Tµ : L1 (Rn ) × L1 (Rn ) → Lp (Rn ), then p ≥ 1/2. Proof. Fix 0 < p, q, r ≤ ∞. By translating µ if necessary, we may assume that there exists a compact set E ⊂ [1, M ]n × [1, M ]n (for some M > 1), such that 0 < µ(E) < ∞.

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n Let x = (x1 , . . . , xn ) in Rn . Taking f (x) = j=1 |xj |−α χ(1,∞)n (x), with n α > 1/p, and g(x) = j=1 |xj |−β χ(1,∞)n (x), with β > 1/q we have, for xj > M + 1, j = 1, . . . , n: n Tµ (f, g)(x) ≥ f (x − y)g(x − z) dµ(y, z) ≥ µ(E) (xj − 1)−(α+β) . E

r

j=1 n

Since Tµ (f, g) ∈ L (R ), this implies that α + β > 1/r, for all α > 1/p and all β > 1/q; i.e., 1/p + 1/q ≥ 1/r. In the endpoint case 1/p + 1/q = 1/r, we prove that the boundedness of the bilinear operator Tµ necessarily implies that the measure µ must be ﬁnite. In fact, this result is valid even under the weaker assumption that Tµ is of weak-type (p, q, r). We study this condition in detail in Section 3 where we give an example showing that, in general and contrary to what happens in the linear case, the ﬁniteness of the measure (or the integrability of the kernel) is not a suﬃcient condition for the boundedness of the associated operator. Proposition 2.2. If µ is a nonnegative regular Borel measure and the operator Tµ : Lp (Rn ) × Lq (Rn ) → Lr,∞ (Rn ) for some 0 < p, q ≤ ∞ satisfying 1/p + 1/q = 1/r, then µ is a ﬁnite measure. In particular, if K ≥ 0 and TK : Lp (Rn ) × Lq (Rn ) → Lr,∞ (Rn ), for some 0 < p, q ≤ ∞ with 1/p + 1/q = 1/r, then K ∈ L1 (Rn × Rn ). Proof. We consider ﬁrst the case 0 < r < ∞. Fix R > 0 such that µ(BR × BR ) > 0, where BR is the ball B(0, R) ⊂ Rn . Then, for every x ∈ BR we have: Tµ (χB2R , χB2R )(x) = µ(B(x, 2R) × B(x, 2R)) ≥ µ(BR × BR ) = λ > 0. Therefore BR Tµ (χB2R , χB2R ) > λ/2 , and |BR | ≤ Tµ (χB2R , χB2R ) > λ/2 2r C r ≤ |B2R |r/p |B2R |r/q λr C r 2r+n = |BR |. λr Hence, for every R > 0, we have that µ(BR × BR ) ≤ 21+n/r C, which proves the result when r < ∞ letting R → ∞. When r = ∞ we have 2 µ(BR × BR ) ≤ Tµ (χB2R , χB2R )L∞ ≤ C χB2R L∞ = C, and the conclusion follows letting R → ∞ as well.

3. Suﬃcient conditions We now study certain suﬃcient conditions for boundedness of operators of the form (1). We start with a couple of observations:

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If K ∈ L1 (Rn × Rn ), then TK : Lp (Rn ) × Lq (Rn ) → Lr (Rn ), where 1 ≤ p, q ≤ ∞ and 1/p + 1/q = 1/r ≤ 1. In fact, this statement can be strengthened as follows: Proposition 3.1. If µ is a nonnegative regular Borel measure and 1/p + 1/q = 1/r ≤ 1, then the following statements are equivalent: (a) Tµ : Lp (Rn ) × Lq (Rn ) → Lr (Rn ). (b) Tµ : Lp (Rn ) × Lq (Rn ) → Lr,∞ (Rn ). (c) µ is a ﬁnite measure. Proof. Obviously (a) implies (b) while the fact that (b) implies (c) is proved in Proposition 2.2. Using Minkowski’s integral inequality, we have: f (· − y)g(· − z)r dµ(y, z) Tµ (f, g)r ≤ Rn ×Rn ≤ f (· − y)p g(· − z)q dµ(y, z) Rn ×Rn

= µ(Rn × Rn )f p gq .

It is interesting that this result is false, in general, when 0 < r < 1. We show that there exists K ≥ 0, K ∈ L1 (in fact K ∈ L1 ∩ L∞ ) such that TK does not map L1 × L1 to L1/2,∞ ; see Theorem 3.4. A second observation is that if a kernel K satisﬁes |K(y, z)| ≤ K1 (y)K2 (z), 1

n

p

n

q

(2) n

r

n

where 0 ≤ Kj ∈ L (R ), then TK : L (R ) × L (R ) → L (R ), whenever 1 ≤ p, q ≤ ∞ and 1/p + 1/q = 1/r. In this case r ≥ 1/2 and K lies in L1 (Rn × Rn ), which is a necessary condition by Proposition 2.2. We now provide a weaker suﬃcient condition than (2), that yields the boundedness of TK in the nontrivial case 0 < r < 1: Theorem 3.2. Suppose that 1/p + 1/q = 1/r ≥ 1 and ϕ is a nonnegative function on R+ × R+ , decreasing in each variable separately and obeying the estimate: (ϕ(2j1 , 2j2 )2j1 n 2j2 n )r < ∞. j1 ∈Z j2 ∈Z

Let K be a function on Rn × Rn that satisﬁes |K(y1 , y2 )| ≤ ϕ(|y1 |, |y2 |). Then TK maps Lp (Rn ) × Lq (Rn ) to Lr (Rn ). Proof. For each j1 , j2 integers we set Kj1 ,j2 (y1 , y2 ) = K(y1 , y2 )χIj1 (y1 )χIj2 (y2 ), where Ijl = {2jl < |yl | ≤ 2jl +1 }. Then we have T (f1 , f2 )(x) ≤

j1 ∈Z j2 ∈Z

j1

j2

ϕ(2 , 2 )

2 l=1

I jl

|fl (x − yl )| dyl .

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We raise this expression to the power r and integrate over Rn . As we can pass the power r inside the sum we obtain that |T (f1 , f2 )(x)|r dx ≤ ϕ(2j1 , 2j2 )r Rn

j1 ∈Z j2 ∈Z

×

R

2 n l=1

Ijl

r |fl (x − yl )| dyl dx

and we apply H¨ older’s inequality to control the previous quantity by

p r/p j1 j2 r ϕ(2 , 2 ) |f1 (x − y1 )| dy1 dx C j1 ∈Z j2 ∈Z

×

Rn

Rn

≤ C

|f2 (x − y2 )| dy2 j1

j2 r

×

Rn

= C

Rn

I j2

dx

ϕ(2 , 2 )

j1 ∈Z j2 ∈Z

r/q

q

I j2

I j1

q

I j1

j1 n(p−1)

|f1 (x − y1 )| dy1 2

j2 n(q−1)

|f2 (x − y2 )| dy2 2

p

r/p dx

r/q dx

(ϕ(2j1 , 2j2 )2j1 n 2j2 n )r (f1 Lp f2 Lq )r

j1 ∈Z j2 ∈Z

≤ C (f1 Lp f2 Lq )r .

Remark 3.3. It is easy to see that the hypothesis on K can be equivalently written as |K(y1 , y2 )|r dy1 dy2 < ∞, n n 1−r Rn Rn (|y1 | |y2 | ) and in this case the monotonicity condition on ϕ is replaced by the condition: whenever |y1 | ≤ |y1 | we have |K(y1 , y2 )| ≥ |K(y1 , y2 )| and whenever |y2 | ≤ |y2 | we have |K(y1 , y2 )| ≥ |K(y1 , y2 )|. Under no extra conditions on K, and for the case 0 < r < 1, no positive results can be obtained, as the following result indicates: Theorem 3.4. There exists a nonnegative function K on Rn × Rn such that, if X is a rearrangement invariant (r.i.) quasi Banach space, then TK : L1 × L1 → X if and only if L∞ is a subspace of X. Proof. We work the details in the case n = 1, although the construction can be easily extended to Rn for n ≥ 2. For a < 0 and r > 0 set fa,r (x) =

1 χ(a−r,a+r) (x) . 2r

Also let a = {(x − a, x) : x ∈ R}

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be the line of slope 1 passing through the point (0, a). Then for almost all (x − a, x) ∈ R2 we have 1 K(y, z) dzdy TK (fa,r , f0,r )(x) = 2 4r (x−a−r,x−a+r) (x−r,x+r) and from this we deduce that TK (fa,r , f0,r )(x) → K(x − a, x)

(3)

as r → 0. In other words, (3) holds for almost every a < 0 and almost every point on the line a with respect to one-dimensional Lebesgue measure. For each k ∈ N, we construct a sequence of disjoint rectangles Rk as in Figure 1 with base length equal to 1/k 3 , height equal to 2k, and longest side parallel to the line a . We arrange that all these rectangles touch each other and are contained in the right angle −|x| ≤ y ≤ |x| on the (x, y) plane. We let P (Rk ) be the intersection of the smallest strip containing the longest side of Rk and the negative y-axis. Set ∞

R=

Rk

k=1

and K = χR . Then K1 = |R| =

∞

2/k 2 < ∞.

k=1

a

R3 R2 R1

P (R1 )

...

P (R2 ) (0, a) u P (R3 )

Figure 1.

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Suppose that for this kernel K the following estimate holds: TK (f, g)X ≤ Cf 1 g1 for all f, g nonnegative in L1 (Rn ). Then for any k ≥ 1, (3) holds √ ∞ functions −3 for almost all − 2 k=1 k < a < 0, with (0, a) in P (Rk ) (in particular for one such a), and for almost all points (x − a, x) in a . Since χ(0,k) (x) ≤ χRk (x − a, x) ≤ K(x − a, x), for all real x, using Fatou’s lemma and (3) we deduce that χ(0,k) X ≤ lim inf TK (fa,r , f0,r )X ≤ Cfa,r 1 f0,r 1 = C, r→0

for every k ∈ N. Thus, the fundamental function ϕX of X (see [1]) is bounded, which is equivalent to saying that L∞ is a subspace of X. Conversely, if L∞ is a subspace of X and K ∈ L∞ , then it is clear that TK : L1 × L1 → L∞ and thus TK maps L1 × L1 to X. Remark 3.5. If K ∈ L∞ , then we have just observed that, trivially, TK : L1 ×L1 → L∞ . Therefore, if K ∈ L1 ∩L∞ , 1/p+1/q ≤ 1 and 0 ≤ θ ≤ 1, using bilinear interpolation [1] for this estimate and Proposition 3.1, we obtain:

TK : Lp /(p −θ) × Lq /(q −θ) → Lpq/(θ(p+q)) . In particular, TK : Lp × Lp → Lp/2 whenever for 2 ≤ p ≤ ∞, and TK : Lp × Lp → Lp /2 whenever 1 ≤ p ≤ 2. Consequently, for K ∈ L1 ∩ L∞ such that TK : L1 × L1 → Lp for some p ≥ 1/2 (cf. Proposition 2.1), the boundedness TK : L1 × L1 → Lq holds for every q in [p, ∞]. It is then an interesting question to determine the least possible value of p in the interval [1/2, ∞], for which such an operator is bounded from L1 × L1 to Lp . We have indicated that there are examples showing that we can have the best possible situation (boundedness on L1/2 when K is a tensor product of two kernels in L1 ) and also the worst case (only bounded on L∞ , as in Theorem 3.4). See also Proposition 4.2. Modiﬁcations of bilinear fractional integrals also provide examples in the intermediate cases.

4. Other examples and estimates Well-known examples of bilinear singular integral operators, such as the bilinear Riesz transforms [6], indicate that boundedness from L1 × L1 to L1/2 may not hold although boundedness from L1 × L1 to L1/2,∞ is valid. These operators have kernels that change sign but the next result shows that there exist positive measures that provide examples of kernels with the same property. This situation should be contrasted with its linear version that fails: if a convolution operator with a positive Borel measure on Rn maps L1 (Rn ) to L1,∞ (Rn ), then the measure is ﬁnite and therefore the operator maps L1 (Rn ) to itself!

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Theorem 4.1. There exists a nonnegative regular ﬁnite Borel measure µ on R × R with the property that Tµ maps L1 × L1 to L1/2,∞ but does not map L1 × L1 to L1/2 . Proof. We ﬁrst observe that if we want Tµ : L1 × L1 → L1/2,∞ , then necessarily µ must be a ﬁnite measure (Proposition 2.2). We choose a positive sequence {λj }j∈Z ∈ 1/2,∞ \ 1/2 and deﬁne µ = j λj δaj , where aj = (j, j) and δaj is the Dirac mass at aj . Clearly λj < ∞ . µ(R × R) =

j

Then, Tµ (f, g)(x) = j λj f (x − j)g(x − j). Let also Dµ (h)(x) = 1/2,∞ λ h(x − j). Using that {λ and [4, Lemma 3.5] we have j }j∈Z ∈ j j 1/2 1/2,∞ that Dµ : L →L , and hence,

Tµ (f, g)1/2,∞ = Dµ (f g)1/2,∞ ≤ Cf g1/2 ≤ Cf 1 g1 . / 1/2 , by [4, Theorem 3.1] (see also [11]), we have that Now, since{λj }j∈Z ∈ Dµ is not of strong-type L1/2 and, as before, Tµ L1 ×L1 →L1/2 ≥ sup f

Tµ (f, f )1/2 Dµ (h)1/2 = sup = ∞. 2 f 1 h1/2 h

We now consider some particular cases of kernels, deﬁned in terms of a special function ϕ. The ﬁrst example is K(y, z) = ϕ(y + z), where ϕ : R n → R+ . Proposition 4.2. Let 1 ≤ α ≤ ∞ and ϕ ∈ Lα (Rn ) be a positive function. Set K(y, z) = ϕ(y + z). Then, TK : Lp (Rn ) × Lq (Rn ) → Lr (Rn ),

(4)

where

1 1 1 1 = + + − 2 ≤ 1. r p q α Moreover, if ϕ ∈ Lα ∩ L∞ , then 0≤

TK : L1 (Rn ) × L1 (Rn ) → Lr (Rn ),

(5)

for every α ≤ r ≤ ∞ and the result is false, in general, if r < α. Proof. The main observation is that TK (f, g)(x) = (f ∗ g ∗ ϕ)(2x), and hence the result is a reformulation of Young’s inequality: TK (f, g)r ≤ f p g ∗ ϕβ , if 1 ≤ p ≤ β and 1/r = 1/p + 1/β − 1. Similarly, g ∗ ϕβ ≤ gq ϕα ,

if 1 ≤ q ≤ α and 1/β = 1/q + 1/α − 1, which proves (4). If p = q = 1 and ϕ ∈ L∞ , then TK : L1 (Rn ) × L1 (Rn ) → Lα (Rn ) that, together with the estimate TK : L1 (Rn ) × L1 (Rn ) → L∞ (Rn ), gives (5).

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To ﬁnish, take r < α, and deﬁne ϕ(t) = t(−1−ε)/α χ(1,∞) (t) ∈ Lα ∩ L∞ , where 0 < ε < α/r − 1. Set f = g = χ(0,1) . Then, if x > 3/2:

x ∞ χ(x−1,x) (y)(z + y)(−1−ε)/α dy dz TK (f, g)(x) = x−1 1−z

x ∞ ≥ χ(x−1,x) (y) dy (z + x)(−1−ε)/α dz x−1 1−z x ≥ (z + x)(−1−ε)/α dz ≥ (2x)(−1−ε)/α . x−1

Therefore TK (f, g)r = ∞. This proves the result if n = 1. The n-dimensional case follows by adapting this idea. Another example of interest comes when the kernel is deﬁned as K(y, z) = ϕ(|y| + |z|), where ϕ : R+ → R+ is a decreasing function. We will study the behaviour of TK at the endpoints p = q = 1 and r = 1/2, for which we give a complete characterization in terms of the Lorentz space L1/2n,1/2 (R+ ): Theorem 4.3. Let ϕ : R+ → R+ be a decreasing function and deﬁne K(y, z) = ϕ(|y| + |z|). Then, TK : L1 (Rn ) × L1 (Rn ) → L1/2 (Rn ) if and only if ϕ ∈ L1/2n,1/2 (R+ ). Proof. Assume that TK : L1 (Rn ) × L1 (Rn ) → L1/2 (Rn ). Set Rk = {(y, z) ∈ Rn × Rn : 2k < |y| + |z| ≤ 2k+1 }, so that |Rk | ≈ 22kn . Fix 2j−1 < |x| ≤ 2j and δ ≤ 2l , with l ≤ j − 2. Consider also the functions f (x) = g(x) = χ{|x|<δ} (x). Then (TK (f, g)(x))1/2 dx ≤ Cδ n . (6) Rn

Observe that {y ∈ Rn : |x − y| < δ} × {z ∈ Rn : |x − z| < δ} ⊂ Rj−1 ∪ Rj ∪ Rj+1 , j+2

j−1

and |y| + |z| ≥ 2|x| − 2δ ≥ 2 . since |y| + |z| ≤ 2δ + 2|x| ≤ 2 Discretizing the operator, and using (7), we obtain: f (x − y)g(x − z)ϕ(|y| + |z|) dydz TK (f, g)(x) = k∈Z

Rk

≥ ϕ(2j+2 )

j+1 k=j−1

= Cn ϕ(2j+2 )δ 2n ≥ Cn ϕ(8|x|)δ 2n .

Rk

f (x − y)g(x − z) dydz

(7)

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Thus, by (6) and the previous estimate: (TK (f, g)(x))1/2 dx ≥ C δ n Cδ n ≥ C {|x|>2δ}

ϕ(8|x|) dx,

{|x|>2δ}

and hence,

∞

ϕ(t) tn

16δ

Letting δ → 0 we ﬁnally obtain:

1/2

ϕ1/2n,1/2 =

∞

dt ≤ C . t

ϕ(t) tn

0

dt < ∞. t

Conversely, since ϕ(|y| + |z|) ≤ ϕ(|y|) and ϕ(|y| + |z|) ≤ ϕ(|z|), we have

ϕ(|y| + |z|) ≤ ϕ(|y|) ϕ(|z|) , and therefore K is bounded from above by the tensor product of two functions in L1 (Rn ), since 1/2 ϕ(|y|) dy = Cϕ1/2n,1/2 < ∞, Rn

which implies the result (see (2)).

Remark 4.4. By Proposition 2.2 we know that the boundedness of TK in the previous theorem would imply that K ∈ L1 (Rn × Rn ). This condition is, in fact, equivalent to ϕ ∈ L1/2n,1 (R+ ): K1 =

ϕ(|y| + |z|) dydz

Rn

Rn ∞ ∞

ϕ(s + t)t

0

= = ≈ =

dt sn−1 ds

∞ ∞ n−1 ϕ(u)(u − s) du sn−1 ds C 0 s u

∞ n−1 n−1 ϕ(u) (u − s) s ds du C 0 ∞0 du ϕ(u)u2n u 0 ϕ1/2n,1 .

= C

n−1

0

Since L1/2n,1/2 (R+ ) L1/2n,1 (R+ ), we observe that Theorem 4.3 gives a stronger condition. We end by giving an analogous version of Proposition 3.1 in the case of linear convolution operators that, surprisingly enough, seems to be missing from the literature. For K ≥ 0, we deﬁne the averaging operator: 1 K(y) dy. A(K)(x, r) = |B(x, r)| B(x,r)

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We observe that A(K)(x, ·)L∞ = M (K)(x), where M is the Hardyr Littlewood maximal function. We use the following notation for the mixed norm space X[Y ]: F X[Y ] denotes the quasinorm in X of the function F (x, ·)Y . We consider ﬁrst the case p = 1: Proposition 4.5. Let K ≥ 0, and TK (f )(x) =

Rn

f (x − y)K(y) dy.

Then, the following statements are equivalent: ∞ (a) A(K) ∈ L1,∞ x [Lr ]. ∞ 1,∞ (b) A(K) ∈ Lr [Lx ]. (c) K ∈ L1 . (d) TK : L1 → L1,∞ . (e) TK : L1 → L1 . Moreover, A(K)L∞ [Lx1,∞ ] ≈ A(K)Lx1,∞ [L∞ ] ≈ K1 . r

r

Proof. It is well known that A(K)L∞ [Lx1,∞ ] ≤ A(K)Lx1,∞ [L∞ ] ≈ K1 , r r i.e., (a) ⇔ (c) ⇒ (b). Taking r > 0 such that B(0,r/2) K(y) dy > 0 implies C n K ≤ A(K)(·, r)1,∞ rn , r ≤ C TK (χB(0,r) ) ≥ K B(0,r/2) B(0,r/2) hence (b) ⇒ (c). Clearly (c) ⇒ (e) ⇒ (d). Finally, if (d) holds, taking f = χB(0,r) , we obtain that TK (f )(x) = Crn A(K)(x, r), thus TK (f )1,∞ = Crn A(K)(·, r)1,∞ ≤ Crn . Therefore (d) ⇒ (b).

Remark 4.6. (i) It is easy to see that if 1 < p < ∞, then we also have that: K ∈ L1 ⇔ TK : Lp → Lp,∞ ⇔ TK : Lp → Lp . This should be compared to the bilinear case (cf. Theorem 4.1), where weaktype estimates do not imply, in general, the strong-type boundedness of the operator. (ii) For an r.i. Banach function space X for which the maximal operator M maps X to itself (e.g., X = Lp , 1 < p ≤ ∞), the equivalences: A(K)Xx [L∞ ≈ A(K)L∞ ≈ KX , r ] r [Xx ] are easy consequences of Fatou’s Lemma. Remark 4.7. The results of this article concerning positive bilinear operators easily adapt to the setting of m-linear positive convolution operators when m ≥ 3. The precise formulation of these statements and their proofs are analogous to the case m = 2 and are omitted.

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References [1] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, Inc. 1988. [2] R. R. Coifman and Y. Meyer, Commutateurs d’int´ egrales singuli`eres et op´erateurs multilin´eaires, Ann. Inst. Fourier (Grenoble) 28 (1978), 177–202. [3] R. R. Coifman and Y. Meyer, Au d´el` a des op´erateurs pseudo-diﬀ´ erentiels, Ast´erisque No. 57, Societ´e Mathematique de France, 1979. [4] L. Colzani, Translation invariant operators on Lorentz spaces, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), 257–276. [5] L. Grafakos and N. Kalton, Some remarks on multilinear maps and interpolation, Math. Ann. 319 (2001), 151–180. [6] L. Grafakos and R. H. Torres, Multilinear Calder´ on–Zygmund theory, Adv. in Math. 165 (2002), 124–164. [7] L. H¨ ormander, Estimates for translation invariant operators in Lp spaces, Acta Math. 104 (1960), 93–140. [8] C. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), 1–15. [9] M. Lacey and C. Thiele, Lp bounds for the bilinear Hilbert transform, p > 2, Ann. of Math. 146 (1997), 693–724. [10] M. Lacey and C. Thiele, On Calder´ on’s conjecture, Ann. of Math. 149 (1999), 475–496. [11] D. M. Oberlin, Translation-invariant operators on Lp (G), 0 < p < 1, Michigan Math. J. 23 (1976), 119–122. Loukas Grafakos Dept. of Mathematics University of Missouri Columbia MO 65211 USA e-mail: [email protected] Javier Soria Dept. Appl. Math. and Analysis University of Barcelona E-08007 Barcelona Spain e-mail: [email protected] Submitted: May 9, 2009.

Integr. Equ. Oper. Theory 66 (2010), 265–282 DOI 10.1007/s00020-010-1748-0 Published online January 28, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

Convergence of Collocation Method with Delta Functions for Integral Equations of First Kind Urve Kangro Abstract. Integral equations of ﬁrst kind with periodic kernels arising in solving partial diﬀerential equations by interior source methods are considered. Existence and uniqueness of solution in appropriate spaces of linear analytic functionals is proved. Rate of convergence of collocation method with Dirac’s delta-functions as the trial functions is obtained in case of uniform meshes. In case of an analytic kernel the convergence rate is exponential. Mathematics Subject Classiﬁcation (2010). Primary 45B05; Secondary 65R20. Keywords. Interior source method, integral equations, scattering.

1. Introduction This paper provides a theory of existence and approximation for a class of ﬁrst kind Fredholm equations arising in two-dimensional electromagnetic and acoustic scattering problems and elsewhere. Usually, the kernels of such equations are weakly singular, and often the integral equations are of second kind, so that solvability is not hard to establish. However in some recent applications it has become necessary to investigate the case of ﬁrst kind equations with analytic kernels and that is the subject of this paper. Integral equations with analytic kernels arise when solving elliptic boundary value problems using interior source methods, where the solution is represented as an integral over some curve diﬀerent from the boundary of the domain. Such methods are proposed especially for electromagnetic scattering problems [1], [2] and many references therein, but also for exterior problems in acoustics [8]. These This work was supported by Estonian Science Foundation under grant No. 5221 and by Estonian Targeted Financing Project SF0180039s08.

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methods have also been called discrete sources methods or ﬁctitious current methods. In [11] Vainikko provided a theoretical framework for analysing a wide class of operator equations which, in fact, includes the weakly singular integral equations mentioned above as a special case. However, Vainikko’s theory, which is set in Sobolev spaces, is not directly applicable to analytic kernels. A main goal of this work is to extend the theory to cover the analytic situation. Our theory also covers integral equations which arise, for example, when using interior source methods for domains with nonsmooth boundaries. In this case the kernel is not analytic and one has to look for solutions in spaces of periodic analytic functionals. In the theory of [11], the kernel is represented as a sum of functions in which the dominant part generally contains the singular behavior. In the theory presented here the dominant part can be analytic. The smoothness of the kernel is often expressed by decay rates of its Fourier coeﬃcients. Here we consider the case of exponential decay of the coeﬃcients, instead of the algebraic decay of [11]. To solve the integral equation we propose to use the collocation method with Dirac’s δ-functions as the trial functions. This method has the advantage of being the simplest method to use (to set up the matrix of the discrete system, one just has to evaluate the kernel at a given set of points; no integration is needed). Also, when the integral equation is obtained by an interior source method from some diﬀerential equation, the resulting approximate solution of the integral equation generates a very simple approximate solution for the diﬀerential equation. For smooth data the convergence is very fast (exponential in the number of variables), and the method converges even for boundaries with corners, if one chooses the interior boundary, the supports of the δ-functions and the collocation points carefully. We will work out the details for the application to scattering problems in [4]. The convergence of the collocation method has been considered e.g. in [6] which considers the dependence of the convergence on the node distribution, but no convergence rates have been obtained. It is pointed out in [9] that interior source methods work best when the solution of the diﬀerential equation can be analytically extended as a solution of the same equation up to the interior boundary. Extendability of solutions of two-dimensional scattering problems is considered in [3]. From the results of current paper it follows that in certain cases the method can also be used without the extendability condition; in fact, it is also possible to use the interior source method in case of nonanalytic boundary, but in this case one looses the exponential convergence. For Laplace equation in interior domains exponential convergence of similar method has been proved in [5]. Dirac’s δ-functions have been used as trial functions in conventional boundary element methods as well (see e.g. [10]). Generally in this case the collocation (on the same mesh) cannot be used. Therefore in [10] splines

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are used for test functions, which makes this method more diﬃcult to use than ours. Also, when using δ-functions in boundary element methods, the approximation of the solution of the corresponding diﬀerential equation is poor near the boundary, because the solution blows up at the mesh points on the boundary. Interior source methods are free of this shortcoming. The discrete equations for the collocation method with δ-functions are actually the same as the equations for the quadrature method with equal weights, which in case of integral equations of second kind or weakly singular integral equations has been well analysed [11],[7]. In contrast, we discuss the case of integral equation of ﬁrst kind with analytic kernel. Also the interpretation of the approximate solution is diﬀerent in our case, and the norms we use to estimate the error are also diﬀerent. The subsequent contents of the paper are as follows: in the next section we present an example to show how equations with analytic kernels arise in potential theory. Then we construct the spaces and operators for which the existence and other results will be obtained. Next we discuss an approximation procedure which is convenient for implementation and obtain an error estimate. The error estimate shows that, in suitable norms, convergence is exponentially fast in terms of the number of discretization nodes.

2. Interior source methods for solving scattering problems In this section we will present a simple example of an interior source method, where a ﬁrst kind integral equation with an analytic kernel appears. Consider a two-dimensional acoustic scattering problem U + k 2 U = 0 in Ω,

(2.1)

U (x) = F (x), x ∈ Γ, x 1 · ∇U + ikU = o as |x| → ∞, |x| |x|

(2.2) (2.3)

where Ω ⊂ R2 is a complement of a compact set (the scattering body) with a smooth boundary Γ, and F (x) is a given incident ﬁeld. An integral representation of a solution of the Helmholtz equation in Ω satisfying the outgoing wave condition (2.3) is given by (1) u(y)H0 (k|x − y|)dsy , x ∈ Ω, (2.4) U (x) = γ

(1) H0

is the zeroth order Hankel function of the ﬁrst kind and γ is a where closed curve with γ Ω = ∅. If u satisﬁes the integral equation (1) H0 (k|x − y|)u(y)dsy = F (x), x ∈ Γ, (2.5) γ

then (2.4) gives the solution of (2.1)–(2.3).

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In conventional potential theory γ = Γ and the equation (2.5) is weakly singular. In interior source methods, however, γ is chosen strictly inside the scattering body. Therefore smoothness of the kernel is determined by smoothness of the boundary and the auxiliary curve γ, and if they are analytic, the kernel is also analytic. Note that the kernel is a function of |x − y| and therefore may be expected to be close to a convolution kernel. In fact, if Γ and γ are concentric circles, then the integral equation is exactly of convolution type (using the polar angles as variables), and in general, it is possible to choose γ and parametrizations of the boundaries so that the dominant part (the most singular part) of the kernel is of convolution type. It may be nontrivial, since one needs to map the boundary into a real line so that the mapping is conformal in some (possibly one-sided) neighborhood of the boundary. This is simple if the boundary has an analytic parametrization. If the boundary is not analytic, but is simply connected, then such a mapping exists, but it may be nontrivial to ﬁnd. For polygons, the Schwarz-Christoﬀel mapping can be used. In general, it can be shown that equation (2.5), under certain assumptions and after a suitable change of variables, can be rewritten as

2π

0

(κ0 (t − s) + κ1 (t − s)a1 (t, s) + a2 (t, s))v(s)ds = f (t),

t ∈ [0, 2π], (2.6)

where κ0 (t − s) = ln[2(cosh η − cos(t − s) )] and κ1 (t − s) = (cosh η − cos(t − s) ) ln[2(cosh η − cos(t − s) )], where η is a parameter characterizing the distance between Γ and γ, and a1 and a2 are 2π-biperiodic (2π-periodic with respect to both arguments), and smoother (in some sense) than κ0 and κ1 . For details see [4]. We note here that the existence and uniqueness of solution for (2.5) can easily be obtained in ordinary function spaces (e.g. in L2 (γ) or in C n (γ)), if the solution of the exterior problem (2.1)-(2.3) can be extended as a solution of the Helmholtz equation (2.1) up to the interior curve γ (under the condition that −k 2 is not an eigenvalue for with Dirichlet boundary conditions in the domain bounded by γ). If the solution of the exterior problem can be extended to a neighborhood of γ, and γ itself is analytic, then the solution of the integral equation is also analytic. For the extension results for the Helmholtz equation see [3], where an algorithm is given to determine the region, where the solution of the Helmholtz equation can be extended.

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3. The integral equation In the following we will consider the integral equation (using the notation of [11]) k Au := Aj u = f, (3.1) j=0

where f is 2π-periodic, and the operators Aj , j = 0, . . . , k are of the form 2π (Aj u)(t) = κj (t − s)aj (t, s)u(s)ds, j = 0, . . . , k. (3.2) 0

Assume that κj , j = 0, . . . , k are 2π-periodic, a0 ≡ 1, and aj , j = 1, . . . , k are 2π-biperiodic. The integrals here may actually stand for linear functionals on certain periodic function spaces, e.g. κ0 may actually be a δ-function. In the case of interior source methods κj and aj are usually smooth functions, but the solution u may only exist in some space of linear functionals on analytic periodic functions. In both these cases the integrals have to be understood as the dual products in appropriate spaces. We will present the smoothness assumptions in terms of Fourier coefﬁcients. We deﬁne the Fourier coeﬃcients of an integrable 2π-periodic function v by 2π 1 v(t)e−int dt, n ∈ Z. vˆ(n) = 2π 0 Then v(t) = vˆ(n)eint . If v is a periodic distribution or a continuous linear n∈Z

functional on a space of periodic analytic functions, then the integral above should be replaced by the corresponding duality pairing. In this case the Fourier coeﬃcients may be exponentially growing, and the convergence of the Fourier series will take place in appropriate spaces of linear functionals. For n ∈ Z, λ, α ∈ R we denote cn (λ, α) = e|n|λ max{1, |n|}α . We assume that there exist η, θ, α1 , α2 , nonnegative λ1 , λ2 and positive constants ε, C, C1 , C2 such that for all m, n ∈ Z, j = 1, . . . , k κ0 (n)| ≤ C2 cn (−η, −θ), C1 cn (−η, −θ) ≤ |ˆ

(3.3)

|ˆ κj (n)| ≤ Ccn (−η, −θ − ε),

(3.4)

|ˆ aj (m, n)| ≤ Ccm (−λ1 , −α1 )cn (−λ2 , −α2 ).

(3.5)

These assumptions are motivated by the example considered in Section 2. Assumption (3.5) guarantees that one can multiply by aj in the function spaces we will be working with; usually aj are smoother than needed. However, there are cases (e.g. using the interior source method for domains with nonanalytic boundaries), where the exact smoothness of aj is needed to determine the smoothness of the solution and the convergence rate of the method. Similar assumptions hold for integral equations obtained by using other integral representations of the solution of the scattering problem (2.1)-(2.3),

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and also for integral equations obtained from other elliptic partial diﬀerential equations in (analytic) domains by representing the solution in an integral form, where the integral is taken over a suitable curve, which does not intersect the domain or the boundary of the domain. Conventional boundary integral equations also fall under these assumptions: they correspond to the case η = 0. We will consider the integral operators Aj on spaces Aλ,α , λ, α ∈ R deﬁned by ∞ int 2 v (n)cn (λ, α))n=−∞ ∈ l Aλ,α = v(t) = vˆ(n)e (ˆ n∈Z

with the norm

v λ,α =

1/2 2

|ˆ v (n)cn (λ, α)|

.

n∈Z

For λ = 0 this is the space of 2π-periodic functions in the Sobolev space H α (0, 2π). For λ > 0 this is the space of 2π-periodic (real) analytic functions which can be extended to be complex analytic in the strip Dλ = {t + iτ | t, τ ∈ R, |τ | < λ} with the traces on the edges of the strip v(· ± iλ) ∈ H α (0, 2π). For λ < 0 the space is the dual space of A−λ,−α . We use lexicographic ordering ≺ for the indices of these spaces, i.e. (λ, α) ≺ (λ , α ) means that either λ < λ , or λ = λ and α < α . Obviously (λ, α) ≺ (λ , α ) if and only if Aλ,α ⊃ Aλ ,α . Recall that a continous linear operator between two Banach spaces is called a Fredholm operator of index 0, if its range is closed and the dimension of its null space and the codimension of its range are ﬁnite and equal. A continuous linear operator is a Fredholm operator of index 0 if and only if it can be represented as a sum of an invertible operator and a compact operator in the same spaces. If A : X → Y is a Fredholm operator of index 0, then the uniqueness of the zero solution of either Au = 0 in X or A∗ u = 0 in Y implies the unique solvability of Au = f for any f ∈ Y . The next theorem gives suﬃcient conditions for the operator A in the integral equation (3.1) to be a Fredholm operator of index 0 in appropriate function spaces. Theorem 3.1. Let the assumptions (3.3)-(3.5) be satisﬁed. Let λ, α satisfy the following conditions: i) (|λ|, max{ 12 , |α|}) ≺ (λ2 , α2 − 12 ), ii) (|λ + η|, max{ 12 , |α + θ|}) ≺ (λ1 , α1 − 12 ). Then the operator A : Aλ,α → Aλ+η,α+θ is a Fredholm operator of index 0.

ˆ 0 (n)ˆ v (n), we have A0 ∈ L(Aλ,α , Aλ+η,α+θ ) and Proof. Since A 0 v(n) = κ −1 A0 ∈ L(Aλ+η,α+θ , Aλ,α ) for any λ, α ∈ R by assumption (3.3). We will show in Section 5 that assumptions (3.4)-(3.5) together with i) and ii) imply that Aj , j = 1, . . . , k are compact operators from Aλ,α to Aλ+η,α+θ . The claim of the theorem then follows.

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Corollary 3.2. Let the assumptions (3.3)-(3.5) be satisﬁed. Let (λ0 , α0 ) satisfy the assumptions i) and ii) of Theorem 3.1 and additionally, either the equation Au = 0 has only the zero solution in Aλ0 ,α0 , or the equation A∗ u = 0 has only the zero solution in A−λ0 −η,−α0 −θ . Then for any (λ, α) satisfying the assumptions i) and ii) of Theorem 3.1, and for any f ∈ Aλ+η,α+θ , the integral equation Au = f has a unique solution in Aλ,α . Proof. First note that the uniqueness of the zero solution of Au = 0 in Aλ0 ,α0 and A∗ u = 0 in A−λ0 −η,−α0 −θ are equivalent. For (λ, α) (λ0 , α0 ) the uniqueness of the zero solution of the homogeneous equation in Aλ,α follows from the uniqueness of the zero solution of the equation in Aλ0 ,α0 . For (λ, α) (λ0 , α0 ) the uniqueness of the zero solution of the adjoint homogeneous equation in A−λ−η,−α−θ follows from the uniqueness of the zero solution of the adjoint equation in A−λ0 −η,−α0 −θ . For any (λ, α) satisfying the assumptions i) and ii) of Theorem 3.1, and for any f ∈ Aλ+η,α+θ , the unique solvability of the integral equation Au = f in Aλ,α now follows. In practice, for checking the uniqueness, it may be preferable to choose (λ0 , α0 ) = (0, 0) or (λ0 , α0 ) = (−η, −θ). Then one has to check the uniqueness of the zero solution of either Au = 0 or A∗ u = 0 in L2 (0, 2π). In case of the interior source methods, often the uniqueness of the solution of the integral equation can be obtained by using the uniqueness results for the solution of the boundary value problem for the diﬀerential equation.

4. The collocation method We look for solutions of the integral equation (3.1) in the form u(s) =

N

cl δ(s − sl ),

sl ∈ [0, 2π]

l=1

, where δ(s − sl ) are Dirac’s δ-functions with supports at sl , and require that the equation (3.1) is exactly satisﬁed at points tm ∈ [0, 2π], m = 1, . . . , N . Then one has to solve the following linear system of equations N k

κj (tm − sl )aj (tm , sl )cl = f (tm ),

m = 1, . . . , N.

(4.1)

j=0 l=1

Note that the collocation method with delta-functions can be used without explicitly knowing aj and κj , j = 0, . . . , k. In fact, let the original equation be γ

K(x, y)u(y)dsy = F (x),

x∈Γ

, with γ and Γ some closed curves. Then the collocation method is N l=1

K(xm , yl )cl = F (xm ),

m = 1, . . . , N

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, with xm , m = 1, . . . , N and yl , l = 1, . . . , N some points on Γ and γ respectively. On the other hand, to choose the points on the curves well, it is necessary to know parametrizations of Γ and γ, which transform the integral equation into a form (3.1). The collocation points xm , m = 1, . . . , N and the supports of the delta-functions yl , l = 1, . . . , N should then be chosen as the images of uniform meshes in the parameter space. Generally, the collocation method is very sensitive on the choice of the collocation points and the location of the supports of the δ-functions. In the following we assume that tl = sl = 2πl N , l = 1, . . . , N . Denote N cl δ(s − sl ) cl ∈ C . XN = l=1

1 −insl 1 −i(n+N p)sl e = 2π e Since the n-th Fourier coeﬃcient of δ(s − sl ) is 2π for any p ∈ N, the Fourier coeﬃcients of elements of XN are periodic with period N . Linear independence of the δ-functions now implies ∞ ins ˆ(n + N p) = u ˆ(n), p ∈ Z . XN = u = u ˆ(n)e u n=−∞

We will also use notations N N ZN = n ∈ Z − < n ≤ , 2 2

YN =

n∈ZN

dn eint dn ∈ C .

Then the approximate equation (4.1) can be written as QN AuN = QN f,

uN ∈ XN .

(4.2)

where QN is the trigonometric interpolation operator from Aλ+η,α+θ onto YN . The interpolation operator is well deﬁned if (λ + η, α + θ) (0, 12 ). The convergence of the collocation method is given in the following theorem. Theorem 4.1. Let the assumptions of Corollary 3.2 be satisﬁed. Additionally, let κ0 satisfy κ ˆ 0 (n) = Ccn (−η, −θ), (4.3) for some C = 0, and let λ, α satisfy the following conditions: i) (λ, α) ≺ (0, − 12 ); ii) (λ + η, α + θ) (0, 12 ). Let f ∈ Aλ∗ +η,α∗ +θ be given, where λ∗ , α∗ satisfy the assumptions i) and ii) of Theorem 3.1, and let (λ, α) ≺ (λ∗ , α∗ ). Then for N large enough the discrete equation (4.2) is uniquely solvable and its solution uN satisﬁes

uN − u λ,α ≤ C inf u − v λ,α v∈XN

≤ C u λ∗ ,α∗ max eλN N α , e(λ−λ∗ )N/2 N α−α∗ ,

(4.4)

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and

AuN − f λ+η,α+θ ≤ C f λ∗ +η,α∗ +θ max eλN N α , e(λ−λ∗ )N/2 N α−α∗ ,

(4.5) where u ∈ Aλ∗ ,α∗ is the unique solution of Au = f , and C, C and C do not depend on N . Remark 4.2. Conditions i) and ii) imply that (η, θ) (0, 1). It means that κ0 must be smooth enough for A0 to transform δ-functions into at least continuous functions, otherwise the method cannot be applied. Condition i) guarantees that XN ⊂ Aλ,α and condition ii) implies Aλ+η,α+θ ⊂ C[0, 2π]. Remark 4.3. The assumption (4.3) seems very restrictive. Nevertheless, it can be satisﬁed (redeﬁning κ0 and the other parts, if necessary), if the most singular part of the original integral equation behaves as |x − y|β if β is not a nonnegative even number, or |x − y|β ln |x − y| if β is a nonnegative even number, and x and y are points on (possibly diﬀerent) smooth curves, which can be parametrized “in the same way” (by a conformal mapping of the region involved, see [4]). This covers integral equations arising from solving two-dimensional elliptic diﬀerential equations. But (4.3) can not be satisﬁed κ0 (n)). In this case serious if the most singular part is odd (then κ ˆ 0 (−n) = −ˆ problems with the stability of the collocation method (4.1) may occur. The condition (4.3) can actually be weakened to: there is an increasing sequence of natural numbers Nk , k ∈ N, for which κ ˆ (n + N p) (4.6) 0 k ≥ Ccn (−η, −θ) ∀n ∈ ZNk p∈Z with some C > 0. In this case one should use for discretization only spaces XNk , YNk with dimensions Nk , k ∈ N. The same convergence results will hold with Nk substituted for N in (4.4) and (4.5). For example, if κ0 is odd, then for even N the method does not work, but (4.6) may still hold for Nk = 2k + 1. Outline of proof of Theorem 4.1. We use a basic result in the theory of projection methods (see e.g. Theorems 13.6 and 13.7 in [7]): Theorem 4.4. Let X and Y be Banach spaces, S : X → Y a bounded linear operator with a bounded inverse and B : X → Y a compact operator with S + B injective. Let f ∈ (S + B)(X) be given. Let XN ⊂ X and YN ⊂ Y with dim XN = dim YN = N be given and let QN : Y → YN be a projection operator. Assume that the ﬁnite dimensional operators SN := QN S : XN → YN −1 are invertible for N large enough and SN QN S : X → X are uniformly bounded. Then for N large enough the discrete equation QN (S+B)uN = QN f is uniquely solvable and its solution uN satisﬁes

uN − u X ≤ C inf u − v X , v∈XN

where u is the unique solution of (S + B)u = f .

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k We take X = Aλ,α , Y = Aλ+η,α+θ , S = A0 and B = j=1 Aj . Compactness of Aj , j = 1, . . . , k is proved in Section 5. Boundedness of the interpolation projector in Aλ+η,α+θ is proved in Section 6. We also have to show that for N large enough the operators A0,N = QN A0 : XN → YN are invertible and A−1 0,N QN A0 : Aλ,α → Aλ,α are uniformly bounded. This is also done in Section 6. To obtain the rate of convergence, one has to compute the number inf v∈XN u − v λ,α , which is estimated in Section 7. The convergence of AuN immediately follows from the fact that A ∈ L(Aλ,α , Aλ+η,α+θ ).

5. Compactness of Aj In this section we will prove that under the assumptions of Theorem 3.2, Aj , j = 1, . . . , k are compact operators from Aλ,α to Aα+θ,λ+η . First we will prove a compact embedding result for the spaces. Lemma 5.1. If (λ, α) ≺ (γ, β), then the space Aγ,β is compactly embedded into the space Aλ,α . Proof. Let I : Aγ,β → Aλ,α be the natural injection operator (i.e. it preserves the Fourier coeﬃcients). For N ∈ N deﬁne IN : Aγ,β → Aλ,α by IN v =

N

vˆ(n)eint .

n=−N

Clearly IN are compact operators. Since |ˆ v (n)e|n|λ |n|α |2

(I − I N )v 2λ,α = |n|>N

≤

sup e|n|(λ−γ) |n|(α−β)

n>N

2

|ˆ v (n)e|n|γ |n|β |2 ,

|n|>N

we have

(I − I N ) ≤ sup e|n|(λ−γ) |n|(α−β) → 0 n>N

for N → ∞,

hence I is a compact operator.

To prove compactness of Aj , j = 1, . . . , k we show that it maps the space Aλ,α into a space of “smooth enough” functions. First a result on boundedness of the multiplication operator is needed. ∞

Lemma 5.2. Let v and q be such that (ˆ v (n)cn (|λ|, |α|))n=−∞ ∈ lq with q = 1 1 if α = 0, q = 2 if |α| > 1/2 and q ∈ 1, 1−|α| otherwise. Then multiplication by v is a bounded linear operator in Aλ,α .

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Proof. Let u ∈ Aλ,α . Assume ﬁrst that α ≥ 0. Then for all m, n ∈ Z we have eλ|n| ≤ eλ|n−m| e|λm| and max{|n|, 1}α ≤ C(α) (max{|n − m|, 1}α + max{|m|, 1}α ) , cn (λ, α) ≤ C(α) e|λm| cn−m (λ, α) + eλ|n−m| cm (|λ|, α)

hence and

uv 2λ,α

=

c2n (λ, α)

n∈Z

⎛

m∈Z

2 u ˆ(n − m)ˆ v (m)

2 ≤ 2C 2 (α) ⎝ cn−m (λ, α)ˆ u(n − m)e|λm| vˆ(m) n∈Z m∈Z 2 ⎞ λ|n−m| e u ˆ(n − m)cm (|λ|, α)ˆ v (m) ⎠ . + n∈Z m∈Z

To estimate these two sums we use a variant of Young’s inequality, namely ∞ fn−m gm ≤ (fn ) lp (gn ) lp , m=−∞ l2

1 p

1 p

3 2.

To estimate the ﬁrst sum above we use where p, p ∈ [1, 2] with + = p = 2, p = 1 and note that |λn| v (n)) lq . e vˆ(n) 1 ≤ C (cn (|λ|, |α|)ˆ l

For α > 12 the second sum can be estimated similarly, exchanging the roles of u and v and using the fact that λ|n| ˆ(n) 1 ≤ max{1, |n|}−α l2 (cn (λ, α)ˆ u(n)) l2 ≤ C u λ,α . e u l

For α = 0 the second sum is the same as the ﬁrst sum so the same estimate holds. For 0 < α ≤ 12 we use Young’s inequality with p = q and the corresponding p and note that in this case by H¨ older’s inequality we have p λ|n| ˆ(n) = max{1, |n|}−pα (cn (λ, α)ˆ u(n))p e u lp

n∈Z

≤

n∈Z

p/2 2

(cn (λ, α)ˆ u(n)) p

pα − 1−p/2

1− p2

max{1, |n|}

n∈Z

= C u λ,α because mate

pα > 1 by assumption. In all cases we have obtained the esti1 − p/2

uv λ,α ≤ C (cn (|λ|, |α|)ˆ v (n)) lq u λ,α .

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For α < 0 note that for any u ∈ Aλ,α and w ∈ A−λ,−α we have uv, w = u, vw, where ·, · denotes the duality pairing between Aλ,α and A−λ,−α . Hence the multiplication operator is bounded in Aλ,α if and only if it is bounded in A−λ,−α . This completes the proof of the lemma. Corollary 5.3. If v satisﬁes |ˆ v (n)| ≤ Ccn (−γ, −β), then multiplication by vis 1 for all λ, α which satisfy |λ|, max{ , |α|} ≺ a bounded linear operator in A λ,α 2 1 γ, β − 2 . Proof. For |λ| < γ the assumptions of Lemma 5.2 are obviously satisﬁed. For |λ| = γ and α = 0 or |α| > 12 they are also easy to check. For |λ| = γ and 0 < |α| ≤ 12 note that the assumption of the corollary reduces to β > 1 and 1 1 therefore one can ﬁnd q ≥ 1 such that β−|α| < q < 1−|α| . With such q, the assumptions of Lemma 5.2 are satisﬁed. Lemma 5.4. Let the assumptions (3.4)-(3.5) together with i) and ii) of Theorem 3.1 be satisﬁed. Let ε1 be such that 0 < ε1 ≤ ε, and if |λ + η| = λ1 , then additionally ε1 < α1 − 12 − α − θ. Then Aj , j = 1, . . . , k are bounded linear operators from Aλ,α to Aλ+η,α+θ+ε1 and hence compact operators from Aλ,α to Aλ+η,α+θ . Proof. From ii) it follows that if |λ + η| = λ1 , then α1 − 12 − α − θ > 0, so that it is always possible to choose ε1 satisfying the assumptions. We can estimate the Fourier coeﬃcients of Aj u as follows: κ ˆ j (n)ˆ a(k − n, n − l)ˆ u(l) |(Aj u)(k)| = l,n∈Z ≤C ck−n (−λ1 , −α1 )cn (−η, −θ − ε)cn−l (−λ2 , −α2 )|ˆ u(l)|. l,n∈Z

The last expression can be considered as the Fourier coeﬃcients of the composition of the following three operators applied to |ˆ u(n)|eint : i) multiplication by

n∈Z

ii) convolution with

n∈Z

iii) multiplication by

n∈Z

n∈Z ins

cn (−λ2 , −α2 )e

,

cn (−η, −θ − ε)eins , and cn (−λ1 , −α1 )eint .

The convolution is a bounded linear operator from Aλ,α to Aλ+η,α+θ+ε , hence also to Aλ+η,α+θ+ε1 . By Corollary 5.3, under the assumptions i) and ii), the multiplication operators are bounded in Aλ,α and correspondingly 1 A . Here ε was chosen so that |λ + η|, max{ , |α + θ + ε1 |} ≺ 1 2 λ+η,α+θ+ε 1 λ1 , α1 − 12 to satisfy the assumptions of Corollary 5.3. It follows that Aj , j = 1, . . . , k are also bounded linear operators from Aλ,α to Aλ+η,α+θ+ε1 .

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6. Properties of the interpolation operator and A0,N In this section ﬁrst the uniform boundedness of QN in suitable spaces is proved. Then, for large enough N , invertibility of A0,N = QN A0 : XN → YN and uniform boundedness of A−1 0,N QN A0 : Aλ,α → Aλ,α is shown. First we need an auxiliary result. Lemma 6.1. Let (γ, β) ≺ 0, − 12 . Then c2n+N p (γ, β) ≤ Cc2N (γ, β)e−2γ|n| ∀N ≥ 1, ∀n ∈ ZN , p∈Z,p=0

with C depending only on γ and β. Proof. For n ∈ ZN we have

∞

c2n+N p (γ, β) =

p=1

p∈Z,p=0

≤N

2β 2γ(N −|n|)

e

e2γ(n+N p) (n + N p)2β + e2γ(N p−n) (N p − n)2β

∞

2γN (p−1)

e

2 max

p=1

1 p+ 2

2β 2β 1 , p− 2

≤ CN 2β e2γ(N −|n|) = Cc2N (γ, β)e−2γ|n| , since the last sum can be estimated by a constant not dependending on N . Next, the boundedness of QN in our spaces can be proved. Lemma 6.2. If (γ, β) 0, 12 , then the interpolation projectors QN are uniformly bounded in Aγ,β . Proof. Note that for p ∈ Z and n ∈ ZN we have hence for g ∈ Aγ,β

QN ei(n+N p)t = QN eint = eint , with (γ, β) 0, 12 we have Q g(n + N p). N g(n) = p∈Z

Let PN be the orthogonal projection onto YN . For any g ∈ Aγ,β we have 2

QN g − PN g 2γ,β = c2n (γ, β) g(n + N p) p∈Z,p=0 n∈ZN ⎞ ⎛ 1 2 ≤ c2n (γ, β) ⎝ c2n+N p (γ, β) | g (n + N p)| ⎠ c2n+N p (γ, β) n∈ZN p∈Z,p=0 p∈Z,p=0 ⎞ ⎛ 1 ⎠ g − PN g 2γ,β . ≤ max ⎝c2n (γ, β) n∈ZN c2n+N p (γ, β) p∈Z,p=0

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1 = c2n+N p (−γ, −β), it follows from Lemma 6.1 that c2n+N p (γ, β) (we are using the lemma with (−γ, −β) instead of (γ, β)) Since

QN − PN 2γ,β ≤ Cc2N (−γ, −β) max e2γ|n| c2n (γ, β). n∈ZN

For N large enough the maximum here is obtained at |n| = [N/2]. By substituting it into the estimate above, we get an uniform bound on QN − PN γ,β and hence the interpolation projectors QN are uniformly bounded in Aγ,β . 6.3. Actually from the proof it follows that for g ∈ Aγ,β with (γ, β) Remark 0, 12 we have

QN g − g γ,β ≤ QN g − PN g γ,β + PN g − g γ,β ≤ C PN g − g γ,β , which implies convergence of QN g to g for any g ∈ Aγ,β . Now we prove invertibility of QN A0 in ﬁnite dimensional subspaces. Lemma 6.4. Assume that (4.3) and assumptions i) and ii) of Theorem 4.1 are satisﬁed. Then the operators A0,N = QN A0 : XN → YN are invertible and A−1 0,N QN A0 are uniformly bounded in Aλ,α . Proof. Since for w ∈ XN A0,N w(t) =

n∈ZN

=

n∈ZN

for g ∈ YN we have

which is deﬁned if

⎝

⎞ κ ˆ 0 (n + N p)w(n ˆ + N p)⎠ eint

p∈Z

⎛ ⎝

⎞ int κ ˆ 0 (n + N p)⎠ w(n)e ˆ ,

p∈Z

⎞

⎛

A−1 0,N g(t) =

⎛

⎜ ⎟ i(n+N m)t ⎟e ⎜ gˆ(n) , ⎠ ⎝ κ ˆ 0 (n + N p) m∈Z n∈ZN p∈Z

κ ˆ 0 (n + N p) = 0 for n ∈ ZN (from this one can see that

p∈Z

some condition guaranteeing that the sum is not “too small” is needed, e.g. condition (4.3) or its weakened form (4.6) ). To show the uniform boundedness of A−1 0,N QN A0 ﬁrst note that A0 is bounded from Aλ,α to Aλ+η,α+θ and, by Lemma 6.2, QN is bounded in Aλ+η,α+θ . We only have to prove that for any g ∈ YN we have A−1 0,N g λ,α ≤ κ0 (n)| for n ∈ ZN , and C g λ+η,α+θ . By (4.3) we have κ ˆ 0 (n + N p) > |ˆ p∈Z

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hence 2

A−1 0,N g λ,α

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279

2 gˆ(n) = c2n+mN (λ, α) κ ˆ 0 (n + N p) m∈Z n∈ZN p∈Z gˆ(n) 2 ≤ c2n+mN (λ, α) κ ˆ 0 (n) n∈ZN m∈Z ⎛ 2 ⎞ cn+N m (λ, α) gˆ(n) 2 ⎜ m∈Z ⎟ 2 ⎜ ⎟ ≤ max ⎝ cn (λ, α) ⎠ n∈ZN c2n (λ, α) κ ˆ 0 (n) n∈Z

≤ max

n∈ZN

1+

CN 2α e2λ(N −|n|) max{1, |n|}2α e2λ|n|

2 2

A−1 0 g λ,α ≤ C g λ+η,α+θ

(we used Lemma 6.1 and the fact that for N large, the maximum here is obtained at |n| = [N/2].

7. The approximation properties of XN The rate of convergence is determined by inf v∈XN u − v λ,α , which depends on the regularity of the solution. Under assumptions of Theorem 4.1 the solution u belongs to Aλ∗ ,α∗ . We can estimate the rate of convergence in spaces Aλ,α with (λ, α) ≺ (λ∗ , α∗ ). Lemma 7.1. Let u ∈ Aλ∗ ,α∗ be given and let (λ, α) ≺ (λ∗ , α∗ ) and (λ, α) ≺ (0 − 12 ). Then

inf u − v λ,α ≤ C u λ∗ ,α∗ max eλN N α , e(λ−λ∗ )N/2 N α−α∗ . v∈XN

Proof. Let PN denote the orthogonal projection to YN . Then inf u − v λ,α ≤ u − PN u λ,α + inf PN u − v λ,α .

v∈XN

v∈XN

Estimate these terms separately:

u − PN u 2λ,α = | u(n)|2 c2n (λ, α) n∈ZN

≤ sup

n∈ZN

c2n (λ, α) | u(n)|2 c2n (λ∗ , α∗ ) c2n (λ∗ , α∗ )

(λ−λ∗ )N

≤ Ce

n∈ZN

N

2(α−α∗ )

u − PN u 2λ∗ ,α∗ .

To estimate inf v∈XN PN u − v λ,α note that v for which the inﬁmum is obtained, can be written down explicitly: its Fourier’ coeﬃcients are u (n)c2n (λ, α) vˆ(n) = , c2n+N k (λ, α) k∈Z

n ∈ ZN ,

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and vˆ(n + N p) = vˆ(n), n ∈ ZN , p ∈ Z. Hence inf PN u − v 2λ,α = PN u 2λ,α − v 2λ,α

v∈XN

=

| u(n)|2 c2n (λ, α) −

n∈ZN

=

| u(n)|2 c2n (λ, α)

n∈ZN

≤

| u(n)|2 c4n (λ, α) c2n+N k (λ, α) n∈ZN

k∈Z

c2n+N k (λ, α)

k∈Z,k=0

c2n+N k (λ, α)

k∈Z

| u(n)|2 c2N (λ, α)e−2λ|n|

n∈ZN

e−2λ|n| | u(n)|2 c2n (λ∗ , α∗ ) n∈ZN c2 n (λ∗ , α∗ ) n∈Z N

2 2λN 2α (λ−λ∗ )N 2(α−α∗ ) N ,e N ≤ C u λ∗ ,α∗ max e ≤ c2N (λ, α) max

(the last inequality follows from the fact that for large N the maximum over n is obtained at |n| = [N/2] or at a place, which does not depend on N and can therefore be absorbed in the constant). In the last estimate the ﬁrst term under the maximum will be used when u is smooth enough to get the optimal convergence of the method in Aλ,α , i.e (λ∗ , α∗ ) (−λ, 0), the second term expresses the loss in the convergence rate if the solution is not smooth enough. This completes the proof of Theorem 4.1.

8. Conclusions We saw that even for integral equations of ﬁrst kind with analytic kernels with the dominant part of convolution type it is possible to choose spaces in which the problem is well posed. The collocation method on uniform meshes and with δ-functions as the trial functions is then exponentially convergent in suitable norms. The choice of the particular method is motivated by simplicity of the discrete equations. Also, when solving diﬀerential equations by an interior source method, the resulting solution of the diﬀerential equation is of very simple form and easy to calculate at an arbitrary point. Exponential convergence of the method means that an accurate solution can be calculated with relatively few degrees of freedom in discrete equations. Since the problem is ill-posed in usual function spaces, the resulting discrete equations are usually very badly conditioned and the dependence of the condition number on the distance of the interior boundary and the physical boundary is exponential. On the other hand, the convergence rate also gets better exponentially (η being roughly a measure of the distance between the boundaries). Hence the distance between the boundaries is of

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crucial importance. In our numerical experiments, when the interior boundary was chosen so that the solution of the exterior problem was analytically extendable to some distance across the interior boundary, then no additional conditioning was needed in solving the discrete equations. In case the singularities of the extension of the solution are not enclosed inside the interior boundary one can use FFT to solve the integral equtions and cut down the size of matrices representing Aj , j = 1, . . . , k. The discrete version of A0 is diagonal after FFT and therefore easy to invert. In case where the boundary is not analytic, it is very hard to ﬁnd a suitable interior boundary for the collocation method to work. In this case we propose to use, instead of the collocation method, a least squares method, where one chooses more points on the physical boundary, and solves the resulting overdetermined system in the least squares sense. This method is not very sensitive on the choice of the interior boundary, nor on the choice of the supports of the δ-functions, and there is no need to transform the integral equation into form (3.1). The method is convergent on very mild conditions. On the other hand, it is very hard to obtain results on the rate of convergence of this method in cases where the collocation method is not convergent (it is easy to see that if the collocation method is convergent, the convergence rate is at least the same). Some numerical results will be presented in [4].

References [1] Am. Boag, Y Leviatan, Al. Boag, Analysis of electromagnetic scattering using a current model method. Comp. Phys. Comm., 68 (1991), 331-345. [2] A. Doicu, Y. A. Eremin, T. Wriedt, Acoustic & Electromagnetic Scattering Analysis Using Discrete Sources. Academic Press, 2000. [3] R. Kangro, U. Kangro, R.A. Nicolaides, Extendability of Solutions of Helmholtz’s Equation to the Interior of a Two Dimensional Scatterer. Quarterly of Applied Math., 58, 3 (2000), 591–600. [4] U. Kangro, Convergence of the interior source method with point matching for two dimensional scattering problems: the case of analytic boundary. To be submitted. [5] M. Katsurada, Charge Simulation Method Using Exterior Mapping Functions. Japan J. Indust. Appl. Math., 11 (1994), 47-61. [6] A. I. Kleev and A. B. Manenkov, The convergence of point matching techniques. IEEE Trans. Ant. and Prop., 37, 1, (1989), 50–54. [7] R. Kress, Linear Integral Equations. Springer-Verlag 1999. [8] R. Kress, A. Mohsen, On the simulation source technique for exterior problems in acoustics. Math. Meth. in the Appl. Sci., 8, (1986), 585-597. [9] Y. Leviatan, Analytic continuation considerations when using generalized formulations for scattering problems. IEEE Trans. Ant. Prop., 38, (1990), 1259– 1263. [10] K. Ruotsalainen, J. Saranen, Some boundary element methods using Dirac’s distributions as trial functions. SIAM J. Numer. Anal., 24, 4 (1987), 816-827.

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[11] J. Saranen, G. Vainikko, Periodic Integral and Pseudodiﬀerential Equations with Numerical Approximation. Springer Monographs in Mathematics, Springer-Verlag 2002. Urve Kangro University of Tartu J. Liivi 2 50409 Tartu Estonia e-mail: [email protected] Received: December 19, 2008.

Integr. Equ. Oper. Theory 66 (2010), 283–292 DOI 10.1007/s00020-010-1745-3 Published online January 26, 2010 © Birkhäuser Verlag Basel/Switzerland 2010

Integral Equations and Operator Theory

New Formulae for the Wave Operators for a Rank One Interaction Serge Richard and Rafael Tiedra de Aldecoa Abstract. We prove new formulae for the wave operators for a Friedrichs scattering system with a rank one perturbation, and we derive a topological version of Levinson’s theorem for this model. Mathematics Subject Classiﬁcation (2010). Primary 47A40; Secondary 81U15. Keywords. Friedrichs model, wave operators, Levinson’s theorem.

1. Introduction and main results Let us consider the Hilbert space H := L2 (R) with norm · and scalar product ·, ·, and let H0 ≡ X be the self-adjoint operator of multiplication by the variable, i.e. (H0 f )(x) = xf (x) for any f ∈ D(H0 ) ≡ L2 R, (1+x2 ) dx . For u ∈ H, we also consider the rank one perturbation of H0 deﬁned by Hu f := H0 f + u, f u,

f ∈ D(H0 ).

It is well known that the wave operators Ω± := s- limt→±∞ eiHu t e−iH0 t exist and are asymptotically complete, and that the scattering operator S := Ω∗+ Ω− is a unitary operator in H. In fact, S ≡ S(X) is simply an operator of multiplication by a function S : R → T, with T the set of complex numbers of modulus 1. A rather explicit formula for the wave operators for this model was proposed in [8]. Its expression involves singular integrals that have to be manipulated with some care. In this Note, we propose a simpler formula for the wave operators, and put into light a straightforward consequence of it. However, we stress that contrary to [8], our formula and its corollary hold only if some additional (but weak) hypotheses on u are imposed. To state our results, and in particular to have an explicit formula of the Hilbert transform in terms of the generator of dilations (see Equation (2.3) below), we need to introduce the even / odd representation of H. Given any function m on R, we write me and mo for the even part and the odd part of

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2 2 m. We also set H := L + ; C ) and introduce the unitary map U : H → H f(R 1 given on any f ∈ H, f2 ∈ H , x ∈ R by √ f 1 f U f := 2 e and U ∗ 1 (x) := √ [f1 (|x|) + sgn(x)f2 (|x|)] . fo f2 2

Now, observe that if m is a function on R and A the generator of dilations in H, then we have me (X+ ) mo (X+ ) A+ 0 ∗ ∗ U m(X)U = and U AU = , mo (X+ ) me (X+ ) 0 A+ where X+ is the operator of multiplication by the variable in L2 (R+ ), and A+ the generator of dilations in L2 (R+ ), namely (eitA+ f )(x) := et/2 f (et x) for f ∈ L2 (R+ ), x ∈ R+ . In the sequel we assume that the vector u satisﬁes the following assumption. Assumption 1.1. The function u ∈ H is H¨ older continuous with exponent α > 0. Furthermore, if x0 ∈ R satisﬁes u(x0 ) = 0 and 1 − R dy |u(y)|2 (x0 − y)−1 = 0, then there exists an exponent α > 1/2 such that

|u(x) − u(y)| ≤ Const. |x − y|α for all x, y near x0 .

This assumption implies that u is bounded and satisﬁes lim|x|→∞ u(x) = 0. Moreover, it is known that under Assumption 1.1, the operator Hu has at most a ﬁnite number of eigenvalues of multiplicity one [1, Sec. 2] (see also [6, 11]). Clearly, Assumption 1.1 is satisﬁed if u ∈ H is H¨older continuous with exponent α > 1/2. Our main result is the following representation of the wave operator Ω− in H . Theorem 1.2. Let u satisfy Assumption 1.1. Then, one has 1 0 1 −φ(A+ ) ∗ 1 U Ω− U = +2 0 1 1 −φ(A+ ) So (X+ ) Se (X+ ) − 1 + K, × Se (X+ ) − 1 So (X+ )

(1.1)

where φ(A+ ) := tanh(πA+ ) + i[cosh(πA+ )]−1 and K is a compact operator in H . Let us immediately mention that a similar formula holds for Ω+ . Indeed, by using Ω+ = Ω− S(X) one gets 1 0 φ(A+ ) 1 ∗ 1 +2 U Ω+ U = 0 1 1 φ(A+ ) Se (X+ ) − 1 So (X+ ) × + K , So (X+ ) Se (X+ ) − 1

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where K is a compact operator in H . We also note that the proof of Theorem 1.2 will make clear why the minimal hypothesis u ∈ H is not suﬃcient in order to prove the claim. We now state a corollary of the above theorem which corresponds to a topological version of Levinson’s Theorem: Corollary 1.3. Let u satisfy Assumption 1.1. Then S(±∞) = 1 and the following equality holds: ω(S) = − number of eigenvalues of Hu , where ω(S) is the winding number of the continuous map S : R → T. Such a result was already known for more general perturbations but under stronger regularity conditions [2, 4] (see also [6, 14] for general information on the Friedrichs model). Our result does require neither the diﬀerentiability of the scattering matrix nor the diﬀerentiability of u. Nonetheless, if S is continuously diﬀerentiable, then the winding number can also be expressed in terms of an integral involving the (on-shell) time delay operator, which is the logarithmic derivative of the scattering matrix [13]. Remark 1.4. The authors emphasize that Corollary 1.3 is a straightforward consequence of formula (1.1), even if its proof requires the algebraic framework presented in Section 3. They do not doubt that for smooth u this result can also be obtained via more analytical technics, but one of their motivations was to show that Levinson’s Theorem can directly be inferred from the explicit formula (1.1), without any further analysis (see [10] and references therein for similar proofs of Levinson’s Theorem for other scattering systems). The content of this Note is the following. In Section 2 we prove Formula (1.1) and derive some auxiliary results. In Section 3 we give a description of the algebraic framework involved in the proof of the Corollary 1.3, which is proved at the end of the section.

2. Derivation of the new formula We start by recalling some notations and results borrowed from [1] and [8]. We shall always suppose that u satisﬁes Assumption 1.1. For x ∈ R and ε > 0 we set

|u(y)|2 ε I± (x) := dy . x − y ± iε R ε By Privalov’s theorems, the limit I± (x) := limε0 I± (x) exists for all x ∈ R. Furthermore, the set of x such that I± (x) = 1 is ﬁnite [1, p. 396]. In consequence the expression [1 − I± (x)]−1 is well deﬁned for almost every x ∈ R, and the domain D± of [1 − I± (X)]−1 in H is dense.

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Let F denote the Fourier transform in H, namely

1 (F f )(x) := √ dy e−ixy f (y), f ∈ H ∩ L1 (R). 2π R Given a Borel function m on R, we set m(D) := F ∗ m(X)F . Finally, χ(−∞,0) stands for the characteristic function for the half-line (−∞, 0). We are now in a position to recall the formula [8, eq. (56b)] for Ω− . One has Ω− = 1 − 2πiu(X)χ(−∞,0) (D)u(X)[1 − I+ (X)]−1 on D+ (note that we use a convention for the sign ± of the wave operators Ω± which diﬀers from the one of [8]). So, if K := −2πi[u(X), χ(−∞,0) (D)]u(X)[1− I+ (X)]−1 , then one gets on D+ Ω− − 1 = −2πiu(X)χ(−∞,0) (D)u(X)[1 − I+ (X)]−1 = χ(−∞,0) (D) − 2πi|u(X)|2 [1 − I+ (X)]−1 + K = χ(−∞,0) (D){S(X) − 1} + K,

(2.1)

by using [8, Eq. (66b)] in the last equality. In the next lemma, we determine the action of χ(−∞,0) (D) in H . For that purpose, we deﬁne φ ∈ C([−∞, ∞]; T) by φ(x) := tanh(πx) + i[cosh(πx)]−1 for all x ∈ R. Lemma 2.1. One has U χ(−∞,0) (D)U ∗ = Φ(A+ ), where 1 −φ(A+ ) Φ(A+ ) := 12 . 1 −φ(A+ ) Proof. The usual Hilbert transform H on R satisﬁes sgn(D) = iH. Thus (2.2) χ(−∞,0) (D) = 12 1 − sgn(D) = 12 (1 − iH). Using the expression for iH in terms of the generator of dilations in H given in [10, Lemma 3], one gets U iHU ∗ 0 = tanh(πA+ ) + i[cosh(πA+ )]−1

(2.3) tanh(πA+ ) − i[cosh(πA+ )]−1 . 0

The claim follows then from (2.2) and (2.3).

We now recall some results on the scattering matrix. Lemma 2.2. Let u satisfy Assumption 1.1. Then the map S belongs to C([−∞, ∞]; T) and satisﬁes S(±∞) = 1. Proof. The continuity of S follows from [1, Thm. 1.(i)]. The equalities S(±∞) = 1 follow from the formula S(x) − 1 = −2πi|u(x)|2 [1 − I+ (x)]−1 together with Lemma 1.(a) of [1] and the fact that lim|x|→∞ |u(x)|2 = 0. The last lemma deals with the remainder term K of Formula (1.1).

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Lemma 2.3. Let u satisfy Assumption 1.1. Then the operator [u(X), χ(−∞,0) (D)]u(X)[1 − I+ (X)]−1 , deﬁned on D+ , extends to a compact operator in H. Proof. (i) Deﬁne for all x ∈ R the function ψ(x) := u(x)[1 − I+ (x)]−1 . We know that u is bounded and that lim|x|→∞ u(x) = 0. We also know from [1, Lemma 1.(a)] that I+ is H¨older continuous and that lim|x|→∞ I+ (x) = 0. So, outside any neighbourhood of the ﬁnite set of points where I+ equals 1, the function ψ is bounded. Furthermore, Assumption 1.1 and [1, Lemma 1.(c)], imply that ψ is locally square integrable (see also [8, p. 2423]). Therefore, ψ can be written as ψ = ψ∞ + ψ2 , with ψ∞ ∈ L∞ (R) and ψ2 ∈ L2 (R) with support in a small neighbourhood of the points where I+ equals 1. We now show the compacity of the operator [u(X), χ(−∞,0) (D)]ψ∞ (X) and of the operator [u(X), χ(−∞,0) (D)]ψ2 (X). (ii) Choose a function ϕ1 ∈ C ∞ (R) and a function ϕ2 ∈ L∞ (R) with compact support such that ϕ1 + ϕ2 = χ(−∞,0) . Then [u(X), ϕ1 (D)] is compact due to [3, Thm. C], and [u(X), ϕ2 (D)] is Hilbert-Schmidt due to [12, Thm. 4.1]. So [u(X), χ(−∞,0) (D)]ψ∞ (X) = [u(X), ϕ1 (D)]ψ∞ (X) + [u(X), ϕ2 (D)]ψ∞ (X) is a compact operator. (iii) For each f ∈ H and almost every x ∈ R, deﬁne the operator

i u(x) − u(y) ψ2 (y)f (y). dy (K0 f )(x) := 2π R y−x From the Assumption 1.1 we know that

|u(y + x) − u(y)| ≤ Const. |x|α ,

α > 1/2

for each y ∈ supp(ψ2 ) and each x ∈ R with |x| small enough. In particular, there exists δ > 0 such that

2

i u(x) − u(y)

2

ψ2 (y)

dx dy 4π 2π y−x R2

|u(y + x) − u(y)|2 = dy dx |ψ2 (y)|2 x2 R R

|u(y + x) − u(y)|2 = dy dx |ψ2 (y)|2 x2 R R\[−δ,δ]

δ |u(y + x) − u(y)|2 + dy dx |ψ2 (y)|2 2 x −δ R

δ 4u2∞ 2 ≤ dy dx |ψ2 (y)| + Const. dy dx |x|2(α −1) |ψ2 (y)|2 2 x R R\[−δ,δ] R −δ < ∞.

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Thus, K0 is a Hilbert-Schmidt operator. Furthermore, we have for f ∈ D+ and almost every x ∈ R i [u(X), χ(−∞,0) (D)]ψ2 (X)f (x) = − [u(X), H]ψ2 (X)f (x) 2

u(x) − u(y) i ψ2 (y)f (y) dy =− 2π R x−y = (K0 f )(x). Therefore, the operator [u(X), χ(−∞,0) (D)]ψ2 (X) extends to a HilbertSchmidt operator. Proof of Theorem 1.2. The operator K extends to a compact operator due to Lemma 2.3. So Equation (2.1) holds on H, and the claim follows from Lemma 2.1. Remark 2.4. The proof of Corollary 1.3 relies on the fact that the range of the wave operators is the orthocomplement of the subspace spanned by the eigenvectors of Hu . Since the wave operators are complete, such a property holds if and only if Hu has no singularly continuous spectrum. Now, by using the characterization of the singular spectrum recalled in [5, p. 299] and by taking into account Lemmas 1 and 2 of [1] (which are valid since u satisﬁes Assumption 1.1), one easily gets that the singular spectrum of Hu only consists of a ﬁnite set. So Assumption 1.1 implies the absence of singularly continuous spectrum for Hu .

3. Algebraic framework This section is dedicated to the presentation of the algebraic framework leading to Corollary 1.3. Since a similar construction already appears in [10] for the proof of Levinson’s theorem in one dimensional potential scattering, we only sketch the construction and refer to this reference for more details. We start by giving the deﬁnition of the Mellin transform associated with the generator of dilations A+ in L2 (R+ ) (see [9, Sec. 2] for a general presentation when the operator acts in L2 (Rn )). Let V : L2 (R+ ) → L2 (R) be deﬁned by (V f )(x) := ex/2 f (ex ) for x ∈ R, and remark that V is a unitary map with adjoint V ∗ given by (V ∗ g)(x) = x−1/2 g(ln x) for x ∈ R+ . Then, the Mellin transform M : L2 (R+ ) → L2 (R) is deﬁned by M := F V . Its main property is that it diagonalizes the generator of dilations, namely, M A+ M ∗ = X. Formally, one also has M ln(X+ )M ∗ = −D. Let us now recall from Remark 2.4 that under the Assumption 1.1 the wave operators Ω± are isometries with range projection 1 − Pp , where Pp is the projection onto the subspace spanned by the ﬁnite number N of eigenvectors of Hu . In particular, Ω− is a Fredholm operator with index(Ω− ) = − Tr(Pp ) = −N . Furthermore, we recall that any Fredholm operator F in H is invertible modulo a compact operator, that is, its image q(F ) in the Calkin algebra B(H)/K(H) is invertible.

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Now, assume that Ω− belongs to a norm-closed subalgebra E of B(H) containing K(H). Moreover, assume that E/K(H) is isomorphic to C S; M2 (C) , the algebra of continuous functions over the circle with values in the 2 × 2 matrices. Then, viewing q(Ω− ) as such a function, we can take pointwise its determinant to obtain a non-vanishing function over the circle. The winding number of that latter function can be related to the index of Ω− ; this is essentially the content of Corollary 1.3. The choice of E is suggested by the formula obtained in Theorem 1.2. Indeed, we consider the closure E in B(H ) of the algebra generated by elements of the form ϕ(A+ )ψ(X+ ), where ϕ is a continuous function on R with values in M2 (C) which converges at ±∞, and ψ is a continuous function on Stated diﬀerR+ with values in M2 (C) which converges at 0 and at +∞. ently, ϕ ∈ C R; M2 (C) with R = [−∞, ∞], and ψ ∈ C R+ ; M2 (C) with R+ = [0, ∞]. Let J be the norm closed algebra generated by ϕ(A+ )ψ(X+ ) with functions ϕ and ψ for which the above limits vanish. Then, J is an ideal in E , and the are obtained ifψ(X+ ) is replaced by η(ln(X+ )) same algebras with η ∈ C R; M2 (C) or η ∈ C0 R; M2 (C) , respectively. These algebras have already been studied in [7] in a diﬀerent context. The authors constructed them in terms of the operators X and D on L2 (R, E), with E an auxiliary Hilbert space, possibly of inﬁnite dimension. In that situation, ϕ and η are norm continuous functions on R with values in K(E). The isomorphism between our algebras and the algebras introduced in [7, Sec. 3.5], with E = C2 , is given by the Mellin transform M , or more precisely by M ⊗ 1, where 1 is identity operator in M2 (C). For that reason, we shall freely use the results obtained in that reference, and refer to it for the proofs. For instance, it is proved that J = K(H ), and an explicit description of the quotient E /J is given, which we specify now in our context. To describe the quotient E /J , we consider the square := R+ × R, whose boundary ∂ is the union of four parts: ∂ ≡ B1 ∪ B2 ∪ B3 ∪ B4 , with It B1 := {0} × R, B2 := R+ × {∞}, B3 := {∞} × R, and B4 := R+ × {−∞}. is proved in [7, Thm. 3.20] that E /J is isomorphic to C ∂; M2 (C) . This algebra can be seen as the subalgebra of C R; M2 (C) ⊕ C R+ ; M2 (C) ⊕ C R; M2 (C) ⊕ C R+ ; M2 (C) (3.1) given by elements γ ≡ (γ1 , γ2 , γ3 , γ4 ) which coincide at the corresponding end points, that is, γ1 (∞) = γ2 (0), γ2 (∞) = γ3 (∞), γ3 (−∞) = γ4 (∞), and γ4 (0) = γ1 (−∞). Furthermore, for any ϕ ∈ C R; M2 (C) and ψ ∈ C R+ ; M2 (C) , the image of the operator ϕ(A+ )ψ(X+ ) through the quotient map q : E → C ∂; M2 (C) is given by γ1 = ϕψ(0), γ2 = ϕ(∞)ψ, γ3 = ϕψ(∞) and γ4 = ϕ(−∞)ψ. From what precedes we deduce that the subalgebra E of B(H), deﬁned operators by E := U ∗ E U , contains the ideal of compact on H and that the quotient E/K(H) is isomorphic to C ∂; M2 (C) ∼ = C S; M2 (C) . We are thus in the framework deﬁned above, and the for any invertible element

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γ of C ∂; M2 (C) , the winding number of its pointwise determinant is a well-deﬁned quantity. So we are ready to give the proof of Corollary 1.3. Proof of Corollary 1.3. We know from Theorem 1.2 and Lemma 2.2 that U Ω− U ∗ ∈ E , or equivalently that Ω− ∈ E. Due to Formula (1.1), the element γ belonging to (3.1) and associated with q(Ω− ) is given by suitable restrictions of the function Γ : R+ × R → M2 (C), where s (x) − 1 so (x) Γ(x, y) := 1 + Φ(y) e se (x) − 1 so (x) se (x) − φ(y)so (x) + 1 so (x) − φ(y)[se (x) − 1] . = 12 se (x) − φ(y)so (x) + 1 so (x) − φ(y)[se (x) − 1] Namely, γ1 = Γ(0, · ), γ2 = Γ( · , +∞), γ3 = Γ(+∞, · ), and γ4 = Γ( · , −∞). The pointwise determinants of these functions are easily calculated by using the identity φ(±∞) = ±1: one gets det γ1 (y) = se (0), det γ2 (x) = s(−x), det γ3 (y) = 1 and det γ4 (x) = s(x). The precise relation between the winding number of the map det γ : ∂ → T and the index of Ω− has been described in [10, Prop. 7]. However, the algebra corresponding to E in that reference was deﬁned in terms of the operators A+ and B+ = 12 ln (D2 )+ which satisfy the relation [iA+ , B+ ] = −1. In our case, the algebra E has been constructed with the operators A+ and ln(X+ ) which satisfy the relation [iA+ , ln(X+ )] = 1. Therefore, in order to apply [10, Prop. 7] in our setting, one previously needs to apply the au tomorphism of C R; M2 (C) deﬁned by η(x) := η(−x) for all x ∈ R, or := ψ(x−1 ) equivalently the automorphism of C R+ ; M2 (C) deﬁned by ψ(x) j assofor all x ∈ R+ . Therefore the pointwise determinants of the function γ 1 (y) = 1, det γ 2 (x) = s(−x−1 ), det γ 3 (y) = se (0) ciated with q(Ω− ) are det γ and det γ 4 (x) = s(x−1 ). Now, [10, Prop. 7] reads ω(det γ ) = index(Ω− ) = −N , where N is the number of eigenvalues of Hu . The convention used in that reference for the calculation of the winding number implies that the contribution of x → 4 (x) is det γ 2 (x) is from x = 0 to x = +∞ and the contribution of x → det γ from x = +∞ to x= 0. This corresponds to the calculation of the winding number of x → det S(x) , from x = −∞ to x = +∞. Since the contributions 3 are null because these terms are constant, the claim is of det γ 1 and det γ proved. Acknowledgment S. Richard is supported by the Swiss National Science Foundation. R. Tiedra de Aldecoa is partially supported by N´ ucleo Cient´ıﬁco ICM P07-027-F “Mathematical Theory of Quantum and Classical Magnetic Systems” and by the Chilean Science Foundation Fondecyt under the Grant 1090008.

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References [1] P. Alsholm, Inverse scattering theory for perturbations of rank one, Duke Math. J. 47 no. 2 (1980), 391–398. [2] V. S. Buslaev, Spectral identities and the trace formula in the Friedrichs model, in Spectral theory and wave processes, 43–54. Consultants Bureau Plenum Publishing Corporation, New York, 1971. [3] H. O. Cordes, On compactness of commutators of multiplications and convolutions, and boundedness of pseudodiﬀerential operators, J. Funct. Anal. 18 (1975), 115–131. [4] T. Dreyfus, The determinant of the scattering matrix and its relation to the number of eigenvalues, J. Math. Anal. Appl. 64 no. 1 (1978), 114–134. [5] E. M. Dyn kin, S. N. Naboko, S. I. Yakovlev, A ﬁniteness bound for the singular spectrum in a selfadjoint Friedrichs model, St. Petersburg Math. J. 3 no. 2 (1992), 299–313. [6] L. D. Faddeev, On a model of Friedrichs in the theory of perturbations of the continuous spectrum, in American Mathematical Society Translations. Series 2, Vol. 62: Five papers on functional analysis, 177–203, American Mathematical Society, Providence, R.I. 1967. [7] V. Georgescu, A. Iftimovici, C ∗ -algebras of quantum Hamiltonians, in Operator Algebras and Mathematical Physics, Conference Proceedings: Constant¸a (Romania) July 2001, 123–167, Theta Foundation, 2003. [8] M. A. Grubb, D. B. Pearson, Derivation of the wave and scattering operators for an interaction of rank one, J. Mathematical Phys. 11 (1970), 2415–2424. [9] A. Jensen, Time-delay in potential scattering theory, some ”geometric” results, Comm. Math. Phys. 82 no. 3 (1981/82), 435–456. [10] J. Kellendonk, S. Richard, On the structure of the wave operators in one dimensional potential scattering, Mathematical Physics Electronic Journal 14 (2008), 1–21. [11] B. S. Pavlov, S. V. Petras, The singular spectrum of a weakly perturbed multiplication operator, Funkcional. Anal. i Priloˇzen. 4 1970 no. 2, 54–61. Translation in Functional Anal. Appl. 4 (1970), 136–142. [12] B. Simon, Trace ideals and their applications, Mathematical Surveys and Monographs 120, American Mathematical Society, Providence, RI, second edition, 2005. [13] R. Tiedra de Aldecoa, Time delay for dispersive systems in quantum scattering theory, Rev. Math. Phys. 21 no. 5 (2009), 675-708. [14] D. R. Yafaev, Mathematical scattering theory. General theory, Translations of Mathematical Monographs, 105. American Mathematical Society, Providence, RI, 1992.

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Serge Richard Department of Pure Mathematics and Mathematical Statistics Centre for Mathematical Sciences University of Cambridge Cambridge CB3 0WB United Kingdom On leave from : Universit´e de Lyon Universit´e Lyon 1, CNRS, UMR5208 Institut Camille Jordan 43 blvd du 11 novembre 1918 69622 Villeurbanne-Cedex France e-mail: [email protected] Rafael Tiedra de Aldecoa Facultad de Matem´ aticas Pontiﬁcia Universidad Cat´ olica de Chile Av. Vicu˜ na Mackenna 4860 Santiago Chile e-mail: [email protected] Submitted: February 22, 2009. Revised: October 6, 2009.

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Integr. Equ. Oper. Theory 66 (2010), 293–325 DOI 10.1007/s00020-010-1750-6 Published online February 16, 2010 © Birkhäuser/Springer Basel AG 2010

Integral Equations and Operator Theory

Holomorphic Semi-almost Periodic Functions Alexander Brudnyi and Damir Kinzebulatov Abstract. We study the Banach algebras of bounded holomorphic functions on the unit disk whose boundary values, having, in a sense, the weakest possible discontinuities, belong to the algebra of semi-almost periodic functions on the unit circle. The latter algebra contains as a special case an algebra introduced by Sarason in connection with some problems in the theory of Toeplitz operators. Mathematics Subject Classiﬁcation (2010). Primary 30H05, Secondary 46J20. Keywords. Approximation property, semi-almost periodic functions, maximal ideal space.

1. Introduction We study the Banach algebras of holomorphic semi-almost periodic functions, i.e., bounded holomorphic functions on the unit disk D ⊂ C whose boundary values belong to the algebra SAP (∂D) ⊂ L∞ (∂D) of semi-almost periodic functions on the unit circle ∂D. A function f ∈ L∞ (∂D) is called semi-almost periodic if for any s ∈ ∂D and any ε > 0 there exist functions fk : ∂D → C (k ∈ {−1, 1}) and arcs γk with s being their right (if k = −1) or left (if k = 1) endpoint with respect tox the counterclockwise orientation of ∂D such that the functions x → fk seike , −∞ < x < 0, k ∈ {−1, 1}, are restrictions of Bohr’s almost periodic functions on R (see Deﬁnition 2.1 below) and sup |f (z) − fk (z)| < ε,

z∈γk

k ∈ {−1, 1}.

The graph of a real-valued semi-almost periodic function discontinuous at a single point has a form as depicted in Figure 1. The Algebra SAP (∂D) contains as a special case an algebra introduced by Sarason [23] in connection with some problems in the theory of Toeplitz operators. Our primary interest in holomorphic semi-almost periodic functions Research of the ﬁrst author was supported in part by NSERC.

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Figure 1

was motivated by the problem of description of the weakest possible boundary discontinuities of functions in H ∞ (D), the Hardy algebra of bounded holomorphic functions on D. (Recall that a function f ∈ H ∞ (D) has radial limits almost everywhere on ∂D, the limit function f |∂D ∈ L∞ (∂D), and f can be recovered from f |∂D by means of the Cauchy integral formula.) In the general form this problem is as follows (see also [7]): Given a continuous function Φ : C → C to describe the minimal Banach subalgebra HΦ∞ (D) ⊂ H ∞ (D) containing all elements f ∈ H ∞ (D)∗ such that Φ(f )|∂D is piecewise Lipschitz having ﬁnitely many ﬁrst-kind discontinuities. Here H ∞ (D)∗ is the group of invertible elements of H ∞ (D). Clearly, each HΦ∞ (D) contains the disk-algebra A(D) (i.e., the algebra of holomorphic functions continuous up to the boundary). Moreover, if Φ(z) = z, Re(z) or Im(z), then the Lindel¨ of theorem, see, e.g., [13], implies that HΦ∞ (D) = A(D). In contrast, if Φ is constant on a closed simple curve which does not encompass 0 ∈ C, then HΦ∞ (D) = H ∞ (D). (This result is obtained by consequent applications of the Carath´eodory conformal mapping theorem, the Mergelyan theorem and the Marshall theorem, see, e.g., [13].) In [6] we studied the case of Φ(z) = |z| and showed that HΦ∞ (D) coincides with the algebra of holomorphic semi-almost periodic functions SAP (∂D) ∩ H ∞ (D). In the present paper we continue the investigation started in [6]. Despite the fact that our results concern the particular choice of Φ(z) = |z|, the methods developed here and in [6] can be applied further to a more general class of functions Φ. Let bD be the maximal ideal space of the algebra SAP (∂D) ∩ H ∞ (D), i.e., the set of all nonzero homomorphisms SAP (∂D)∩H ∞ (D) → C equipped with the Gelfand topology. The disk D is naturally embedded into bD. In [6] we proved that D is dense in bD (the so-called corona theorem for SAP (∂D)∩ H ∞ (D)). We also described the topological structure of bD. In the present

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paper we reﬁne and extend some of these results. In particular, we introduce Bohr-Fourier coeﬃcients and spectra of functions from SAP (∂D), describe ˇ Cech cohomology groups of bD and establish projective freeness of certain subalgebras of SAP (∂D) ∩ H ∞ (D). Recall that a commutative ring R with identity is called projective free if every ﬁnitely generated projective R-module is free. Equivalently, R is projective free iﬀ every square idempotent matrix F with entries in R (i.e., such that F 2 = F ) is conjugate over R to a matrix of the form Ik 0 , 0 0 where Ik stands for the k × k identity matrix. Every ﬁeld F is trivially projective free. Quillen and Suslin proved that if R is projective free, then the rings of polynomials R[x] and formal power series R[[x]] over R are projective free as well (see, e.g., [18]). Grauert proved that the ring O(Dn ) of holomorphic functions on the unit polydisk Dn is projective free [14]. In turn, it was shown in [8] that the triviality of any complex vector bundle of ﬁnite rank over the connected maximal ideal space of a unital semi-simple commutative complex Banach algebra is suﬃcient for its projective freeness. We employ this result to show that subalgebras of SAP (∂D) ∩ H ∞ (D) whose elements have their spectra in non-negative or non-positive semi-groups are projective free. Note that if a unital semi-simple commutative complex Banach algebra A is projective free, then it is Hermite, i.e., every ﬁnitely generated stably free A-module is free. Equivalently, A is Hermite iﬀ any k × n matrix, k < n, with entries in A having rank k at each point of the maximal ideal space of A can be extended to an invertible n × n matrix with entries in A, see [9]. (Here the values of elements of A at points of the maximal ideal space are deﬁned by means of the Gelfand transform.) Finally, we prove that SAP (∂D) ∩ H ∞ (D) has the approximation property. (This result strengthen the approximation theorem of [6].) Recall that a Banach space B is said to have the approximation property if for every compact set K ⊂ B and every ε > 0 there is an operator T : B → B of ﬁnite rank so that T x − xB < ε

for every

x ∈ K.

(Throughout this paper all Banach spaces are assumed to be complex.) Although it is strongly believed that the class of spaces with the approximation property includes practically all spaces which appear naturally in analysis, it is not known yet even for the space H ∞ (D) (see, e.g., the paper of Bourgain and Reinov [3] for some results in this direction). The ﬁrst example of a space which fails to have the approximation property was constructed by Enﬂo [10]. Since Enﬂo’s work several other examples of such spaces were constructed, for the references see, e.g., [20]. Many problems of Banach space theory admit especially simple solutions if one of the spaces under consideration has the approximation property. One of such problems is the problem of determination whether given two Banach algebras A ⊂ C(X), B ⊂ C(Y ) (X and Y

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are compact Hausdorﬀ spaces) their slice algebra S(A, B) := {f ∈ C(X × Y ) : f (·, y) ∈ A for all y ∈ Y, f (x, ·) ∈ B for all x ∈ X} coincides with A⊗B, the closure in C(X ×Y ) of the symmetric tensor product of A and B. For instance, this is true if either A or B have the approximation property. The latter is an immediate consequence of the following result of Grothendieck. Let A ⊂ C(X) be a closed subspace, B be a Banach space and AB ⊂ CB (X) := C(X, B) be the Banach space of all continuous B-valued functions f such that ϕ(f ) ∈ A for any ϕ ∈ B ∗ . By A ⊗ B we denote the completion of the symmetric tensor product of A and B with respect to norm m m ak ⊗ bk := sup ak (x)bk with ak ∈ A, bk ∈ B. (1.1) x∈X k=1

k=1

B

Theorem 1.1 ([15]). The following statements are equivalent: 1) A has the approximation property; 2) A ⊗ B = AB for every Banach space B. Our proof of the approximation property for SAP (S) ∩ H ∞ (D) is based on Theorem 1.1 and on a variant of the approximation theorem in [6] for Banach-valued analogues of algebra SAP (S) ∩ H ∞ (D). The paper is organized as follows. Section 2 is devoted to the algebra of semi-almost periodic functions SAP (∂D). In Section 3 we formulate our main results on the algebra of holomorphic semi-almost periodic functions SAP (∂D) ∩ H ∞ (D). All proofs are presented in Section 4. The results of the present paper have been announced in [7].

2. Preliminaries on semi-almost periodic functions We ﬁrst recall the deﬁnition of a Bohr almost periodic function on R. In what follows, by Cb (R) we denote the algebra of bounded continuous functions on R endowed with sup-norm. Deﬁnition 2.1 (see, e.g., [2]). A function f ∈ Cb (R) is said to be almost periodic if the family of its translates {Sτ f }τ ∈R , Sτ f (x) := f (x + τ ), x ∈ R, is relatively compact in Cb (R). The basic example of an almost periodic function is given by the formula m x → cl eiλl x , cl ∈ C, λl ∈ R. l=1

Let AP (R) be the Banach algebra of almost periodic functions endowed with sup-norm. The main characteristics of an almost periodic function f ∈ AP (R) are its Bohr-Fourier coeﬃcients aλ (f ) and the spectrum spec(f ) deﬁned in terms of the mean value T 1 M (f ) := lim f (x)dx. (2.1) T →+∞ 2T −T

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Speciﬁcally, aλ (f ) := M (f e−iλx ),

λ ∈ R.

(2.2)

[2]. These Then aλ (f ) = 0 for at most countably many values of λ, see, e.g., ∞ iλl x values constitute the spectrum spec(f ) of f . In particular, if f = l=1 cl e ∞ (cl = 0 and l=1 |cl | < ∞), then spec(f ) = {λ1 , λ2 , . . . }. One of the main results of the theory of almost periodic functions states that each function m f ∈ AP (R) can be uniformly approximated by functions of the form l=1 cl eiλl x with λl ∈ spec(f ). Let Γ ⊂ R be a unital additive semi-group (i.e., 0 ∈ Γ). It follows easily from the cited approximation result that the space APΓ (R) of almost periodic functions with spectra in Γ forms a unital Banach subalgebra of AP (R). We will use the following result. Theorem 2.2. APΓ (R) has the approximation property. Next, we recall the deﬁnition of a semi-almost periodic function on ∂D introduced in [6]. In what follows, we consider ∂D with the counterclockwise orientation. For s := eit , t ∈ [0, 2π), let γsk (δ) := {seikx : 0 ≤ x < δ < 2π},

k ∈ {−1, 1},

(2.3)

be two open arcs having s as the right and the left endpoints (with respect to the orientation), respectively. Deﬁnition 2.3 ([6]). A function f ∈ L∞ (∂D) is called semi-almost periodic if for any s ∈ ∂D, and any ε > 0 there exist a number δ = δ(s, ε) ∈ (0, π) and functions fk : γsk (δ) → C, k ∈ {−1, 1}, such that functions x f˜k (x) := fk seikδe , −∞ < x < 0, k ∈ {−1, 1}, are restrictions of some almost periodic functions from AP (R), and sup |f (z) − fk (z)| < ε,

z∈γsk (δ)

k ∈ {−1, 1}.

By SAP (∂D) we denote the Banach algebra of semi-almost periodic functions on ∂D endowed with sup-norm. It is easy to see that the set of points of discontinuity of a function in SAP (∂D) is at most countable. For S being a closed subset of ∂D we denote by SAP (S) the Banach algebra of semialmost periodic functions on ∂D that are continuous on ∂D \ S. (Note that the Sarason algebra introduced in [23] is isomorphic to SAP ({z0 }), z0 ∈ ∂D.) Example ([6]). A function g deﬁned on R (R + iπ) is said to belong to the space AP (R (R + iπ)) if the functions g(x) and g(x + iπ), x ∈ R, belong to AP (R). The space AP (R (R + iπ)) is a function algebra (with respect to sup-norm). Given s ∈ ∂D consider the map ϕs : ∂D \ {−s} → R, ϕs (z) := 2i(s−z) s+z , and deﬁne a linear isometric embedding Ls : AP (R (R + iπ)) → L∞ (∂D) by the formula (Ls g)(z) := (g ◦ Log ◦ ϕs )(z), (2.4)

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where Log(z) := ln |z| + iArg(z), z ∈ C \ R− , and Arg : C \ R− → (−π, π) stands for the principal branch of the multi-function arg. Then the range of Ls is a subspace of SAP ({−s, s}). Theorem 2.4 ([6]). For every s ∈ ∂D there exists a homomorphism of Banach algebras Es : SAP (∂D) → AP (R (R + iπ)) of norm 1 such that for each f ∈ SAP (∂D) the function f − Ls (Es f ) ∈ SAP (∂D) is continuous and equal to 0 at s. Moreover, any bounded linear operator SAP (∂D) → AP (R (R + iπ)) satisfying this property coincides with Es . The functions f−1,s (x) := (Es f )(x) and f1,s (x) := (Es f )(x+iπ), x ∈ R, are used to deﬁne the left (k = −1) and the right (k = 1) mean values Msk (f ) of a function f ∈ SAP (∂D) over s (cf. Remark 2.6 below). Precisely, we put Msk (f ) := M (fk,s ). Similarly, we deﬁne the left (k = −1) and the right (k = 1) Bohr-Fourier coeﬃcients and spectra of f over s by the formulas akλ (f, s) := aλ (fk,s ) and specks (f ) := {λ ∈ R : akλ (f, s) = 0}. It follows immediately from the properties of the spectrum of an almost periodic function on R that specks (f ) is at most countable. Let Σ : S × {−1, 1} → 2R be a set-valued map which associates with each s ∈ S, k ∈ {−1, 1} a unital semi-group Σ(s, k) ⊂ R. By SAP Σ (S) ⊂ SAP (S) we denote the Banach algebra of semi-almost periodic functions f with specks (f ) ⊂ Σ(s, k) for all s ∈ S, k ∈ {−1, 1}. By the deﬁnition homomorphism Es of Theorem 2.4 sends each f ∈ SAPΣ (S) to the pair of functions f−1,s , f1,s such that fk,s ∈ APΣ(s,k) , k ∈ {−1, 1}. For a unital semi-group Γ ⊂ R by bΓ (R) we denote the maximal ideal space of algebra APΓ (R). (E.g., for Γ = R the space bR := bR (R), commonly called the Bohr compactiﬁcation of R, is a compact abelian topological group viewed as the inverse limit of compact ﬁnite-dimensional tori. The group R admits a canonical embedding into bR as a dense subgroup.) Let bSΣ (∂D) be the maximal ideal space of algebra SAP Σ (S) and S : bSΣ (∂D) → ∂D rΣ

be the map transpose to the embedding C(∂D) → SAP Σ (S). The proof of the next statement is analogous to the proof of Theorem 1.7 in [6]. Theorem 2.5. (1) The map transpose to the restriction of homomorphism Es to SAPΣ (S) determines an embedding hsΣ : bΣ(s,−1) (R) bΣ(s,1) (R) → bSΣ (∂D) S −1 whose image coincides with (rΣ ) (s).

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Figure 2

S S −1 (2) The restriction rΣ : bSΣ (∂D) \ (rΣ ) (S) → ∂D \ S is a homeomorphism. (For an m point set S and each Σ(s, k), s ∈ S, k ∈ {−1, 1}, being a group, the maximal ideal space bSΣ (∂D) is the union of ∂D \ S and 2m Bohr compactiﬁcations bΣ(s,k) (R) that can be viewed as (ﬁnite or inﬁnite dimensional) tori; see Figure 2).

Remark 2.6. There is an equivalent way to deﬁne the mean value of a semialmost periodic function. Speciﬁcally, it is easily seen that for a semi-almost periodic function f ∈ SAP (∂D), k ∈ {−1, 1}, and a point s ∈ S the left (k = −1) and the right (k = 1) mean values of f over s are given by the formulas bn

t 1 k Ms (f ) := lim f seike dt, n→∞ bn − an a n where {an }, {bn } are arbitrary sequences of real numbers converging to −∞ such that lim (bn − an ) = +∞. n→∞

The Bohr-Fourier coeﬃcients of f over s can be then deﬁned by the formulas k akλ (f, s) := Msk (f e−iλ logs ), where logks (seikx ) := ln x,

0 < x < 2π,

k ∈ {−1, 1}.

The next result encompasses the basic properties of the mean value and the spectrum of a semi-almost periodic function.

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Theorem 2.7. (1) For each s ∈ S, k ∈ {−1, 1} the mean value Msk is a complex continuous linear functional on SAP (∂D) of norm 1. (2) A function f ∈ SAP Σ (S) if and only if for each s ∈ S and k ∈ {−1, 1} the almost periodic functions f˜k in Deﬁnition 2.3 can be chosen from AP Σ(s,k) (R). (3) The “total spectrum“ s∈S, k=±1 specks (f ) of a function f ∈ SAP (S) is at most countable.

3. Holomorphic semi-almost periodic functions: Main results 3.1. Let Cb (T ) denote the complex Banach space of bounded continuous functions on the strip T := {z ∈ C : Im(z) ∈ [0, π]} endowed with sup- norm. Deﬁnition 3.1 (see, e.g., [2]). A function f ∈ Cb (T ) is called holomorphic almost periodic if it is holomorphic in the interior of T and the family of its translates {Sx f }x∈R , Sx f (z) := f (z + x), z ∈ T , is relatively compact in Cb (T ). We denote by AP H(T ) the Banach algebra of holomorphic almost periodic functions endowed with sup-norm. Any function in AP H(T ) is uniformly continuous on T . The mean value of a function f ∈ AP H(T ) is deﬁned by the formula T 1 f (x + iy)dx ∈ C (3.1) M (f ) := lim T →+∞ 2T −T (M (f ) does not depend on y, see, e.g., [2]). Further, the Bohr-Fourier coeﬃcients of f are deﬁned by aλ (f ) := M (f e−iλz ),

λ ∈ R.

(3.2)

Then aλ (f ) = 0 for at most countably many values these values form ∞of λ, iλ lz c e , with cl = 0, the spectrum spec(f ) of f . For instance, if f = l=1 l ∞ |c | < ∞, then spec(f ) = {λ , λ , . . . }. Similarly to the case of func1 2 l=1 l tions from AP (R) each f ∈ AP H(T ) can be uniformly approximated by m functions of the form l=1 cl eiλl z with λl ∈ spec(f ). Let Γ ⊂ R be a unital additive semi-group. The space AP HΓ (T ) of holomorphic almost periodic functions with spectra in Γ forms a unital Banach algebra. Analogously to Theorem 2.2 one has Theorem 3.2. AP HΓ (T ) has the approximation property. The functions in SAPΣ (S)∩H ∞ (D) are called holomorphic semi-almost periodic. Example ([6]). For s ∈ ∂D consider the map 2i(s − z) . s+z Here H+ is the upper half-plane. Then ϕs maps D conformally onto H+ and ∂D \ {−s} diﬀeomorphically onto R (the boundary of H+ ) so that ϕs (s) = 0. ¯ \ {−s} → H ¯ +, ϕs : D

ϕs (z) :=

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Let T0 be the interior of the strip T . Consider the conformal map Log : H+ → T0 , z → Log(z) := ln |z|+iArg(z), where Arg : C\R− → (−π, π) is the principal branch of the multi-function arg. The function Log is extended to a homeomorphism of H+ \{0} onto T . Let g ∈ AP H(T ). Then the function (Ls g)(z) := (g ◦ Log ◦ ϕs )(z),

z ∈ D,

∞

belongs to SAP ({−s, s}) ∩ H (D). Proposition 3.3. Suppose that f ∈ SAP (S) ∩ H ∞ (D). Then 1 spec−1 s (f ) = specs (f ) =: specs (f )

and, moreover, λπ 1 a−1 λ (f, s) = e aλ (f, s)

λ ∈ specs (f ).

for each

(3.3)

(Recall that the choice of the upper indices ±1 is determined by the orientation of ∂D.) Using Proposition 3.3 and the Lindel¨ of theorem (see, e.g., [13]) one obtains that SAPΣ (S) ∩ H ∞ (D) = SAPΣ (S ) ∩ H ∞ (D), where S := {s ∈ S : Σ(s, −1) ∩ Σ(s, 1) = {0}} and

Σ (s, k) := Σ(s, −1) ∩ Σ(s, 1) for k = −1, 1, s ∈ S . In what follows we assume that Σ(s, −1) = Σ(s, 1) =: Σ(s) and each Σ(s), s ∈ S, is non-trivial. iλ

Example. If g(z) := e π z , z ∈ T , then Ls g = eλh , where h is a holomorphic functions whose real part Re(h) is the characteristic function of the closed arc going in the counterclockwise direction from the initial point at s to the 2 endpoint at −s, and such that h(0) = 12 + i ln π . Thus specs (eλh ) = {λ/π}. Indeed, in this case the restriction of g ◦ Log to R is equal to x → eλ(χR+ (x)+

i ln |x| ) π

,

where χR+ is the characteristic function of R+ . In turn, the restriction of the λ(χR+ + pullback eλh ◦ ϕ−1 s to R coincides with e required result.

i ln |x| ) π

as well. This implies the

3.2. The main result of this section is Theorem 3.4. SAPΣ (S) ∩ H ∞ (D) has the approximation property. Our proof of Theorem 3.4 is based on the equivalence established in Theorem 1.1 and on an approximation result for Banach-valued analogues of algebra SAP (S) ∩ H ∞ (D) formulated below. Speciﬁcally, for a Banach space B we deﬁne SAPΣB (S) := SAPΣ (S) ⊗ B. Using the Poisson integral formula we can extend each function from SAPΣB (S) to a bounded B-valued harmonic function on D having the same

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∞ sup-norm. We identify SAPΣB (S) with its harmonic extension. Let HB (D) be the Banach space of bounded B-valued holomorphic functions on D equipped with sup-norm. By (SAPΣ (S) ∩ H ∞ (D))B we denote the Banach space of all continuous B-valued functions f on the maximal ideal space bS (D) of algebra SAP (S) ∩ H ∞ (D) such that ϕ(f ) ∈ SAPΣ (S) ∩ H ∞ (D) for any ϕ ∈ B ∗ . In what follows we naturally identify D with a subset of bS (D).

Proposition 3.5. Let f ∈ (SAPΣ (S) ∩ H ∞ (D))B . Then ∞ f |D ∈ SAPΣB (S) ∩ HB (D).

Let ASΣ be the closed subalgebra of H ∞ (D) generated by the disk-algebra A(D) and the functions of the form geλh , where Re(h)|∂D is the characteristic function of the closed arc going in the counterclockwise direction from the initial point at s to the endpoint at −s such that s ∈ S, πλ ∈ Σ(s) and g(z) := z + s, z ∈ D (in particular, geλh has discontinuity at s only). The next result combined with Proposition 3.5 and Theorem 1.1 implies Theorem 3.4. ∞ (D) = ASΣ ⊗ B. Theorem 3.6. SAPΣB (S) ∩ HB

As a corollary we obtain Corollary 3.7. SAPΣ (S) ∩ H ∞ (D) = ASΣ . This immediately implies the following result. Theorem 3.8. SAPΣ (S) ∩ H ∞ (D) is generated by algebras SAPΣ|F (F ) ∩ H ∞ (D) for all possible ﬁnite subsets F of S. 3.3. The algebras SAPΣ (S) ∩ H ∞ (D) are preserved under the action of the group Aut(D) of biholomorphic automorphisms D → D. More precisely, each ¯ →D ¯ (denoted by the same κ ∈ Aut(D) is extended to a diﬀeomorphism D symbol). We denote by κ∗ : H ∞ (D) → H ∞ (D) the pullback by κ, and put κ∗ S := κ(S), (κ∗ Σ)(s, ·) := Σ(κ(s), ·). Theorem 3.9. The linear operator κ∗ maps SAPκ∗ Σ (κ∗ S) ∩ H ∞ (D) isometrically onto SAPΣ (S) ∩ H ∞ (D). We conclude this section with a result on the tangential behavior of functions from SAP (∂D) ∩ H ∞ (D). Theorem 3.10. Let {zn }n∈N ⊂ D and {sn }n∈N ⊂ ∂D converge to a point s0 ∈ ∂D. Assume that |zn − sn | = 0. (3.4) lim n→∞ |s0 − sn | Then for every f ∈ SAP (∂D) ∩ H ∞ (D) the limits lim f (zn ) and lim f (sn ) n→∞ n→∞ do not exist or exist simultaneously and in the latter case they are equal.

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Remark 3.11. This result implies that the extension (by means of the Gelfand transform) of each f ∈ SAP (∂D) ∩ H ∞ (D) to the maximal ideal space of H ∞ (D) is constant on a nontrivial Gleason part containing a limit point of a net in D converging tangentially to ∂D. In turn, one can easily show that if s ∈ S and the minimal subgroup of R containing Σ(s) is not isomorphic to Z, then the algebra SAPΣ (S) ∩ H ∞ (D) separates points of each nontrivial Gleason part containing a limit point of a net in D converging non-tangentially to s (we refer to [13] for the corresponding deﬁnitions). In the next two sections we formulate some topological results about the maximal ideal spaces of algebras SAPΣ (S) ∩ H ∞ (D). 3.4. Let bSΣ (D) denote the maximal ideal space of SAPΣ (S) ∩ H ∞ (D). The inclusion SAPΣ|F1 (F1 ) ∩ H ∞ (D) ⊂ SAPΣ|F2 (F2 ) ∩ H ∞ (D)

if

F1 ⊂ F2

determines a continuous map of maximal ideal spaces F1 2 ωFF12 : bF Σ|F (D) → bΣ|F (D). 2

1

{bF Σ|F (D) ;

ω}F ⊂S ; #F <∞ forms the inverse limiting system. From The family Theorem 3.8 we obtain Theorem 3.12. bSΣ (D) is the inverse limit of {bF Σ|F (D) ; ω}F ⊂S ; #F <∞ . Let ¯ aSΣ : bSΣ (D) → D

(3.5)

be the continuous surjective map of maximal ideal spaces transpose to the embedding A(D) → SAPΣ (S) ∩ H ∞ (D). (Recall that the maximal ideal ¯ By bΓ (T ) we denote space of the disk-algebra A(D) is homeomorphic to D.) the maximal ideal space of algebra AP HΓ (T ) and by ιΓ : T → bΓ (T ) the continuous map determined by evaluations at points of T . (Observe that ιΓ is not necessarily an embedding.) The proof of the next statement can be obtained by following closely the arguments in the proof of Theorem 1.14 in [6]. In its formulation we assume that the corresponding algebras are deﬁned on their maximal ideal spaces by means of the Gelfand transforms. Theorem 3.13. (1) For each s ∈ S there exists an embedding isΣ : bΣ(s) (T ) → bSΣ (D) whose image is (aSΣ )−1 (s) such that the pullback (isΣ )∗ maps algebra SAPΣ (S) ∩ H ∞ (D) surjectively onto AP HΣ(s) (T ). Moreover, the composition of the restriction map to R (R + iπ) and (isΣ ◦ ιΣ(s) )∗ coincides with the restriction of homomorphism Es to SAPΣ (S) ∩ H ∞ (D) (cf. Theorem 2.4). ¯ \ S is a homeomorphism. (2) The restriction aSΣ : bSΣ (D) \ (aSΣ )−1 (S) → D

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Figure 3

Since SAPΣ (S) ∩ H ∞ (D) separates the points on D, the evaluation at points of D determines a natural embedding ι : D → bSΣ (D). One has the following commutative diagram of maximal ideal spaces considered in the present paper, where the ‘dashed’ arrows stand for embeddings in the case Σ(s, −1) = Σ(s, 1) are (non-trivial) groups for all s ∈ S, and for continuous maps otherwise. Theorem 3.14 (Corona Theorem). ι(D) is dense in bSΣ (D) iﬀ each Σ(s), s ∈ S, is a group. Recall that the corona theorem is equivalent to the following statement: for any collection of functions f1 , . . . , fm ∈ SAPΣ (S) ∩ H ∞ (D) such that max |fk (z)| ≥ δ > 0

1≤k≤m

for all z ∈ D

there exist functions g1 , . . . , gm ∈ SAPΣ (S) ∩ H ∞ (D) such that f1 g 1 + · · · + fm g m = 1 Our next result shows that

bSΣ (D),

on D.

S = ∅, is not arcwise connected.

Theorem 3.15. Assume that F : [0, 1] → bSΣ (D) is continuous. Then either ¯ \S F ([0, 1]) ⊂ D or there exists s ∈ S such that F ([0, 1]) ⊂ (aSΣ )−1 (s). Remark 3.16. From Theorem 3.15 one obtains straightforwardly a similar statement with [0, 1] replaced by an arcwise connected topological space. 3.5. Let KΣS be the Shilov boundary of algebra SAPΣ (S)∩H ∞ (D), that is, the minimal closed subset of bSΣ (D) such that for every f ∈ SAPΣ (S) ∩ H ∞ (D) sup |f (z)| = max |f (ϕ)|, z∈D

S ϕ∈KΣ

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where f is assumed to be extended to bSΣ (D) by means of the Gelfand transform. For a non-trivial semigroup Γ ⊂ R by clΓ (R + iπ) and clΓ (R) we denote closures of ιΓ (R + iπ) and ιΓ (R) in bΓ (T ) (the maximal ideal space of AP HΓ (T )). One can easily show that these closures are homeomorphic to is the minimal subgroup of R containing Γ. bΓ (R), where Γ We retain notation of Theorem 3.13. Theorem 3.17. KΣS

=

isΣ

clΣ(s) (R) ∪ clΣ(s) (R + iπ) ∪ ∂D \ S.

s∈S

Remark 3.18. If each Σ(s), s ∈ S, is a group, then the Shilov boundary KΣS is naturally homeomorphic to the maximal ideal space bSΣ (∂D) of algebra SAPΣ (S), cf. Theorem 2.5. ˇ Next, we formulate a result on the Cech cohomology groups of bSΣ (D). Let bT (S) := s∈S bΣ(s) (T ). According to Theorem 3.13 there exists a natural embedding isΣ : bΣ(s) (T ) → bSΣ (D) whose image is (aSΣ )−1 (s). Then the map I : bT (S) → (aSΣ )−1 (S),

I(ξ) := isΣ (ξ)

for

ξ ∈ bΣ(s) (T ),

is a bijection. Theorem 3.19. (1) The map transpose to the composition I

bT (S) −→ (aSΣ )−1 (S) → bSΣ (D) induces an isomorphism of groups k H k bSΣ (D), Z ∼ H bΣ(s) (T ), Z , =

k ≥ 1.

s∈S

(2) Suppose that each Σ(s) is a subset of R+ or R− . Then H k bSΣ (D), Z = 0, k ≥ 1, and SAPΣ (S) ∩ H ∞ (D) is projective free. ˆ is its dual, If G is a compact connected abelian topological group and G k k ˆ ∼ then H (G, Z) = ∧Z G, k ≥ 1 (see, e.g., [16]). Using Pontryagin duality one obtains Corollary 3.20. Assume that each Σ(s) is a group. Then H k (bSΣ (D), Z) ∼ ∧kZ Σ(s), k ≥ 1. = s∈S

In particular, if for a ﬁxed n ∈ N each Σ(s) is isomorphic to a subgroup of Qn , then H k (bSΣ (D), Z) = 0 for all k ≥ n + 1. Finally, we describe the set of connected components of the group of invertible matrices with entries in SAPΣ (S) ∩ H ∞ (D). Let GLn (A) denote the group of invertible n × n matrices with entries in a unital Banach algebra A. By [GLn (A)] we denote the group of connected

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components of GLn (A), i.e., the quotient of GLn (A) by the connected component containing the unit In ∈ GLn (A) (this is clearly a normal subgroup of GLn (A)). We set n GΣ(s) (T ) := GLn AP HΣ(s) (T ) and GnΣ (S) := GLn SAPΣ (S)∩H ∞ (D) . Theorem 3.21. The map transpose to the composition I

bT (S) −→ (aSΣ )−1 (S) → bSΣ (D) induces an isomorphism of groups [GnΣ (S)] ∼ =

[GnΣ(s) (T )]. s∈S

In particular, if each Σ(s), s ∈ S, is a subset of R+ or R− , then GnΣ (S) is connected. Remark 3.22. According to a result of Arens [1], [GnΣ(s) (T )], s ∈ S, can be identiﬁed with the group [bΣ(s) (T ), GLn (C)] of homotopy classes of continuous maps bΣ(s) (T ) → GLn (C). Moreover, if Σ(s) is a group, then bΣ(s) (T ) is homotopically equivalent to bΣ(s) (R), the maximal ideal space of algebra APΣ(s) (R) (see the proof of Corollary 3.20). In this case bΣ(s) (R) is the inverse limit of a family of ﬁnite-dimensional tori. Then [bΣ(s) (T ), GLn (C)] is isomorphic to the direct limit of groups of homotopy classes of maps from tori to GLn (C). As follows from the classical results of Fox [12], the latter can be expressed as a direct sum of certain homotopy groups of the unitary group Un ⊂ GLn (C).

4. Proofs 4.1. Proofs of Theorems 2.2 and 3.2 We will prove Theorem 3.2 only (the proof of Theorem 2.2 is similar). We refer to the book of Besicovich [2] for the corresponding deﬁnitions and facts from the theory of almost periodic functions. Proof. Let K ⊂ AP HΓ (T ) be a compact subset. For a given ε > 0 consider an 3ε -net {f1 , . . . , fl } ⊂ K. Let

ν νr ν1 νr −i n1 β1 +···+ n β t r r 1 K(t) := 1− ... 1 − e n1 nr |ν1 |≤n1 ,...,|νr |≤nr

be a Bochner-Fejer kernel such that for all 1 ≤ k ≤ l sup |fk (z) − Mt {fk (z + t)K(t)}| ≤

z∈Σ

ε . 3

(4.1)

Here β1 , . . . , βr are linearly independent over Q and belong to the union of spectra of functions f1 , . . . , fl , ν1 , . . . , νr ∈ Z, n1 , . . . , nr ∈ N, and T 1 Mt {fk (z + t)K(t)} := lim fk (z + t)K(t) dt T →∞ 2T −T

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are the corresponding Bochner-Fejer polynomials belonging to AP HΓ (T ) as well (clearly, the spectrum of the function z → Mt {fk (z +t)K(t)} is contained in spec(fk )). We deﬁne a linear operator T : AP HΓ (T ) → AP HΓ (T ) from the deﬁnition of the approximation property by the formula (T f )(z) := Mt {f (z + t)K(t)},

f ∈ AP HΓ (T ).

(4.2)

Then T is a bounded linear projection

onto a ﬁnite-dimensional subspace i

ν1

β +···+ νr β

z

nr r generated by functions e n1 1 , |ν1 | ≤ n1 , . . . , |νr | ≤ nr . Moreover, since K(t) ≥ 0 for all t ∈ R and Mt {K(t)} = 1, the norm of T is 1. Finally, given f ∈ K choose k such that f − fk AP HΓ (T ) ≤ ε3 . Then we have by (4.1)

T f − f AP HΓ (T ) ≤ T (f − fk )AP HΓ (T ) + T fk − fk AP HΓ (T ) + fk − f AP HΓ (T ) < ε.

4.2. Proof of Theorem 2.7 (1) The result follows directly from Remark 2.6. (2) The fact that for f ∈ SAPΣ (S) the functions f˜k in Deﬁnition 2.3 can be chosen from APΣ(s,k) (R) follows from Theorem 2.4 and the deﬁnition of spectra of elements of SAP (∂D). Let us show the validity of the converse statement. Let f ∈ SAP (S) and s ∈ S. Assume that for any ε > 0 the functions f˜k in Deﬁnition 2.3 (for f and s) can be chosen in APΣ(s,k) (R). Let ρ be a smooth cut-oﬀ function equals 1 in a neighbourhood of s contained in the open set γs−1 (δ)∪{s} ∪γs1(δ) and 0 outside of this set. Let us consider a function fˆ on ∂D \ {s} that coincides with ρf1 on γs1 (δ) and with ρf−1 on γs−1 (δ) and equals 0 outside of these arcs. By the deﬁnition of spectra of functions in SAP (∂D) the function fˆ belongs to SAPΣ (S). Next, Theorem 2.7 (1) implies that |akλ (fˆ, s) − akλ (f, s)| = |akλ (fˆ, s) − akλ (ρf, s)| < ε,

k ∈ {−1, 1}.

Since ε > 0 is arbitrary, the latter inequality shows that if akλ (f, s) = 0, then λ ∈ Σ(s, k), as required. (3) First, let us show that the set T (f ) of points of discontinuity of a function f ∈ SAP (∂D) is at most countable. For each s ∈ ∂D deﬁne ⎛ ⎞ cs (f ) := lim ⎝ ε→0+

sup

s ,s ∈ ei(t−ε) ,ei(t+ε)

⎠ |f (s ) − f (s )| ,

(f ). For n ∈ N, we put where s = eit . One has cs (f ) = 0 if and only if s ∈ T ∞ Tn (f ) := {s ∈ ∂D : cs (f ) ≥ n1 }, so that T (f ) = n=1 Tn (f ). Suppose that T (f ) is uncountable, then there exists n ∈ N such that Tn (f ) is inﬁnite. Since 1 ∂D is compact, Tn (f ) has a limit point eit0 . Choosing ε < 2n in Deﬁnition it0 it0 2.3 (for f and e ) from the fact that e is a limit point of Tn (f ) one obtains that the required functions fk do not exist, a contradiction. According to statement (2) fk,s ≡ const for all points s ∈ ∂D at which f is continuous. Therefore due to the previous statement specks (f ) is {0} or

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∅ for all but at most countably many values of s ∈ S. Since for each s ∈ S the spectrum specks (f ) is at most countable, the required result follows. 4.3. Proof of Proposition 3.3 Let g ∈ AP H(T ), put g1 (x) := g(x), g2 (x) := g(x + iπ), x ∈ R. It follows easily from the approximation result for algebra AP H(T ) cited in Section 3.1 that spec(g1 ) = spec(g2 ) and for each λ ∈ spec(g1 ) aλ (g1 ) = eλπ aλ (g2 ). Suppose that f ∈ SAP (S) ∩ H ∞ (D). Then Theorem 3.13 (1) implies that for each s ∈ S and k ∈ {−1, 1} the functions fk,s are the boundary values of the function (isΣ ◦ ιΣ(s) )∗ (f ) ∈ AP H(T ). 4.4. Proofs of Proposition 3.5 and Theorem 3.6 Our proof of Theorem 3.6 is based on the equivalence established in Theorem 1.1. We ﬁrst formulate the B-valued analogues of the deﬁnitions of almost periodic and semi-almost periodic functions for B being a complex Banach space. Let CbB (R) and CbB (T ) denote the Banach spaces of B-valued bounded continuous functions on R and T , respectively, with norms f := sup f (x)B . x

CbB (R)

is said to be almost periodic if the Deﬁnition 4.1. 1) A function f ∈ family of its translates {Sτ f }τ ∈R , Sτ f (x) := f (x + τ ), x ∈ R, is relatively compact in CbB (R). 2) A function f ∈ CbB (T ) is called holomorphic almost periodic if it is holomorphic in the interior of T and the family of its translates {Sx f }x∈R is relatively compact in CbB (T ). Let AP B (R) and AP H B (T ) denote the Banach spaces of almost periodic and holomorphic almost periodic functions on R and T , respectively, endowed with sup-norms. Remark 4.2. Since AP (R) and AP H(T ) have the approximation property, it m iλl x b e (x ∈ R, follows from Theorem 1.1 that the functions of the form l l=1 iλl z B bl ∈ B, λl ∈ R) and m b e (z ∈ T, b ∈ B, λ ∈ R) are dense in AP (R) l l l l=1 B and AP H (T ), respectively. As in the case of scalar almost periodic functions, a Banach-valued almost periodic function f ∈ APB (R) is characterized by its Bohr-Fourier coeﬃcients aλ (f ) and the spectrum spec(f ), deﬁned in terms of the mean value T 1 M (f ) := lim f (x)dx ∈ B. (4.3) T →+∞ 2T −T Namely, we deﬁne aλ (f ) := M (f e−iλx ), λ ∈ R.

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It follows from the above remark and the properties of scalar almost periodic functions that aλ (f ) = 0 in B for at most countably many values of λ. ∞ iλl x These values constitute the spectrum spec(f ) of f . E.g., if f = l=1 bl e ∞ ( l=1 bl < ∞ and all bl = 0), then spec(f ) = {λ1 , λ2 , . . . }. Similarly, one deﬁnes the mean-values and the spectra for functions from AP H B (T ). Let L∞ B (∂D) be the Banach space of B-valued bounded measurable functions on ∂D equipped with sup-norm. Deﬁnition 4.3. A function f ∈ L∞ B (∂D) is called semi-almost periodic if for any s ∈ ∂D and any ε > 0 there exist a number δ = δ(s, ε) ∈ (0, π) and functions fk : γsk (δ) → B, γsk (δ) := {seikx : 0 ≤ x < δ < 2π}, k ∈ {−1, 1}, such that functions x x → fk seikδe , −∞ < x < 0, k ∈ {−1, 1}, are restrictions of B-valued almost periodic functions from AP B (R) and sup f (z) − fk (z)B < ε,

z∈γsk (δ)

k ∈ {−1, 1}.

Analogously to the scalar case, for a closed subset S ⊂ ∂D, we denote by SAP B (S) ⊂ L∞ B (∂D) the Banach space of semi-almost periodic functions that are continuous on ∂D \ S, so that SAP (S) := SAP C (S). Let SAP (S) ⊗ B denote the completion in L∞ B (∂D) of the symmetric tensor product of SAP (S) and B. Proposition 4.4. SAP B (S) = SAP (S) ⊗ B. The statement is an immediate consequence of Theorem 1.1 and the following two facts: each function f ∈ SAP B (S) admits a norm preserving extension to the maximal ideal space bS (∂D) of the algebra SAP (S) as a continuous B-valued function, and C(bS (∂D)) has the approximation property. The ﬁrst fact follows straightforwardly from the deﬁnitions of SAP B (S) and bS (∂D) (see [6]) and the existence of analogous extensions of functions in AP B (R) to bR, while the second fact is valid for any algebra C(X) on a compact Hausdorﬀ topological space X (it can be proved using ﬁnite partitions of unity of X). Next, we introduce the Banach space AP B (R (R + iπ)) := AP (R (R + iπ)) ⊗ B of B-valued almost periodic functions on R (R + iπ), see Example 2. Also, B ∞ for each s ∈ S we deﬁne a linear isometry LB s : AP (R(R+iπ)) → LB (∂D) by the formula (cf. (2.4)) (LB s g)(z) := (g ◦ Log ◦ ϕs )(z),

g ∈ AP B (R (R + iπ)).

(4.4)

Now, using Proposition 4.4 we prove a B-valued analog of Theorem 2.4. Theorem 4.5. For every point s ∈ ∂D there exists a bounded linear operator EsB : SAP B (∂D) → AP B (R (R + iπ))

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B of norm 1 such that for each f ∈ SAP B (∂D) the function f − LB s (Es f ) ∈ B SAP (∂D) is continuous and equal to 0 at s. Moreover, any bounded linear operator SAP B (∂D) → AP B (R (R + iπ)) satisfying this property coincides with EsB .

Proof. According to Proposition 4.4, it suﬃces to deﬁne m the required operator EsB on the space of functions of the form f = l=1 bl fl , where bl ∈ B, fl ∈ SAP (∂D). In this case we set EsB (f ) :=

m

bl Es (fl ),

l=1

where Es is the operator from Theorem 2.4. Let B1 denote the unit ball in B ∗ . Then according to Theorem 2.4 we have sup |ϕ(EsB (f )(z))| EsB (f ) = sup EsB (f )(z)B = z∈∂D z∈∂D, ϕ∈B1 m = sup ϕ(bl )Es (fl )(z) z∈∂D, ϕ∈B1 l=1 m ≤ sup ϕ(bl )fl (z) = f . z∈∂D, ϕ∈B1 l=1

Therefore, the operator EsB is continuous and of norm 1 on a dense subspace of SAP B (∂D). Moreover, for any function f from this subspace we have B B (by Theorem 2.4), f − LB s (Es (f )) ∈ SAP (∂D) is continuous and equal to B 0 at s. Extending Es by continuity to SAP B (∂D) we obtain the operator satisfying the required properties. Its uniqueness follows from the uniqueness of operator Es . B B (x) := (EsB f )(x) and f1,s (x) := (EsB f )(x+iπ), Using the functions f−1,s x ∈ R, belonging to APB (R) we deﬁne the left (k = −1) and the right (k = 1) mean values of f ∈ SAP B (∂D) over s:

Msk (f ) := M (fk,s ) ∈ B. Then using formulas similar to those of the scalar case we deﬁne the BohrFourier coeﬃcients akλ (f, s) ∈ B and the spectrum specks (f ) of f over s. It follows straightforwardly from the properties of the spectrum of a B-valued almost periodic function on R that specks (f ) is at most countable. B By SAP Σ (S) ⊂ SAP B (S) we denote the Banach algebra of semi-almost periodic functions f with specks (f ) ⊂ Σ(s, k) for all s ∈ S, k ∈ {−1, 1}. Note that SAPΣB (S) = SAPΣ (S) ⊗ B, i.e., this deﬁnition is equivalent to the one used in Section 3.2 (the proof is obtained easily from Deﬁnition 4.3, using an appropriate partition of unity on ∂D and Theorems 1.1 and 3.2, see [6] for similar arguments). Also, a statement analogous to Theorem 3.3 holds for SAPΣB (S) ∩ ∞ ∞ HB (D). Namely, if f ∈ SAPΣB (S) ∩ HB (D), then spec1s (f ) = spec−1 s (f ) =: specs (f ).

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Proof of Proposition 3.5. We must show that if f ∈ (SAPΣ (S) ∩ H ∞ (D))B on the maximal ideal space bS (D) of algebra SAP (S) ∩ H ∞ (D), then the ∞ (D). restriction f |D ∈ SAPΣB (S) ∩ HB B S Indeed, since f ∈ C (b (D)) and C(bS (D)) has the approximation property, f ∈ C(bS (D)) ⊗ B by Theorem 1.1. Next, C(bS (D)) is generated by algebra SAP (S) ∩ H ∞ (D) and its conjugate. Therefore f can be uniformly approximated on bS (D) by a sequence of B-valued polynomials in variables from algebras SAP (S) ∩ H ∞ (D) and its conjugate. This easily implies that f |∂D is well deﬁned and belongs to SAP B (S). In fact, f |∂D ∈ SAPΣB (S) because k φ(f )k ∈ SAP Σ (S) and the Bohr-Fourier coeﬃcients of ∗f satisfy aλ (ϕ(f ), s) = ϕ aλ (f, s) for any s ∈ S, k ∈ {−1, 1} and φ ∈ B . Further, by the definition f |D is such that ϕ(f ) ∈ H ∞ (D) for any ϕ ∈ B ∗ . This shows that ∞ f ∈ HB (D). For the proof of Theorem 3.6 we require some auxiliary results. Let AP C(T ) be the Banach algebra of functions f : T → C uniformly continuous on T and almost periodic on each horizontal line. We deﬁne AP C B (T ) := AP C(T ) ⊗ B. The proof of the next statement is analogous to the proof of Lemma 4.3 in [6]. Lemma 4.6. Suppose that f1 ∈ AP B (R), f2 ∈ AP B (R+iπ). Then there exists a function F ∈ AP C B (T ) harmonic in the interior of Σ whose boundary values are f1 and f2 . Moreover, F admits a continuous extension to the maximal ideal space bT of AP H(T ). The proof of the next statement uses Lemma 4.6 and is very similar to the proof of Lemma 4.2 (for B = C) in [6], so we omit it as well. Suppose that s := eit and γsk (δ) are arcs deﬁned in (2.3). For δ ∈ (0, π) we set γ1 (s, δ) := Log(ϕs (γs1 (δ))) ⊂ R and γ−1 (s, δ) := Log(ϕs (γs−1 (δ))) ⊂ R + iπ (see Example 3). Lemma 4.7. Let s ∈ S. Suppose that f ∈ SAP B ({−s, s}). We put fk =f |γsk (π) and deﬁne on γk (s, π) −1 hk := fk ◦ ϕ−1 , s ◦ Log

k ∈ {−1, 1}.

Then for any ε > 0 there exist δε ∈ (0, π) and a function H ∈ AP C B (T ) harmonic in the interior T0 of T such that sup z∈γk (s,δε )

hk (z) − H(z)B < ε,

k ∈ {−1, 1}.

Let s ∈ ∂D and Us be the intersection of an open disk of some radius ¯ \ s. We call such Us a circular neighbourhood of s. ≤ 1 centered at s with D Deﬁnition 4.8. We say that a bounded continuous function f : D → B is almost-periodic near s if there exist a circular neighbourhood Us , and a function f¯ ∈ AP C B (T ) such that ¯ ¯ f (z) = (LB z ∈ Us . (4.5) s f )(z) := f ◦ Log ◦ ϕs (z),

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In what follows for Σ : S × {−1, 1} → 2R such that Σ(s) = R for each s ∈ S we omit writing Σ in aSΣ , isΣ , bSΣ (D) etc., see Section 3.4 for the corresponding deﬁnitions. In the proof of Theorem 1.8 of [6] (see Lemmas 4.4, 4.6 there) we established, cf. Theorem 3.13, (1) Any scalar harmonic function f on D almost periodic near s admits a ¯s ) ⊂ bS (D) for some circular neighcontinuous extension fs to (aS )−1 (U bourhood Us . (2) For any s ∈ S and any g ∈ AP H(T ) the holomorphic function g˜ := Ls g on D almost periodic near s is such that g˜s ◦ is coincides with the extension of g to bT . More generally, Lemma 4.6, statements (1), (2) and the fact that AP B (R) = AP (R) ⊗ B (see Theorems 1.1 and 2.2) imply (3) Any B-valued harmonic function f on D almost periodic near s admits ¯s ) ⊂ bS (D) for some circular a continuous extension fsB to (aS )−1 (U neighbourhood Us . (4) For any s ∈ S and any g ∈ AP H B (T ) the B-valued holomorphic function g˜ := LB ˜sB ◦ is coincides s g on D almost periodic near s is such that g with the extension of g to bT . ∞ (D) and s ∈ ∂D. There is a bounded Lemma 4.9. Let f ∈ SAPΣB (S) ∩ HB ˆ B-valued holomorphic function f on D almost periodic near s such that for any ε > 0 there is a circular neighbourhood Us;ε of s so that

sup ||f (z) − fˆ(z)||B < ε.

z∈Us;ε

B ¯ ¯ Moreover, fˆ = LB s f for some f ∈ AP HΣ(s) (T ).

Proof. Assume, ﬁrst, that s ∈ S. By Lemma 4.7, for any n ∈ N there exist a number δn ∈ (0, π) and a function Hn ∈ AP C B (T ) harmonic on T0 such that 1 (4.6) sup fk (z) − Hn (z)B < , k ∈ {−1, 1}. n z∈γk (s,δn ) Using the Poisson integral formula for the bounded B-valued harmonic function f − hn , hn := LB s Hn := Hn ◦ Log ◦ ϕs , on D we easily obtain from (4.6) that there is a circular neighbourhood Vs;n of s such that 2 (4.7) sup f (z) − hn (z)B < . n z∈Vs;n ˆ n to (aS )−1 (s) ∼ According to (3) each hn admits a continuous extension h = ˆ n }n∈N to bT . Moreover, (4.7) implies that the restriction of the sequence {h ˆ ∈ C B ((aS )−1 (s)) (aS )−1 (s) forms a Cauchy sequence in C B ((aS )−1 (s)). Let h be the limit of this sequence. Further, for any functional φ ∈ B ∗ the function φ ◦ f belongs to the admits a continuous extension fφ algebra SAP (S) ∩ H ∞ (D) and therefore to (aS )−1 (s) such that on (is )−1 (aS )−1 (s) the function fφ ◦ is belongs to

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ˆ for any φ ∈ B ∗ . Then it follows AP H(T ). Now, (4.7) implies that fφ = φ ◦ h s ˆ ◦ i ∈ AP H B (T ). Therefore by (4) we ﬁnd from Theorems 1.1 and 3.2 that h ˆ a bounded B-valued holomorphic function fˆ on D of the same sup-norm as h S −1 ˆ almost periodic near s such that its extension to (a ) (s) coincides with h. Next, by the deﬁnition of the topology of bS (D), see [6], Lemma 4.4 (a), we obtain that for any ε > 0 there is a number N ∈ N such that for all n ≥ N , ε sup fˆ(z) − hn (z)B < . 2 z∈Vs;n Now, choose n ≥ N in (4.7) such that the right-hand side there is < ε2 . For this n we set Us;ε := Vs;n . Then the previous inequality and (4.7) imply the required inequality sup ||f (z) − fˆ(z)||B < ε. z∈Us;ε

Further, if s ∈ S, then, by deﬁnition, f |∂D is continuous at s. In this case as the function fˆ we can choose the constant B-valued function equal to f (s) on D. Then the required result follows from the Poisson integral formula for f − fˆ. By deﬁnition, fˆ is determined by formula (4.5) with an f¯ ∈ AP H B (T ). B (T ). To this end it suﬃces to prove that Let us show that f¯ ∈ AP HΣ(s) ¯ ϕ(f ) ∈ AP HΣ(s) (T ) for any ϕ ∈ B ∗ . Indeed, it follows from the last inequality that the extension of ϕ(f ) ∈ SAPΣ (S) ∩ H ∞ (D) to (aSΣ )−1 (s) coincides with ϕ(f¯). By the deﬁnition of spectrum of a semi-almost periodic function, this implies that specs (ϕ(f¯)) ⊂ Σ(s). Now, we are ready to prove Theorem 3.6. The inclusion ASΣ ⊂ SAPΣ (S)∩H ∞ (D) follows from Example 3. Indeed, for s ∈ S assume that the holomorphic function eλh ∈ H ∞ (D) is such that Re(h)|∂D is the characteristic function of the closed arc going in the counterclockwise direction from the initial point at s to the endpoint at −s and such that πλ ∈ Σ(s). Then Example 3 implies that eλh ∈ SAP ({s, −s}) ∩ H ∞ (D) λ and specs (eλh ) = { πλ }. In particular, (z + s)e π h ∈ SAPΣ|{s} ({s}) ∩ H ∞ (D), as required. Let us prove the opposite inclusion. (A) Consider ﬁrst the case S = F , where F = {si }m i=1 is a ﬁnite subset of ∞ ∂D. Let f ∈ SAPΣB (F ) ∩ HB (D). Then according to Lemma 4.9 there exists B a function fs1 ∈ AP HΣ(s (T ) such that the bounded B-valued holomorphic 1) function gs1 − f , where gs1 := fs1 ◦ Log ◦ ϕs1 , on D is continuous and equals 0 at s1 . B Let us show that gs1 ∈ A{s1 ,−s1 } ⊗ B. Since fs1 ∈ AP HΣ(s (T ), by 1) B Theorems 1.1 and 3.2 it can be approximated in AP HΣ(s1 ) (T ) by ﬁnite sums of functions of the form beiλz , b ∈ B, λ ∈ Σ(s1 ), z ∈ T . In turn, gs1 can be approximated by ﬁnite sums of functions of the form beiλLog◦ϕs1 , b ∈ B.

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As was shown in [6], eiλLog◦ϕs1 ∈ A{s1 ,−s1 } . Hence, gs1 ∈ A{s1 ,−s1 } ⊗ B. We deﬁne gs (z)(z + s1 ) . gˆs1 = 1 2s1 Then, since the function z → (z + s1 )/(2s1 ) ∈ A(D) and equals 0 at −s1 , and gs1 ∈ A{s1 ,−s1 } ⊗ B, the function gˆs1 ∈ A{s1 } ⊗ B. Moreover, by the {s } construction of gˆ1 and the deﬁnition of the spectrum gˆs1 ∈ AΣ(s1 1 ) ⊗ B. By deﬁnition, the diﬀerence gˆs1 − f is continuous and equal to zero at z1 . Thus, B ∞ (F \ {s1 }) ∩ HB (D). gˆs1 − f ∈ SAPΣ| F \{s } 1

{s }

We proceed in this way to get functions gˆsk ∈ AΣ(sk k ) ⊗ B, 1 ≤ k ≤ m, such that m gˆsk ∈ AB (D), f− k=1

where AB (D) is the Banach space of B-valued bounded holomorphic functions on D continuous up to the boundary. As in the scalar case using the Taylor expansion at 0 of functions from AB (D) one can easily show that AB (D) = A(D) ⊗ B. Therefore, f ∈ AF Σ ⊗ B. (B) Let us consider the general case of S ⊂ ∂D being an arbitrary ∞ (D). As follows from Lemma 4.9 and the closed set. Let f ∈ SAPΣB (S) ∩ HB arguments presented in part (A), given an ε > 0 there exist points sk ∈ ∂D, {s } functions fk ∈ AΣ(sk k ) ⊗ B and circular neighbourhoods Usk (1 ≤ k ≤ m) m such that {Usk }m k=1 forms an open cover of ∂D \ {sk }k=1 and f (z) − fk (z)B < ε on Usk ,

1 ≤ k ≤ m.

(4.8)

¯s . Since S is closed, for sk ∈ S we may assume that fk is continuous in U k m Let us deﬁne a B-valued 1-cocycle {ckj }k,j=1 on intersections of the sets in {Usk }m k=1 by the formula ckj (z) := fk (z) − fj (z),

z ∈ Usk ∩ Usj .

(4.9)

Then (4.8) implies supk,j,z ||ckj (z)||B < 2ε. Let A ⊂⊂ ∪m k=1 Usk be an open annulus with outer boundary ∂D. Using the argument from the proof of Lemma 4.7 in [6] one obtains that if the width of the annulus is suﬃciently small, then there exist B-valued functions ci holomorphic on Usi ∩ A and ¯si ∩ A¯ satisfying continuous on U sup z∈Usi ∩A

||ci (z)||B ≤ 3ε

(4.10)

and such that ci (z) − cj (z) = cij (z),

z ∈ Usi ∩ Usj ∩ A.

For such A let us deﬁne a function fε on A¯ \ {si }m i=1 by formulas fε (z) := fi (z) − ci (z),

¯ z ∈ Usi ∩ A.

(4.11)

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According to (4.9) and (4.11), fε is a bounded continuous B-valued function on A¯ \ {si }m i=1 holomorphic in A. Furthermore, since ci is continuous on ¯si ∩ A, ¯ and fi ∈ A{si } ⊗ B for si ∈ S, and fi ∈ AB (D) otherwise, fε |∂D ∈ U Σ(si ) B SAPΣ| (F ), where F = {si }m i=1 ∩ S. Also, from inequalities (4.8) and (4.10) F we obtain (4.12) sup f (z) − fε (z)B < 4ε. z∈A

Next, as in [6] we consider a 1-cocylce subordinate to a cover of the unit disk D consisting of an open annulus having the same interior boundary as A and the outer boundary {z ∈ C : |z| = 2}, and of an open disk centered at 0 not containing A but intersecting it by a nonempty set. Resolving this cocycle1 one obtains a B-valued holomorphic function Fε on D such that for an absolute constant Cˆ > 0 ˆ sup f (z) − Fε (z)B < Cε z∈D

B ∞ and by deﬁnition Fε ∈ SAPΣ| (F ) ∩ HB (D), where F = {s1 , . . . , sm } ∩ S. F The latter inequality and part (A) of the proof show that the complex vector space generated by spaces AF Σ|F ⊗ B for all possible ﬁnite subsets F ⊂ S ∞ (D). Since by deﬁnition the closure of all such is dense in SAPΣB (S) ∩ HB S B ∞ S AF Σ|F ⊗ B is AΣ ⊗ B, we obtain the required: SAPΣ (S) ∩ HB (D) = AΣ ⊗ B.

4.5. Proofs of Theorems 3.9 and 3.10 Proof of Theorem 3.9. Corollary 1.6 in [6] states that κ∗ |∂D : C(∂D) → C(∂D), the pullback by κ|∂D , maps SAP (κ∗ S) isomorphically onto SAP (S). Following closely the arguments in its proof, one obtains even more: κ∗ maps SAPκ∗ Σ (κ∗ S) isomorphically onto SAPΣ (S). Since κ∗ preserves H ∞ (D), the required result follows. Proof of Theorem 3.10. Let f ∈ SAP (∂D) ∩ H ∞ (D). According to Lemma 4.9 there exists a function fs ∈ AP H(T ) such that the diﬀerence h := f − Fs , where Fs := fs ◦Log◦ϕs , see (4.5), is continuous and equal to 0 at s. Therefore, it suﬃces to prove the assertion of the theorem for Fs . Let {zn } ⊂ T0 and {sn } ⊂ R ∪ (R + iπ) be the images of sequences {zn } and {sn } under the mapping Log◦ϕs (see Example 3). By the hypotheses of the theorem we have zn , sn → ∞ and |zn − sn | → 0 as n → ∞ (this follows from condition (3.4)). Since any function in AP H(T ) is uniformly continuous (see Section 3), the latter implies the required result. 1 There

is a misprint in [6] at this place: the inequality maxi ∇ρi L∞ (C) ≤ Cw(B∩A) < Cε

for smooth radial functions ρ1 , ρ2 , must be replaced by maxi ∇ρi L∞ (C) ≤

C . w(B∩A)

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4.6. Proofs of Theorems 3.14 and 3.15 Proof of Theorem 3.14. Below we identify ι(D) ⊂ bSΣ (D) with D, see Section 3.4. By Theorem 3.13, the maximal ideal space bSΣ (D) is ¯ \ S) s∈S is (bΣ(s) (T )) (D Σ (here isΣ : bΣ(s) (T ) → (aSΣ )−1 (s) is a homeomorphism). For each s ∈ S one has the natural map ιΣ(s) : T → bΣ(s) (T ) (determined by evaluations at points of T ). Also, the argument of the proof of Theorem 1.12 in [6] implies that the ¯ closure of D in bSΣ (D) contains (as a dense subset) (D\S) s∈S isΣ (ιΣ(s) (T )) . Thus in order to prove the theorem, it suﬃces to show that ιΣ(s) (T ) is dense in bΣ(s) (T ) if and only if Σ(s) is a group. We will use the following result. Theorem 4.10 ([21]). Suppose that Γ is the intersection of an additive subgroup of R and R+ . Then the image of the upper half-plane H+ in the maximal ideal space bΓ (T ) is dense. Observe that in this case each element of AP HΓ (T ) is extended to a holomorphic almost periodic function on H+ by means of the Poisson integral. Therefore the evaluations at points of H+ of the extended algebra determine the map H+ → bΓ (T ) of the theorem. First, assume that Σ(s) is a group. We have to show that ιΣ(s) (T ) is dense in bΣ(s) (T ). Assume the opposite. Then there exists ξ ∈ bΣ(s) (T ) and a neighbourhood of ξ U (λ1 , . . . , λm , ξ, ε) := {η ∈ bΣ(s) (T ) : |η(eiλk z ) − ck | < ε, 1 ≤ k ≤ m}, where λ1 , . . . , λm ∈ Σ(s), ck := ξ(eiλk z ), such that U (λ1 , . . . , λm , ξ, ε) ∩ cl ιΣ(s) (T ) = ∅, cf. the proof of Theorem 2.4 in [6]. Therefore, max |eiλk z − ck | ≥ ε > 0

1≤k≤m

for all z ∈ T.

(4.13)

Without loss of generality we may assume that ck = 0 and λk > 0, i.e., eiλk z − ck ∈ AP HΣ(s)∩R+ (T ). (For otherwise we replace eiλk z − ck with −iλk z e−iλk z −c−1 −c−1 k . Here e k ∈ AP HΣ(s)∩R+ (T ) since Σ(s) is a group. Also, (4.13) will be satisﬁed, possibly with a diﬀerent ε > 0.) Note that eiλk z − ck is not invertible in AP HΣ(s) (T ), since ξ(eiλk z − ck ) = 0. Therefore, since each function eiλk z − ck is periodic (with period λ2πk ), it has a zero in T . Since solutions of the equation eiλk z = ck are of the form zk = −

i ln |ck | Argck + 2πl + , λk λk

l ∈ Z,

all zeros of eiλk z − ck belong to T . Hence, in virtue of inequality (4.13), one has max |eiλk z − ck | ≥ ε˜ > 0 for all z ∈ H+ . 1≤k≤m

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Hence, by Theorem 4.10, there exist functions g1 , . . . , gm ∈ AP HΣ(s)∩R+ (H+ ) such that m gk (z)(eiλk z − ck ) = 1 for all z ∈ H+ . k=1

In particular, the above identity holds on T . This gives a contradiction with the assumption ξ(eiλk z − ck ) = 0, 1 ≤ k ≤ m. Now, assume that Σ(s) is not a group, i.e., it contains a non-invertible element λ0 . Suppose that ιΣ(s) (T ) is dense in bΣ(s) T . Then, since the modulus of f1 := eiλ0 z is bounded from below on T by a positive number, there exists g1 ∈ AP HΣ(s) (T ) such that f1 g1 ≡ 1. Therefore, g1 = e−iλ0 z ∈ AP HΣ(s) (T ), i.e., −λ0 ∈ Σ(s), a contradiction. Proof of Theorem 3.15. For the proof we will need the following auxiliary result. Let Γ ⊂ R be a nontrivial additive semi-group. For a subset X ⊂ T by X∞ we denote the set of limit points of ιΓ (X) in bΓ (T ) \ ιΓ (T ). Lemma 4.11. Let G ∈ C([0, 1), T ) be such that the closure of G([0, 1)) in T is non-compact. Then the set G∞ contains more than one element. Proof. If there exists a horizontal line R + ic, 0 ≤ c ≤ π, such that lim distT G(t), R + ic = 0, t→1−

then clearly (R + ic)∞ = G∞ . Moreover, (R + ic)∞ is inﬁnite (e.g., it contains a subset homeomorphic to interval [0, 1]). In the case that such a line does not exist, one can ﬁnd two closed substrips T1 , T2 ⊂ T , T1 ∩ T2 = ∅, such that the closures in T of both G([0, 1)) ∩ T1 and G([0, 1)) ∩ T2 are noncompact. Then (G([0, 1)) ∩ T1 )∞ and (G([0, 1)) ∩ T2 )∞ are nonempty, while (T1 )∞ ∩ (T2 )∞ = ∅. This implies the required statement. Now, we are ready to prove the theorem. Suppose on the contrary that for a continuous map F : [0, 1] → bSΣ (D) the conclusion of the theorem is not ¯ \S valid. First, assume that there exists a point c ∈ [0, 1) such that F (c) ∈ D ¯ \ S. Then, because bS (D) \ (D ¯ \ S) is a compact set (here but F ([0, 1]) ⊂ D Σ ¯ \ S with a subset of bS (D)), passing to a subinterval, we naturally identify D Σ ¯ \S if necessary, we may assume without loss of generality that F [0, 1) ⊂ D and F (1) ∈ (aSΣ )−1 (s) for a certain s ∈ S. Deﬁne G(t) := (Log ◦ ϕs ) F (t) ⊂ T, t ∈ [0, 1) (cf. Example 2). Then G satisﬁes conditions of Lemma 4.11 for Γ = Σ(s). Next, consider an f ∈ SAPΣ (S). According to Lemma 4.9 there exists a (unique) function fs ∈ AP HΣ(s) (T ) such that the diﬀerence f − Fs , where Fs := fs ◦ Log ◦ ϕs , is continuous and equal to 0 at s. This yields lim fs (G(t)) − f (F (t)) = 0. t→1−

¯ The latter implies that the set of limit points of F ([0, 1)) in bSΣ (D)\(D\S) is in one-to-one correspondence with the set of limit points G∞ of ιΣ(s) (G([0, 1)))

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in bΣ(s) (T ) \ ιΣ(s) (T ). By our assumption the set of limit points of F ([0, 1)) ¯ \ S) consists of the point F (1). This contradicts the assertion in bSΣ (D) \ (D ¯ \ S. of Lemma 4.11. Hence, in this case F ([0, 1]) ⊂ D ¯ \ S). Let s ⊂ S be such that In the second case, F ([0, 1]) ⊂ bSΣ (D) \ (D F ([0, 1]) ∩ (aSΣ )−1 (s) = ∅. Consider the continuous map {s}

ωs : bSΣ (D) → bΣ|{s} (D), transpose to the embedding SAPΣ|{s} ({s}) ∩ H ∞ (D) ⊂ SAPΣ (S) ∩ H ∞ (D). {s}

According to the case considered above, if ωs ◦ F : [0, 1] → bΣ|{s} (D) is such ¯ \ {s} for some c ∈ [0, 1), then (ωs ◦ F )([0, 1]) ⊂ D ¯ \ {s} that (ωs ◦ F )(c) ∈ D S −1 which contradicts the assumption F ([0, 1]) ∩ (aΣ ) (s) = ∅. Thus {s} ¯ \ {s}) = (a{s} )−1 (s). (ωs ◦ F )([0, 1]) ⊂ bΣ|{s} (D) \ (D Σ|{s}

−1 (s). This implies that F ([0, 1]) ⊂ aSΣ

4.7. Proof of Theorem 3.17 Since SAPΣ (S) ∩ H ∞ (D) is generated by algebras SAPΣ|F (F ) ∩ H ∞ (D) for F all possible ﬁnite subsets F of S, the inverse limit of {KΣ| ; ω}F ⊂S ; #F <∞ F of the corresponding Shilov boundaries coincides with KΣS (see Section 3.4 for the corresponding notation). Therefore to establish the result it suﬃces to prove that

F s iΣ|F clΣ(s) (R) ∪ clΣ(s) (R + iπ) ∪ ∂D \ F. (4.14) KΣ|F = s∈F

Since each point of ∂D \ F is a peak point for A(D) (which is a subset F of SAPΣ |F (F ) ∩ H ∞ (D)), ∂D \ F ⊂ KΣ| . Next, the closure of ∂D \ F in F ∞ bF (D) (the maximal ideal space of SAP Σ |F (F ) ∩ H (D)) coincides with Σ|F the right-hand side of (4.14), see the proof of Theorem 1.14 in [6]. Thus the F right-hand side of (4.14) is a subset of KΣ| . Finally, Theorem 1.14 of [6] F ∞ implies that for each f ∈ SAPΣ |F (F ) ∩ H (D), |f | attains its maximum on the set in the right-hand side of (4.14). This produces the required identity. One can easily show that the inverse limit of the family of sets in the right-hand sides of equations (4.14) coincides with

s iΣ clΣ(s) (R) ∪ clΣ(s) (R + iπ) ∪ ∂D \ S. s∈S

The proof of the theorem is complete. 4.8. Proofs of Theorems 3.19, 3.21 and Corollary 3.20 Proof of Theorem 3.19. (1) Consider ﬁrst the case of S being a ﬁnite subset of ∂D. For s ∈ S we deﬁne U1 := bSΣ (D) \ (aSΣ )−1 (s).

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Let U2 be the union of (aSΣ )−1 (s) and a circular neighbourhood of s whose ¯ Both U1 , U2 are open in bS (D) and U1 ∩ U2 = closure is a proper subset of D. Σ S −1 U2 \(aΣ ) (s) is the circular neighbourhood of s. Since U1 ∩U2 is contractible, H k (U1 ∩ U2 , Z) = 0, k ≥ 1. Let us show that for any k ∈ Z, H k (U2 , Z) ∼ = H k (bΣ(s) (T ), Z).

(4.15)

To this end consider a sequence V1 ⊃ V2 ⊃ . . . of circular neigbourhoods of ¯ s such that ∩∞ k=1 Vk = {s} and V1 = U1 ∩ U2 . We set ˆk := Vk ∪ (aS )−1 (s). U Σ S −1 ˆ ˆ (s) Let ιm l : Um → Ul , m ≥ l, be the corresponding embedding. Then (aΣ ) ˆ is the inverse limit of the family {Uj ; ι}j∈N . It is well known (see, e.g., [4], ˇ Chapter II, Corollary 14.6) that the direct limit of Cech cohomology groups k ˆ k S −1 H (Ul , Z) with respect to this family gives H ((aΣ ) (s), Z). Note also that ˆ1 := U2 . Thus the maps ιl1 induce ˆl is a deformation retract of U each U ˆl , Z), l ∈ N. Since (aS )−1 (s) ∼ isomorphisms H k (U2 , Z) ∼ = H k (U = bΣ(s) (T ), Σ these facts imply (4.15). Next, consider the Mayer-Vietoris sequence corresponding to the cover {U1 , U2 } of bSΣ (D):

· · · → H k−1 (bSΣ (D), Z) → H k (U1 ∩ U2 , Z) → H k (U1 , Z) ⊕ H k (U2 , Z) → H k (bSΣ (D), Z) → . . . . By the above results H k (U1 ∩ U2 , Z) = 0 and H k (U2 , Z) ∼ = H k (bΣ(s) T, Z), k ≥ 1. Therefore, H k (bSΣ (D), Z) = H k (U1 , Z) ⊕ H k bΣ(s) (T ), Z , k ≥ 1. Proceeding further inductively (i.e., applying similar arguments to U1 etc.) and using the fact that H k (bSΣ (D) \ S, Z) = 0, k ≥ 1, we obtain that H k (bSΣ (D), Z) = H k bΣ(s) (T ), Z . s∈S

Now, if S ⊂ ∂D is an arbitrary closed subset, then since bSΣ (D) is the inverse limit of bF Σ|F (D) for all possible ﬁnite subsets F ⊂ S, by the cited result in [4] H k (bSΣ (D), Z) is the direct limit of H k (bF Σ|F (D), Z). Based on the case considered above we obtain that this limit is isomorphic to k b H (T ), Z . Σ(s) s∈S This proves the ﬁrst part of the theorem. (2) As is shown in [5], if Γ ⊂ R+ or Γ ⊂ R− , then bΓ (T ) is contractible. Therefore under hypotheses of the theorem H k bΣ(s) (T ), Z = 0 for all s ∈ S. The required result now follows from (1), i.e., H k (bSΣ (D), Z) = 0 for all k ≥ 1. Further, according to [8] the connectedness of bSΣ (D) and the topological triviality of any complex vector bundle of a ﬁnite rank over bSΣ (D) are suﬃcient for projective freeness of SAPΣ (S) ∩ H ∞ (D).

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Clearly bSΣ (D) is connected. For otherwise, according to the Shilov theorem on idempotents, see [24], SAPΣ (S) ∩ H ∞ (D) contains a function f not equal identically to 0 or 1 on D such that f 2 = f , a contradiction. Next, we show that any ﬁnite rank complex vector bundle ξ over bSΣ (D) is topologically trivial. Since bSΣ (D) is the inverse limit of the system {bF Σ|F (D) ; ω}F ⊂S ; #F <∞ , see Section 3.4, ξ is isomorphic (as a topological bundle) to pullback to bSΣ (D) of a bundle on some bF Σ|F (D) with F ⊂ S ﬁnite, see, e.g., [11] and [17]. Therefore it suﬃces to prove the statement for S ⊂ ∂D being a ﬁnite subset. In this case, for each s ∈ S by the contractibility of (aSΣ )−1 (s) ∼ = bΣ(s) (T ) (see [5]) we have that the restriction of ξ to (aSΣ )−1 (s) is topologically trivial. Using a ﬁnite open cover {Ui }1≤i≤m of (aSΣ )−1 (s) such that ξ|Ui ∼ = U i × Cn , n = rankC ξ, for each i, we extend (by the Urysohn lemma) global continuous sections tj : (aSΣ )−1 (s) → ξ, 1 ≤ j ≤ n, determining the trivialization of ξ over (aSΣ )−1 (s) to each Ui . Then using a continuous partition of unity subordinate to a ﬁnite reﬁnement of {Ui }1≤i≤m we glue together these extensions to get global continuous sections t˜j , 1 ≤ j ≤ n, of ξ on a neighbourhood Us of (aSΣ )−1 (s) in bSΣ (D) such that t˜j |(aSΣ )−1 (s) = tj for each j. Since sections tj , 1 ≤ j ≤ n, are linearly independent at each point of (aSΣ )−1 (s), diminishing, if necessary, Us we obtain that sections t˜j , 1 ≤ j ≤ n, are linearly independent at each point of Us . Thus ξ is topologically trivial on Us . Also, by the deﬁnition of the topology on bSΣ (D) without loss of generality we may assume that Us \ (aSΣ )−1 (s) is a circular neighbourhood of s. Suppose that S = {s1 , . . . , sk }. We cover bSΣ (D) by sets Uj := Usj , ¯ \ V , where V ⊂ ∪k Us and 1 ≤ j ≤ k, described above and by U0 := D j j=1 V ∩ Usj is a circular neighbourhood of sj distinct from Usj \ (aSΣ )−1 (sj ), 1 ≤ j ≤ k. Since U0 is contractible, ξ|U0 is topologically trivial. Using trivializations of ξ on Uj , 0 ≤ j ≤ k, we obtain that ξ is deﬁned by a 1-cocycle {cij } with values in GLn (C) deﬁned on intersections Ui ∩Uj , 0 ≤ i < j ≤ k. In turn, ˜j }k of D ¯ such that by the deﬁnition of sets Uj , there is an acyclic cover {U j=0 S −1 ˜ ˜ j }k (aΣ ) (Uj ) = Uj , 0 ≤ j ≤ k. Thus there exists a cocycle {˜ cij } on {U j=0 S such that c˜ij ◦ aΣ = cij for all i, j. This cocycle determines a continuous vec¯ trivial on each U ˜i , 0 ≤ i ≤ k, such that (aS )∗ ξ˜ = ξ. Since tor bundle ξ˜ on D Σ ¯ is contractible, ξ˜ is topologically trivial. Hence ξ is topologically trivial as D well. Proof of Corollary 3.20. Let G ⊂ R be an additive subgroup and let AP CG (T ) ⊂ AP C(T ) be the algebra of uniformly continuous almost periodic functions on T having their spectrum in G. Here the spectrum of a function in AP C(T ) is the union of the spectra of its restrictions to each horizontal k line in T (see [2]). The vector space of functions j=1 cj (y)eiλj x , x + iy ∈ T , cj ∈ C([0, π]), λj ∈ G, k ∈ N, is dense in AP CG (T ) and, hence, the maximal ideal space M (AP CG (T )) of AP CG (T ) is homeomorphic to bG (R) × [0, π]. On the other hand, AP HG (T ) ⊂ AP CG (T ) and the extension of AP HG (T )

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to M (AP CG (T )) separates the points of M (AP CG (T )). Since the image of T in bG (T ) is dense (see the proof of Theorem 3.14), the latter implies that bG (T ) ∼ = M (AP CG (T )). Hence, taking G := Σ(s), s ∈ S, we obtain ∼ H k (bΣ(s) (R), Z). H k (bΣ(s) (T ), Z) = Since bΣ(s) (R) is a compact connected abelian group, the required statements follow from the remark before the formulation of the corollary and Theorem 3.19 (1). Proof of Theorem 3.21. In what follows we assume that uniform algebras are deﬁned on their maximal ideal spaces via the Gelfand transforms. We will require the following auxiliary result. Lemma 4.12. Assume that a set-valued map Σ as in Section 3.1 is deﬁned on {−s, s} and f ∈ SAPΣ ({−s, s}) ∩ H ∞ (D). Consider the function z+s Hs f (z) := f , z ∈ D. 2 Then Hs f ∈ SAPΣ|{s} ({s}) ∩ H ∞ (D) and [(isΣ|{s} ◦ ιΣ(s) )∗ Hs f ](z) = [(isΣ ◦ ιΣ(s) )∗ f ](z − ln 2),

z∈T

(4.16)

(see Theorem 3.13 for notations). This result states that the map Hs : SAPΣ ({−s, s}) ∩ H ∞ (D) → SAPΣ|{s} ({s}) ∩ H ∞ (D) is a bounded linear operator which induces under the identiﬁcation of the {−s,s} −1 ﬁbre (aΣ ) (s) with bΣ(s) (T ) by isΣ the map AP HΣ(s) (T ) → AP HΣ(s) (T ) deﬁned by h(z) → h(z − ln 2), z ∈ T , h ∈ AP HΣ(s) (T ). Proof. Clearly Hs f is holomorphic on D and continuous on ∂D \ {s}. Let us consider the function g(z) := [(Hs f ) ◦ (Log ◦ ϕs )−1 ](z) − [f ◦ (Log ◦ ϕs )−1 ](z − ln 2),

z ∈ T.

Next, we have (Log ◦ ϕs )−1 (z) + s se2z − (Log ◦ ϕs )−1 (z − ln 2) = →0 2 (2i + ez )(4i + ez ) as Re(z) → −∞. Since by the deﬁnition of SAPΣ ({−s, s}) ∩ H ∞ (D) the function f ◦ (Log ◦ ϕs )−1 is uniformly continuous on T , the last expression implies that g(z) → 0 as Re(z) → −∞. But Re(z) → −∞ if and only if (Log ◦ ϕs )−1 (z) → s. Therefore the function g ◦ Log ◦ ϕ is continuous in a circular neighbourhood of s and equals 0 at s. Since the pullback of the function [f ◦ (Log ◦ ϕs )−1 ](z − ln 2), z ∈ T , by (Log ◦ ϕs )−1 belongs to SAPΣ ({−s, s}) ∩ H ∞ (D) (it is obtained as the composition of f with a M¨ obius transformation preserving points −s and s), the function Hs f ∈ SAPΣ|{s} ({s}) ∩ H ∞ (D). Now, the identity (4.16) follows from the fact that (g ◦ Log ◦ ϕ)(s) = 0 by the deﬁnition of isΣ .

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Corollary 4.13. Let f ∈ GnΣ(s) (T ) (see Section 3.5 for notations). Consider the function F := Hs [Kf (Log ◦ ϕs )], Then F ∈

GnΣ|{s} ({s})

Kf (z) := f (z − ln 2),

where

z ∈ T.

and (isΣ|{s} ◦ ιΣ(s) )∗ F = f.

Proof. The implication F ∈ GnΣ|{s} ({s}) follows from the proof of Lemma 4.12 because the pullback by Log ◦ ϕs maps the space AP HΣ(s) (T ) isometrically into SAP ({−s, s}) ∩ H ∞ (D) so that specs of each of the pulled back function is a subset of Σ(s). The second statement of the corollary follows directly from (4.16) because (isΣ|{s} ◦ ιΣ(s) )∗ (h ◦ Log ◦ ϕs ) = h for any h ∈ AP HΣ(s) (T ) by Theorem 3.13 (1).

We are ready to prove the theorem. First we will consider the case when S = {s1 , . . . , sm } is a ﬁnite subset of ∂D. By the deﬁnition of connected components of GLn (A), where A is a Banach algebra, the map f → ((isΣ|1 {s

◦ ιΣ(s1 ) )∗ f, . . . , (isΣ|m{s

1}

induces a homomorphism ΨS : [GnΣ (S)] →

m}

si ∈S

◦ ιΣ(sm ) )∗ f ),

f ∈ GnΣ (S),

[GnΣ(si ) (T )].

We will show that ΨS is an isomorphism. n Suppose that (g1 , . . . , gm ) ∈ si ∈S GΣ(si ) (T ) represents an element n [g] ∈ si ∈S [GΣ(si ) (T )]. Then according to Corollary 4.13 for an element g˜ := Hs1 [K(Log ◦ ϕs1 )∗ g1 ] · · · Hsm [K(Log ◦ ϕsm )∗ gm ] ∈ GnΣ (S) and each l ∈ {1, . . . , m} we have (isΣ|l {s } ◦ ιΣ(sl ) )∗ g˜ = c1l · · · cl−1l · gl · cl+1l · · · cml , l

where every cjl is an invertible matrix. Since the matrix-function on the righthand side is homotopic to gl , for the element [˜ g] ∈ [GnΣ (S)] representing g˜, g ]) = [g]. Hence ΨS is a surjection. we obtain ΨS ([˜ To prove that ΨS is an injection, we require a modiﬁcation of the construction of Corollary 4.13. So suppose that Fsl = Hsl [K(Log◦ϕsl )∗ f ], where f ∈ GnΣ(sl ) (T ). By the deﬁnition, Fsl (sj ), j = l, are well-deﬁned invertible matrices. Let M be a matrix-function with entries from A(D) such that M (sj ) = Log(Fsl (sj )), j = l, and M (sl ) = 0. (Here the logarithm of an invertible matrix c is a matrix c˜ such that exp(˜ c) = c.) Then we have n (1) F˜sl := Fsl · exp(−M ) ∈ GΣ|{s } ({sl }) and satisﬁes l

(isΣ|l {s } ◦ ιΣ(sl ) )∗ F˜sl = f l

and F˜sl (sj ) = In ,

(here In is the unit n × n matrix);

j = l

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(2) F˜sl is homotopic to Fsl . Statement (2) follows from the fact that exp(−M ) belongs to the connected component containing In . Now, suppose that f ∈ GnΣ (S) is such that every matrix-function gl := (isΣ|l {s } ◦ ιΣ(sl ) )∗ f , l ∈ {1, . . . , m}, belongs to the connected compol nent of GnΣ(sl ) (T ) containing the unit matrix In , (i.e., [f ] ∈ Ker(ΨS )). We set ˜s , ˜ := G Gs := Hs [K(Log ◦ ϕs )∗ gl ], G := Gs , G l

l

l

l

1≤l≤m

l

1≤l≤m

˜ s is constructed from Gs as F˜s from Fs . where each G l l l l According to property (1), ˜ = gl , (isΣ|l {s } ◦ ιΣ(sl ) )∗ G l

for l ∈ {1, . . . , m}.

˜ is homotopic to G. Observe also that Moreover, property (2) implies that G each Gsl is homotopic to In (because gl satisﬁes this property and so the required homotopy is deﬁned as the image of the homotopy between gl and In under the continuous map Hsl ◦ K ◦ (Log ◦ ϕsl )∗ ) and therefore G and are homotopic to In . Finally, according to our construction f · G ˜ −1 is an G ¯ invertible matrix with entries from A(D). Since D is contractible, each such a matrix is homotopic to In . These facts imply that f is homotopic to In , that is [f ] = 1 ∈ [GnΣ (S)], where [f ] stands for the connected component containing f ∈ GnΣ (S). So ΨS is an injection, which completes the proof of the theorem in the case of a ﬁnite S. To prove the result in the general case we require the following lemma. Lemma 4.14. For every f ∈ GnΣ (S) there exists f¯ ∈ GnΣ|F (F ), where F ⊂ S is ﬁnite, such that f¯ ∈ [f ]. Proof. Let MΣn (S) be the Banach algebra of n × n matrix-functions with entries in SAPΣ (S)∩H ∞ (D) equipped with the norm h := supz∈D h(z)2 , h ∈ MΣn (S), where · 2 is the 2 operator norm on the complex vector space Mn (C) of n × n matrices. According to Corollary 3.7, f can be approximated n (F ) for some ﬁnite subsets F ⊂ S. Since in MΣn (S) by functions from MΣ| F the connected component [f ] is open (because GnΣ (S) ⊂ MΣn (S) is open), the latter implies the required statement: there exists f¯ ∈ GnΣ|F (F ), where F ⊂ S is ﬁnite, such that f¯ ∈ [f ]. This lemma implies that [GnΣ (S)] is the direct limit of the family F ⊂ S, #F < ∞}. Therefore we can deﬁne a homomorphism [GnΣ(s) (T )] ΨS : [GnΣ (S)] →

{[GnΣ|F (F )];

s∈S

as the direct limit of homomorphisms ΨF described above. Then ΨS is an isomorphism because each ΨF is an isomorphism on each image. This proves the ﬁrst statement of the theorem.

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The second statement follows from the fact that if Σ(s) ⊂ R+ or R− , then the maximal ideal space bΣ(s) (T ) of Banach algebra AP HΣ(s) (T ) is contractible [5]. Then the result of Arens [1] implies that GnΣ(s) (T ) is connected and therefore [GnΣ(s) (T )] is trivial. From here and the ﬁrst statement of the theorem we obtain that [GnΣ (S)] is trivial, or equivalently, that GnΣ (S) is connected. Acknowledgment We are grateful to S. Favorov, S. Kislyakov and O. Reinov for useful discussions.

References [1] R. Arens, To what extent does the space of maximal ideals determine the algebra? in “Function Algebras“ (Birtel, ed.), Scott-Foresman, Chicago, 1966. [2] A. S. Besicovich, Almost periodic functions. Dover Publications, 1958. [3] J. Bourgain and O. Reinov, On the approximation properties for the space H ∞ . Matemathische Nachrichten 122 (1983), 19–27. [4] G. E. Bredon, Sheaf theory. Second edition. Graduate Texts in Mathematics 170, Springer-Verlag, New York, 1997. [5] A. Brudnyi, Contractibility of maximal ideal spaces of certain algebras of almost periodic functions. Integral Equations and Operator Theory 52 (2005), 595–598. [6] A. Brudnyi and D. Kinzebulatov, On uniform subalgebras of L∞ on the unit circle generated by almost periodic functions. Algebra and Analysis 19 (2007), 1–33. [7] A. Brudnyi and D. Kinzebulatov, On algebras of holomorphic functions with semi-almost periodic boundary values. C. R. Math. Rep. Acad. Sci. Canada, in press. [8] A. Brudnyi and A. Sasane, Suﬃcient conditions for projective freeness of Banach algebras, J. Funct. Anal. 257 (2009), 4003–4014. [9] P.M. Cohn, From Hermite rings to Sylvester domains. Proc. Amer. Math. Soc. 128 (2000), 1899–1904. [10] P. Enﬂo, A counterexample to the approximation property in Banach spaces. Acta Math. 130 (1973), 309–317. [11] S. Eilenberg and N. Steenrod, Foundations of algebraic topology. Princeton University Press, Princeton, New Jersey, 1952. [12] R. Fox, Homotopy groups and torus homotopy groups. Ann. Math. 49 (1948), 471–510. [13] J. Garnett, Bounded analytic functions. Academic Press, 1981. [14] H. Grauert, Holomorphe Funktionen mit Werten in komplexen Lieschen Gruppen. (German) Mathematische Annalen, 133 (1957), 450–472. [15] A. Grothendieck, Products tensoriels toplogiques et espaces nucl´eaires. Memoirs Amer. Math. Society 16, 1955. [16] K. Hofmann and O. Mostert, Cohomology Theories for Compact Abelian Groups. Springer-Verlag, 1973.

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[17] D. Husemoller, Fibre bundles. Springer-Verlag, New York, 1994. [18] T.Y. Lam, Serre’s Conjecture. Lecture Notes in Mathematics 635, SpringerVerlag, 1978. [19] V. Ya. Lin, Holomorphic ﬁberings and multivalued functions of elements of a Banach algebra. Func. Anal. Appl., 7, 1973, English translation. [20] J. Lindenstrauss, Some open problems in Banach space theory. S´eminaire Choquet 18 (1975), 1–9. [21] L. Rodman and I. Spitkovsky, Almost periodic factorization and corona problem. Indiana Univ. Math. J. 47 (1998), 243–281. [22] W. Rudin, Fourier Analysis on Groups. Interscience Publishers, 1962. [23] D. Sarason, Toeplitz operators with semi-almost periodic kernels. Duke Math J. 44 (1977), 357–364. [24] G. E. Shilov, On decomposition of a commutative normed ring in a direct sums of ideals. Mat. Sb. 32(74):2 (1953), 353–364. Alexander Brudnyi Department of Mathematics and Statistics Math Science Building, Room 476 612 Campus Place University of Calgary 2500 University Drive NW Calgary, AB Canada T2N 1N4 e-mail: [email protected] Damir Kinzebulatov Department of Mathematics University of Toronto Bahen Centre 40 St. George St. Toronto, ON Canada M5S 2E4 e-mail: [email protected] Submitted: November 09, 2009. Revised: January 01, 2010.

Integr. Equ. Oper. Theory 66 (2010), 327–365 DOI 10.1007/s00020-010-1753-3 Published online February 23, 2010 © Birkhäuser/Springer Basel AG 2010

Integral Equations and Operator Theory

Estimates for Solutions of a Parameter-Elliptic Multi-Order System of Diﬀerential Equations R. Denk and M. Faierman Abstract. This paper is concerned with a boundary value problem deﬁned over a bounded region of Euclidean space, and in particular it is devoted to the establishment of a priori estimates for solutions of a parameter-elliptic multi-order system of diﬀerential equations under limited smoothness assumptions. In this endeavour we extend the results of Agranovich, Denk, and Faierman pertaining to a priori estimates for solutions associated with a parameter-elliptic scalar problem, as well as the results of various other authors who have extended the results of Agranovich et. al. from the scalar case to parameter-elliptic systems of operators which are either of homogeneous type or have the property that the diagonal operators are all of the same order. In addition, we extend some results of Kozhevnikov and Denk and Volevich who have also dealt with sytems of the kind under consideration here, in that one of the works of Kozhevnikov deals only with 2 × 2 systems, while the other, as well as the work of the last two authors, do not cover Dirichlet boundary conditions. Mathematics Subject Classiﬁcation (2010). Primary 35J55; Secondary 35S15. Keywords. Parameter-ellipticity, multi-order systems, a priori estimates.

1. Introduction This paper is concerned with a boundary problem deﬁned over a bounded region in Rn , and in particular is devoted to the establishment of a priori estimates for solutions of a parameter-elliptic multi-order system of diﬀerential equations under limited smoothness assumptions. To elaborate on what was just said, let us now explain in more detail our two main objectives.

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With regards to the ﬁrst, we point out that very general results were derived in [ADF] pertaining to the spectral theory for scalar, non-selfadjoint elliptic boundary problems involving diﬀerential operators under limited smoothness assumptions and under a parameter-ellipticity condition. In particular, a method was developed there for deriving results pertaining to the eigenvalue asymptotics even under the limited conditions imposed. In [DFM] the techniques of [ADF] were used to extend the results for the scalar problem to that for a problem involving a parameter-elliptic system of diﬀerential operators of homogeneous type, and subsequently this result was extended in [F] to the case where only the diagonal operators of the system all had to be of the same order. It is important to mention at this point that all the spectral results derived in the above works depended fundamentally upon the establishment of a priori estimates for solutions of the boundary problem under consideration. Furthermore, in the above works, the a priori estimates could be established using standard methods, but when one attempts to extend the results of [ADF] to fully parameter-elliptic multi-order systems of diﬀerential operators, one ﬁnds that the standard methods are no longer adequate, and new techniques must be introduced. Accordingly, with all of this in mind, let us turn to the problem under N consideration here. Let N ∈ N with N > 1 and let {sj }N 1 and {tj }1 denote sequences of integers satisfying s1 ≥ s2 ≥ ... ≥ sN , t1 ≥ t2 ≥ ... ≥ tN ≥ 0, and put mj = sj + tj for j = 1, ..., N . We suppose that m1 = m2 = ... = mk1 > mk1 +1 = ... = mkd−1 > mkd−1 +1 = ... = mkd > 0, where kd = N , put m j = mkj for j = 1, ..., d, and let Ir denote the (kr − kr−1 ) × (kr − kr−1 ) identity matrix for r = 1, ..., d, where k0 = 0. In the sequel we will use the notation I to denote the × unit matrix for ∈ N and also impose kr conditions which will ensure that for r = 1, ..., d, the sum j=1 mj is even; N we henceforth denote this sum by 2Nr . Then with {σj } , N = Nd , denoting

1

a sequence of integers such that max{σj }N 1 < sN , we shall be concerned here with the boundary problem A(x, D)u(x) − λu(x) = f (x) in Ω, , Bj (x, D)u(x) = gj (x) on Γ for j = 1, . . . , N

(1.1) (1.2)

where Ω is a bounded region in Rn , n ≥ 2, with boundary Γ, u(x) = T (u1 (x), . . . , uN (x))T , and f (x) = (f1 (x), . . . , fN (x)) are N × 1 matrix T denotes transpose, the gj (x) are scalar functions functions deﬁned in Ω, deﬁned on Γ, A(x, D) is an N ×N matrix operator whose entries Ajk (x, D) are linear diﬀerential operators deﬁned on Ω of order not exceeding sj + tk , is a 1 × N and deﬁned to be 0 if sj + tk < 0, and Bj (x, D), 1 ≤ j ≤ N matrix operator whose entries Bj,k (x, D) are linear diﬀerential operators deﬁned on Γ of order not exceeding σj + tk , and deﬁned to be 0 if σj + tk < 0. Our assumptions concerning the problem (1.1), (1.2) will be made precise in Section 2.

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To motivate the second objective of this paper, let us point out that the ﬁrst investigation into the spectral theory for a fully parameter-elliptic multi-order system of operators was investigated by Kozhevnikov [K1]. In this paper the author deals with a system of pseudodiﬀerential operators acting over a compact manifold without boundary; and by introducing the so-called Kozhevnikov conditions, the author is able to establish a priori estimates for solutions as well as various other spectral results. In subsequent works [K2], [K3] the author deals with a genuine boundary problem under inﬁnitely smooth conditions involving a parameter-elliptic multi-order system of diﬀerential operators acting over a bounded region in Rn and a system of diﬀerential operators deﬁned on the boundary. In particular, in [K2] the author restricts himself to a 2 × 2 system of diﬀerential operators and a system of boundary operators which can be expressed in the form a a lower triangular block matrix, that is, in the terminology of this paper, we now 2 have N = 2, A(x, D) = (Ajk (x, D))j,k=1 , B(x, D) = (Bjk (x, D))j=1,...,µ/2 = k=1,2 2 2 ord A (x, D). By (Tjk (x, D))j,k=1 , with T12 (x, D) = 0, where µ = jj j=1 introducing various conditions related to parameter-ellipticity, a result pertaining to the resolvent operator is established. In [K3] the author removes the restriction of [K2] that N = 2, but now requires that the system of boundary operators admits a representation in the form of an upper triangular block matrix. Furthermore, he requires that A(x, D) be of Petrovskii type. Then under various conditions, including those of [K1], the author derives some results pertaining to the resolvent operator. The problem considered in [K1] was also dealt with by Denk, Mennicken, and Volevich [DMV], but now in more detail. And by introducing conditions equivalent to those of Kozhevnikov and using the method of Newton polygons, the authors establish a priori estimates for solutions as well as various spectral results. In a subsequent work [DV], Denk and Volevich deal with a genuine boundary problem of the form (1.1), (1.2), but under the assumption that the operators Bjk (x, D), if not identically zero, contain only top order terms and that these orders must satisfy a special condition. Then by appealing to the Kozhevnikov conditions as well as to the conditions of VishikLyusternik [VL], the authors establish a priori estimates for solutions of the boundary problem (1.1), (1.2) for the case where Ω = Rn+ and Γ = Rn−1 . What is novel in this latter work is the introduction of the Vishik-Lyusternik conditions, as this enables one to deal with multi-order systems of operators in the traditional way, that is, by means of contour integration. Unfortunately, because of the restrictions imposed, both papers [K3] and [DV] are not even able to deal with the important problem where sj = tj = tj for j = 1, . . . , N, (here {tj }N 1 denotes a monotonic decreasing sequence of positive integers) and the boundary conditions are of Dirichlet type (see [ADN, Section 2], [G, p.448]). Of course the paper [K2] does include Dirichlet boundary conditions, but the requirement that N = 2 is very restrictive.

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Thus the ﬁrst objective of this paper is to extend the results of [ADF] concerning solutions of the boundary problem for the scalar case to the nonscalar problem under consideration here, while our second objective is to extend the results of [K3] and [DV] by establishing a priori estimates for solutions of the boundary problem (1.1), (1.2) under boundary conditions which do include those of Dirichlet. And in this endeavour we shall make use of the conditions of both Kozhevnikov and Vishik-Lyusternik (see Deﬁnitions 2.4 and 2.6 below). Our main result is then given in Theorem 2.8 below. Let us also mention at this point that from a consideration of length, we have limited ourselves in this paper solely to the proof of the suﬃciency part of Theorem 2.8, that is to say, that the conditions cited in Deﬁnition 2.6 ensure the validity of this theorem. The converse of this result, that is, the proof of the necessity part of the theorem, will be left for a later work. Finally, let us outline the contents of the paper. In Section 2 we introduce some terminology, deﬁnitions, and assumptions concerning the boundary problem (1.2), (1.2) which we require for our work and then present the main result of this work, namely Theorem 2.8 below. In Section 3 we restrict ourselves to the case where all the operators involved have constant coeﬃcients and which act either over Rn , without boundary conditions, or over Rn+ , with boundary conditions deﬁned on Rn−1 . All the preliminary results required for the proof of Theorem 2.8 are established here, with the main result being Proposition 3.6. These results are then used in Section 4 to prove Theorem 2.8.

2. Preliminaries In this section we are going to introduce some terminology, deﬁnitions, and assumptions concerning the boundary problem (1.1), (1.2) which we require for our work and then state the main result of this paper, namely Theorem 2.8 below. Accordingly, we let x = (x1 , . . . , xn ) = (x , xn ) denote a generic point n in R and use the notation Dj = −i∂/∂xj , D = (D1 , . . . .Dn ), Dα = D1α1 · · · Dnαn = D α Dnαn , and ξ = ξ1α1 · · · ξnαn for ξ = (ξ1 , . . . , ξn ) = (ξ ,nξn ) ∈ n R , where α = (α1 , . . . , αn ) = (α , αn ) is a multi-index whose length j=1 αj is denoted by |α|. Diﬀerentiation with respect to another variable, say y ∈ Rn , instead of x will be indicated by replacing D, Dα , Dα , and Dnαn by Dy , Dyα , Dyα , and Dyαnn , respectively. For 1 < p < ∞, s ∈ N0 = N ∪ { 0 }, and G an open set in R , ∈ N, we let Wps (G) denote the Sobolev space of order s related to Lp (G) and denote the norm in this space by · s,p,G , where 1/p α p us,p,G = |D u(x)| dx for u ∈ Wps (G). In addition we shall |α|≤s G

use norms depending upon a parameter λ ∈ C\{ 0 }, namely for 1 ≤ j ≤ d, we let (j) j u0,p,G for u ∈ Wps (G). |||u|||s,p,G = us,p,G + |λ|s/m

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˚ s (G) denote the closure of C ∞ (G) in W s (G). We also let W p 0 p In the sequel we shall also at times deal with the Bessel-potential space Hps (G) for 0 ≥ s ∈ Z and equipped with norms depending upon the parameter (j)

λ. Namely for u ∈ Hps (G) and 1 ≤ j ≤ d, we introduce the norms |||u|||s,p,G =

(j) (j) F −1 ξ, λsj F u0,p,Rn if G = Rn and |||u|||s,p,G = inf |||v|||s,p,Rn otherwise, where the inﬁmum is taken over all v ∈ Hps (Rn ) for which u = v G, F denotes

1 2 2 j the Fourier transformation in Rn (x → ξ) and ξ, λj = |ξ|2 + |λ| m (see [GK, Section 1], [T, p. 177]). Lastly, let Rn+ = { x ∈ Rn |xn > 0 }, R+ = { t ∈ R|t > 0 }, and let Rn− and R− be deﬁned similarly. Assume for the moment that the boundary Γ of Ω (see (1.1), (1.2)) is of class C m−1,1 for some m ∈ N , and let s be an integer satisfying 1 ≤ s ≤ m. Then for G = Ω or G = Rn+ , the vectors u ∈ Wps (G) have boundary values s−1/p (∂G) v = u∂G and we denote the space of these boundary values by Wp and denote by · s−1/p,p,∂G the norm in this space, where vs−1/p,p,∂G

s−1/p

= inf us,p,G for v ∈ Wp (∂G) and the inﬁmum is taken over those u ∈ Wps (G) for which u∂G = v (see also [ADF, Section 2] and [Gr, p.20] for s−1/p

further deﬁnitions of Wp (∂G)). In addition we shall use norms depending upon a parameter λ ∈ C\{ 0 }, namely for 1 ≤ j ≤ d, (j)

j |||v|||s−1/p,p,∂G = vs−1/p,p,∂G +|λ|(s−1/p)/m v0,p,∂G

for v ∈ Wps−1/p (∂G),

where · 0,p,∂G denotes the norm in Lp (∂G). Turning now to the boundary problem (1.1), (1.2), let us write α Ajk (x, D) = ajk for x ∈ Ω and 1 ≤ j, k ≤ N, α (x)D |α|≤sj +tk

Bjk (x, D) =

α . (2.1) bjk for x ∈ Γ and k = 1, . . . , N, j = 1, . . . , N α (x)D

|α|≤σj +tk

Observing that the orders of the operators Ajk (x, D), Bjk (x, D) remain N N unchanged if we replace the sequences {sj }N 1 , {tj }1 , and {σj }1 by {sj − N † N † N † σ † }N 1 , {tj + σ }1 , and {σj − σ }1 , respectively, where σ = max{σj }1 + 1, we see that by making such substitutions if necessary, there is no loss of . Likewise, by generality in henceforth supposing that σj < 0 for j = 1, . . . , N N N N N replacing the sequences {sj }1 , {tj }1 , and {σj }1 by {sj − sN }N 1 , {tj + sN }1 , and {σj − sN }N 1 }, respectively, if necessary, we may also henceforth suppose that sN ≥ 0. Assumption 2.1. It will henceforth be supposed that tj ≥ 0 and sj ≥ 0 for . j = 1, . . . , N , and that σj < 0 for j = 1, . . . , N Assumption 2.2. It will henceforth be supposed that: (1) Γ is of class C κ0 −1,1 ∩

jk sj C s1 , where κ0 = max t1 , max{−σj }N 1 ; (2) for each pair j, k, aα ∈ C (Ω) jk for |α| ≤ sj + tk if sj > 0, while if sj = 0, then aα ∈ L∞ (Ω) for |α| < sj + tk

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0 jk −σj −1,1 and ajk (Γ) α ∈ C (Ω) for |α| = sj + tk ; (3) for each pair j, k, bα ∈ C for |α| ≤ σj + tk .

For ξ ∈ Rn let

N ˚ ξ) = A ˚jk (x, ξ) A(x, for x ∈ Ω, j,k

˚jk (x, ξ) ˚ ξ) = B B(x, for x ∈ Γ, j=1,...,N k=1,...,N

˚jk (x, ξ) (resp. B ˚jk (x, ξ)) consists of those terms in Ajk (x, ξ) (resp. where A ˚j (x, ξ) Bjk (x, ξ)), which are just of order sj +tk (resp. σj +tk ). We denote by B ˚ ξ). Then in the sequel we shall also require the following the j-th row of B(x, notation. For x ∈ Ω and ξ ∈ Rn , let kr (r) ˚jk (x, ξ) for 1 ≤ r ≤ d, A11 (x, ξ) = A (r) A12 (x, ξ)

˚jk (x, ξ) = A

j,k=1

j=1,...,kr k=kr +1,...,N

,

(r) A21 (x, ξ)

N (r) ˚jk (x, ξ) and A22 (x, ξ) = A

˚jk (x, ξ) = A j=kr +1,...,N , k=1,...,kr

j,k=kr +1

for 1 ≤ r ≤ d − 1.

Also for x ∈ Γ, ξ ∈ Rn , and 1 ≤ r ≤ 1 , ≤ d, let ˚jk (x, ξ) B (r,) (x, ξ) = B j=N−1 (1−δr, )+1,...,N , (r,) B1 ,1 (x, ξ)

k=1,...,N

˚jk (x, ξ) = B j=N−1 (1−δr, )+1,...,N , k=1,...,k1

and

(r,) ˚jk (x, ξ) B1 ,2 (x, ξ) = B j=N−1 (1−δr, )+1,...,N , k=k1 +1,...,N

where δr, is the Kronecker delta. In addition we let Ir,0 = diag(0 · I1 , . . . , 0 · Ir−1 , Ir ) for r = 2, . . . , d and I1,0 = I1 . Note that when x0 ∈ Γ we can rewrite the boundary problem (1.1), (1.2) in terms of a local coordinate system at x0 wherein x0 → 0 and ν → en , where ν denotes the exterior normal to Γ at x0 and (e1 , . . . , en ) denotes the standard basis in Rn . Then supposing that this has been done, we shall in the sequel be concerned with the boundary problem ˚ D)u(x) − λ u(x) = f (x) for x ∈ Rn , A(0, + ˚j (0, D)u(x) = gj (x ) at xn = 0 for j = 1, . . . , N, B and corresponding to this boundary problem we deﬁne the associated sym˚j,k (0, ξ), A(r) (0, ξ), B (r,) (0, ξ), and B (r,) (0, ξ) in pre˚ ξ), B ˚j,k (0, ξ), A bols B(0, j,k 1 ,j cisely the same way their analogues were deﬁned in the original coordinate system.

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Deﬁnition 2.3. Let L be a closed sector in the complex plane with vertex at the origin. Then the operator A(x, D) − λ IN will be called parameter (r) elliptic in L if det A11 (x, ξ) − λIr,0 = 0 for x ∈ Ω, ξ ∈ Rn \ {0}, and λ ∈ L, r = 1, . . . , d. In the sequel we let C± = {z ∈ C, Im z > < 0}. Deﬁnition 2.4. Suppose that the operator A(x, D)−λ IN is parameter-elliptic in the sector L introduced above. Let x0 be an arbitrary point of Γ and assume that the boundary problem (1.1), (1.2) has been rewritten in a local coordinate system associated with x0 in the manner just explained. Then the operator A(x, D) − λ IN will be called properly parameter-elliptic in L if the following conditions are satisﬁed. (r) (1) The polynomial det A11 (0, ξ , z) − λIr,0 has precisely Nr zeros lying in C+ for ξ ∈ Rn−1 \ {0} and λ ∈ L, r = 1, . . . , d. (r) (2) The polynomial det A11 (0, 0, z) − λIr,0 has precisely Nr − Nr−1 zeros lying in C+ for λ ∈ L \ {0}, r = 2, . . . , d. Remark 2.5. Referring to Condition (1) of Deﬁnition 2.4, we know from [AV, (r) Section 2] that det A11 (0, ξ , z) − λIr,0 has precisely Nr zeros in C+ if r = 1 or if r > 1 and n > 2. In the sequel, when proper parameter-ellipticity is supposed, it will be assumed that this is also the case when r > 1 and n = 2. Turning next to Condition (2) of the deﬁnition, it is clear that the number of zeros of the determinant in C+ (resp. C− ) does not depend upon λ. Hence it follows from an expansion of the determinant in powers of z and λ that r is odd, kr − kr−1 is even, Condition (2) always holds if m r is even or if m and there is a λ ∈ L \ {0} such that −λ ∈ L. Lastly we mention at this point that it is also clear from what was said above that Condition (2) is always satisﬁed if the operator A(x, D) is essentially upper triangular at x0 (see Deﬁnition 2.7 below). Deﬁnition 2.6. Let L denote the sector introduced in Deﬁnition 2.3 above. Then we say that the boundary problem (1.1), (1.2) is parameter-elliptic in L if A(x, D) − λ IN is properly parameter-elliptic in L and the following conditions are satisﬁed. Let x0 be an arbitrary point of Γ and suppose that the boundary problem (1.1), (1.2) has been rewritten in a local coordinate system associated with x0 , as explained above. According to the notation introduced above, let (r,r) ˚jk (0, ξ , Dn ) 1 ≤ r ≤ d, Br,1 (0, ξ , Dn ) = B j=1,...,Nr , (r,) Br, (0, ξ , Dn )

k=1,...,k

r ˚jk (0, ξ , Dn ) = B j=N−1 +1,...,N ,

k=1,...,k

Then

1 ≤ r < d, r < ≤ d.

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(1) the boundary problem on the half-line A11 (0, ξ , Dn )v(xn ) − λIr,0 v(xn ) = 0 for xn > 0, (r)

(r,r)

Br,1 (0, ξ , Dn )v(xn ) = 0 at xn = 0, |v(xn )| → 0 as xn → ∞, has only the trivial solution for ξ ∈ Rn−1 \ {0}, λ ∈ L and 1 ≤ r ≤ d; (2) the boundary problem on the half-line () A11 (0, 0, Dn )v(xn ) − λ I,0 v(xn ) = 0 for xn > 0, (r,)

B,1 (0, 0, Dn )v(xn ) = 0 at xn = 0, |v(xn )| → 0 as xn → ∞, has only the trivial solution for λ ∈ L \ {0}, 1 ≤ r < d and r < ≤ d. For our purposes we need to introduce some further terminology. To this end we henceforth let π1 (j) = r if 1 ≤ j ≤ N and kr−1 < j ≤ kr , and π2 (j) = r if 1 ≤ j ≤ Nd and Nr−1 < j ≤ Nr , where N0 = 0. In addition 1 2 2 1 1 j we let ξ = 1 + |ξ|2 2 , ξ = 1 + |ξ |2 2 , and ξ , λj = |ξ |2 + |λ| m for 1 ≤ j ≤ d. We also require the following deﬁnition. Deﬁnition 2.7. Let x0 ∈ Γ. Then we say that the operator A(x, D) is essen0 tially upper triangular at x0 if ajk α (x ) = 0 for |α| = sj + tk , k−1 < j ≤ k , 1 ≤ k ≤ k−1 , = 2, . . . , d. Likewise we say that the operator B(x, D) = 0 (Bjk (x, D))j=1,...,N is essentially upper triangular at x0 if bjk α (x ) = 0 for k=1...,N

|α| = σj + tk , N−1 < j ≤ N , 1 ≤ k ≤ k−1 , = 2, . . . , d. We are now in a position to state the main result of this paper, namely Theorem 2.8 below, and which will be proved in Section 4. In this theorem we will require the further assumption, which will be made precise in Deﬁnition 3.13 below, that the operators A(x, D) and B(x, D) are compatible at each point of Γ. Hence for the moment let us state that this condition will always be satisﬁed if B(x, D) is of Dirichlet type on Γ or if the operators A(x, D) and B(x, D) are essentially upper triangular at every point of Γ. Theorem 2.8. Suppose that the boundary problem (1.1), (1.2) is parameterelliptic in L. Suppose also that the operators A(x, D)and B(x, D) are compatible at every point of Γ. In addition, suppose that B(x, D) is essentially upper triangular at every point of Γ. Then there exists a constant λ0 = λ0 (p) > 1 such that for λ ∈ L with |λ| ≥ λ0 , the boundary problem (1.1), (1.2) has

N

N t −s a unique solution u ∈ j=1 Wp j (Ω) for every f ∈ j=1 Hp j (Ω) and g = 1 T N −σj − p g1 , . . . , gN ∈ j=1 Wp (Γ), and the a priori estimate ⎞ ⎛ N N N (π1 (j)) (π (j)) (π (j)) (2.2) |||uj |||tj ,p,Ω ≤C⎝ |||fj |||−s1j ,p,Ω + |||gj |||−σ2j − 1 ,p,Γ ⎠ j=1

j=1

j=1

p

holds, where the constant C does not depend upon the fj , gj , and λ.

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Remark 2.9. The proof of Theorem 2.8 will depend upon the results of Section 3 and those of [AV], and as a consequence of these results it will also follow that the estimate (2.2) is 2-sided, i.e., an estimate reverse to (2.2) holds. Indeed, we know from [AV] that we can cover Ω by a ﬁnite number of open sets {Uk }n1 1 , where Uk ∩ Γ = ∅ if k ≤ n0 for some n0 < n1 , and Uk ⊂ Ω 1 for k > n0 . If {φk }N 1 denotes a partition of unity subordinate to the covering n1 {Uk }1 such that supp φk ∩ Γ = ∅ for k ≤ n0 and supp φk ∩ Γ = ∅ otherwise, (π (j)) where supp denotes support, then a norm equivalent to the norm |||fj |||−s1j ,p,Ω deﬁned above is given by n0 n1 (π (j)) (π (j)) |||φk fj |||−s1j ,p,Rn + |||φk fj |||−s1j ,p,Rn , +

k=1

where the norm

(π (j)) |||φk fj |||−s1j ,p,Rn +

k=n0 +1

is taken in local coordinates. Since similar (π (j))

(π (j)) , 1 j − p ,p,Γ

1 statements can be made for both |||uj |||tj ,p,Ω and |||gj |||−σ2

the asser-

tion concerning the 2-sidedness of (2.2) follows directly from the results of Section 3.

3. The Constant Coeﬃcient Case In this section we are going to establish some results concerning the existence of and a priori estimates for solutions of a boundary problem involving constant coeﬃcient systems which is related to (1.1), (1.2). These results will then be used in Section 4 to prove Theorem 2.8. To this end, let x0 ∈ Ω and let us ﬁx our attention upon the diﬀerential equation ˚ 0 , D)u(x) − λ u(x) = f (x) for x ∈ Rn and λ ∈ L \ {0}. A(x (3.1) Then we have the following two results, where here and below we let uj and fj , 1 ≤ j ≤ N , denote the components of u and f , respectively.

N t Proposition 3.1. Suppose that u ∈ j=1 Wp j (Rn ) and that (3.1) holds. Then

N N −s (π1 (j)) (π1 (j)) N f ∈ j=1 Hp j (Rn ) and j=1 |||fj |||−sj ,p,Rn ≤ C j=1 |||uj |||tj ,p,Rn , where the constant C does not depend upon u and λ. Proposition 3.2. Suppose that the operator A(x, D) − λ IN is parameter −sj elliptic in L and that f ∈ N (Rn ). Then there exists the constant j=1 Hp 0 λ > 0 such that for λ ∈ L with |λ| ≥ λ0 , the diﬀerential equation (3.1) has

N t a unique solution u ∈ j=1 Wp j (Rn ) and N j=1

(π (j))

1 |||uj |||tj ,p,R n ≤ C

N

(π (j))

|||fj |||−s1j ,p,Rn ,

j=1

where the constant C does not depend upon f and λ. We will only prove Proposition 3.2 as the proof of Proposition 3.1 follows directly from the deﬁnition and the Mikhlin-Lizorkin multiplier theorem.

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Proof of Proposition 3.2. Under our assumptions we know from [DMV] and ˚ 0 , ξ) − λ IN is invertible [K1] that for ξ ∈ Rn and λ ∈ L with |λ| ≥ λ0 , A(x and d 2(N −N ) ˚ 0 , ξ) − λ IN ≥ C ξ, λ j j−1 , det A(x j

j=1

where the constant C does not depend upon ξ and λ. Furthermore, if we −1 N ˚ 0 , ξ) − λ IN = ( aj,k (ξ, λ))j,k=1 , then the aj,k (ξ, λ) are rational put A(x functions of their arguments, while it also follows from the references just cited that for any multi-index α whose entries are either 0 or 1, −m

j k aj,k (ξ, λ)| ≤ Cξsj +tk ξ, λπ1 (j) ξ, λ−m |ξ α Dξα π1 (k)

for all ξ ∈ Rn whose components are all non-zero, where the constant C does not depend upon ξ and λ. Now observe that under a Fourier transformation (3.1) becomes ˚ 0 , ξ)F u(ξ) − λ F u(ξ) = F f (ξ). A(x Furthermore, in light of what was said above, we conclude that this equation has a unique solution in the space of tempered distributions on Rn given −1 ˚ 0 , ξ) − λ IN F f (ξ). Hence all of the assertions of the by F u(ξ) = A(x proposition follow immediately from this last result, the deﬁnitions of the terms involved, and the Mikhlin-Lizorkin multiplier theorem. Let us suppose from now on that x0 ∈ Γ. Then assuming that the boundary problem (1.1), (1.2) has been rewritten in terms of the local coordinates at x0 as explained in the text preceding Deﬁnition 2.3, let us ﬁx our attention upon the problem in the half-space ˚ D)u(x) − λ u(x) = f (x) for x ∈ Rn and λ ∈ L \ {0}. A(0, +

(3.2)

−s

Then from a consideration of the pairing between Hp j (Rn+ ), equipped with (π (j)) ˚ sj (Rn+ ), equipped with the norm the norm ||| · |||−s1j ,p,Rn , and its dual W p (π (j))

+

1 ||| · |||sj ,p = p/(p − 1) (see [GK, Theorem 1.1]), we can ,Rn , 1 ≤ j ≤ d, p + easily derive the following analogue of Proposition 3.1.

N t Proposition 3.3. Suppose that u ∈ j=1 Wp j (Rn+ ) and that (3.2) holds. Then

N N N −s (π1 (j)) (π1 (j)) f ∈ j=1 Hp j (Rn+ ) and ≤ C j=1 |||uj |||tj ,p,R n , where j=1 |||fj |||−sj ,p,Rn + + the constant C does not depend upon u and λ.

Proposition 3.4. Suppose that the operator A(x, D) − λ IN is parameter N −s elliptic in L and that f ∈ j=1 Hp j (Rn+ ). Then there exists a λ0 > 0 such that for λ ∈ L with |λ| ≥ λ0 , the diﬀerential equation (3.2) has a solution N

tj (π1 (j)) (π1 (j)) n ≤C N , where u∈ N j=1 Wp (R+ ) and j=1 |||uj |||tj ,p,Rn j=1 |||fj |||−sj ,p,Rn + + the constant C does not depend upon f and λ.

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Proof. It follows from [T, Lemma 2.9.3, p.218] and [GK, Eqn.(1.27)] that

N −s there is a f ∈ j=1 Hp j (Rn ) such that fRn+ = f and N

|||fj |||−s1j ,p,Rn ≤ C (π (j))

j=1

N

(π (j))

|||fj |||−s1j ,p,Rn , +

j=1

where the constant C does not depend upon f and λ. Hence if u denotes n the solution of (3.1) when f there is replaced by f and u = u R+ , then the assertion of the proposition follows directly from Proposition 3.2. Let us next ﬁx our attention upon the boundary problem ˚ D)u(x) − λ u(x) = 0 for x ∈ Rn , A(0, +

(3.3)

˚j (0, D)u(x) = gj (x ) at xn = 0, j = 1, . . . , N , B

(3.4)

with λ ∈ C \ {0}.

N t Proposition 3.5. Suppose that u ∈ j=1 Wp j (Rn+ ) and that (3.4) holds. Then 1 T N −σj − p ∈ j=1 Wp (Rn−1 ) and g = g1 , . . . , gN N j=1

(π (j)) |||gj |||−σ2j − 1 ,p,Rn−1 p

≤C

N N

(π (j))

1 |||uk |||tk ,p,R n,

(3.5)

+

j=1 k=1

where the constant C does not depend upon u and λ. Furthermore, if B(x, D) is essentially upper triangular at x0 , then we may replace (3.5) by N

(π (j)) 1 n−1 j − p ,p,R

|||gj |||−σ2

j=1

≤C

N

(π (j))

1 |||uj |||tj ,p,R n. +

j=1

and let µ(j) = π2 (j) if B(x, D) is essentially upper Proof. Let 1 ≤ j ≤ N 0 triangular at x and let µ(j) = 1 otherwise. Then it follows from [ADF, Proposition 2.2] that (π (j)) 1 n−1 j − p ,p,R

|||gj |||−σ2

N ≤ C1

(π2 (j)) 0 α bjk (x )D u k α

−σj ,p,Rn +

k=1 |α|=σj +tk N

≤ C2

−σj

π (j) 2 Dα uk −σj ,p,Rn+ + |λ| m Dα uk 0,p,Rn+

k=µ(j) |α|=σj +tk N (π2 (j)) uk tk ,p,Rn+ + |||uk |||tk ,p,R n

≤ C3

+

k=µ(j)

≤ 2 C3

N k=µ(j)

(π (j))

2 |||uk |||tk ,p,R n,

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where the constants Cj do not depend upon u and λ. Hence all the assertions of the proposition follow from this last result. We now come to the main result of this section. Proposition 3.6. Suppose that the boundary problem (1.1), (1.2) is parameterelliptic in L. Suppose also that the operators A(x, D) and B(x, D) are compatible at x0 . Then there exists a constant λ0 = λ0 (p) > 1 such that for λ ∈ L with |λ| ≥ λ0 , the boundary problem (3.3), (3.4) has a unique solution 1

N −σj − p tj n T u∈ N (Rn−1 ), and ) ∈ j=1 Wp j=1 Wp (R+ ) for every g = (g1 , . . . , gN the a priori estimate N

(π (j))

1 |||uj |||tj ,p,R n ≤ C

N

+

j=1

(π (j))

|||gj |||−σ2j − 1 ,p,Rn−1 p

j=1

holds, where the constant C does not depend upon the gj and λ. As a consequence of Propositions 3.4, 3.5, and 3.6 as well as from a standard argument, we obtain the following result. Proposition 3.7. Suppose that the hypotheses of Proposition 3.6 hold. Suppose also that the operator B(x, D) is essentially upper triangular at x0 . Then the exists a constant λ0 = λ0 (p) > 1 such that for λ ∈ L with |λ| ≥ λ0 , the

tj n boundary problem (3.2), (3.4) has a unique solution u ∈ N j=1 Wp (R+ ) for 1

N

−σj − p −s N every f ∈ j=1 Hp j (Rn+ ) and g ∈ j=1 Wp (Rn−1 ), and the a priori estimate ⎞ ⎛ N N N (π1 (j)) (π (j)) (π (j)) |||uj |||tj ,p,Rn ≤ C ⎝ |||fj |||−s1j ,p,Rn + |||gj |||−σ2j − 1 ,j,p,Rn−1 ⎠ +

j=1

+

j=1

j=1

p

holds, where the constant C does not depend upon f, g, and λ. In order to prove Proposition 3.6, some preliminary results are required. Accordingly, let us ﬁx λ ∈ L with |λ| > 1 suﬃciently large and let { j }d1 denote a sequence of numbers satisfying 0 < j < 1, j = 1, . . . , d (the magnitudes of λ and the j will be speciﬁed below). Also for r = 0, . . . , d, let us introduce functions ψr ∈ C ∞ (Rn−1 ) satisfying 0 ≤ ψr (ξ ) ≤ 1,

1 supp ψ0 ∈ ξ ∈ Rn−1 : |ξ | ≤ 34 1 |λ|1/m 1 and ψ0 (ξ ) = 1 for |ξ | ≤ 14 1 |λ|1/m ,

n−1 1 1/m r r+1 supp ψr ∈ ξ ∈ R : 4 r |λ| ≤ |ξ | ≤ 34 r+1 |λ|1/m r r+1 and ψr (ξ ) = 1 for 34 r |λ|1/m ≤ |ξ | ≤ 14 r+1 |λ|1/m , 1 ≤ r < d,

n−1 1/m d 1 supp ψd ∈ ξ ∈ R : |ξ | ≥ 4 d |λ| d , and ψd (ξ ) = 1 for |ξ | ≥ 34 d |λ|1/m d while in addition r=0 ψr (ξ ) = 1 and each ψr is a Fourier multiplier of type (p, p) whose norm is bounded by a constant not depending upon λ and the j . Then in the sequel we will require the following three results.

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Proposition 3.8. Suppose that the boundary problem (1.1),(1.2) is parameterelliptic in L. Suppose also that λ ∈ L with |λ| > 1 and that 0 ≤ |ξ | ≤ 1 7 0 m 1 . Then we can choose the numbers 0 suﬃciently small and λ suf8 1 |λ| ˚ ξ , z) − λ IN has ﬁciently large so that for 1 ≤ 0 and |λ| ≥ λ0 , det A(0,

) zeros, say z (0) (ξ , λ) Nd , lying in C+ , and satisfying precisely Nd (= N j j=1 (0)

(0)

Im zj (ξ , λ) ≥ C1 ξ , λ1 , |zj (ξ , λ)| ≤ C2 ξ , λ1 , j = 1, . . . , N1 , (0)

(0)

Im zj (ξ , λ) ≥ C1 ξ , λ ,

|zj (ξ , λ)| ≤ C2 ξ , λ , j = N−1 + 1, . . . , N

for = 2, . . . , d, and where C2 ξ , λ < C1 ξ , λ+1 for = 1 . . . , d − 1, and the Cj denote constants not depending upon ξ and λ. Proposition 3.9. Suppose that the boundary problem (1.1), (1.2) is parameterelliptic in L. Suppose also that 1 ≤ r < d, that λ ∈ L with |λ| > 1, and that 1

1 1 m r 8 r |λ| 0

r+1 ≤ |ξ | ≤ 78 r+1 |λ| m . Then for ﬁxed r we can choose the numbers 0 suﬃciently small and λ suﬃciently large so that for r+1 ≤ 0 and |λ| ≥

˚ ξ , z) − λ IN has precisely Nd zeros, say z (r) (ξ , λ) Nd , lying λ0 , det A(0, j j=1 in C+ and satisfying

(r)

Im zj (ξ , λ) ≥ C1 ξ , λr , (r)

Im zj (ξ , λ)| ≥ C1 ξ , λ ,

(r)

|zj (ξ , λ)| ≤ C2 ξ , λr , j = 1, . . . , Nr , (r)

|zj (ξ , λ)| ≤ C2 ξ , λ , j = N−1 , . . . , N

for = r + 1, . . . , d, and where C2 ξ , λ < C1 ξ , λ+1 for = r, . . . , d − 1, and the Cj denote positive constants not depending upon ξ and λ. Proposition 3.10. Suppose that the boundary problem (1.1), (1.2) is para1 d meter-elliptic in L. Suppose also that λ ∈ L with |λ| > 1 and that 18 d |λ| m ≤ 0 number λ suﬃciently large |ξ | < ∞. Then for ﬁxed d we can choose the ˚ ξ , z) − λ IN has precisely Nd zeros, say so that for |λ| ≥ λ0 , det A(0, (d) Nd zj (ξ , λ) j=1 , lying in C+ and satisfying (d)

Im zj (ξ , λ) ≥ C1 ξ , λd ,

(d)

|zj (ξ , λ)| ≤ C2 ξ , λd for j = 1, . . . , Nd ,

where the Cj denote constants not depending upon ξ and λ. Since the proofs of Propositions 3.8 and 3.10 are similar to that of Proposition 3.9, we will only prove this latter proposition. Proof of Proposition 3.9. To begin with let us observe that (r) (r) A11 (0, ξ , z) − λ Ikr A12 (0, ξ , z) ˚ A(0, ξ , z) − λ IN = (r) (r) A21 (0, ξ , z) A22 (0, ξ , z) − λ IN −kr and that A11 (0, ξ , z) − λ Ikr = A11 (0, ξ , z) − λIr,0 − λ(Ikr − Ir,0 ). (r)

(r)

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(r) Then as a consequence of our hypotheses we know that det A11 (0, ξ , z) − λIr,0 has precisely Nr zeros lying in C+ and that there is a closed contour γr+ (ξ , λ) ⊂ C+ containing all these zeros in its interior such that for z ∈ γr+ (ξ , λ), Im z ≥ C1 ξ , λr , |z| ≤ C2 ξ , λr , and (r) r r ≤ det A11 (0, ξ , z) − λIr,0 ≤ C4 ξ , λ2N , C3 ξ , λ2N r r

where the constants Cj do not depend upon ξ , z, and λ. Furthermore, it is easy to show that for z ∈ γr+ (ξ , λ), (r) (r) det A11 (0, ξ , z) − λ Ikr − det A11 (0, ξ , z) − λIr,0

≤C

mi(k) |λ|1/mi(k)

kr−1

ξ , λr

=1 1≤i(1)<...
r ξ , λ2N , r

where the constant C does not depend upon ξ , z, and λ, while by employing the Laplace method of expanding a determinant, we can alsoshow that, apart ˚ ξ , z) − from a constant not depending upon ξ , z, and λ, the term det A(0, (r) N −k r is bounded by the sum λ IN - det A11 (0, ξ , z) − λ Ikr λ r −1 N −k N −kr −µ r+1 det A(r) (0, ξ , z) − λ Ikr |λ|µ ξ , zm

11

+

µ=0

⎛ q

⎝ξ , z

µ=1

q

⎛

miµ

⎞ ν

ξ , z

µ=1

j1 <...<jν q

× ⎝ξ , z

µ=1

miµ

ν

ξ , z

µ=1

mjµ

|λ|kr −q−ν ⎠

(3.6)

⎞ mjµ

|λ|N −kr −q−ν ⎠ .

j1 <...< jν

1 Here ξ , z = |ξ |2 + |z|2 2 , q indicates that the summation is over those q satisfying 1 ≤ q ≤ kr (resp. 1 ≤ q ≤ N − kr ) if 2kr ≤ N (resp. 2kr > N ), and {iµ }qµ=1 and {jµ }νµ=1 denote distinct sequences of integers satisfying 1 ≤ i1 < . . . < iq ≤ kr and 1 ≤ j1 < . . . < jν ≤ kr , 1 ≤ ν ≤ kr − q, respectively, while {iµ }qµ=1 and { jµ }νµ=1 denote distinct sequences of integers satisfying j1< . . . < jν ≤ N, 1 ≤ ν ≤ N −kr −q, kr +1 ≤ i1 < . . . < iq ≤ N and kr +1 ≤ respectively, and where the summation j1 <...<jν in (3.6) is to be replaced by 1 if q = kr and the summation j1 <... r. Then we can write () () A12 (0, ξ , z) A11 (0, ξ , z) − λ Ik ˚ A(0, ξ , z) − λ IN = (3.7) () () A22 (0, ξ , z) − λ IN −k A21 (0, ξ , z)

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if < d, and where (3.7) must be modiﬁed in an obvious way if = d. In addition we have () () () A11 (0, ξ , z) − λ Ik = A11 (0, 0, z) − λ I,0 + A11 (0, ξ , z)

− A11 (0, 0, z) − λ(Ik − I,0 ). () Then as a consequence of our hypotheses we know that det A11 (0, 0, z) − λI,0 has precisely N − N−1 zeros lying in C+ and that there is a closed + contour γr, (λ) ⊂ C+ containing all these zeros in its interior such that for z ∈ 2N () 1 1 + , |z| ≤ C2 |λ| m and C3 |λ| m ≤ | det A11 (0, 0, z) − γr, (λ), Im z ≥ C1 |λ| m 2N λ I,0 | ≤ C4 |λ| m , where the constants Cj do not depend upon λ. Further+ (λ), more, we can show that for z ∈ γr, () () |det A11 (0, ξ , z) − λ Ik − det A11 (0, 0, z) − λI,0 | 2N 1 1 −m −1 ≤ C |λ| m + δ,r+1 r+1 |λ| m , where δ,r+1 denotes the Kronecker delta and the constant C does not depend upon ξ , z,and λ, and that, apart from upon a constant not depending ˚ ξ , z) − λ IN − det A() (0, ξ , z) − λ Ik λN −k | is ξ , z and λ, |det A(0, 11 bounded by the expression (3.6) if we replace r there by . It follows again from Rouch´e’s theorem that if r+1 is chosen suﬃciently small and |λ| suf ˚ ξ , z) − λ IN has precisely N − N−1 zeros ﬁciently large, then det A(0, + contained in γr, (λ). Hence since was chosen arbitrary, this completes the proof of the proposition. Turning again to Propositions 3.8-3.10 and supposing henceforth that the boundary problem (1.1), (1.2) is parameter-elliptic in L and that 0 has been chosen suﬃciently small and λ0 suﬃciently large so that the conclusions of these propositions hold, let us suppose that 0 ≤ r ≤ d and that 0 ≤ 1 1 1 m r+1 1 r ≤ |ξ | ≤ 7 |ξ | ≤ 78 1 |λ| m if r = 0, 18 r |λ| m if 1 ≤ r < d, 8 r+1 |λ| 1

d and 18 d |λ| m ≤ |ξ | < ∞ if r = d. Then it follows from [V] that the set of solutions of the diﬀerential equation

˚ ξ , Dn )u(xn ) − λ u(xn ) = 0 for xn > 0 A(0,

(3.8)

which decay exponentially at ∞ form a vector space of dimension Nd . Furthermore, this vector space is precisely the vector space spanned by the columns of the matrix −1 ˚ ξ , z) − λ IN IN , zIN , . . . , z m1 −1 IN dz, (3.9) eixn z A(0, γr+ (ξ ,λ)

if r = d, while if r < d, then it is precisely the direct sum of the vector space (of dimension Nr ) spanned by the columns of (3.9) and the N − r vector

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spaces (of dimension N − N−1 , = r + 1, . . . , d) spanned by the columns of each of the matrices + γr, (λ)

−1 ˚ ξ , z) − λ IN IN , zIN , . . . , z m1 −1 IN dz eixn z A(0,

for = r + 1, . . . , d. Note that here the contours γ0+ (ξ , λ) (resp. γd+ (ξ , λ)) + and the γ0, (λ) are deﬁned in an analogous fashion to the way they were deﬁned in the proof of Proposition 3.9 for the case 0 < r < d. Proposition 3.11. Suppose that the boundary problem (1.1),(1.2) is parameterelliptic in L and that λ ∈ L. Then we can choose the constants 0 and λ0 of Propositions 3.8–3.10 suﬃciently small and suﬃciently large, respectively, so that the following results hold. 1 1 , then If 0 ≤ |ξ | ≤ 78 1 |λ| m rank

−1 ˚ ξ , z) − λ IN B (1,1) (0, ξ , z) A(0,

γ + (ξ ,λ)

0 × IN , zIN , . . . , z m1 −1 IN dz = N1 , rank

and

−1 ˚ ξ , z) − λ IN B (1,)(0, ξ , z) A(0,

(3.10)

+ γ0, (λ)

× IN , zIN , . . . , z m1 −1 IN dz = N − N−1

for = 2, . . . , d;

1

r ≤ |ξ | ≤ if 1 ≤ r ≤ d and 18 r |λ| m |ξ | < ∞ if r = d, then

rank

1

7 m r+1 8 r+1 |λ|

if r < d or

1

1 m d 8 d |λ|

≤

−1 ˚ ξ , z) − λ IN B (r,r)(0, ξ , z) A(0,

γr+ (ξ ,λ)

× IN , zIN , . . . , z m1 −1 IN dz = Nr , rank

and B

(r,)

+ γr, (λ)

−1 ˚ ξ , z) − λ IN (0, ξ , z) A(0,

(3.11)

× IN , zIN , . . . , z m1 −1 IN dz = N − N−1

for = r + 1, . . . , d if r < d.

Proof. We will only prove the proposition for the case 1 ≤ r < d; the remaining cases can be similarly treated. Accordingly, referring to the proof of Proposition 3.9 for notation, we see that for z ∈ γr+ (ξ , λ) we have the

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343

representation ˚ ξ , z) − λ IN = A(0,

(r) 0 A11 (0, ξ , z) − λ Ir,0 0 −λ IN −kr (r) λ(Ir,0 − Ikr ) A12 (0, ξ , z) + (r) (r) A21 (0, ξ , z) A22 (0, ξ , z)

(3.12)

Furthermore, it follows from the proof of Proposition 3.9 that for each ﬁxed (r) triple (ξ , z, λ), the operator A11 (0, ξ , z) − λ Ir,0 : Ckr → Ckr is bounded and invertible, while if we denote by ajk (ξ , z, λ) the entry in the j-th row and (r) −1 k-th column of A11 − λIr,0 , then ajk (ξ , z, λ) is a rational function of its r arguments and satisﬁes the quasi-homogeneity condition ajk (ρξ , ρ z, ρm λ) = ρ−tj −sk ajk (ξ , z, λ) for ρ > 0. From this it is clear that | ajk (ξ , z, λ)| ≤ −t −s Cξ , λr j k , where here and for the remainder of this proof C denotes a generic constant that does not depend upon the variables indicated, and that (r) the entry in the j-th row and k-th column of λ(Ir,0 − Ikr ) A11 (0, ξ , z) − −1 λ Ir,0 is λ ajk (ξ , z, λ) if j ≤ kr−1 and is 0 otherwise. Thus if we let r λ, and assume henceforth that ρ = ξ , λr , η = ρ−1 ξ , ζ = ρ−1 z, µ = ρ−m |λ| has been chosen suﬃciently large, it is now a simple matter to show that if we put −1 (r) H11 (ξ , z, λ) = λ Ir,0 − Ikr A11 (0, ξ , z) − λIr,0

= H11 (η , ζ, µ, ρ) kr r ajk (η , ζ, µ) diag ρ−t1 , . . . , ρ−tkr−1 , 0Ir = µρm j,k=1 × diag ρ−s1 , . . . , ρ−skr , then the matrix Ikr + H11 (ξ , z, λ) is invertible and admits the representation −1 11 (ξ , z, λ), where (Ikr + H11 (ξ , z, λ)) = Ikr + H 11 (ξ , z, λ) = H 11 (η , ζ, µ, ρ) H

kr jk (η , ζ, µ, ρ) = diag(ρs1 , . . . , ρskr ) H 11

diag ρ−s1 , . . . , ρ−skr

j,k=1

jk (η , ζ, µ, ρ) is a rational function of its arguments which satisﬁes the and H 11 m j jk (η , ζ, µ, ρ) ≤ C|λ|1− m r estimate H if j ≤ kr−1 and equals 0 otherwise. 11

Next let us put H(ξ , z, λ) =

H11 (ξ , z, λ) −H12 (ξ , z, λ) H21 (ξ , z, λ) −H22 (ξ , z, λ),

(r)

where H12 (ξ , z, λ) = λ−1 A12 (0, ξ , z),

−1 (r) (r) , H21 (ξ , z, λ) = A21 (0, ξ , z) A11 (0, ξ , z) − λIr,0

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(r)

and H22 (ξ , z, λ) = λ−1 A22 (0, ξ , z). Thus it follows from the foregoing results and (3.12) that ˚ ξ , z) − λ IN A(0, = (IN + H(ξ , z, λ))

(r) A11 (0, ξ , z) − λ Ir,0 0

0

−λ IN −kr ,

,

(3.13)

while a factorization in the sense of Schur (see [G, Theorem 3.16, p.302]) gives Ikr 0 IN + H(ξ , z, λ) = −1 IN −kr H21 (ξ , z, λ) Ikr + H11 (ξ , z, λ) 0 Ikr + H11 (ξ , z, λ) × 0 IN −kr − H † (ξ , z, λ) −1 Ikr − Ikr + H11 (ξ , z, λ) H12 (ξ , z, λ) × , 0 IN −kr where −1

H † (ξ , z, λ) = H22 (ξ , z, λ) − H21 (ξ , z, λ) (Ikr + H11 (ξ , z, λ))

H12 (ξ , z, λ)

= H † (η , ζ, µ, ρ)

N † (η , ζ, µ, ρ) = diag(ρskr +1 , . . . , ρsN ) Hjk

j,k=kr +1

diag(ρ−skr +1 , . . . , ρ−sN ),

† and where Hjk (η , ζ, µ, ρ) is a rational function of its arguments and m r+1 † r+1 −1 H (η , ζ, µ, ρ) ≤ C m m r . r+1 + |λ| jk

Thus if we henceforth suppose that 0 has been chosen suﬃciently small and λ0 suﬃciently large, then IN −kr − H † (ξ , z, λ) is invertible and we have −1 † (ξ , z, λ), where IN −kr − H † (ξ , z, λ) = IN −kr + H † (ξ , z, λ) = H † (η , ζ, µ, ρ) H N † (η , ζ, µ, ρ) = diag (ρskr +1 , . . . , ρsN ) H jk

j,k=kr +1

diag ρ−skr +1 , . . . , ρ−sN

† (η , ζ, µ, ρ) is a rational function of its arguments and and H jk m r+1 r+1 −1 † (η , ζ, µ, ρ)| ≤ C m m r + |λ| |H . r+1 jk ˚ ξ , z) − λ IN is An immediate consequence of this last result is that A(0, invertible and −1 −1 (r) A (0, ξ , z) − λ I 0 r,0 ˚ ξ , z) − λ IN 11 A(0, = 0 −λ−1 IN −kr (3.14) × IN + H (ξ , z, λ) ,

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where

H11 (ξ , z, λ) H12 (ξ , z, λ) , (ξ , z, λ) H22 (ξ , z, λ) H21 11 (ξ , z, λ) + Ikr + H 11 (ξ , z, λ) H12 (ξ , z, λ)H (ξ , z, λ), (ξ , z, λ) = H H11 21 † 11 (ξ , z, λ) H12 (ξ , z, λ) IN −kr + H (ξ , z, λ) , H12 (ξ , z, λ) = Ikr + H † (ξ , z, λ) H21 (ξ , z, λ) Ikr + H 11 (ξ , z, λ) , H21 (ξ , z, λ) = − IN −kr + H H (ξ , z, λ) =

† (ξ , z, λ). (ξ , z, λ) = H H22

Furthermore, we can write H11 (ξ , z, λ) = H11 (η , ζ, µ, ρ) jk = diag (ρs1 , . . . , ρskr ) H11 (η , ζ, µ, ρ)

diag ρ−s1 , . . . , ρ−skr , j,k=1,...,kr

jk (η , ζ, µ, ρ) is a rational function of its arguments and where H11 m r+1 m r−1 m r+1 jk r r , |H11 (η , ζ, µ, ρ)| ≤ C r+1 + |λ|−1+ m + |λ|1− m H21 (ξ , z, λ) = H21 (η , ζ, µ, ρ)

jk (η , ζ, µ, ρ) j=kr +1,...,N = diag (ρskr +1 , . . . , ρsN ) H21 × diag ρ−s1 , . . . , ρ−skr ,

k=1,...,kr

jk (η , ζ, µ, ρ) is a rational function of its arguments and where H21 jk (η , ζ, µ, ρ)| ≤ C. |H21

Let us next observe that under our assumptions concerning ξ and λ, (η , µ) ∈ Σr , where

2 1

r = 1, | r Σr = ( η ,µ ) ∈ Rn−1 × L : | η |2 + | µ| m η | ≥ 18 r | µ| m − 12 r = ( ⊂Σ η , µ ) ∈ Rn−1 × L : 1 + 64 ≤ | η | ≤ 1, 0 ≤ | µ | ≤ 1 .

2 r

r , det A(r) (0, η , ζ) − It also follows from Deﬁnition 2.6 that for ( η , µ ) ∈ Σ 11 µ Ir,0 has precisely Nr zeros lying in C+ and that there is a closed contour γr+ ⊂ C+ , not depending upon η and µ , containing all these zeros in its interior. And it is to be understood henceforth that the contour γr+ (ξ , λ) introduced above is obtained from γr+ by dilation by the factor ρ. Furthermore, as a consequence of Deﬁnition 2.6 we also know from [V] that if we ﬁx our r , then there exists matrices J ) ∈ Σ attention upon a particular pair ( η , µ and Z(ζ) depending upon η and µ , such that the matrix −1 (r,r) (r) η , µ ) = Br,1 (0, η , ζ) A11 (0, η , ζ) − µ Ir,0 JZ(ζ)dζ (3.15) K1 ( γr+

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has rank Nr . Here J denotes a kr × Nr matrix with the property that each of its columns has precisely one non-zero component, namely 1, and Z(ζ) = diag ζ q(1) , . . . , ζ q(Nr ) , where the q(j) denote non-negative integers not exceeding m1 − 1. Thus det K1 ( η , µ ) > c (3.16) for some c > 0. As a consequence of (3.15), (3.16) we see that for the values of ξ and λ under consideration here, the rank of the matrix −1 (r,r) (r) Br,1 (ξ , z) A11 (0, ξ , z) − λ Ir,0 JZ(z)dz (3.17) γr+ (ξ ,λ)

is Nr and that (3.17) can be written in the form ρ diag (ρσ1 , . . . , ρσNr ) K1 (η , µ) diag ρq(1)−sk(1) , . . . , ρq(Nr )−sk(Nr ) , (3.18) where 1 ≤ k(j) ≤ kr for 1 ≤ j ≤ Nr . Hence referring to (3.10), let us now use this fact to show that if 0 is chosen suﬃciently small and λ0 suﬃciently large, then the rank of −1 J (r,r) ˚ B (0, ξ , z) A(0, ξ , z) − λ IN Z(z)dz (3.19) 0 γr+ (ξ ,λ) is Nr . Indeed, it follows from (3.14) that the expression (3.19) can be expressed as the sum of the expression (3.18) plus σ1 σNr K2 (η , ζ, µ, ρ)JZ(ζ)dζ ) ρ diag (ρ , . . . , ρ γr+ × diag ρq(1)−sk(1) , . . . , ρq(Nr )−sk(Nr ) , where 2 (η , ζ, µ, ρ) K

−1 (r,r) (r) = B11 (0, η , ζ) A11 (0, η , ζ) − µ Ir,0 H11 (η , ζ, µ, ρ) (r,r)

r −1 ) B11 (0, η , ζ) diag (ρmkr +1 , . . . , ρmN ) H21 (η , ζ, µ, ρ) − (µ ρm

is an Nr × Nr matrix function whose entry in the j-th row and k-th column, kjk (η , ζ, µ, ρ) is a rational function of its arguments and m r+1 m r−1 m r+1 −1 1− m m r r |kjk (η , ζ, µ, ρ)| ≤ C r+1 + |λ| + |λ| . (3.20) It follows immediately from these results that if we suppose that 0 is chosen suﬃciently small and λ0 suﬃciently large, then the matrix (3.19) has rank Nr , which proves the proposition for the case under consideration. + Suppose next that r < ≤ d and that z ∈ γr, (λ). Then as a consequence () of our hypotheses we know that A11 (0, 0, z) − λ I,0 is invertible and if we −1 k () () () let A11 (0, 0, z) − λ I,0 = ajk (z, λ) , then ajk (z, λ) is a rational j,k=1

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functional function of its arguments and satisﬁes the quasi-homogeneity con() () dition ajk (ρ z, ρm λ) = ρ−tj −sk ajk (z, λ) for ρ > 0. Let us now introduce the 2 () matrix function H () (ξ , z, λ) = Hjk (ξ , z, λ) , where j,k=1

() () () H11 (ξ , z, λ) = A11 (0, ξ , z) − A11 (0, 0, z) + λ(I,0 − Ik −1 () , × A11 (0, 0, z) − λ I,0

()

and if < d, then H12 (ξ , z, λ) = λ−1 A12 (ξ , z),

−1 () () () , H21 ((ξ , z, λ) = A21 (ξ , z) A11 (0, 0, z) − λ I,0 ()

()

and H22 (ξ , z, λ) = λ−1 A22 (ξ , z). Then proceeding as in the previous case, we can write ()

()

H11 (ξ , z, λ) = H11 (η , ζ , µ , ρ ) k sk () jk = diag ρs 1 , . . . , ρ H11 (η , ζ , µ , ρ )

j,k=1

−sk 1 diag ρ−s , . . . , ρ ,

1

−m −1 where ρ = |λ| m , η = ρ−1 λ, and the coeﬃcient ξ , ζ = ρ z, µ = ρ () jk H11 (η , ζ , µ , ρ ) is a rational functional of its arguments satisfying () jk m −m −1 H , (3.21) (η , ζ , µ , ρ ) ≤ C δ + ρ r+1, r+1 11

where here and below C denotes a generic constant which does not depend upon any of the variables indicated. Hence it follows that for 0 suﬃciently () small and λ0 suﬃciently large, Ik + H11 (ξ , z, λ) is invertible and we have

−1 () () (η , ζ , µ , ρ ) Ik + H11 (ξ , z, λ) = diag (ρs 1 , . . . , ρs ) Ik + H −sk 1 , , . . . , ρ × diag ρ−s

() (η , ζ , µ , ρ ) = where H 11

k () jk (η , ζ , µ , ρ ) H 11

and the coeﬃcient

j,k=1

() jk (η , ζ , µ , ρ ) is a rational functional function of its arguments and H 11 is bounded in modulus by the expression on the right side of (3.21). Suppose next that < d. Then proceeding as we did in our previous case, let us now put ()

H † () (ξ , z, λ) = H22 (ξ , z, λ)

−1 () () () H12 (ξ , z, λ). − H21 (ξ , z, λ) Ik + H11 (ξ , z, λ)

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Then we can write H † () (ξ , z, λ) = H † () (η , ζ , µ , ρ ) s N k +1 † () Hjk (η , ζ , µ , ρ ) = diag ρ , . . . , ρs N j,k=k +1 −s k +1 −sN , × diag ρ , . . . , ρ † ()

where it follows from its deﬁnition that Hjk (η , ζ , µ , ρ ) is a rational func−1+

m +1

m tion of its arguments and is bounded in modulus by C|λ| . Hence if we 0 † () suppose that λ is suﬃciently large, then IN −k − H (ξ , z, λ) is invertible, and if we write −1 † () (ξ , z, λ), = IN −k + H IN −k − H † () (ξ , z, λ)

and † () (ξ , z, λ) = H † () (η , ζ , µ , ρ ) H N s k +1 † () (η , ζ , µ , ρ ) H = diag ρ , . . . , ρs N jk j,k=k +1 −s k +1 −sN , × diag ρ , . . . , ρ then H jk (η , ζ , µ , ρ ) is a rational function of its arguments and is bounded † ()

−1+

m +1

s −s

m in modulus by C|λ| ρ j k . In light of these deﬁnitions, we can now proceed as in the ﬁrst part of the proof to show that for < d, −1 −1 () A11 (0, 0, z) − λ I,0 0 ˚ = A(0, ξ , z) − λ IN 0 −λ−1 IN −k (3.22) () × IN + H (ξ , z, λ) ,

where

2 () , H () (ξ , z, λ) = Hjk (ξ , z, λ) j,k=1 () () (ξ , z, λ) + Ik + H () (ξ , z, λ) H11 (ξ , z, λ) = H 11 11 ()

()

× H12 (ξ , z, λ)H21 (ξ , z, λ), () () (ξ , z, λ) H () (ξ , z, λ) IN −k + H † () (ξ , z, λ) , H12 (ξ , z, λ) = Ik + H 11 12 () () † () H21 (ξ , z, λ) = − IN −k + H (ξ , z, λ) H21 (ξ , z, λ) () (ξ , z, λ) , × Ik + H 11 () † () (ξ , z, λ), H22 (ξ , z, λ) = IN −k + H

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and where these results are to be modiﬁed in an obvious way if = d. Note also that ()

()

H11 (ξ , z, λ) = H11 (η , ζ , µ , ρ ) k sk () H11,jk (η , ζ , µ , ρ ) = diag ρs 1 , . . . , ρ

j,k=1

−sk 1 diag ρ−s , . . . , ρ ,

()

where H11,jk (η , ζ , µ , ρ ) is a rational functional function of its arguments m +1 m −1 −1+ m 1− m and is bounded in modulus by C |λ| + |λ| + r+1 , and if < d, then ()

()

H21 (ξ , z, λ) = H21 (η , ζ , µ , ρ ) s () k +1 = diag ρ , . . . , ρs N H21,jk (η , ζ , µ , ρ ) j=k +1,...,N −sk 1 × diag ρ−s , . . . , ρ ,

k=1,...,k

()

where H21,jk (η , ζ , µ , ρ ) is a rational function of its arguments and is bounded in modulus by C. Let us next our assumptions concerning ξ and observe that under

λ, µ ∈ L1 = µ ∈ L | | µ| = 1 . It also follows from Deﬁnition 2.4 that () I,0 has precisely N − N−1 zeros lying in for µ ∈ L1 , det A (0, 0, ζ) − µ 11

+ C+ and that there is a closed contour γr, ⊂ C+ , not depending upon µ , containing all these zeros in its interior. And it is to be understood henceforth + + that the contour γr, (λ) introduced above is obtained from γr, by dilation by the factor ρ . Furthermore, as consequence of Deﬁnition 2.6 it also follows from [V] that if we ﬁx our attention upon a particular µ ∈ L1 , then there exist matrices J () and Z () (ζ ), depending upon µ , such that −1 () (r,) () K1 (0, µ ) = B,1 (0, 0, ζ ) A11 (0, 0, ζ ) − µ I,0 J () Z () (ζ )dζ (3.23) + γr,

has rank N − N−1 . Here J () is a k × (N − N−1 )matrix having the same q(N −N ) q(1) properties as the matrix J of (3.15), Z () (ζ ) = diag ζ , . . . , ζ −1 , and the q(j) are deﬁned as in the text following (3.15). Then det K () (0, µ ) > c() (3.24) 1 for some constant c() > 0. As a consequence of (3.23) and (3.24) we see that for the values of ξ and λ under consideration here the rank of the matrix −1 (r,) () B,1 (0, z) A11 (0, 0, z) − λ I,0 J () Z () (z)dz (3.25) + γr, (λ)

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is N − N−1 and that (3.25) can be written in the form σ σN −N N +1 () ρ diag ρ −1 , . . . , ρ −1 K1 (0, µ ) q(N −N−1 )−sk(N −N−1 ) q(1)−sk(1) × diag ρ , . . . , ρ ,

IEOT

(3.26)

where 1 ≤ k(j) ≤ k for 1 ≤ j ≤ N − N−1 . Hence referring to (3.11), let us now use these facts to show that if 0 is suﬃciently small and λ0 is suﬃciently large, then the rank of the matrix −1 () J ˚ ξ , z) − λ IN Z () (z)dz B (r,) (0, ξ , z) A(0, (3.27) + 0 γ (λ) r,

is N − N−1 . Indeed it follows from (3.22) that the expression (3.27) can be expressed as the sum of the expression (3.26) plus the expression σ σN N−1 +1 () () () ρ diag ρ K2 (η , ζ , µ , ρ )J Z (ζ )dζ , . . . , ρ + γr,

q(N −N−1 )−sk(N −N−1 ) q(1)−sk(1) , . . . , ρ × diag ρ where () (η , ζ , µ , ρ ) K 2 −1 (r,) (r,) () = B,1 (0, η , ζ ) − B11 )(0, 0, ζ ) A11 (0, 0, ζ ) − µ I,0 −1 (r,) () () + B,1 (0, η , ζ ) A11 (0, 0, ζ ) − µ I,0 H11 (η , ζ , µ , ρ ) (r,)

()

−1 − (1 − δ,d )(µ ρm B,2 (0, η , ζ )H21 (η , ζ , µ , ρ ) )

is an (N − N−1 ) × (N − N−1 ) matrix function whose entry in the j-th row () and k-th column, kjk (η , ζ , µ , ρ ), is a rational function of its arguments and 1 1 −m () −1 |kjk (η , ζ , µ , ρ )| ≤ C δr+1, r+1 + |λ| m (3.28) m −1 m +1 1− m m −1 . + |λ| + (1 − δ,d )|λ| It follows immediately from these results that if we suppose that 0 and λ0 are suﬃciently large and small, respectively, then the matrix (3.27) has rank N − N−1 , and this completes the proof of the proposition. In light of Proposition 3.11 and from what was said in the text preceding that proposition, we are now in a position to present some results pertaining to the solutions of (3.8). To this end let us remark for later use that the contour γr+ and the set Σr for r = 0 and r = d are deﬁned in an analogous manner to the way they were in the proof of Proposition 3.11, except that

2 1 = 1 . A similar remark we now take Σ0 = (ξ , λ) ∈ Rn−1 × L : |ξ |2 + |λ| m + holds for γ0, .

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Proposition 3.12. Suppose that the hypotheses of Proposition 3.11 hold and that 0 and λ0 have been chosen small enough and large enough, respectively, so that the conclusions of that proposition hold. Suppose also that either 1 , or (i) r = 0 and |ξ | ≤ 78 1 |λ|1/m 1 1/m r r+1 (ii) 1 ≤ r < d and 8 r |λ| ≤ |ξ | ≤ 78 r+1 |λ|1/m , or 1 1/m d ≤ |ξ | < ∞. (iii) r = d and 8 d |λ| linearly independent solutions, Then the diﬀerential equation (3.8) has N

{w(r,ν) (ξ , xn , λ)}N ν=1 , which decay exponentially at ∞ and satisfy B (1,) (0, ξ , Dn )W (0,) (ξ , 0, λ) = IN −N−1

for = 1, . . . , d,

if r = 0, B (r,) (0, ξ , Dn )W (r,) (ξ , 0, λ) = IN −N †

for = r, . . . , d,

(3.29)

r,−1

if 1 ≤ r < d,

B (d,d)(0, ξ , Dn )W (d,d) (ξ , 0, λ) = INd ,

† where Nr,−1 = (1 − δmax{r,1}, )N−1 , W (0,) (ξ , xn , λ) denotes the N × (N − N−1 ) matrix function whose columns are precisely the w(0,ν) (ξ , xn , λ) for ν = N−1 + 1, . . . , N , while for r > 0, W (r,) (ξ , xn , λ) denotes the N × (N − † Nr,−1 ) matrix function whose columns are precisely the w(r,ν) (ξ , xn , λ) for

† ν = Nr,−1 + 1, . . . , N . Furthermore, we have the representations (here we let r† = max{r, 1}) † † ˚ ξ , z) − λIN −1 G(r,r ) (ξ , z, λ)dz, W (r,r ) (ξ , xn , λ) = eixn z A(0, + γ (ξ ,λ) r ˚ ξ , z) − λIN −1 G(r,) (ξ , z, λ)dz, W (r,) (ξ , xn , λ) = eixn z A(0, + γr, (λ)

for = r† + 1, . . . , d if r < d,

(3.30)

where for r† ≤ ≤ d (r,)

G

(r,) G (ηr, , ζr, , µr, , ρr, ) (ξ , z, λ) = , 0 IN −k

(r,) (η , ζr, , µr, , ρr, ) = ρ−1 diag(ρs1 , . . . , ρsk )K (r,) (η , ζr, , µr, , ρr, ) G r, r, r, r, r, −σ

× diag(ρr,

1/m

ρr,r† = ξ , λr† , ρr, = |λ| m µr, = ρ− r, λ, and

N

† +1 r,−1

†

for > r ,

−σN

, . . . , ρr, ηr,

=

(r,) , ζr, , µr, , ρr, ) = Kjk (ηr, , ζr, , µr, , ρr, ) K (r,) (ηr, (r,)

),

ρ−1 r, ξ ,

ζr, = ρ−1 r, z,

, j=1,...,k † k=Nr,−1 +1,...,N

, ζr, , µr, , ρr, ) is a ﬁnite sum of a such that for each pair j, k, Kjk (ηr, product of a power of ζr, and an expression of rational type in ηr, , ζr, , µr, and ρr, (i.e., a rational function of terms which are integrals over

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+ δr† , γr+ +(1−δr†, )γr, of rational functions of the components of ηr, , ζr, , µr, and ρr, ) and is bounded in modulus by a constant depending only upon ηr,r † † † and µr,r† (∈ Σr ) if = r and only upon ρr, (∈ L1 ) if > r , and lastly the block 0 IN −k is to be omitted when = d.

Proof. We will only prove the proposition for the case 1 ≤ r < d as the proofs of the other cases are similar. Accordingly, returning to the proof of Proposition 3.11 and employing the notation of that proposition, let Λ(ξ , λ) denote the expression (3.19). Then we know from the proof of Proposition 3.11 that Λ(ξ , λ) = ρ diag ρσ1 , . . . , ρσNr K1 (η , µ) + K2 (η , µ, ρ) × diag ρq(1)−sk(1) , . . . , ρq(Nr )−sk(Nr ) , 2 (η , ζ, µ, ρ)JZ(ζ)dζ and where we refer to the where K2 (η , µ, ρ) = γr+ K text following (3.15) for notation. Hence bearing in mind (3.16) and (3.20), it is an immediate consequence of Proposition 3.11 that Λ(ξ , λ) is invertible and Λ(ξ , λ)−1 = ρ−1 diag ρsk(1) −q(1) , . . . , ρsk(Nr ) −qN (r) × (INr + K3 (η , µ, ρ)K1 (η , µ)−1 diag ρ−σ1 , . . . , ρ−σNr , where K3 (η , µ, ρ) is an Nr × Nr matrix function deﬁned by −1 INr + K3 (η , µ, ρ) = INr + K1 (η , µ)−1 K2 (η , µ, ρ) and where each entry of K3 (η , µ, ρ) is a function of its arguments of rational type and is bounded in modulus by the expression on the right-hand side of (3.20). Hence it follows that ˚ ξ , z) − λIN −1 B (r,r)(0, ξ , z) A(0, γr+ (ξ ,λ)

×

(1) (η , ζ, µ, ρ) + G (2) (η , ζ, µ, ρ) G dz = INr , 0IN −kr

where (1) (η , ζ, µ, ρ) = ρ−1 diag(ρs1 , . . . , ρskr )JZ(ζ)K1 (η , µ)−1 G × diag(ρ−σ1 , . . . , ρ−σNr ), (2) (η , ζ, µ, ρ) = ρ−1 diag(ρs1 , . . . , ρskr )JZ(ζ)K3 (η , µ, ρ)K1 (η , µ)−1 G × diag(ρ−σ1 , . . . , ρ−σNr ).

(3.31)

Suppose next that r < ≤ d and let Λ() (ξ , λ) denote the expression (3.27). Then we know that σN +1 σN () () Λ() (ξ , λ) = ρ diag ρ −1 , . . . , ρ K1 (µ ) + K2 (η , µ , ρ ) q(1)−sk(1) q(N −N−1 )−sk(N −N−1 ) , , . . . , ρ × diag ρ

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()

where K2 (η , µ , ρ ) =

+ γr,

353

() (η , ζ , µ , ρ )J () Z () (ζ )dζ . Hence bearing K 2

in mind (3.24) and (3.28), it follows that Λ() (ξ , λ) is invertible and Λ() (ξ , λ)−1 sk(N −N ) −q(N −N−1 ) s(1)−q(1) = ρ−1 , . . . , ρ −1 diag ρ () −σN +1 () × IN −N−1 + K3 (η , µ , ρ ) K1 (µ )−1 diag ρ −1 , . . . , ρ−σN , ()

where K3 (η , µ , ρ ) is an (N −N−1 )×(N −N−1 ) matrix function deﬁned by −1 () () () IN −N−1 + K3 (η , µ , ρ ) = IN −N−1 + K1 (µ )−1 K2 (η , µ , ρ ) ()

and each entry of K3 (η , µ , ρ ) is a function of its arguments of rational type and is bounded in modulus by the expression on the right-hand side of (3.28). Hence it follows that ˚ ξ , z) − λIN −1 B (r,) (0, ξ , z) A(0, + γr, (λ)

(1,) (2,) (η , ζ , µ , ρ ) dz = IN −N , × G (ζ , µ , ρ ) + G −1 where (1,) (ζ , µ , ρ ) = ρ−1 diag(ρs1 , . . . , ρsk )J () Z () (ζ )K () (µ )−1 G 1 −σN−1 +1

× diag(ρ (2,) (η , ζ , µ , ρ ) G

−σN

, . . . , ρ

),

sk () s1 () () = ρ−1 Z (ζ )K3 (η , ζ , µ , ρ ) diag(ρ , . . . , ρ )J −σN −σN +1 () × K1 (µ )−1 diag(ρ −1 , . . . , ρ ).

, ζr,r , and µr,r , respectively, Finally if we rewrite ρ, η , ζ, and µ as ρr,r , ηr,r and put K r,r (ηr,r , ζr,r , µr,r , ρr,r ) = JZ(ζr,r )(INr + K3 (ηr,r , µr,r , ρr,r ))K1 (ηr,r , µr,r )−1 , , ζr, , and µr, , while if for > r we rewrite ρ , η , ζ , and µ as ρr, , ηr, respectively, and put , ζr, , µr, , ρr, ) K (r,) (ηr, ()

()

= J () Z () (ζr, )(IN −N−1 + K3 (ηr, , µr, , ρr, ))K1 (µr, )−1 ,

then all the assertions of the proposition for the case under consideration here follow from the foregoing results. Let us ﬁx r, 0 ≤ r < d, and referring again to Proposition 3.12 for notation, let us denote by W (r) (ξ , xn , λ) the N × Nd matrix function whose

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columns are precisely the w(r,ν) (ξ , xn , λ). For r† ≤ j ≤ d, let † d Ir,j (ξ , λ) = B (r ,) (0, ξ , Dn )W (r,1 ) (ξ , 0, λ)

,1 =j d

† (r,1 ) (ξ , 0, λ) Ir,j (ξ , λ) = B(r ,) (0, ξ , Dn )W

,1 =j

, ,

(3.32)

where −σN † +1 † † −σN B(r ,) (0, ξ , Dn ) = diag ρr, r,−1 , . . . , ρr, B (r ,) (0, ξ , Dn ), σN † +1 σ (r,1 ) (ξ , 0, λ) = W (r,1 ) (ξ , 0, λ) diag ρ r,1 −1 , . . . , ρ N1 . W r,1 r,1 Then we can write ˚ ξ , Dn )W (r) (ξ , 0, λ) = Ir,r† (ξ , λ) B(0, σN −σN 1 = diag ρσ1 1 , . . . , ρNdd Ir,r† (ξ , λ) diag ρ−σ , . . . , ρNd d , 1

where ρν =

(3.33)

for 1 ≤ ν ≤ Nr† , for N−1 < ν ≤ N , = r† + 1, . . . , d.

ρr,r† ρr,

We remark at this point that as a consequence of Proposition 3.14 be† (r,1 ) (ξ , 0, λ) is a low, it will be seen that for = 1 , B(r ,) (0, ξ , Dn )W † † (N − Nr,−1 ) × (N1 − Nr,1 −1 ) matrix function whose entries are products of powers of ρr, /ρr,1 and an expression of rational type in the components of ηr, , µr,1 , and ρr,1 . 1 Deﬁnition 3.13. Suppose that the boundary problem (1.1), (1.2) is parameterelliptic in L and that λ ∈ L. Suppose also that x0 ∈ Γ and that the boundary problem has been rewritten in terms of a ﬁxed local coordinate system at x0 (see the paragraph preceding Deﬁnition 2.3 and [AV, p. 63]). In addition, suppose that with respect to this boundary problem all hypotheses of Proposition 3.12 hold. Then bearing in mind the deﬁnitions of the various terms introduced above, we say that the operators A(x, D) and B(x, D) = (B1 (x, D), . . . , BN (x, D))T are compatible at x0 if for each r satisfying 0 ≤ r < d and for each pair of integers , 1 satisfying r† ≤ 1 < d, (r,1 ) (ξ , 0, λ) is 1 + 1 ≤ ≤ d, each entry of the matrix B(r,) (0, ξ , Dn )W bounded in modulus by a constant depending only upon (ηr,r † , µr,r † ) ∈ Σr and µr, ∈ L1 for > r. Proposition 3.14. Suppose that the hypotheses of Proposition 3.12 hold and that the operators A(x, D) and B(x, D) are compatible at x0 . Suppose also that 0 ≤ r < d. Then for 0 suﬃciently small and λ0 suﬃciently large the matrix function Ir,r† (ξ , λ) of (3.33), and hence the matrix function Ir,r† (ξ , λ) are invertible and we have σN −σNd , Ir,r† (ξ , λ)−1 = diag ρσ1 , . . . , ρ d Ir,r† (ξ , λ)−1 diag ρ−σ1 , . . . , ρ 1

Nd

1

Nd

where | det Ir,r† (ξ , λ)| > 12 , while the entries of Ir,r† (ξ , λ)−1 are rational functions of the ρr, and expressions of rational type in ηr, , µr, , and ρr,

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for ≥ r† and are bounded in modulus by a constant depending only upon (ηr,r † , µr,r † ) ∈ Σr and µr, ∈ L1 for > r. Lastly, the operators A(x, D) and B(x, D) are always compatible at x0 if (i) the boundary conditions (1.2) are of Dirichlet type at x0 or if (ii) the operators A(x, D) and B(x, D) are both essentially upper triangular at x0 . Proof. We will only prove the proposition for the case 1 ≤ r < d; the case r = 0 can be similarly treated. Accordingly, let us ﬁx our attention upon (3.32) and suppose that = 1 . Then it follows from the proofs of Proposition 3.11 and 3.12 that (r,) (ξ , 0, λ) B(r,)(0, ξ , Dn )W ρ −σ † ρ −σN +1 r, N r, r,−1 = diag ,..., ρr,1 ρr,1 (r,1 ) k1 (r,) × , ζr,1 ) ajk (ηr,1 , ζr,1 , µr,1 ) j,k=1 B1 ,1 (ηr, 1 γ r,1

(r, ) × Ik1 + H11 1 (ηr, , ζr,1 , µr,1 , ρr,1 ) 1 mk +1 m (r,) N , ζr,1 ) diag ρr,11 , . . . , ρm − (µr,1 ρr,11 )−1 B1 ,2 (ηr, r,1 1 (r,1 )

× H21

(3.34)

(ηr, , ζr,1 , µr,1 , ρr,1 ) K (r,1 ) (ηr, , ζr,1 , µr,1 , ρr,1 ) dζr,1 , 1 1

+ where, referring to Proposition 3.11 for notation, γ r, denotes γr+ if 1 = r 1 (r,1 )

+ and γr, otherwise, ajk 1

(ηr, , ζr,1 , µr,1 ) denotes ajk (η , ζ, µ) if 1 = r 1

( ) and ajk1 (ζ1 , µ1 ) otherwise, notes Hjk (η , ζ, µ, ρ) if 1 = r

(r,1 )

(ηr, , ζr,1 , µr,1 , ρr,1 ) 1 (1 ) Hjk (η1 , ζ1 , µ1 , ρ1 ) otherwise. (r,)

and ﬁnally Hjk and

de-

(0, ξ , Dn ) Suppose that > 1 . Then by hypothesis the entries of B (r,1 ) (ξ , 0, λ) are bounded in modulus by a constant depending only on W , µr, ) ∈ Σr and µr, ∈ L1 for > r. Let us now show that this bound(ηr,r edness condition is always satisﬁed if the boundary conditions (1.2) are of Dirichlet type at x0 . Indeed, if this latter condition holds, then every entry (r,) (r,) of B1 ,1 (ηr, , ζr,1 ) is 0, while the only non-zero entries of B1 ,2 (ηr, , ζr,1 ) 1 1 are those lying in rows N − ν, ν = 0, . . . , N − N−1 − 1, and columns k − ν, ν = 0, . . . , k − k−1 − 1. Then recalling from Section 1 that we now have sj = tj = tj for j = 1, . . . , N , it follows from the foregoing results and (3.34) that the entry in the (N − ν)-th row and the (k − ν)-th column of (r,1 ) (ξ , 0, λ) is bounded in modulus by B(r,)(0, ξ , Dn )W tk

− 12 − t

C|λ|

k 1

†

− ν2 ( t1 − t 1 ) k

k 1

|λ|

tk t k 1

+ δr,1 r+1 ,

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where 0 ≤ ν † ≤ tk −1 and the constant C depends only upon (ηr,r , µr,r ) ∈ Σr and µr, ∈ L1 . Our claim concerning Dirichlet boundary conditions are an immediate consequence of this last result. Note also from (3.34) that when A(x, D) and B(x, D) are essentially (r,) , ζr,1 ) (resp. every entry upper triangular at x0 , then every entry of B1 ,1 (ηr, 1 (r,1 ) (η , ζr,1 , µr,1 , ρr,1 )) is 0, and hence the entries of B(r,)(0, ξ , Dn ) of H 21

r,1

(r,1 ) (ξ , 0, λ) are all 0. Thus the boundedness condition also holds under W the cited conditions concerning A(x, D) and B(x, D). Let us now consider the case 1 > . Then it is a simple matter to deduce that for this case each entry of (3.34) is bounded in modulus by m +1 1 1 1 −m −1 m 1 + δ 1 , (3.35) C |λ| m r+1,1 r+1 + (1 − δ1 ,d )|λ| where the constant C depends only upon µr, ∈ L1 . Returning again to (3.32), let us next introduce the block row and column matrices d (1,2) (r,1 ) (ξ , 0, λ) Ir,j (ξ , λ) = B(r,j) (0, ξ , Dn )W , 1 =j+1

d (2,1) (r,j) (ξ , 0, λ) Ir,j (ξ , λ) = B(r,) (0, ξ , Dn )W

,

=j+1

respectively. Then a factorization in the sense of Schur gives 0 INj −N † r,j−1 Ir,j (ξ , λ) = (2,1) Ir,j (ξ , λ) INd −Nj 0 INj −N † r,j−1 × (2,1) (1,2) 0 Ir,j+1 (ξ , λ) − Ir,j (ξ , λ)Ir,j (ξ , λ) (1,2) INj −N † Ir,j (ξ , λ) r,j−1 . × 0 INd −Nj (2,1) Since we already know that each entry of Ir,j (ξ , λ) is bounded in mod ulus by a constant depending only upon (ηr,r , µr,r ) ∈ Σr , that each en(1,2) try of Ir,j (ξ , λ) is bounded in modulus by the quantity (3.35), and that Ir,d (ξ , λ) = I , an inductive argument involving Schur factoriza† Nd −Nr,d−1

tions shows that we may choose 0 suﬃciently small and λ0 suﬃciently large so that all the assertions of the proposition concerning Ir,r (ξ , λ) hold. This completes the proof of the proposition for the case under consideration. As a consequence of Proposition 3.14, we are now in a position to present the last result required for the proof of Proposition 3.6. To this end we require some further notation. Accordingly, for 0 ≤ r ≤ d and λ ∈ C \ {0} let (r) (r) (r) χλ ∈ C ∞ (Rn−1 ) such that 0 ≤ χλ (ξ ) ≤ 1, χλ = 1 for 0 ≤ |ξ | ≤ (r) 3 1/m 1 1 and vanishes for |ξ | ≥ 78 1 |λ|1/m if r = 0, χλ (ξ ) = 1 for 4 1 |λ| 1 1/m r r+1 r ≤ |ξ | ≤ 34 r+1 |λ|1/m and vanishes for |ξ | ≤ 18 r |λ|1/m and for 4 r |λ|

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357

(r)

r+1 d |ξ | ≥ 78 r+1 |λ|1/m if 1 ≤ r < d, χλ (ξ ) = 1 for 14 d |λ|1/m ≤ |ξ | < ∞ and (r) 1 d vanishes for |ξ | ≤ 8 d |λ|1/m if r = d, while χλ (ξ ) is a Fourier multiplier of type (p, p) whose norm is bounded by a constant not depending upon λ, r , and r+1 . We also let S(Rn−1 ) denote the Schwartz space of rapidly decreasing functions on Rn−1 and F the Fourier transform in Rn−1 (x → ξ ).

Proposition 3.15. Suppose that the boundary problem (1.1)–(1.2) is parameter-elliptic in L. Suppose that x0 ∈ Γ and that the boundary problem (1.1), (1.2) is rewritten in terms of the local coordinates at x0 . In addition, suppose that the operators A(x, D) and B(x, D) are compatible at x0 and for −1 g ∈ S(Rn−1 )N and 0 ≤ r ≤ d, let g (r) (x ) = (F ψr (ξ )F g)(x ). Then there exists a λ0 = λ0 (p) > 0 such that for λ ∈ L with |λ| ≥ λ0 , the boundary problem ˚ D)u(x) − λu(x) = 0 A(0, ˚ D)u(x) = g B(0, (r)

for xn > 0, , (r)

(x )

(r)

has a solution u(r) (x) = (u1 (x), . . . , uN (x))T ∈ priori estimate N

(r) (π (j))

1 |||uj |||tj ,p,R n ≤ C +

j=1

N j=1

(3.36)

at xn = 0,

N j=1

(3.37)

t

Wp j (Rn+ ) and the a

(r) (π (j)) 1 n−1 j − p ,p,R

|||gj |||−σ2

(3.38)

holds, where the constant C does not depend upon g and λ. Proof. We will only prove the proposition for the case 1 ≤ r < d; the remaining cases can be treated similarly. Accordingly, under Fourier transform with respect to x , the boundary problem (3.36), (3.37) is transformed into the boundary problem: (3.8), ˚ ξ , Dn )F u(ξ , xn ) = F g (r) (ξ ) B(0,

at xn = 0.

(3.39)

Then assuming that 0 and λ0 have been chosen so that the conclusions of Propositions 3.12 and 3.14 hold, it follows from these propositions that for a ﬁxed λ ∈ L with |λ| ≥ λ0 , this boundary problem has a solution in the space of vector functions decaying exponentially at ∞, u (r) (ξ , xn , λ), which 1 7 1/m r 1/m r+1 is unique for 8 r |λ| ≤ |ξ | ≤ 8 r+1 |λ| and is given in block row matrix form by u (r) (ξ , xn , λ) =

d

−tN 1 diag(ρ−t r, , . . . , ρr, )

=r

×

+ γr,

eiρr, xn ζr, G (r,) (η r , ζr, , µr , ρr )dζr,

−σN 1 × diag ρ−σ , . . . , ρNd d g (r) (ξ ), 1

(3.40)

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where η r = (ηr,r , . . . , ηr,d ), µr = (µr,r , . . . , µr,d ), ρr = (ρr,r , . . . , ρr,d ),

G

(r,)

(η r , ζr, , µr , ρr )

=

k (r,) (r,) , ζr, , µr, ) j,k=1 (Ik + H11 (·)K (r,) (·) aj,k (ηr, −λ−1 H21

(r,)

(·)K (r,) (·)

() × Ir,r (η r , µr , ρr )−1 , () † Ir,r (η r , µr , ρr )−1 denotes the matrix whose rows are precisely rows Nr,−1 + −1 1 to Nr, of Ir,r (η , µ , ρ ) , and for brevity we have written (·) for the r

r

r

, ζr, , µr, , ρr, ). arguments (ηr,

Referring again to the texts preceding (3.15) and (3.23), respectively, for r for µ η ∈ Rn−1 | ( ηr , µ ) ∈ Σ ∈ L1 } and let ( η0 , µ0 ) notation, let Sr ( r ) = { denote an arbitrary point of Sr ( r ) × L1 . Then it follows from Deﬁnition 2.6 that there is a neighbourhood of ( η0 , µ0 ) in Rn−1 × C, say U ( η0 , µ0 ), such η0 , µ0 ) ∩ (Sr ( r ) × L1 ), ( η ,µ ) ∈ Σr and all assertions that for ( η , µ) ∈ U ( r /2 concerning (3.15)–(3.18) hold with µ = (1 − | η |2 )m µ, ξ = ρr,r η , z = ρr,r ζ m r , while for r < ≤ d, all assertions concerning (3.23)–(3.26) and µ = ρr,r µ hold with µ = µ, z = ρr, ζ, and λ = ρr, µ. Since Sr ( r ) × L1 is compact, it is covered by a ﬁnite number of such opens sets, say {Uj }M j=1 , and we let {φj ( η , µ)}M denote a partition of unity subordinate to this covering of j=1 Sr ( r ) × L1 . r ( r ) = { | Let S η ∈ S ( ) η | < 1} and Σ(λ0 ) = {(ξ , λ) ∈ Rn−1 × r r 7 0 1 1/m r r+1 L |λ| ≥ λ , 8 r |λ| ≤ |ξ | ≤ 8 r+1 |λ|1/m }. Then we observe that each 0 ηr,r , µ) ∈ Uj ∩(Sr ( r )∩L1 ) pair (ξ , λ) ∈ Σ(λ ) uniquely determines the point ( = ρ−1 for some j, where ηr,r r,r ξ ,

7 8 r+1 |≤ | ηr,r , 7 r −2/m r+1 1/2 ( 8 r+1 )2 + |λ|2/m

µ =

λ |λ| ,

and ρr,r is deﬁned as before. Conversely, the set of all such pairs M j , where (ξ , λ) is contained in j=1 U

r r /2 j = (ξ , λ) ∈ Rn−1 × L ξ = τ η , |λ| = τ m U (1 − | η |2 )m µ 1/m r λ0 for τ ≥ , and ( η , µ) ∈ Uj ∩ (Sr ( r ) × L1 ) . 2 m /2 (1 − | η| ) r An immediate consequence of these considerations is that if we henceforth (r,) j , agree to replace G (r,) (·) in (3.40) by Gj (·) whenever (ξ , λ) ∈ Σ(λ0 ) ∩ U (r,) (r) M (r,) and let G (·) = j=1 φj (ηr,r , µr,r+1 )Gj (·)χλ (ξ ), where for brevity we

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have written (·) for (η r , ζr, , µr , ρr ), then we obtain a solution u(r) (ξ , xn , λ) =

d

−tN 1 diag(ρ−t r, , . . . , ρr, )

=r

×

+ γr,

(3.41)

−σN 1 ρ−σ eiρr, xn ζr, G(r,) (η r , ζr, µr , ρr )dζr, diag( , . . . , ρNd d )g (r) (ξ ) 1

of (3.8), (3.39) for all values of ξ ∈ Rn−1 and λ ∈ L1 with |λ| ≥ λ0 with the property that each entry of (r,) G(r,) (η r , ζr, , µr , ρr ) = Gjk (η r , ζr, , µr , ρr ) j=1,...,N k=1,...,Nd

has derivatives of all orders with respect to the components of ξ which are continuous in ξ and λ, and, in addition, for any multi-index α whose entries are either 0 or 1,

α

|ξ α D ξ Gj,k (η r , ζr, , µr , ρr )| ≤ C

(resp. |λ|−1 C) if j ≤ k (resp. j > k ),

where the constant C does not depend upon ξ , λ, and α . In light of (3.41) we see that the proposition will be proved for the case (r) (π1 (j)) under consideration here if we can show that |||uj,k, |||tj ,p,R n is bounded by + the expression on the right-hand side of (3.38) for every j, k, and satisfying 1 ≤ j ≤ N , 1 ≤ k ≤ Nd , and r ≤ ≤ d with π2 (k) = , where −tj (r) (r,) k (r) uj,k, (ξ , xn , λ) = ρr, eiρr, xn ζr, Gj,k (η r , ζr, , µr , ρr )dζr, ρ−σ r, gk (ξ ). + γr,

To this end let us observe that (r)

(π (j))

1 |||uj,k, |||tj ,p,R n ≤ C +

tj −1 m F ξ , λπ1 (j) ρ−σk −m m=0

×

γ+

(r,) (r) eixn ρζ ζ tj −m Gj,k (η r , ζ, µr , ρr )dζ F gk (ξ )

0,p,Rn +

,

+ , and where C where for brevity we have put ρ = ρr, , ζ = ζr, , γ + = γr, (r) denotes a constant not depending upon g and λ. Hence if we put Λ(ξ , λ) = ξ if = r, Λ(ξ , λ) = ξ , λ if > r, and −1 (r) h(x, λ) = F e−Λ(ξ ,λ)xn F gk (x) for x ∈ Rn+

(see [ADF, Proposition 2.3]), argue as in [V, Section 5], and appeal to Minkowski’s inequality, then we deduce that 2 ∞ ∞ p 1/p (r) (π1 (j)) |||uj,k, |||tj ,p,R dxn Jν (xn , τ, λ)dτ , (3.42) n ≤ C +

ν=1

xn =0

τ =0

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where J1 (xn , τ, λ) = (xn + τ )−1 F

−1 −σk

ρ

F h(s , τ )J1 (xn , τ, ξ , λ)0,p,Rn−1 ,

J2 (xn , τ, λ) = (xn + τ )−1 F

−1 −σk −1

ρ

F Dτ h(s , τ )J2 (xn , τ, ξ , λ)0,p,Rn−1 ,

(3.43) (3.44)

J1 (xn , τ, ξ , λ) −m (r,) = ρζ(xn + τ )eiρζ(xn +τ ) ξ , λm Gjk (η r , ζ, µr , ρr )ζ tj −m dζ, π1 (j) ρ γ+

J2 (xn , τ, ξ , λ) −m (r,) = ρζ(xn + τ )eiρζ(xn +τ ) ξ , λm Gjk (η r , ζ, µr , ρr )ζ tj −1−m dζ, π1 (j) ρ γ+

and the constant C does not depend upon g (r) and λ. Let us ﬁrstly ﬁx our attention upon the case where π2 (k) ≤ r, so that now we have = r in (3.42)–(3.44). Then with the use of the Mikhlin-Lizorkin multiplier theorem we can show that for 1 ≤ ν ≤ 2, ρ −σk −ν+1 (r) Jν (xn , τ, ξ , λ)χλ (ξ ) ξ is a Fourier multiplier of type (p, p) with norm bounded by a constant not depending upon g (r) , xn , τ , and λ. Hence it follows from (3.42)–(3.44) and [ADF, Proposition 2.3] that (r)

π (j)

(r)

(r) π (k) 1 n−1 , k − p ,p,R

2 |||uj,k, |||tj1,p,Rn ≤ Cr gk −σk − p1 ,p,Rn−1 ≤ Cr |||gk |||−σ +

where the constant Cr does not depend upon g (r) and λ. Turning next to the case where π2 (k) = > r, we can again show that for 1 ≤ ν ≤ 2 −σk −ν+1 ρ (r) Jν (xn , τ, ξ , λ)χλ (ξ ) ξ , λπ2 (k) is a Fourier multiplier of type (p, p) with norm bounded by a constant not depending upon g (r) , xn , τ , and λ. Thus it follows from (3.42)–(3.44) and [ADF, Proposition 2.3] that (r)

π (j)

(r) π (k) 1 n−1 , k − p ,p,R

2 |||uj,k, |||tj1,p,Rn ≤ C |||gk |||−σ +

and the constant C has the same properties as the constant C above; and this completes the proof of the proposition. Proof of Proposition 3.6. In proving the proposition there is no loss of gen erality in supposing that g ∈ S(Rn−1 )N since the result for general g ∈ 1

N −σj − p (Rn−1 ) will then follow from a standard approximation procej=1 Wp dure. Accordingly, under a Fourier transformation with respect to x (x → ξ ), the boundary problem (3.3), (3.4) is transformed into the boundary

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problem (3.8), (3.39). Then assuming that 0 and λ0 have been chosen so that the conclusions of Propositions 3.12 and 3.14 hold, it follows from [DV, Theorem 3.5] and the above considerations that for a ﬁxed λ ∈ L with |λ| ≥ λ0 this boundary problem has a unique solution u(ξ , xn , λ) in the space of vector functions decaying exponentially at ∞, and hence d u(ξ , xn , λ) = r=0 u(r) (ξ , xn , λ). The assertions of the proposition follow directly from the results of Proposition 3.15 and the Mikhlin-Lizorkin multiplier theorem.

4. Proof of Theorem 2.8 We suppose henceforth that the hypotheses of Theorem 2.8 hold. Moreover, by employing a standard extension procedure we can also assume that the jk n ajk α (x) and bα (x) of (2.1) are deﬁned on all of R , are compactly supported, and satisfy the same smoothness assumptions as asserted in Assumption 2.2 for Ω and Γ, respectively. Lastly for x0 ∈ Ω and δ > 0, let Bδ (x0 ) denote the open ball in Rn with centre x0 and radius δ. Then in order to prove the theorem we require some further results. Proposition 4.1. For any > 0 and x0 ∈ Ω there exists a δ > 0 and a λ0 > 0 such that for λ ∈ L with |λ| ≥ λ0 N N N (π1 (j)) ˚jk (x0 , D) uk Ajk (x, D) − A ≤ ||uj ||tj ,p,Ω (4.1) j=1

k=1

for every u ∈

−sj ,p,Ω

N

tj j=1 Wp (Ω)

j=1 0

such that supp u ⊂ Bδ (x ) ∩ Ω.

Proof. Let us ﬁx a j, 1 ≤ j ≤ N , and suppose ﬁrstly that sj > 0. Then we have two cases to consider: (1) x0 ∈ Ω and (2) x0 ∈ Γ. Let x0 ∈ Ω and suppose that Bδ (x0 ) ⊂ Ω . Then for 1 ≤ k ≤ N we have α jk 0 ˚jk (x0 , D) uk = Ajk (x, D) − A ajk α (x) − aα (x ) D uk |α|=sj +tk

+

α ajk α (x)D uk .

(4.2)

|α|<sj +tk

α jk 0 jk α It is clear that for a ﬁxed α, ajk α (x) − aα (x ) D uk resp. aα (x)D uk is a distribution on Rn of order sj (resp. ≤ sj ) whose support is contained in that of uk . Hence it follows from [GK, Section 1], [T, Theorem 2.6.1, p.198] that for |α| = sj + tk , α (π1 (j)) jk 0 ||| ajk α (x) − aα (x ) D uk |||−sj ,p,Ω (π (j))

jk 0 α 1 ≤ |||ajk α (x) − aα (x )D uk |||−sj ,p,Rn jk 0 α = sup aα (x) − ajk α (x ) D uk , ζ ζ

jk 0 = sup Dα−β uk , Dβ ajk α (x) − aα (x ) ζ , ζ

(4.3)

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−s

where ·, · denotes the pairing between Hp j (Rn ) and its dual, the supre(π1 (j)) mum is taken over all ζ ∈ C0∞ (Rn ) satisfying |||ζ|||sj ,p ,Rn ≤ 1, and |β| = sj . Note that here duality between the spaces indicated is to be understood in the sense that the spaces are now equipped not with their standard norms, but with equivalent norms, namely their parameter dependent norms (π (j)) (π1 (j)) ||| · |||−s1j ,p,Rn and ||| · |||sj ,p ,Rn , respectively (see [GK, Eqn. (1.9)]). We conclude from (4.3) that the expression on the left side of this equation does not exceed uk tk ,p,Rn ρζsj ,p ,Rn + Cζsj −1,p ,Rn , where ρ = supx∈Bδ (x0 ) |ajk α (x) − jk 0 aα (x )|, and here and below C denotes a generic constant that does not depend upon uk , λ, and ζ.Thus it follows that the expression on the left side of (4.3) does not exceed

−m

C|λ|

1 π1 (j)

+ ρ uk tk ,p,Rn . On the other hand

if |α| < sj + tk , then similar arguments show that (π (j))

−m

α 1 |||ajk α (x)D uk |||−sj ,p,Ω ≤ C|λ|

1 π1 (j)

uk tk ,p,Rn .

Suppose next that x0 ∈ Γ and let {U, φ} be a chart on Γ such that x ∈ U, φ(x0 ) = 0, and φ is a diﬀeomorphism of class C κ0 −1,1 ∩ C s1 mapping U onto an open set in Rn with φ(U ∩ Ω) ⊂ Rn+ and φ(U ∩ Γ) ⊂ Rn−1 . We suppose henceforth that B2δ (x0 ) ⊂ U . Then for 1 ≤ k ≤ N we have the representation from which it follows that for a ﬁxed α (4.2), jk jk 0 α α, ajk α (x) − aα (x ) D uk resp. aα (x)D uk is a distribution on Ω of order sj (resp. ≤ sj ) whose support is contained in that of uk . Further(π (j)) (π (j)) more, it is clear from the deﬁnition that |||v|||−s1j ,p,Ω ≥ |||v|||−s1j ,p,Ω∩U for both jk 0 α jk α (1) v = aα (x) − ajk α (x ) D uk and (2) v = aα (x)D uk . To obtain a reverse estimate, let us ﬁrstly ﬁx our attention upon the v of case (1), choose −s (π (j)) (π (j)) w ∈ Hp j (Rn ) such that wΩ∩U = v and |||w|||−s1j ,p,Rn ≤ 2|||v|||−s1j ,p,Ω∩U , and let ψ ∈ C0∞ (Rn ) such that ψ(x) = 1 for x ∈ Bδ (x0 ) and supp ψ ⊂ U . The −s ψ w ∈ Hp j (Rn ), ψ wΩ = v, and hence 0

(π (j))

(π (j))

(π (j))

(π (j))

|||v|||−s1j ,p,Ω ≤ |||ψ w|||−s1j ,p,Rn ≤ C|||w|||−s1j ,p,Rn ≤ 2C|||v|||−s1j ,p,Ω∩U , where we have used an argument similar to that used in (4.3). Likewise we can show that (π (j))

(π (j))

1 1 α jk α |||ajk α (x)D uk |||−sj ,p,Ω ≤ C|||aα (x)D uk |||−sj ,p,Ω∩U .

By means of the mapping φ we pass to the local coordinates at can now jk 0 α x0 , and in these local coordinates ajk α (x) − aα (x ) D uk can be written as α jk α jk 0 aα (y) − φ∗ ajk ajk k α (x) − aα (x ) D uk = α (0) Dy u β + ajk k , α (y)Dy u 0<|β|<|α|

where y = φ(x). Furthermore, from the results given in [GK, Section 1], [T, Theorem 4.8.1, p.332] and the proofs of Lemma 4.2 and Theorems 4.7.2

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and 4.8.2 of [T, pp.310-311 and 332-333] it is not diﬃcult to deduce that α (π1 (j)) jk 0 ||| ajk α (x) − aα (x ) D uk |||−sj ,p,Ω∩U α (π1 (j)) hjk 0 ≤ C|||φ∗ ajk α (x) − aα (x ) D uk |||−sj ,p,φ(Ω∩U) α (π1 (j)) jk 0 ≤ C|||φ∗ ajk α (x) − aα (x ) D uk |||−sj ,p,Rn + (4.4) α jk jk ≤ C sup Dy u k , aα (y) − aα (0) ψζ ζ

sup Dyβ u k , ajk β (y)ψζ ,

+

0<|β|<|α|

ζ

where the constant C does not depend upon uk , ·, · denotes the pairing −s ˚ sj (Rn ), equipped with their paramebetween Hp j (Rn+ ) and its dual W + p (π (j))

(π (j))

1 ter dependent norms ||| · |||−s1j ,p,Rn and ||| · |||sj ,p ,Rn , respectively, the supre + ∞ n (π+1 (j))

mum is taken over the set ζ ∈ C0 (R+ )|||ζ|||sj ,p ,Rn ≤ 1 , ψ = φ∗ (ψ), +

and ψ ∈ C0∞ (Rn ) such that ψ(x) = 1 for x ∈ Bδ (x0 ), supp ψ ⊂ B2δ (x0 ). From this last result it is not diﬃcult the expression on the to deduce that −

1

π (j) 1 left side of (4.4) does not exceed C ρ + C1 |λ| m uk tk ,p,Ω∩U , where jk jk 0 ρ = supx∈B2δ (x0 ) aα (x) − aα (x ) and the constant C1 does not depend

(π (j))

1 α upon uk and λ. Since a similar result holds for |||ajk α (x)D uk |||−sj ,p,Ω∩U for |α| < sj + tk , the proof of the proposition is complete for the case sj > 0; the proof for the case sj = 0 follows from a standard argument.

Arguments similar to those used in the proof of Proposition 4.1 can also be used to prove the following two results. Proposition 4.2. For any > 0 and x0 ∈ Ω there exits a δ, 0 < δ 0< jk 0 dist{x0 , Γ}, and a λ0 > 0 such that if supp ajk (x) − a (x ) ⊂ Bδ (x ) α α 0 for |α| = sj + tk , j, k = 1, . . . , N , and λ ∈ L with |λ| ≥ λ , then the estimate N N N ˚jk (x, D) − A ˚jk (x0 , D) uk |||(π1 (j)) n ≤ ||| A uk tk ,p,Rn −sj ,p,R j=1 k=1

holds for every u ∈

j=1

N

tj n j=1 Wp (R ).

We wish now to consider the analogue of Proposition 4.2 for the case x0 ∈ Γ. Accordingly, let us turn again to the chart {U, φ} introduced in the proof of Proposition 4.1. Then by means of the mapping φ we can now pass to local coordinates at x0 , and in the local coordinates each of the operators α jk (y, D) = Ajk (x, D) can be written as A ajk α (y)Dy for y ∈ φ(U ). |α|≤sj +tk We are interested in the case where for some ball Bδ (0) ⊂ φ(U ), ajk α (y) = jk aα (0) for y ∈ φ(U ) \ Bδ (0) and for all α satisfying |α| = sj + tk . Because o

jk (y, Dy ), the principal of this assumption we can now extend the operator A

364

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jk (y, Dy ), to an operator on all of Rn by putting part of A ajk ajk α (y) = α (0) n for y ∈ R \ φ(U ) and |α| = sj + tk . Proposition 4.3. For any > 0 there exists a δ, 0 < δ < dist{0, ∂φ(U )}, ajk ajk and a λ0 > 0 such that if supp α (y) − α (0) ⊂ Bδ (0) for |α| = sj + tk , 0 j, k = 1, . . . , N and λ ∈ L with |λ| ≥ λ , then the estimate N N N o o Dy ) − A(0, Dy ) uk |||(π1 (j)) n ≤ ||| A(y, uk tk ,p,Rn+ −sj ,p,R +

j=1 k=1

holds for every u ∈

N

j=1

tj n j=1 Wp (R+ ).

Finally, the proof of the theorem follows directly from the results of Section 3, Propositions 4.1 - 4.3, and the arguments used in the proofs of Theorems 4.1 and 5.1 of [AV].

References [ADN]

[ADF]

[AV]

[DFM]

[DMV] [DV]

[F] [Gr] [G] [GK]

[K1]

S. Agmon, R. Douglis, and L. Nirenberg: Estimates near the boundary for solutions of elliptic partial diﬀerential equations satisfying general boundary conditions II. Comm. Pure Appl. Math. 17 (1964), 35–92. M. S. Agranovich, R. Denk, and M. Faierman: Weakly smooth nonselfadjoint elliptic boundary problems. In: Advances in partial diﬀerential equations: Spectral theory, microlocal analysis, singular manifolds, Math. Top. 14 (1997), 138–199. M. S. Agranovich and M.I. Vishik: Elliptic problems with a parameter and parabolic problems of general form. Russ. Math. Surveys 19 (1964), 53–157. R. Denk, M. Faierman, and M. M¨ oller: An elliptic boundary problem for a system involving a discontinuous weight. Manuscripta Math. 108 (2001), 289–317. R. Denk, R. Mennicken, and L. Volevich: The Newton polygon and elliptic problems with parameter. Math. Nachr. 192 (1998), 125–157. R. Denk and L. Volevich: Elliptic boundary value problems with large parameter for mixed order systems. Amer. Math. Soc. Transl.(2) 206 (2002), 29–64. M. Faierman: Eigenvalue asymptotics for a boundary problem involving an elliptic system. Math. Nachr. 279 (2006), 1159–1184. P. Grisvard: Elliptic problems in nonsmooth domains.Pitman, London, 1992. G. Grubb: Functional calculus of pseudodiﬀerential boundary problems 2nd edn.. Birkh¨ auser, Boston, 1996. G. Grubb and N.J. Kokholm: A global calculus of parameter-dependent pseudodiﬀerential boundary problems in Lp Sobolev spaces. Acta Math. 171 (1993), 1–100. A. N. Kozhevnikov: Spectral problems for pseudodiﬀerential systems elliptic in the Douglis-Nirenberg sense, and their applications. Math. USSR Sobornik 21 (1973), 63–90.

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A. N. Kozhevnikov: Parameter-ellipticity for mixed order boundary problems. C. R. Acad. Sci. Paris S´ er. I Math. 324 (1997), 1361–1366. A. N. Kozhevnikov: Parameter-ellipticity for mixed order systems in the sense of Petrovskii. Commun. Appl. Anal. 5 (2001), 277–291. H. Triebel: Interpolation theory, function spaces, diﬀerential operators. North-Holland, Amsterdam, 1978. M. I. Vishik and L. A. Lyusternik: Regular degeneration and boundary layer for linear diﬀerential equations with small parameter. Amer. Math. Soc. Transl.(2) 20 (1962), 239–264. L. R. Volevich: Solvability of boundary value problems for general elliptic systems. Amer. Math. Soc. Transl.(2), 67 (1968), 182–225.

R. Denk Department of Mathematics and Statistics University of Konstanz D-78457 Konstanz Germany e-mail: [email protected] M. Faierman School of Mathematics and Statistics The University of New South Wales UNSW Sydney NSW 2052 Australia e-mail: [email protected] Submitted: September 17, 2009. Revised: January 8, 2010.

Integr. Equ. Oper. Theory 66 (2010), 367–395 DOI 10.1007/s00020-010-1756-0 Published online March 10, 2010 © Birkhäuser / Springer Basel AG 2010

Integral Equations and Operator Theory

Products of Toeplitz Operators on a Vector Valued Bergman Space Robert Kerr Abstract. We give a necessary and a suﬃcient condition for the boundedness of the Toeplitz product TF TG∗ on the vector valued Bergman space L2a (Cn ), where F and G are matrix symbols with scalar valued Bergman space entries. The results generalize those in the scalar valued Bergman space case [13]. We also characterize boundedness and invertibility of Toeplitz products TF TG∗ in terms of the Berezin transform, generalizing results found by Zheng and Stroethoﬀ for the scalar valued Bergman space [17]. Mathematics Subject Classiﬁcation (2010). Primary 47B35. Keywords. Bergman space, Vector valued functions, Toeplitz operator.

1. Introduction 1.1. Notation 1 For a measurable function f : D → Cn with ( D f (z)pCn dA(z)) p < ∞, we say that f ∈ Lp (D, Cn ). The vector-valued Bergman space Lpa (D, Cn ) is the intersection of Lp (D, Cn ), with the analytic Cn - valued functions on D with the usual identiﬁcation of functions which only diﬀer on sets of measure 1 0. The norm is given by f Lpa(Cn ) = ( D f (z)pCn dA(z)) p , where dA is normalized Lebesgue measure on the unit disk D. In the case p = 2 this the inner product given by f, g = space becomes a Hilbert pspace with 2 f (z), g(z) dA(z). L and L are Banach spaces for 1 ≤ p < ∞. For n a C D details see for example [2]. On the scalar valued Bergman space L2a , the Toeplitz operator with symbol f ∈ L2 is the densely deﬁned operator Tf v = P (f v), where P is the orthogonal projection from L2 into L2a and v is a polynomial. The Toeplitz This work was completed with the support of the EPSRC. Part of this work was carried out while visiting the Fields Institute as part of the Thematic Program on New Trends in Harmonic Analysis.

368

Robert Kerr

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operator is a multiplication operator composed with an orthogonal projection. The Bergman projection is explicitly given by the following integral; P f (w) = f, Kw =

D

f (w) dA(w), (1 − zw)2

1 where Kw (z) = (1−zw) 2 is the reproducing kernel of the Bergman space 2 L2 (D). So using this explicit form we can deﬁne a Toeplitz operator on a dense subset of L2a , the polynomials, with symbol in L2 rather than L∞ . We can also see that with a symbol f ∈ L2 and v ∈ L2a , Tf v(w) is well deﬁned point-wise for each w ∈ D. In [11] Sarason conjectured that a product of Toeplitz operators (deﬁned densely in an appropriate way for analytic functions f and g) Tf Tg on 2 (w) was uniformly |2 (w)|g| the Hardy Space H 2 is bounded if and only if |f bounded on the disc, f(w) being the Poisson integral of f . This turned out to be false, [8], but a suﬃcient condition is found in [21]. Another conjecture by Sarason dealt with in [9, 10, 13, 14, 15, 16] and [17] was as to when the densely deﬁned operator Tf Tg is bounded on L2a for f, g ∈ L2a ? The question was originaly posed by Sarason in [11] and a conjecture in section 8 of [13] more explicitly resembles Sarason’s Hardy space case conjecture. This time it is conjectured that the Toeplitz product Tf Tg is bounded for analytic f and g on the Bergman space L2a if and 2 (w) is uniformly bounded, where f is the Berezin transonly if |f |2 (w)|g| form of f . The question is investigated in various diﬀerent cases, such as the weighted Bergman space with standard weights and the Bergman space on the unit ball and polydisk. These papers prove results that approximate to the Bergman space version of Sarason’s conjecture as stated in section 8 of [13]. The purpose of this paper is to investigate products of Toeplitz operators on a Bergman space of vector-valued functions. In the case of the vector valued Bergman space L2a (Cn ), we deﬁne the Toeplitz operator to be the densely deﬁned composition of multiplication with a matrix valued function and the orthogonal projection from L2 (Cn ) into L2a (Cn ). So in this case the symbol F will be a matrix of L2 functions and TF v = P (F v), where v is a bounded analytic Cn valued function. If

⎛

f11

⎜ F =⎜ ⎝f21 .. .

f12 .. .

⎞ ... ⎟ ⎟ ⎠

and v = (v1 , v2 , . . . , vn ), where fij ∈ L2 and vi ∈ H ∞ , then

TF v = P (F v) = P

n i=1

f1i vi ,

n i=1

f2i vi , . . . ,

n i=1

fni vi

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⎛

Tf11

⎜ =⎜ ⎝Tf21 .. .

Tf12 .. .

369

⎞ ... ⎟ ⎟ v, ⎠

where each Tfij is a densely deﬁned Toeplitz operator on the scalar Bergman space L2a . When looking at products of these Toeplitz operators analagous to the treatment in [13] we have products of the form TF TG∗ , where F and G are square matrices of scalar valued Bergman space L2a functions. 1.2. Main Theorems The ﬁrst two main theorems follow, one giving a suﬃcient condition for the Toeplitz product TF TG∗ to be bounded and the other a necessary condition. Both are conditions involving the Berezin transform; 2 Deﬁnition 1.1. The Berezin transform of a matrix A with L entries is the matrix-valued function B(A), where B(A)(w) = (A ◦ φw )(z)dA(z),w ∈ D, composition here being composition with each matrix entry. Here, φw is the w−z M¨ obius transform z → 1−wz . We should also note here that B(A)(w) = 2 2 (1−|w| ) A(z) |1−wz|4 dA(z) by a change of variables. Deﬁning the normalized re-

producing kernel kw (z) to be

Kw (z) Kw ,

we obtain |kw (z)|2 =

(1−|w|2 )2 |1−wz|4 .

Here is our ﬁrst main result: Theorem 1.2. If for some > 0 the trace of the matrix B((F ∗ F )

2+ 2

)(w)B((G∗ G)

2+ 2

)(w)

is uniformly bounded for all w ∈ D, then the Toeplitz product TF TG∗ is bounded L2a (Cn ) → L2a (Cn ). We also have the following condition: If there exists > 0 such that 1 2+ ∗ ∗ 2+ 2 2 2 (tr(G(z)F (x) F (x)G(z) )) |kw (z)| dA(z) |kw (x)| dA(x) D

D

is uniformly bounded, then the Toeplitz product TF TG∗ : L2a (Cn ) → L2a (Cn ) is bounded. Here is the necessary condition: Theorem 1.3. If the product of Toeplitz operators TF TG∗ is bounded, then the trace of the matrix B(F ∗ F )(w)B(G∗ G)(w) is uniformly bounded for w ∈ D. The next theorem is the other main result presented here, involving a characterization of bounded and invertible Toeplitz products. Theorem 1.4. The Toeplitz product TF TG∗ is bounded and invertible if and only if the trace of the matrix B(F ∗ F )(w)B(G∗ G)(w) is uniformly bounded and there exists η > 0 with (F G∗ GF ∗ )(z) > ηI for all z ∈ D. This last inequality is a matrix inequality.

370

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2. Bounded Toeplitz Products 2.1. A Suﬃcient Condition (Proof of Theorem 1.2) The technique in [13] for showing a suﬃcient condition on the boundedness of a Toeplitz product involves an inner product formula that easily generalizes to the vector valued case. So for g, f ∈ L2a (Cn ) f (z), g(z)Cn dA(z) = 3 (1 − |z|2 )2 f (z), g(z)Cn dA(z) f, gL2a (Cn ) = D D 1 (1 − |z|2 )2 f (z), g (z)Cn dA(z) + 2 D 1 + (1 − |z|2 )3 f (z), g (z)Cn dA(z). 3 D So to estimate the norm of TG TF∗ , we will look at the inner product TG TF∗ u, vL2 (Cn ) with v, u ∈ L2a (Cn ) in the form just given. a Let us start by estimating the term TF ∗ (u)(w), TG∗ (v)(w) Cn . Deﬁnition 2.1. For f, g ∈ L2 (D) , deﬁne the rank 1 operator f ⊗ g : L2 (D) → L2 (D) by (f ⊗ g)h = h, g f for h ∈ L2 (D). Also for F, G ∈ Mn×n (L2 (D)), deﬁne the operator F ⊗ G : L2 (D, Cn ) → L2 (D, Cn ) by ⎞ ⎛ ... i f1i ⊗ g2i i f1i ⊗ gni i f1i ⊗ g1i ⎜ ⎟ ... ⎟ ⎜ i f2i ⊗ g1i ⎟ ⎜ . . .. .. (F ⊗ G)h = ⎜ ⎟h ⎟ ⎜ ⎠ ⎝ f ⊗ g f ⊗ g . . . f ⊗ g ni 1i ni 2i ni ni i i i for h ∈ L2 (D, Cn ). Theorem 2.2. TF ∗ (u)(w), TG∗ (v)(w) Cn 1 = (Gkw ⊗ F kw )u(z), v(z)Cn dA(z), (1 − |w|2 )2 D where kw is the normalized reproducing kernel. Proof. TF ∗ (u)(w), TG∗ (v)(w)Cn ∗ ∗ = F (z)u(z)Kw (z)dA(z), G (ζ)v(ζ)Kw (ζ)dA(ζ) D

D

Cn

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∗ G(ζ)Kw (ζ) (F (z)Kw (z)) u(z), v(ζ) Cn dA(z)dA(ζ) D D 1 ∗ = G(ζ)kw (ζ) (F (z)kw (z)) u(z), v(ζ) Cn dA(z)dA(ζ) 2 2 (1 − |w| ) D D 1 (Gkw ⊗ F kw u)(ζ), v(ζ)Cn dA(ζ). = (1 − |w|2 )2 D =

Lemma 2.3. (F ⊗ G)(F ⊗ G)∗ op ∼ tr{(F ⊗ G)(G ⊗ F )} n n n n

=

fqr , fql L2 gml , gmr L2 .

q=1 m=1 r=1 l=1

Proof. Firstly note that (F ⊗ G)∗ = G ⊗ F . As (F ⊗ G)(G ⊗ F ) is of ﬁnite rank the trace of (F ⊗ G)(G ⊗ F ) will be an equivalent norm. We can express F⊗ G as a matrix of operators on the scalar Bergman space with the entries n [ l=1 fil ⊗ gjl ]i,j . We can then express (F ⊗ G)(G ⊗ F ) in a similar manner;

n

n

m=1

=

fil ⊗ gml

l=1

n

m=1

n

gml ⊗ fjl

l=1

n n

i,j

·, fjl L2 gml , gmr L2 fir

r=1 l=1

. i,j

√ Noting that we have as an orthonormal basis el,m = (0, . . . , 0, z l l + 1, 0, . . .) i.e. a vector with each coordinate 0 apart from the mth entry which is the lth orthonormal basis element of the scalar valued Bergman space. So the trace of the operator (F ⊗ G)(G ⊗ F ) will be (F ⊗ G)(G ⊗ F )ep,q , ep,q p,q

=

∞ n z p 1 + p, fql q=1 p=1 m,r,l

L

gml , gmr L2 2

D

fqr (z)z p

1 + pdA(z).

√ We can write each fij as a power series s=1 as,ij z s 1 + s and thus this trace becomes

n n ∞ n n ap,ql gml , gmr L2 ap,qr ∞

q=1 p=1 m=1

r=1 l=1

and thus by Parseval’s identity the expression for the trace becomes;

n n n n fqr , fql L2 gml , gmr L2 . q=1 m=1

r=1 l=1 1

Theorem 2.4. Gkw ⊗ F kw op ≈ (tr(B(G∗ G)(w)B(F ∗ F )(w))) 2 .

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Proof. We can see that Gkw ⊗ F kw op is equivalent to the square root of the trace of the operator (Gkw ⊗ F kw )(Gkw ⊗ F kw )∗ and hence Lemma 2.3 implies that this is equal to n n

q=1 m=1

n n fqr , fql |kw |2 L2 gml , gmr |kw |2 L2

r=1 l=1 ∗

= tr(B(G G)(w)B(F ∗ F )(w)).

Deﬁnition 2.5. The operator P0 deﬁned on Lp (D) is the operator that sends f (z) f ∈ L2 to the function given by (P0 f )(w) = D |1−wz| 2 dA(z). Elements from the following two theorems are borrowed from Theorem 3.2 in [13]. Lemma 2.6. If we have a scalar valued integrable function h and a scalar valued Bergman space function v then for each w ∈ D, 1 2+ 2 xh(x)|v(x)| 2+ |kw (x)| (P0 |v|δ )(w) |h(x)| dA(x) dA(x) ≤ 2 3 2 (1 − xw) 1 − |w| D D Here, for > 0, δ =

1 δ

.

2+ 1+ .

Proof. By H¨older’s inequality, |h(x)1 − xwv(x)| xh(x)|v(x)| dA(x) dA(x) ≤ 3 (1 − xw) |1 − xw|4 D D 1 2+ 1δ |h(x)|2+ |1 − xw|δ |v(x)|δ dA(x) dA(x) ≤ 4 |1 − xw|4 D |1 − xw| D " 1δ 1 ! 2+ 2 2 1+ δ |k (x)| |1 − |w| | |v(x)| w = |h(x)|2+ dA(x) , dA(x) 2 1 − |w|2 D D |1 − xw| |1 − xw| +1 and our result follows from the fact that (1 − |w|)(1 + |w|) (1 − |w|)(1 + |w|) 1 − |w|2 ≤ ≤ |1 − wz| |1 − wz| |1 − |wz (1 − |w|)(1 + |w|) ≤ ≤ 1 + |w| < 2. |1 − |w Let us now take a look at TF ∗ (u) (w), TG∗ (v) (w)Cn .

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Theorem 2.7. Let w ∈ D. Then |TF ∗ (u) (w), TG∗ (v) (w)Cn | 1 2+ |kw (z)|2 |kw (x)|2 ∗ ∗ 2+ 2 (tr(G(z)F (x) F (x)G(z) )) dA(z) dA(x) ≤C 1 − |w|2 1 − |w|2 D D 1 1 × (P0 uδCn )(w) δ (P0 vδCn )(w) δ # 1 $ 2+ 2+ 2+ 1 ∗ ∗ tr(B((F F ) 2 )(w)B((G G) 2 )(w)) ≤C (1 − |w|2 )2 1 1 × (P0 uδCn )(w) δ (P0 vδCn )(w) δ where C is a constant, > 0 and

1 δ

=1−

1 2+ .

Proof. First note that for a function u ∈ L2a (D, Cn ) u, Kw = u (w),

so that | TF ∗ (u) (w), TG∗ (v) (w)Cn | (z), G∗ (x)v(x)K (x) = F ∗ (z)u(z)Kw dA(z)dA(x) w Cn Kw (z) Kw (x) dA(z)dA(x) . G(z)F ∗ (x)Cn op u(z)v(x) ≤ 1 − wz 1 − wx w (x) Then, using Lemma 2.6 with h(z) = G(z)F ∗ (x)Cn op v(x) K 1−wx , we arrive at the following inequality, | TF ∗ (u) (w), TG∗ (v) (w)Cn | 1 ! " 2+ Kw (x) 2+ |kw (z)|2 ∗ ≤ 2 dA(z) G(z)F (x)Cn op v(x) D 1 − wx 1 − |w|2 D 1 × (P0 uδ )(w) δ dA(x) 1 2+ 2 Kw (x) 2+ |kw (z)| ∗ dA(x) n (G(z)F (x)C op ) dA(z) v(x) = D 1 − |w|2 1 − wx D 1 δ δ × (P0 u )(w) , 1 . where > 0 and 1δ = 1 − 2+ Again, using Lemma 2.6, but this time with 1 2+ 2 2+ |kw (z)| ∗ h(x) = (G(z)F (x)Cn op ) dA(z) , 1 − |w|2 D

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we see that | TF ∗ (u) (w), TG∗ (v) (w)Cn | 1 2+ 2 |kw (x)|2 2+ |kw (z)| ∗ n (G(z)F (x)C op ) dA(z) dA(x) ≤ 4 D 1 − |w|2 1 − |w|2 D 1 1 δ δ δ δ (P0 u )(w) × (P0 v )(w) |kw (z)|2 ∗ ∗ 2+ 2 ≤4 (C tr(G(x)F (z) F (z)G(x) )) dA(z) 1 − |w|2 D D 1 1 1 |kw (x)|2 × dA(x) 2+ (P0 uδCn )(w) δ (P0 vδCn )(w) δ . 2 1 − |w| (This is what we want but we can go a step further and get something that looks even more similar to the analogous result in the scalar case.) Letting the 4 be absorbed into the constant C and using the inequality on matrix norms from [1], theorem IX.2.10 on page 258, we see that | TF ∗ (u) (w), TG∗ (v) (w)Cn | 2 2+ |kw (z)| ∗ ∗ 2 n ≤ (C(G(x)F (z) F (z)G(x) )C op ) dA(z) 1 − |w|2 D D 1 1 1 |kw (x)|2 δ δ δ δ 2+ (P (P dA(x) u v × n )(w) n )(w) 0 0 C C 1 − |w|2 2 2+ |kw (z)| 1 1 ∗ ∗ ∗ 2 2 2 ≤ (C((G G(x)) F F (z)(G G(x)) )Cn op ) dA(z) 1 − |w|2 D D 1 1 1 |kw (x)|2 δ δ δ δ 2+ × (P (P dA(x) u v n )(w) n )(w) 0 0 C C 1 − |w|2 2+ 2+ 2+ |kw (z)|2 ∗ ∗ ∗ 4 2 4 ≤ C(G G(x)) (F F (z)) (G G(x)) Cn op dA(z) 1 − |w|2 D D 1 1 1 |kw (x)|2 × dA(x) 2+ (P0 uδCn )(w) δ (P0 vδCn )(w) δ 2 1 − |w| 1 & % & & 2+ % % 2+ 2+ 1 ∗ ∗ ≤ C tr B (F F ) 2 (w)B (G G) 2 (w) (1 − |w|2 )2 1 1 × (P0 uδCn )(w) δ × (P0 vδCn )(w) δ where B is the Berezin transform and C is a constant that is possibly diﬀerent from line to line. Now let us use the estimates from theorems 2.2 and 2.7 in the inner product formula. Taking our inner product formula

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TF ∗ (u), TG∗ (v)L2 (Cn ) = a

D

TF ∗ (u), TG∗ (v)Cn dA(z)

(1 − |z|2 )2 TF ∗ (u), TG∗ (v)Cn dA(z) 1 + (1 − |z|2 )2 TF ∗ (u), TG ∗ (v)Cn dA(z) 2 D 1 + (1 − |z|2 )3 TF ∗ (u), TG ∗ (v)Cn dA(z), 3 D

=3

D

let us take the term 12 D (1 − |z|2 )2 TF ∗ (u), TG ∗ (v)Cn dA(z) and estimate its modulus; 1 (1 − |z|2 )2 TF ∗ (u), TG ∗ (v)Cn dA(z) 2 D % 1 % % & % & && 2+ 2+ 2+ 1 (1 − |w|2 )2 C tr B (F ∗ F ) 2 (w)B (G∗ G) 2 (w) ≤ 2 D (1 − |w|2 )2 1 1 × P0 uδCn (w) δ P0 vδCn (w) δ dA(w)| 1 % % % & % & && 2+ 2+ 2+ 1 ≤ sup C tr B (F ∗ F ) 2 (w)B (G∗ G) 2 (w) 2 w∈D 1 1 P0 uδCn (w) δ P0 vδCn (w) δ dA(w). × D

By Cauchy-Schwarz, this expression will be less than or equal to 1 % % % & % & && 2+ 2+ 2+ 1 sup C tr B (F ∗ F ) 2 (w)B (G∗ G) 2 (w) 2 w∈D 12 12 2 2 δ δ δ δ P0 uCn (w) dA(w) P0 vCn (w) dA(w) × .

D

D

Now, as the operator P0 is Lp bounded for p > 1, [6], this expression will be less than or equal to half the supremum over all w ∈ D of %

1 & % & && 2+ % % 2+ 2+ C tr B (F ∗ F ) 2 (w)B (G∗ G) 2 (w) )uL2 (Cn ) vL2 (Cn ) .

Estimating the term 1 (1 − |z|2 )3 TF ∗ (u), TG ∗ (v)Cn dA(z) 3 D from the inner product formula is similar. Finally, let us estimate 3 D (1 − |z|2 )2 TF ∗ (u), TG∗ (v)Cn dA(z). We can see from 2.2 that

376

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2 2 (1 − |z| ) TF ∗ (u), TG∗ (v)Cn dA(z) D (1 − |w|2 )2 Gk ⊗ F k u, v dA(z)dA(w) = n w w C D (1 − |w|2 )2 D ≤ Gkw ⊗ F kw op dA(w)uL2a Cn vL2a Cn D 1

≤ sup (tr B(G∗ G)(w)B(F ∗ F )(w)) 2 uL2aCn vL2a Cn . w∈D

Now we just use H¨older’s inequality to get an expression similar to the one in the previous estimate. tr(B(G∗ G)(w)B(F ∗ F )(w)) ∗ 2 = tr( G(x) G(x)|kw (x)| dA(x) F (z)∗ F (z)|kw (z)|2 dA(z)) D D ∗ 2 = tr(G(x) G(x)|kw (x)| F (z)∗ F (z)|kw (z)|2 )dA(x)dA(z) D D = tr {G(x)∗ G(x)F (z)∗ F (z)} |kw (z)|2 |kw (x)|2 dA(x)dA(z) D

D

≤

D

D

∗

∗

(tr {G(x) G(x)F (z) F (z)})

2+ 2

2 2+ |kw (z)| |kw (x)| dA(x)dA(z)

2

2

This is then less than or equal to % # $& 2+ 2+ C tr (G(x)∗ G(x)) 2 (F (z)∗ F (z)) 2 D

D

2 $ 2+ × |kw (z)|2 |kw (x)|2 dA(x)dA(z)

by Theorem IX.2.10 on page 258 of [1] and similar steps as before. This ﬁnal expression is then equal to 2+ 2+ ∗ ∗ 2 2 ((G(x) G(x)) (F (z) F (z)) tr D

D

2 $ 2+ × |kw (z)|2 |kw (x)|2 dA(x)dA(z)) 2 # # $ # $ $ 2+ 2+ 2+ = tr(B (G(x)∗ G(x)) 2 (w)B (F (z)∗ F (z)) 2 (w)) .

Note that here we can use the same reasoning if 2+ 1 (C tr(G(z)F (x)∗ F (x)G(z)∗ )) 2 |kw (z)|2 dA(z) |kw (x)|2 dA(x) 2+ D

D

is uniformly bounded for some > 0; then our Toeplitz product TF TG∗ will be bounded. This condition is seemingly stronger and less aesthetic than the other one but it will be used later on when dealing with Toeplitz products that

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are also invertible. Note that these last inequalities show that the suﬃcient condition is stronger than the necessary condition. 2.2. A Necessary Condition Proof of Theorem 1.3. In [9] (see also [13] for a diﬀerent approach) Park shows that for functions f and g in the scalar Bergman space L2a the operator f ⊗ g deﬁned by f ⊗ gh = h, g f with h ∈ L2a is equal to Tf Tg − 2Tz Tf Tg Tz + Tz2 Tf Tg Tz2 . Using this result in the vector valued case, we can see that ⎞ ⎛ f1i ⊗ g1i f1i ⊗ g2i . . . ⎟ ⎜ .. ⎟ . F ⊗G=⎜ ⎠ ⎝ f2i ⊗ g1i .. . = TF TG∗ − 2Tz TF TG∗ Tz + Tz2 TF TG∗ Tz2 . Let us estimate the norm of the operator (F ◦ φw ) ⊗ (G ◦ φw ), where F ◦ φw is the matrix-valued function ⎛ ⎞ f11 ◦ φw f12 ◦ φw . . . ⎜ ⎟ .. ⎜f21 ◦ φw ⎟. . ⎝ ⎠ .. . Noting that the operator (F ◦ φw ) ⊗ (G ◦ φw ) is of ﬁnite rank we can take as an equivalent norm the square root of the trace of the operator (F ◦ φw ⊗ G ◦ φw )(G ◦ φw ⊗ F ◦ φw ) by Lemma 2.3. By the same Lemma we can see that this will be equal to

n n n n fqr ◦ φw , fql ◦ φw L2 gml ◦ φw , gmr ◦ φw L2 q=1 m=1

=

r=1 l=1

n n

n n

q=1 m=1

r=1 l=1

B(fqr fql )(w)B(gml gmr )(w) ,

which is equal to the trace of the matrix B(F ∗ F )(w)B(G∗ G)(w). Let Uw be the unitary operator on our vector valued L2 space given by Uw f = (f ◦ φw )kw . It is well known that TF ◦φw Uw = Uw TF . Hence TF ◦φw = Uw TF Uw∗ , and thus 1

{tr (B(F ∗ F )(w)B(G∗ G)(w))} 2 = CF ◦ φw ⊗ G ◦ φw op = TF ◦φw TG∗ ◦φw − 2Tz TF ◦φw TG∗ ◦φw Tz + Tz2 TF ◦φw TG∗ ◦φw Tz2 op = Uw TF Uw∗ Uw TG∗ Uw∗ − 2Tz Uw TF Uw∗ Uw TG∗ Uw∗ Tz + Tz2 Uw TF Uw∗ Uw TG∗ Uw∗ Tz2 op = (Uw TF TG∗ Uw∗ − 2Tz Uw TF TG∗ Uw∗ Tz + Tz2 Uw TF TG∗ Uw∗ Tz2 )op = (Uw TF TG∗ Uw∗ − 2Uw Tφw TF TG∗ Tφ Uw∗ + Uw Tφ2w TF TG∗ Tφ2 Uw∗ )op . w

w

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We can now use the triangle inequality on the operator Uw (TF TG∗ − 2Tφw TF TG∗ Tφw + Tφ2w TF TG∗ Tφ2 )Uw∗ as in [13] to get our result, w using that Tφw ≤ 1. In the following we will be working with square matrices F and G with entries from the scalar valued Bergman space L2a (D). Where it is not explicitly stated otherwise, this will be the case. When we refer to a matrix being less than another matrix, F < G, we mean in the sense of L¨owner partial ordering of matrices. See [1], [19] and [20] for details on this.

3. Bounded and Invertible Toeplitz Products 3.1. A reverse H¨ older inequality We will now develop some of the theory needed to show a reverse H¨older inequality used to characterize the matrices of analytic functions, F and G, such that the Toeplitz product TF TG∗ is bounded and invertible on the vector valued Bergman space. Compare this next lemma with Lemma 4.3 in [17]. Lemma 3.1. If F (z) is invertible for all z ∈ D, then (F ∗ (w)F (w)) ≤ B(F ∗ F )(w), (F ∗ (z)F (z)) ≤ ηs B(F ∗ F )(w) and (F −1 (w)F ∗−1 (w)) ≤ B(F −1 F ∗−1 )(w) (F −1 (z)F −1∗ (z)) ≤ ηs B(F −1 F ∗−1 )(w), when z ∈ D(w, s) the pseudohyperbolic disk with radius 0 < s < 1 and centre w and ηs is a constant dependent only on s. Proof. Let e be an arbitrary vector. Then for F ∈ L2a (Cn ), F (u)e, F (u)e = F (z)Ku (z)dA(z)e, F (z)Ku (z)dA(z)e ' '2 ' ' ' = ' F (z)Ku (z)dA(z)e' ' n C 2 ∗ ≤ F e, F e dA(z)Ku L2 = F (z)F (z)dA(z)e, e Ku 2L2 . So if u ∈ D(0, s), then ∗ ∗ 2 F (u)F (u) ≤ F (z)F (z)dA(z)Ku 2 ≤ F ∗ (x)F (x)dA(z)

1 . (1 − s2 )2

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If z ∈ D(w, s), then z = φw (u) for some u ∈ D(0, s). Thus F ∗ (z)F (z) = F ∗ (φw (u))F (φw (u)) ≤ F ∗ (φw (x))F (φw (x))dA(z) = B(F ∗ F )(w)

1 (1 − s2 )2

1 . (1 − s2 )2

Now let us show that F −1 (w)F ∗−1 (w) ≤ B(F −1 F ∗−1 )(w) : F ∗−1 (w)e, F ∗−1 (w)e = F ∗−1 (φw (0))e, F ∗−1 (φw (0))e −1 ∗−1 (φw (z))dA(z)e, e = F (φw (z))dA(z) F

and we arrive at the conclusion that F −1 (w)F ∗−1 (w) ≤ B(F −1 F ∗−1 )(w) in a similar manner as before. So for z ∈ D(w, s) we know that F −1 (w)F ∗−1 (w) ≤ B(F −1 F ∗−1 )(w) 1 and F ∗ (z)F (z) ≤ B(F ∗ F )(w) (1−s 2 )2 . The other inequalities follow from ap plying the same procedure to F −1 F ∗−1 instead of F ∗ F. Lemma 3.2. If there exists η such that F (z)G(z)∗ G(z)F (z)∗ > ηI for all z ∈ D and tr(B(G∗ G)(w)B(F ∗ F )(w)) is uniformly bounded on D, then 1

1

B(F −1 (F ∗ )−1 )(w)) 2 B(F ∗ F )(w) 2 is uniformly bounded on D. Proof. Let us suppose that F (w)G(w)∗ G(w)F (w)∗ > ηI for all w ∈ D. Then B(G∗ G)(w) ≥ G(w)∗ G(w) ≥ η(F (w)∗ F (w))−1 . The key inequality here is G(w)∗ G(w) ≥ η(F (w)∗ F (w))−1 , as this implies that B(G∗ G)(w) ≥ ηB((F ∗ F )−1 )(w) and so 1

1

(B(F ∗ F )(w)) 2 B(G∗ G)(w)(B(F ∗ F )(w)) 2 1

1

≥ η(B(F ∗ F )(w)) 2 B((F ∗ F )−1 (w))(B(F ∗ F )(w)) 2 . 1

1

Thus, as (B(F ∗ F )(w)) 2 B(G∗ G)(w)(B(F ∗ F )(w)) 2 < M for all w, tr(B(F −1 (F ∗ )−1 )(w)B(F ∗ F )(w)) 1 1 1 ≤ C(B(F ∗ F )(w)) 2 B(G∗ G)(w)(B(F ∗ F )(w)) 2 < ηCM η and so 1

1

B(F −1 (F ∗ )−1 )(w)) 2 B(F ∗ F )(w) 2 2 1

1

= (B(F −1 (F ∗ )−1 )(w)) 2 B(F ∗ F )(w)(B(F −1 (F ∗ )−1 )(w)) 2 ≤ C tr(B(F −1 (F ∗ )−1 )(w)B(F ∗ F )(w)) 1 1 1 ≤ C(B(F ∗ F )(w)) 2 B(G∗ G)(w)(B(F ∗ F )(w)) 2 , η where C and η are constants independent of w.

380

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IEOT

Deﬁnition 3.3. A dyadic rectangle Qj,k,l is a subset of the unit disk of the form z = reiθ : (k − 1)2−j ≤ r ≤ k2−j , (l − 1)21−j π ≤ θ ≤ l21−j π , where j, k, l are non negative integers and k, l ≤ 2j .

Figure 1. Two nested dyadic rectangles in the unit disk. Lemma 3.4. There exists 0 < r < 1 such that for all dyadic rectangles Q with positive distance to the boundary Q ⊂ D(zQ , r). Here, D is the pseudohyperbolic disk and zQ is the centre of the dyadic rectangle Q.

Proof. This is just Proposition 4.4 in [17]. Compare this next lemma with Lemma 4.5 in [17]. Lemma 3.5. If 1

1

sup B(F −1 (F ∗ )−1 )(w)) 2 B(F ∗ F )(w) 2 < ∞, w∈D

then

sup

Q:dyadic

1 |Q|

12 12 1 −1 ∗−1 (F F )dA(z) (F F )dA(z) < ∞. |Q| Q Q

∗

Proof. If the dyadic rectangle Q is the whole disk, then as F ∗ F dA(z) = B(F ∗ F )(0) D

and

D

F −1 F ∗−1 dA(z) = B(F −1 F ∗−1 )(0),

Vol. 66 (2010) we see that

Toeplitz Products on L2a(Cn)

381

' 12 12 ' ' ' ' ' ∗ −1 ∗−1 F F dA(z) F F dA(z) ' ' ' D ' D 1

1

= B(F −1 (F ∗ )−1 )(0) 2 B(F ∗ F )(0) 2 . Now let us suppose that our dyadic rectangle Q has a positive distance from the boundary. By Lemma 3.4 our rectangle Q will be strictly contained in a pseudohyperbolic disk D(zQ , R), zQ being the centre of our dyadic rectangle and R being the same for each dyadic rectangle. Thus by Lemma 3.1 (F −1 (z)F −1∗ (z)) ≤ ηB(F −1 F ∗−1 )(zQ ) and (F ∗ (z)F (z)) ≤ ηB(F ∗ F )(zQ ) for all z in our pseudohyperbolic disk D(zQ , R). Here the constant η will only be dependent on R which is the same for all of these dyadic rectangles. Thus, using the fact that if A, B and C are positive matrices such that 1 1 1 1 1 1 1 1 A ≤ B implies C 2 AC 2 < C 2 BC 2 tr(C 2 AC 2 ) < tr(C 2 BC 2 ), we can deduce the following series of inequalities from our hypothesis: ' 12 12 ' ' 1 '2 1 ' ' ∗ −1 ∗−1 (F F )dA(z) (F F )dA(z) ' ' ' |Q| Q ' |Q| Q ' 1 ' 1 2 ' =' (F ∗ F )dA(z) ' |Q| Q 12 ' ' 1 1 ' −1 ∗−1 ∗ × (F F )dA(z) (F F )dA(z) ' ' |Q| Q |Q| Q

12 1 ≤ C tr (F ∗ F )dA(z) |Q| Q 12 1 1 ∗ −1 ∗ × (F F ) dA(z) (F F )dA(z) |Q| Q |Q| Q

12 1 ∗ (F F )dA(z) ≤ C tr |Q| Q 12 1 × ηB(F −1 F ∗−1 )(zQ ) (F ∗ F )dA(z) |Q| Q % 1 = C tr η(B(F −1 F ∗−1 )(zQ )) 2 1& 1 × (F ∗ F )dA(z) η(B(F −1 F ∗−1 )(zQ )) 2 |Q| Q % 1 1& ≤ C tr η(B(F −1 F ∗−1 )(zQ )) 2 {ηB(F ∗ F )(zQ )} (ηB(F −1 F ∗−1 )(zQ )) 2

382

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'% 1 ' ≤ C ' η(B(F −1 F ∗−1 )(zQ )) 2 {ηB(F ∗ F )(zQ )} η(B(F −1 F ∗−1 )(zQ )) ' ' 1 1 '2 ' ≤ C 2 η 4 ' (B(F −1 F ∗−1 )(zQ )) 2 {B(F ∗ F )(zQ )} 2 ' < M.

&' ' '

1 2

Note that C is a constant that possibly changes from line to line and is dependent on the dimension of Cn only. M will be dependent only on the uniform bound of B(F −1 F ∗−1 )(w)B(F ∗ F )(w), the dimension we are working in and the constant R which is the same for each dyadic rectangle not touching the boundary. What happens when we have a dyadic rectangle that touches the boundary but is not the whole disk? We can see that the centre of the rectangle zQ is at a distance of at least 1/2 from the centre, i.e. |zQ | ≥ 1/2. Then F ∗ (z)F (z)|kzQ |2 (z)dA(z) B(F ∗ F )(zQ ) = D c ∗ 2 ≥ F (z)F (z)|kzQ (z)| dA(z) ≥ F ∗ (z)F (z)dA(z) (1 − |zQ |)2 Q Q by Lemma 4.2 in [17]. We can also see in this case that |Q| = 8|zQ |(1 − |zQ |)2 and so 4c B(F ∗ F )(zQ ) ≥ F ∗ (z)F (z)dA(z). |Q| Q We can do the same for F −1 F ∗−1 to get that 4c −1 ∗−1 )(zQ ) ≥ F −1 (z)F ∗−1 (z)dA(z). B(F F |Q| Q We can then combine these and take the trace to see that

12 12 1 ∗ ∗ −1 ∗ tr F F (z)dA(z) (F F ) (z)dA(z) F F (z)dA(z) |Q|2 Q Q Q % 1 1 F ∗ (z)F (z)dA(z) ≤ 4c tr B(F −1 F ∗−1 )(zQ ) 2 |Q| Q 1& × B(F −1 F ∗−1 )(zQ ) 2 % 1 1& ≤ 16c2 tr B(F −1 F ∗−1 )(zQ ) 2 B(F ∗ F )(zQ ) B(F −1 F ∗−1 )(zQ ) 2 1 1 1 ≤ C B(F −1 F ∗−1 )(zQ ) 2 {B(F ∗ F )(zQ )} 2 ) 2 < M , where M is independent of Q.

If for all dyadic rectangles Q and some constant M ' 12 12 ' ' ' 1 1 ' ' ∗ −1 ∗−1 (F F )dA(z) (F F )dA(z) ' < M, ' ' ' |Q| Q |Q| Q we will say that F ∗ F has the matrix A2 condition. See [18] for a similar notion of matrix weights. We will now ﬁnd a characterization of such functions F

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383

in terms of the boundedness of certain averaging operators on the function space L2 (F ∗ F ). ∗ Theorem 3.6. If for F ∈ Mn×n (L2a ) the matrix F F has the A2 condition then 1 the averaging operators, f → χQ |Q| Q f (z)dA(z), are uniformly bounded on a dense subset L2 (Cn ) ∩ L2 (F ∗ F ), Q varying over all dyadic rectangles.

The proof here and of the next theorem follow the reasoning in Lemma 2.1 in [18]. Proof. Let R be the subspace 1 n . R = χQ 1 e : e ∈ C |Q| 2 We can see that the orthogonal projection from L2 (D, Cn ) onto R is given by 1 PQ : f → χQ f (z)dA(z). |Q| Q So we want to show that these projections are uniformly bounded with respect to the L2 (F ∗ F ) norm. Clearly, PQ f L2 (F ∗ F ) sup PQ L2 (F ∗ F ) = . f L2 (F ∗ F ) {f ∈L2 ∩L2 (F ∗ F ):f L2 (F ∗ F ) =0} If we let S denote the orthogonal complement of R in L2 , then f = f1 + f2 , where f1 ∈ R and f2 ∈ S = S ∩ L2 (F ∗ F ). Thus the expression for the norm of the projection will become f1 L2 (F ∗ F ) sup {f1 +f2 ∈L2 ∩L2 (F ∗ F ):f L2 (F ∗ F ) =0} f1 + f2 L2 (F ∗ F ) ⎧ ⎫ χQ 1 1 eL2 (F ∗ F ) ⎨ ⎬ |Q| 2 = sup 1 1 e + f2 L2 (F ∗ F ) ⎭ {e∈Cn : e=0} ⎩ inf {f2 ∈S} χQ |Q| 2 ⎧ ⎫ ⎪ ⎪ ⎪ ⎪ χQ 1 1 eL2 (F ∗ F ) ⎨ ⎬ |Q| 2 . = sup ⎪ {e∈Cn : e=0} ⎪ ⎪ ⎩ distL2 (F ∗ F ) χQ 1 1 e, S ⎪ ⎭ So let us take a look at distL2 (F ∗ F ) χQ distL2 (F ∗ F ) χQ =

sup

1

|Q| 2

1

e, S . 1

|Q| 2

1 1 1 ∗ 2 S e, (F F ) = distL2 (F ∗ F ) 2 χQ 1 |Q| 2 1 1 , (F ∗ F ) 2 χQ 1 e, h |Q| 2

1 e, S

% &⊥ 1 :h=1 h∈ (F ∗ F ) 2 S

|Q| 2

384

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(F ∗ F )−1 exists as we have the A2 condition. Note that % &⊥ % & 1 1 (F ∗ F ) 2 S = (F ∗ F )− 2 R .

Then we can see that 1 1 1 ∗ 2 distL2 (F ∗ F ) χQ =# % sup & (F F ) χQ 1 e, S 1 e, h $ 1 |Q| 2 |Q| 2 h∈ (F ∗ F )− 2 R :h=1 1 1 1 ∗ ∗ − 12 2 =! sup χQ (F F ) χQ 1 e, (F F ) 1 g " |Q| 2 |Q| 2 −1 1 g∈Cn :(F ∗ F )

2

χQ

1 |Q| 2

g≤1

1 1 1 ∗ ∗ − 12 2 = sup χQ (F F ) χQ 1 e, (F F ) 1 g 1 |Q| 2 |Q| 2 (F ∗ F )−1 g,g ≤1} {g∈Cn : |Q| Q 1 1 = sup e, χ g χ Q Q 1 1 1 |Q| 2 |Q| 2 (F ∗ F )−1 g,g ≤1} {g∈Cn : |Q| Q / − 12 0 1 −1 ∗−1 = sup F F χQ h e, |Q| {h∈Cn :h≤1} D ' − 12 ' ' ' 1 ' ' −1 ∗−1 F F e' . =' ' ' |Q| Q

Let us now put this equivalent expression for the distance back into our expression for the norm of the projection in L2 (F ∗ F );

PQ L2 (F ∗ F )→L2 (F ∗ F )

⎧ ⎪ ⎪ ⎪ ⎨

' ' 'χQ '

⎧ ⎪ ⎪ ⎪ ⎨

' ' 'χQ '

' ⎫ ' ⎪ ' ⎪ ⎪ 1 e ⎬ |Q| 2 'L2 (F ∗ F ) = sup ⎪ {e∈Cn :e=0} ⎪ 1 ⎪ ⎪ ⎪ ⎭ ⎩ distL2 (F ∗ F ) χQ 1 e, S ⎪ 1

' ' 1 ' e 1 |Q| 2 '

|Q| 2

2

∗

L (F F ) '# $− 12 ' 1 {e∈Cn :e=0} ⎪ ⎪ −1 ∗−1 ⎪ ⎩' ' |Q| Q F F ⎧ ' $ 12 ' '# 1 ' ⎪ ∗ ' ⎪ e' ⎨ ' |Q| Q F F ' '# = sup 1 $ − ' 1 2 {e∈Cn :e=0} ⎪ ⎪ −1 F ∗−1 ⎩' F ' |Q| Q

=

sup

⎫ ⎪ ⎪ ⎪ ⎬

' '⎪ ⎪ e' ⎭ '⎪ ⎫ ⎪ ⎪ ⎬ ' '⎪ ⎪ e' '⎭

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385

'# $ 12 ' ' 1 ' ⎫ ⎪ ∗ ' e' ⎬ ' |Q| Q F F ' ⎪ '# ' = sup 1 $− 2 ' ⎪ ' 1 {e∈Cn :e=0} ⎪ ⎪ ⎪ −1 ∗−1 ⎩' e' '⎭ ' |Q| Q F F ' 12 12 ' ' 1 ' 1 ' ' ∗ −1 ∗−1 =' F F F F '. ' |Q| Q ' |Q| Q ⎧ ⎪ ⎪ ⎨

1 Lemma 3.7. If the averaging operators g → χQ |Q| g(z)dA(z) are uniformly Q 2 2 bounded on L (|f | ) over dyadic rectangles Q, then |f |2 has the scalar A2 condition. 1 Proof. Again we can see that the averaging operator g → χQ |Q| g(z)dA(z) Q is the projection P : L2 → χQ 1 1 C. We are working as before on the dense |Q| 2

subset L2 (C) ∩ L2 (|f |2 ). If we assume that |f1|2 is bounded then we can show as before that 1 2− 12 1 2− 12 1 1 1 1 1 distL2 (|f |2 ) (χQ χQ z = χQ . 1 z, S ) = D |f |2 D |f |2 |Q| |Q| |Q| 2 where |z| = 1. So if we drop this assumption on for > 0, then we can see that distL2 (|f |2 ) (χQ

1 1

|Q| 2

1 |f |2

but instead use

1 1 z, S ) |Q| 2 1 2− 12 1 1 = lim , →0 |Q| Q |f |2 +

z, S ) = lim distL2 (|f |2 +) (χQ →0

where S is the intersection of the orthogonal complement of χQ 2

1 |f |2 +

2

1

1

|Q| 2

C with

L (|f | ) and |z| = 1. As the norm of our bounded projection P is ' ' ⎫ ⎧ ' ' ⎪ ⎪ ' χQ 1 1 z ' ⎪ ⎪ ⎪ ⎪ ' |Q| 2 ' 2 2 ⎨ ⎬ L (|f | ) sup ⎪ z∈C:z=0,|z|=1 ⎪ ⎪ ⎪ 1 ⎪ ⎭ ⎩ distL2 (|f |2 ) χQ 1 z, S ⎪ |Q| 2

' ' ⎫ ' ' ⎪ ' χQ 1 1 ' ⎪ ⎪ ' |Q| 2 ' 2 2 ⎬ L (|f | ) , = sup ⎪ z∈C:z=0,|z|=1 ⎪ ⎪ ⎪ 1 ⎪ ⎭ ⎩ distL2 (|f |2 ) χQ 1 z, S ⎪ ⎧ ⎪ ⎪ ⎪ ⎨

|Q| 2

we know that distL2 (|f |2 ) χQ

1

1 z, S

|Q| 2

is nonzero for nonzero z and hence

1 2 1 1 <∞ lim 2 →0 |Q| Q |f | +

386

Robert Kerr

IEOT

and so by the Monotone Convergence Theorem 1 2 1 1 <∞ |Q| |f |2 Q

and

1 2− 12 1 1 1 distL2 (|f |2 ) χQ = , 1 z, S 2 |Q| |f | 2 |Q| Q

where |z| = 1. Thus

P L2 (|f |2 )

' ' 'χQ '

' ⎫ ' ⎪ ' ⎪ ⎪ 1 z ⎬ |Q| 2 'L2 (|f |2 ) = sup ⎪ z∈C:z=0,|z|=1 ⎪ ⎪ 1 ⎪ ⎪ ⎩ distL2 (|f |2 ) χQ 1 z, S ⎪ ⎭ ⎧ ⎪ ⎪ ⎪ ⎨

1

|Q| 2

3 4 1 ⎫ 1 2 2 ⎪ |f | ⎬ |Q| Q ⎪ 3 = sup 1 4 1 1 −2 ⎪ z∈C:z=0,|z|=1 ⎪ ⎪ ⎪ ⎩ |Q| Q |f |2 ⎭ !1 2 12 1 2 1 " 1 1 2 1 2 |f | = , |Q| Q |f |2 |Q| Q ⎧ ⎪ ⎪ ⎨

which is uniformly bounded as required.

Compare the next lemma with Lemma 3.6 in [18]. Lemma 3.8. If F ∗ F has the A2 condition, then tr(F ∗ F ) has the scalar A2 condition. Proof. We will show that each element on the diagonal of F ∗ F has the scalar A2 condition. We can then deduce that the sum of these will also have the A2 condition. Firstly we know that if F ∗ F has the A2 condition, then the 1 2 ∗ operators f → χQ |Q| Q f (z)dA(z) are uniformly bounded on L (F F ), f ∈ L2 (Cn ). So if we take g ∈ L2 (D) ∩ L2 (D F ∗ F (0, . . . , 1, . . . , 0), (0, . . . , 1, . . . , 0)), where F ∗ F (0, . . . , 1, . . . , 0), (0, . . . , 1, . . . , 0) is the scalar valued function z → F ∗ (z)F (z)(0, . . . , 1, . . . , 0), (0, . . . , 1, . . . , 0), then 1 g(z)(0, . . . , 1, . . . , 0)dA(z) g(0, . . . , 1, . . . , 0) → χQ |Q| Q is uniformly bounded between L2a (D, Cn ). This implies that 1 g(z)dA(z) g → χQ |Q| Q is uniformly bounded with respect to the scalar measure F ∗ F (0, . . . , 1, . . . , 0), (0, . . . , 1, . . . , 0)Cn ,

Toeplitz Products on L2a(Cn)

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387

which will be whatever diagonal element of F ∗ F we want. Thus by the previous lemma the trace of F ∗ F will have the scalar A2 condition. Compare this next lemma with Lemma 4.6 in [17], Lemma 2.5 in [14] and also 1.7 on page 196 of [12]. Lemma 3.9. If a scalar valued function |f |2 has the A2 condition and for some 0 < δ < 1 then for each dyadic rectangle Q and E ⊂ Q such that |E| ≤ δ|Q| we have that µ(E) ≤ λµ(Q) for some 0 < λ < 1 where dµ = |f |2 dA and λ only depends on δ and the A2 constant of |f |2 . Proof. 2

!

"2 −1

|f f |

|Q/E| =

dA

! ≤

Q/E

" !

2

" −2

|f | dA Q/E

!

"

2

≤

|f |

|f | dA Q/E

!

"

−2

|f |

dA

Q

2

|f | dA C|Q|2

≤

dA

Q/E

Q/E

−1 |f |2 dA .

Q

By our A2 condition on |f |2 this equals −1 |f |2 dA − |f |2 dA C|Q|2 |f |2 dA Q E Q

=C

|f |2 dA

1− E

so we know that

−1

|Q|2 ,

Q

µ(E) |Q/E| ≤ C|Q| 1 − µ(Q) 2

and thus

2

|Q/E|2 µ(Q/E) . ≤C |Q|2 µ(Q)

Now we know that |Q/E| |Q|

|f |2 dA

|E| |Q|

≤ δ < 1 from our hypothesis, which implies that

≥ 1 − δ > 0. So we can now deduce that 0<

1 |Q/E|2 (1 − δ)2 µ(Q/E) ≤ . ≤ 2 C C |Q| µ(Q)

This lets us now see that µ(Q/E) + µ(E) µ(E) (1 − δ)2 µ(Q) = ≥ + 1= µ(Q) µ(Q) µ(Q) C and hence

µ(E) (1 − δ)2 ≤1− . µ(Q) C

388

Robert Kerr

IEOT

The following lemma will be crucial to our application of the A2 condition. Lemma 3.10. If F ∗ F has the A2 condition and J is a strictly positive matrix, then JF ∗ F J will have the A2 condition. The A2 constant of JF ∗ F J will depend on the A2 bound of F ∗ F and the dimension only. Proof. ' 12 12 ' '2 ' 1 1 ' ' ∗ ∗ −1 JF F J (JF F J) ' ' ' ' |I| I |I| I ' 1 12 ' ' ' 1 2 1 1 ' ' ∗ ∗ −1 ∗ =' JF F J (JF F J) JF F J ' ' ' |I| I |I| I |I| I

12 2 1 1 1 ≤ C tr JF ∗ F J (JF ∗ F J)−1 JF ∗ F J |I| I |I| I |I| I 1 1 (F ∗ F )−1 F ∗F = C tr |I| I |I| I 1 ' 1 12 ' ' 1 '2 2 1 ' ' ≤ C ' F ∗F (F ∗ F )−1 ' . ' |I| I ' |I| I C and C depend only on the dimension. Thus the result follows.

Deﬁnition 3.11. The dyadic maximal operator M∆ is deﬁned by 1 |f (z)|dA(z), (M∆ f )(w) = sup w∈Q |Q| Q where the Q are dyadic rectangles and f ∈ L2 . Theorem 3.12 (The Calderon-Zygmund Decomposition Theorem). Let f ∈ L1 (D). If we have t > 0 such that the set Λ = {z ∈ D : M∆ f (z) > t} is not the whole of D, then we can decompose Λ into a disjoint union of dyadic intervals Qi such that t < |Q1i | Qi |f (z)|dA(z) < 8t. Proof. The proof of this is exactly as in [14] and [17].

Compare this next lemma with Proposition 4.11 in [17]. Lemma 3.13. The trace of F ∗ F satisies the following: 1. tr(F ∗ F ) ≤ M∆ tr(F ∗ F ) on D; 42 ∗ ∗ tr(F (z)F (z))dA(z) ≤ M∆ tr(F F )(0) ≤ tr(F ∗ (z)F (z))dA(z). 2. 3 D D Proof. 1. This follows from Proposition 4.11 in [17]. We just need to note that tr(F ∗ F ) is continuous and the proof works as it is.

Vol. 66 (2010)

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389

2. D is a dyadic rectangle containing 0 so 1 M∆ tr(F ∗ F )(0) ≥ tr(F ∗ (z)F (z))dA(z) = tr(F ∗ (z)F (z))dA(z). |D| D D Let us take a dyadic rectangle Q containing 0 which is not the unit disk. We know that Q will be contained in the pseudohyperbolic disk D(0, 12 ). Let e ∈ Cn , then as F ∈ L2a (Cn ), 2 F (u)e, F (u)e = F (u)eCn = F (z)Ku (z)dA(z)e2Cn ≤

2 F (z)e|Ku (z)|dA(z)

F e, F e dA(z)Ku (z)2L2 ∗ = F (z)F (z)dA(z)e, e Ku (z)2L2 . ≤

So F ∗ (u)F (u) ≤ ≤ =

F ∗ (z)F (z)dA(z)Ku 22 F ∗ (x)F (x)dA(x)

1 (1 −

12 2 2 )

2 4 F ∗ (x)F (x)dA(x) 3

on each Q containing 0 which is not D. So

2 4 ∗ ∗ tr(F (u)F (u)) ≤ tr F (x)F (x)dA(x) 3 D for u ∈ Q. Hence 2 4 1 tr(F ∗ (z)F (z))dA(z) ≤ tr(F ∗ (x)F (x))dA(x) |Q| Q 3 D and so M∆ tr(F ∗ F )(0) ≤ (4/3)2

tr(F ∗ (z)F (z))dA(z).

D

The proof of the following theorem follows the lines of of Theorem 2.1 in [14] and Theorem 4.1 in [17]. It contains the key to the proof of Theorem 1.4 i.e. the reverse H¨ older property. Theorem 3.14. If F ∗ F satisﬁes A2 , then there exists > 0 such that (tr(F ∗ (z)F (z)))1+ dA(z) ≤ C (tr(F ∗ (z)F (z)))dA(z)1+ with C and dependent only on the A2 constant.

390

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IEOT

Proof. For each k deﬁne ∗ 4k+1 ∗ Ek = z ∈ D : M∆ (tr(F F ))(z) > 2 (tr(F (z)F (z)))dA(z) . D

By Lemma 3.13 we can see that

M∆ tr(F ∗ (0)F (0)) ≤ (4/3)2 tr(F ∗ (z)F (z))dA(z) D 4k+1 <2 tr(F ∗ (z)F (z))dA(z) D

for all k. So we know that each Ek is not the whole disk (as 0 is not contained in it) and hence we can do a Calderon-Zygmund decomposition. So for each Ek we have a disjoint union of dyadic rectangles Qi whose union is equal to Ek and 1 4k+1 ∗ 2 (tr(F (z)F (z)))dA(z) < tr(F ∗ (z)F (z))dA(z) |Qi | Qi D 4(k+1) (tr(F ∗ (z)F (z)))dA(z). <2 D

Two inequalities we will use from this are; −1 |Qi | < 2−4k−1 (tr(F ∗ (z)F (z)))dA(z) D

and

tr(F ∗ (z)F (z))dA(z)

Qi

tr(F ∗ (z)F (z))dA(z) < |Qi |24(k+1) Qi

(tr(F ∗ (z)F (z)))dA(z).

D

We now take a maximal dyadic rectangle Q in Ek−1 (which is larger than Ek ) and note that |Ek ∩ Q| = |Qi | Qi ⊂Q

<

2

−4k−1

Qi ⊂Q

≤ 2−4k−1

−1 (tr(F (z)F (z)))dA(z) ∗

D

tr(F ∗ (z)F (z))dA(z)

Qi

−1 (tr(F ∗ (z)F (z)))dA(z) tr(F ∗ (z)F (z))dA(z),

D

Q

(where the Qi denote the maximal dyadic rectangles in Ek ) due to the dyadic decomposition of Ek . But as Q is also part of a Calderon-Zygmund decomposition (this time for Ek−1 ) we can also see that Q

tr(F ∗ (z)F (z))dA(z) < |Q|24k

(tr(F ∗ (z)F (z)))dA(z).

D

Putting the last two inequalities together we see that

Toeplitz Products on L2a(Cn)

Vol. 66 (2010) |Ek ∩ Q| <2

−4k−1

D

391

−1 4k (tr(F (z)F (z)))dA(z) |Q|2 (tr(F ∗ (z)F (z)))dA(z) ∗

D

1 = |Q|. 2 We are now in a position to use Lemma 3.9 as tr(F ∗ (z)F (z))) satisﬁes the scalar A2 condition and |Ek ∩ Q| ≤ 12 |Q|. So with 12 being our δ in 3.9, we can deduce that µ(Ek ∩ Q) < λµ(Q) for some 0 < λ < 1 independent of k, with dµ(z) = tr(F ∗ (z)F (z)))dA(z). We can now sum over all maximal dyadic rectangles in Ek−1 and see that µ(Ek ) = µ(Ek ∩ Q) < λ µ(Q) = λµ(Ek−1 ). Q

Q

Let us take a moment here to note that λ depends only on our A2 bound of tr(F ∗ (z)F (z)), (we can see this from Lemma 3.9), and that this A2 bound is controlled by the matrix A2 bound for F ∗ F and the dimension. We have established that for each k ≥ 1, µ(Ek ) < λµ(Ek−1 ) and so k k tr(F ∗ (z)F (z))dA(z) µ(Ek ) < λ µ(E0 ) = λ E0 k ≤λ tr(F ∗ (z)F (z))dA(z). D

Now let us move on and look at D tr(F ∗ (z)F (z))1+ dA(z) for some > 0. From Lemma 3.13 we know that tr(F ∗ F )(z) ≤ M∆ tr(F ∗ F )(z) on the disk, so tr(F ∗ (z)F (z))1+ dA(z) ≤ tr(F ∗ (z)F (z)) {M∆ tr(F ∗ F )(z)} dA(z) D D = tr(F ∗ (z)F (z)) {M∆ tr(F ∗ F )(z)} dA(z) x:M∆ tr(F ∗ F )(x)≤

+

≤

k

D

tr(F ∗ F (z))dA(z)

tr(F ∗ (z)F (z)) {M∆ tr(F ∗ F )(z)} dA(z)

Ek −Ek+1

1+ (4(k+1)+1) ∗ tr(F F )(z) + 2 tr(F F )(z)dA(z) µ(Ek ) ∗

D

k

D

1+ ∗ tr(F F )(z) ≤ D + 2(4(k+1)+1) tr(F ∗ F )(z)dA(z) λk tr(F ∗ F )(z)dA(z)

k

= D

D

D

1+ 1+ tr(F ∗ F )(z) + 2(4(k+1)+1) tr(F ∗ F )(z)dA(z) λk k

D

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Robert Kerr

=

IEOT

1+ tr(F ∗ F )(z) (λ24 )k . 1 + 25

D

k

If we choose such that 0 < λ24 < 1, then this will become 1+ 1 tr(F ∗ F )(z) 1 + 25 . 1 − λ24 D older inequality will hold. Thus, for any 0 < ≤ , our reverse H¨

∗

Corollary 3.15. If F F satisﬁes A2 and J is a positive matrix then there exists > 0 such that (tr(JF ∗ (z)F (z)J))1+ dA(z) ≤ C (tr(JF ∗ (z)F (z)J))dA(z)1+ . The same and constant C hold for all positive matrices J and depends only on the dimension and the A2 constant of F ∗ F .

Proof. This follows from 3.14 and 3.10. 3.2. Proof of Theorem 1.3. Two easy lemmas follow before the proof of the Theorem 1.3.

Lemma 3.16. Let F and G be matrices consisting of Bergman space L2a (D) functions. If F G∗ GF ∗ > ηI and TF TG∗ is bounded, then the Toeplitz product TF TG∗ is invertible. Proof. F ∗ GG∗ F > ηI implies that G∗−1 F −1 F ∗−1 G−1 is bounded and so the operator TG∗−1 F −1 = TG∗−1 TF −1 is bounded. It remains to note that (TF TG∗ )TG∗−1 TF −1 F (kw , 0, 0, . . .) = F (kw , 0, 0, . . .) and TG∗−1 TF −1 (TF TG∗ )(kw , 0, 0, . . .) = (kw , 0, 0, . . .), and that these also hold for (0, . . . , kw , . . .). This implication holds because the linear spans of {F (0, . . . , kw , . . .)} and {(0, . . . , kw , . . .)} form dense subspaces. Lemma 3.17. If the trace of a positive matrix A is less than some constant λ > 0 then A < CI for some constant C > 0 depending only on λ and the dimension, I being the identity matrix.

Proof. Trivial. ∗

Proof of Theorem 1.3. “⇐” From Lemma 3.5 we know that F F satisﬁes our A2 condition. Then by Corollary 3.15, 1 1 (tr(((G∗ G)(x)) 2 (F ∗ F )(z)((G∗ G)(x)) 2 ))1+ dA(z) (3.1) 1 1 ≤ c (tr(((G∗ G)(x)) 2 (F ∗ F )(z)((G∗ G)(x)) 2 ))dA(z)1+ holds for all x ∈ D with some > 0 and a constant c independent of x. Note here that we need to use the fact that G∗ G is strictly positive.

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We can also see that G∗ G satisﬁes our A2 condition, so a similar reverse H¨older inequality will hold; 1 1 &&1+ % % ∗ (F F )(z)dA(z) 2 (G∗ G)(x) (F ∗ F )(z)dA(z) 2 dA(x) tr &1+ % % 1 1& . ≤c tr (F ∗ F )(z)dA(z) 2 (G∗ G)(x) (F ∗ F )(z)dA(z) 2 dA(x) So let us set = min {, } . Thus integrating both sides of the reverse H¨older inequality (3.1) with respect to x, we get 1 1 (tr(((G∗ G)(x)) 2 (F ∗ F )(z)((G∗ G)(x)) 2 ))1+ dA(z)dA(x) $1+ # 1 1 ≤c (tr(((G∗ G)(x)) 2 (F ∗ F )(z)((G∗ G)(x)) 2 ))dA(z) dA(x) && % % 1+ 1 1 dA(x) =c tr ((G∗ G)(x)) 2 (F ∗ F )(z)dA(z)((G∗ G)(x)) 2 % % 61 5 6 1 &&1+ 5 ∗ (F F )(z)dA(z) 2 (G∗ G)(x) (F ∗ F )(z)dA(z) 2 dA(x) =c tr older and so as G∗ G also has the A2 condition, we can use our reverse H¨ inequality again to see that this last expression is less than or equal to # % $1+ 1 1& c tr (F ∗ F )(z)dA(z) 2 (G∗ G)(x) (F ∗ F )(z)dA(z) 2 dA(x) , where as usual c is a constant that possibly changes from line to line. By the M¨ obius invariance of the Berezin transform ([22] page 143) we see that 1 1 (tr(((G∗ G)(x)) 2 (F ∗ F )(z)((G∗ G)(x)) 2 ))1+ |kw (x)|2 |kw (z)|2 dA(z)dA(x) 1

1

≤ c(tr((B(G∗ G)(w)) 2 B(F ∗ F )(w)(B(G∗ G)(w)) 2 ))1+ < cM 1+ . Hence, by Theorem 1.2, we can see that the Toeplitz product TF TG∗ is bounded. The invertibility of this Toeplitz product follows from Lemma 3.16. “⇒” If TF TG∗ is bounded and invertible, we know from Theorem 1.3 that tr(B(F ∗ F )(w)B(G∗ G)(w)) is uniformly bounded and that TF TG∗ is bounded below. Thus, in particular, TF TG∗ kw e, TF TG∗ kw e dA(z) > η kw e, kw e dA(z) = η e, e for all vectors e ∈ Cn . We know that TF TG∗ kw = F (z)G∗ (w)kw (z) and so we deduce that G(w)B(F ∗ F )(w)G∗ (w) > ηI. From the fact that (TF TG∗ )∗ is also bounded below we can see that F (w)B(G∗ G)(w)F ∗ (w) > ηI. From these we deduce the following; B(G∗ G)(w) > ηF −1 (w)F ∗−1 (w) and B(F ∗ F )(w) > ηG−1 G∗−1 (w),

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which lets us see that 1 1 −1 ∗−1 G G (w) 2 B(G∗ G)(w) G−1 G∗−1 (w) 2 1 > η G−1 G∗−1 (w) 2 F −1 (w)F ∗−1 (w) G−1 G∗−1 (w)

1 2

and also 1

1

{B(G∗ G)(w)} 2 B(F ∗ F )(w) {B(G∗ G)(w)} 2 1

1

> η {B(G∗ G)(w)} 2 G−1 (w)G∗−1 (w) {B(G∗ G)(w)} 2 , thus tr(B(G∗ G)(w)B(F ∗ F )(w)) 1

1

= tr({B(G∗ G)(w)} 2 B(F ∗ F )(w) {B(G∗ G)(w)} 2 ) 1

1

> η tr({B(G∗ G)(w)} 2 G−1 G∗−1 {B(G∗ G)(w)} 2 )) 1 1 = η(tr( G−1 G∗−1 (w) 2 B(G∗ G)(w) G−1 G∗−1 (w) 2 )) 1 > η 2 (tr( G−1 G∗−1 (w) 2 F −1 (w)F ∗−1 (w) G−1 G∗−1 (w) ∗

1 2

)).

∗

Thus, as tr(B(G G)(w)B(F F )(w)) is uniformly bounded, tr(G∗−1 (w)F −1 (w)F ∗−1 (w)G−1 (w)) is uniformly bounded, by λ, say, and so G∗−1 (w)F −1 (w)F ∗−1 (w)G−1 (w) < λ I, which gives us that F (w)G∗ (w)G(w)F ∗ (w) > λ1 I. Acknowledgment I wish to thank S. Pott for suggesting this problem and for discussing various ideas relating to this work. I would also like to acknowledge the useful suggestions made by the referee.

References [1] R. Bhatia, Matrix analysis. Springer Verlag, New York, 1997. [2] O. Blasco, Introduction to vector valued Bergman spaces. Function spaces and operator theory 8 (2005), 9–30. [3] J. Duoandikoetxea, Fourier analysis. Translated and revised from the 1995 Spanish original by David Cruz-Uribe. Graduate Studies in Mathematics, 29 American Mathematical Society. [4] P. Duren, A. Schuster, Bergman spaces. Mathematical Surveys and Monographs, 100. American Mathematical Society, 2004. [5] J. B. Garnett, Bounded analytic functions. Revised ﬁrst edition. Graduate Texts in Mathematics, 236 Springer, New York, 2007. [6] H. Hedenmalm, B. Korenblum, K. Zhu, Theory of Bergman spaces. Graduate Texts in Mathematics, 199. Springer-Verlag, New York, 2000. [7] J. Miao, Bounded Toeplitz products on the weighted Bergman spaces of the unit ball. J. Math. Anal. Appl. 346 (2008), no. 1, 305–313. [8] F. Nazarov, A counterexample to Sarason’s conjecture. Preprint

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[9] J. Park, Bounded Toeplitz products on the Bergman space of the unit ball in Cn . Integral Equations Operator Theory 54 (2006), no. 4, 571–584. [10] S. Pott, E. Strouse, Products of Toeplitz operators on the Bergman spaces A2α . Algebra i Analiz 18 (2006), no. 1, 144–161. [11] D. Sarason, Products of Toeplitz operators, in ”Linear and Complex Analysis Problem Book 3,” Part I (V. P. Khavin and N. K. Nikol’skii, Eds.), Lecture Notes Math., 1573 318–319, Springer-Verlag, 1994. [12] E. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press. 1993. [13] K. Stroethoﬀ, D. Zheng, Products of Hankel and Toeplitz operators on the Bergman space, J. Funct. Anal. 169 (1999), 289–313. [14] K. Stroethoﬀ, D. Zheng, Invertible Toeplitz Products. J. Funct. Anal. 195 (2002), 48–70. [15] K. Stroethoﬀ, D. Zheng, Bounded Toeplitz products on the Bergman space of the polydisk. J. Math. Anal. Appl. 278 (2003), no. 1, 125–135. [16] K. Stroethoﬀ, D. Zheng, Bounded Toeplitz products on Bergman spaces of the unit ball. J. Math. Anal. Appl. 325 (2007), no. 1, 114–129. [17] K. Stroethoﬀ, D. Zheng, Bounded Toeplitz Products on Weighted Bergman Spaces. J. Operator Theory 59 (2008), no. 2, 277–308. [18] S. Treil, A. Volberg, Wavelets and the angle between past and future. J. Funct. Anal. 143 (1997), no. 2, 269–308. [19] X. Zhan, Matrix Inequalities. Lecture Notes in Mathematics. Springer, 2002. [20] F. Zhang, Matrix Theory: Basic Results and Techniques. Springer, 1999. [21] D. Zheng, The distribution function inequality and products of Toeplitz operators and Hankel operators. J. Funct. Anal. 138 (1996), no. 2, 477–501. [22] K. Zhu, Operator Theory in Function Spaces. Second edition. Mathematical Surveys and Monographs, 138 American Mathematical Society, 2007. Robert Kerr Department of Mathematics University of Glasgow University Gardens Glasgow G12 8QW UK e-mail: [email protected] Submitted: April 14, 2008. Revised: January 12, 2010.

Integr. Equ. Oper. Theory 66 (2010), 397–424 DOI 10.1007/s00020-010-1752-4 Published online February 16, 2010 © Birkhäuser/Springer Basel AG 2010

Integral Equations and Operator Theory

Abstract Hankel Operators in Kre˘ın Spaces S.A.M. Marcantognini and M.D. Mor´an Abstract. Hankel operators and their symbols, as generalized by V. Pt´ ak and P. Vrbov´ a, are considered in the Kre˘ın space setting. Under a generic assumption, without which the Krein space case may be untreatable, a necessary and suﬃcient condition for the existence of Hankel symbols for a given Hankel operator X is given. A parametric labeling of the Hankel symbols of X by means of Schur class functions is obtained. The proof is established by associating to the data of the problem an isometry V acting on a Kre˘ın space so that there is a bijective correspondence between the symbols of X and the minimal unitary Hilbert space extensions of V . The result includes uniqueness criteria and a Schur like formula. Mathematics Subject Classiﬁcation (2010). Primary: 47B35; Secondary: 47B50, 47A20. Keywords. Hankel operators, Hankel symbols, Kre˘ın spaces.

1. Introduction The operators we deal with are the Kre˘ın space versions of the abstract Hankel operators and symbols introduced by V. Pt´ ak and P. Vrbov´ a in the Hilbert space setting [14, 15, 16]. Their starting point is to consider a pair of contractions T1 on H1 and T2 on H2 , H1 and H2 Hilbert spaces, with corresponding minimal isometric dilations U1 on K1 ⊇ H1 and U2 on K2 ⊇ H2 . With these data an abstract Hankel operator is a bounded linear map X from H1 to H2 satisfying the intertwining relation XT1∗ = T2 X. If K2 is the orthogonal projection form K2 onto H2 then the symbols of X PH 2 are the bounded linear operators Z from K1 to K2 which are solutions of the commutant lifting problem ZU1∗ = U2 Z

K2 and PH Z|H1 = X. 2

(1.1)

The classical Hankel operators and their symbols ﬁt into the abstract scheme considered by Pt´ak and Vrbov´ a. Indeed, if S is the shift operator on the space of the L2 -functions on the unit circle of the complex plane, 2 H 2 is the Hardy space and H− is its orthogonal complement in L2 then, by

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setting T1 := P S ∗ |H 2 and T2 := P− S|H−2 , with P and P− the orthogonal 2 projections form L2 onto H 2 and H− , respectively, we see that a Hankel 2 operator is a bounded linear map X : H 2 → H− such that XT1∗ = T2 X. The celebrated Nehari’s Theorem states that there exists an L∞ -function Φ, a symbol of X, such that Xf = P− Φf for all f ∈ H 2 . Hence the symbols are L∞ -functions with prescribed antianalytic part or, from another point of view, operators that commute with S and have the same ﬁxed component 2 . Since the unitary operators U1 := S ∗ and U2 := S on L2 from H 2 into H− are the corresponding minimal isometric dilations of the contractions T1 and T2 , we can conclude that the symbols of the given Hankel operator X are those linear operators Z : L2 → L2 which are solutions of the commutant lifting problem (1.1) with the above deﬁned data set. Unlike the classical case, in the general situation the commutant lifting problem (1.1) may have no solutions at all. The investigations carried out by Pt´ ak and Vrbov´ a indicate that the problem is solvable whenever X veriﬁes certain boundedness condition that depends on the unitary parts of the WoldVon Neumann decompositions of U1 and U2 . More precisely, for j = 1, 2, let Kj = Rj ⊕Sj be the Wold-von Neumann decomposition of Kj , so that Rj and Sj reduce Uj , Uj |Rj is unitary and Uj |Sj is a unilateral shift with wandering K subspace Lj = Kj Uj Kj , and let PRjj be the orthogonal projection from Kj onto Rj . If X is an abstract Hankel operator for T1 and T2 then X is required to satisfy that there exists β ≥ 0 such that, for all h1 ∈ H1 and h2 ∈ H2 , K1 K2 h1 K1 PR h2 K 2 . |Xh1 , h2 H2 | ≤ β PR 1 2

(1.2)

This boundedness condition is necessary and suﬃcient for the existence of K2 intertwining liftings Z of X (ZU1∗ = U2 Z and PR Z = X) which altogether 2 verify Z = inf β, where β runs over all nonnegative numbers satisfying (1.2) (cfr. [14], [15], [16] and [5]). Since the Wold-Von Neumann decomposition is trivial in the classical case, for S being unitary, the result includes the classical situation. When carrying the notion of abstract Hankel operator over into the Kre˘ın space setting we must take into account that Kre˘ın space isometries present signiﬁcant diﬀerences from their counterparts in the Hilbert space case. For instance, if U is a continuous isometry on a Kre˘ın space K, the subspace R on which U acts as a unitary operator need not satisfy the relation K = R ⊕ R⊥ . A suﬃcient condition for this situation not to occur is the uniform boundedness of the operators U, U 2 , U 3 , · · · . If this is the case then U has a Wold decomposition so that K can be written as the direct sum of two orthogonal subspaces R and S which reduce U such that U |R is unitary and U |S is a unilateral shift. In particular, if U is the minimal isometric dilation of a given continuous contraction T on a Kre˘ın space H, then U has a Wold decomposition when sup T n < ∞. Thus, subject to the assumption that sup T1n , sup T2n < ∞, the theory of Hankel operators on Kre˘ın spaces parallels the Hilbert space theory

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by Pt´ ak and Vorbov´ a. In particular, the necessary and suﬃcient condition (1.2) ﬁnds to some extent its analogue in the Kre˘ın space setting. As counterpart of the Hilbert space case, we consider the problem of describing the symbols Z of any abstract Hankel operator X, for given continuous contractions T1 on a Kre˘ın space H1 and T2 on a Kre˘ın space H2 such that sup T1n , sup T2n < ∞. In dealing with the problem we show that there is a bijective correspondence between the symbols of X and the minimal unitary Hilbert space extensions of a Kre˘ın space isometry V determined by X, T1 and T2 . It turns out that the defect subspaces of the coupling isometry V underlying the data set {X, T1 , T2 } are Hilbert spaces. Since any isometry whose defect subspaces are Hilbert spaces has at least one minimal unitary Hilbert space extension, our approach provides a proof of the existence of symbols for the generalized Hankel operator on hand. The Kre˘ın space extension of the Arov-Grossman functional model yields a complete description of the abstractly indistinguishable minimal unitary Hilbert space extensions of V , as it associates to each minimal unitary Hilbert space extension U of V a function ϑU in a suitable Schur class of operator valued functions, and, to each function ϑ in the Schur class, a model operator Uϑ which gives rise to a minimal unitary Hilbert space extension of V, in such a way that the outlined correspondence is bijective. Then the coupling method combined with the Kre˘ın space extension of the Arov-Grossman model gives in return a bijective correspondence between the symbols of X and the Schur class. We show that the connection between the symbols and the Schur functions can be realized as a parametric description. We also include uniqueness criteria and a Schur like formula (see formula (4.4) in the proof of Theorem 4.1). In the Hilbert space case the methods were developed in [5]. The references given therein complement those listed in the bibliography. The paper is organized in ﬁve sections. Section 1, this section, serves as an introduction. In Section 2 we ﬁx the notation and state some results needed in the rest of the paper. Our main results, along with some comments and remarks, are presented in Sections 3 and 4. We supplement this note with an Appendix containing a sketch of the proof of the Kre˘ın space extension of the Arov-Grossman model.

2. Notation and Preliminaries We follow the standard notation, so N, Z and C are, respectively, the set of natural, integral and complex numbers; D stands for the open unit disk and T for the unit circle, hence, D := {z ∈ C : |z| < 1} and T := ∂D. For any separable Hilbert space H (in the sequel all Hilbert spaces are supposed to be separable) we denote by L2 (H) the class of functions f : T → H which are measurable (strongly or weakly, which comes to be the same due to the separability of H) and such that 2π 1

f (eit ) 2H dt < ∞. 2π 0

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With the pointwise linear operations and the scalar product 2π 1 f, gL2 (H) := f (eit ), g(eit )H dt (f, g ∈ L2 (H)) 2π 0 L2 (H) becomes a (separable) Hilbert space under the interpretation that two functions in L2 (H) are identical if they diﬀer on a set of measure zero. Moreover, L2 (H) = n∈Z Gn (H) where, for each n ∈ Z, Gn (H) is the subspace of those functions f ∈ L2 (H) such that f (eit ) = eint x for some x ∈ H. The elements of H 2 (H) are all the analytic functions u : D → H whose Taylor coeﬃcients are square summable, that is, if u ∼ {un }, in the sense n z un , z ∈ D and {un } ⊆ H, then that u(z) = ∞ n=0 ∞

un 2H < ∞.

n=0

We recall that H 2 (H) is a Hilbert space with the pointwise linear operations and the scalar product u, vH 2 (H) :=

∞

un , vn H

u, v ∈ H 2 (H), u ∼ {un }, v ∼ {vn } .

n=0

As a consequence of Fatou’s Theorem, the radial limit limr↑1 u(reit ) exists almost everywhere. The application that maps each u(z) ∈ H 2 (H) into its radial limit provides an embedding of H 2 (H) into L2 (H) preserving the Hilbert space structures. Via the Poisson integral, ∞ it can be shown that the application maps H 2 (H) onto the subspace n=0 Gn (H) of L2 (H). Therefore we may consider that H 2 (H) and ∞ n=0 Gn (H) amount to the same Hilbert space. In the sequel, if N and M are two Hilbert spaces, then S(N , M) stands for the L(N , M)-Schur class, so that ϑ ∈ S(N , M) if and only if ϑ : D → L(N , M) is an analytic function such that supz∈D ϑ(z) ≤ 1. If ϑ ∈ S(N , M), then limr↑1 ϑ(reit ) exists almost everywhere as a strong limit of operators and determines a contraction operator in L(N , M). With each ϑ ∈ S(N , M) we associate a contraction operator from L2 (N ) into L2 (M) deﬁned by f (eit ) −→ ϑ(eit )f (eit ) (f (eit ) ∈ L2 (N )) and a contraction operator from H 2 (N ) into H 2 (M) deﬁned by u(z) −→ ϑ(z)u(z) (u(z) ∈ H 2 (N )). Due to the identiﬁcation of H 2 (N ) (and H 2 (M)) with the subspace ∞ ∞ n=0 Gn (N ) (and n=0 Gn (M), respectively) the latter operator may be considered as a restriction of the former one. We denote both of them by ϑ. When N = M = H and ϑ(z) ≡ z (z times the identity operator on H) the associated operator is the (forward) shift S. Given ϑ ∈ S(N , M) we can likewise consider the positive square root ∆(eit ) of the positive selfadjoint operator 1 − ϑ(eit )∗ ϑ(eit ) (1 the identity operator on N ).

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The basic reference for vector and operator valued analytic functions is [17]. We refer the reader to the detailed exposition given therein. Although familiarity with operator theory on Kre˘ın spaces is presumed, some background material is summarized in this section. We refer to [1], [4], [6], [9] and [13] as authoritative sources of information about indeﬁnite inner product spaces and operators on them, and to [8] for the treatment of the Kre˘ın space extensions of the Hilbert space notions of defect and Julia operators, minimal isometric dilations and minimal unitary dilations. A Kre˘ın space is a linear space H equipped with an indeﬁnite inner product (a hermitian sesquilinear form) ·, ·H such that there exist two linear subspaces H+ and H− with the following properties: 1) H is the algebraic direct sum of H+ and H− , 2) H+ , H− H = {0}, 3) (H+ , ·, ·H ) and (H− , −·, ·H ) are Hilbert spaces. The standard Hilbert space notation is carried over into the Kre˘ın space setting. So all algebraic and geometric notions are to be taken in their Kre˘ın space versions unless otherwise mentioned. In particular, in any Kre˘ın space (H, ·, ·H ) we use the notion of orthogonal direct sum in the same manner as for Hilbert spaces. A fundamental decomposition of a Kre˘ın space (H, ·, ·H ) is a representation of the form H = H+ ⊕ H− , where H+ and H− are subspaces as those in the above deﬁnition. A fundamental decomposition H = H+ ⊕ H− of a given Kre˘ın space (H, ·, ·H ) induces a Hilbert space inner product ·, ·|H| on H. Namely, if x, y ∈ H and x = x+ + x− , y = y+ + y− , then x, y|H| := x+ , y+ H − x− , y− H . In this situation the operator J deﬁned on H by Jx := x+ − x− whenever x = x+ + x− is called a signature operator or fundamental symmetry for H. Clearly x, y|H| = Jx, yH and x, yH = Jx, y|H| for all x, y ∈ H. In general, a Kre˘ın space has inﬁnitely many fundamental decompositions and therefore inﬁnitely many associated Hilbert space inner products. However, the corresponding norms are equivalent. All topological notions on a Kre˘ın space are to be understood with respect to the norm topology associated with a fundamental decomposition. The symbol · |H| will be used to denote the quadratic norm associated with a fundamental decomposition of H. A vector x of a Kre˘ın space (H, ·, ·H ) is said to be negative if x, xH ≤ 0. A linear subspace E is negative if all its vectors are negative. A negative subspace E is said to be uniformly negative if, for a ﬁxed norm · |H| , there exists a number β > 0 such that, for all x ∈ E, x, xH ≤ −β x 2|H| . A (uniformly) negative subspace E is said to be maximal (with respect to the respective property) if it is not a proper subspace of any subspace of the

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same type. The analogous notions with “negative” replaced by “positive” are deﬁned likewise. We point out that E is maximal (uniformly) negative if and only if E ⊥ is maximal (uniformly) positive. A regular subspace of a Kre˘ın space (H, ·, ·H ) is a closed subspace which is itself a Kre˘ın space in the inner product inherited from H. For a closed subspace E of H to be regular it is necessary and suﬃcient that H = E ⊕ E ⊥ . Therefore, E is regular if and only if E ⊥ is regular. Occasionally we write H E for E ⊥ if E is a regular subspace of H. Amongst the closed negative (positive) subspaces of H only those that are uniformly negative (positive) are regular. In particular, if E is a maximal uniformly negative subspace of H then E is regular and H E is a Hilbert subspace of H. Throughout this note, all Kre˘ın spaces are assumed to be complex and separable. If {Gι }ι∈I is a collection of linear subspaces of a Kre˘ın space H then ι∈I Gι is the smallest closed subspace of H containing all the subspaces Gι . If C, D are Kre˘ın spaces and E = C⊕ D, we shall write the elements of E c as sums c + d, pairs (c, d) or columns with no distinctness. d For the space of all everywhere deﬁned continuous linear operators on the Kre˘ın space H to the Kre˘ın space K we keep the Hilbert space notation, so we denote it by L(H, K) and use L(H) instead of L(H, H). The space L(H, K) has the structure of a Banach space depending on choices of fundamental decompositions H = H+ ⊕ H− and K = K+ ⊕ K− and associated quadratic norms · |H| and · |K|. For any A ∈ L(H, K),

A =

sup x|H| =1

Ax |K| .

We call · an operator norm for L(H, K). Any two operator norms for L(H, K) are equivalent and, thus, deﬁne a unique uniform topology for L(H, K). The strong operator topology for L(H, K) is similarly deﬁned. By 1 we indicate either the scalar unit or the identity operator, depending on context. For each A ∈ L(H, K) there is a unique A∗ ∈ L(K, H) such that Ax, yK = x, A∗ yH for all x ∈ H and y ∈ K. A projection P ∈ L(H) satisﬁes P = P 2 = P ∗ . The regular subspaces of H are those that are the ranges of projections. If E is a regular subspace of H we write PEH to indicate the orthogonal projection from H onto E.

∈ L(D,

H), By a defect operator for T ∈ L(H, K) we mean an operator D ∗

D

∗. where D is a Kre˘ın space, such that D has zero kernel and 1 − T T = D

The Kre˘ın space D is called a defect space for T . An operator T ∈ L(H, K) is a contraction if, for all x ∈ H, T x, T xK ≤ x, xH . Any defect space for a contraction T is a Hilbert space. The defect operators for a contraction T are abstractly indistinguishable in the sense that if

∈ L(D,

H) and D

∈ L(D

, H) are two defect operators for T then there D

=D

Φ.

→D

such that D exists a unitary operator Φ : D

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We term isometry any linear operator V deﬁned on a linear subspace D of a Kre˘ın space H to a Kre˘ın space K such that V x, V yK = x, yH for all x, y ∈ D. We call D the domain of V . Its range is R := V D. An isometry V is a weak isomorphism if D is dense in H and R is dense in K. An isometric isomorphism or unitary operator from H onto K is an isometry V such D = H and R = K. We remark that an isometry need not necessarily be continuous. An isometry with regular domain is continuous if and only if its range is regular. In particular, any unitary operator is continuous. As for a weak isomorphism, we point out that it can be extended to a unitary operator if either the domain or the range contains a maximal uniformly negative (positive) subspace. If V is an isometry acting on a Kre˘ın space H such that D and R are regular subspaces of H, the defect subspaces of V are N := H D and M := H R. Given a Kre˘ın space contraction T ∈ L(H), the 2 × 2 operator matrix T 0 H H UT := ∗ : →

D S H 2 (D) H 2 D) is a minimal isometric dilation of T . That is, UT is an isometry everywhere KT n 2

n deﬁned on the Kre˘ın space ∞ KTn := H ⊕ H (D) such that T = PH UT |H , for all n ∈ N, and KT = n=0 UT H. If K is another Kre˘ın space containing H K n as regular subspace and U ∈ L(K) is an isometry such that T n = PH U |H , ∞ n for all n ∈ N, and K = n=0 U H, then the weak isomorphism deﬁned by U n h :→ UTn h (h ∈ H, n ∈ N∪{0}) can be extended by continuity to a unitary operator τ ∈ L(K, KT ) satisfying τ |H = 1 and τ U = UT τ . Hence UT ∈ L(KT ) is the essentially unique minimal isometric dilation of T . In the above discussion a straightforward computation gives ⎤ ⎡ 0 Tn ⎦ , n ∈ N. k ∗ n−1−k UTn = ⎣n−1

T S D Sn k=0

We get that, for all n ∈ N and all h ∈ H, n−1 2

∗ T n−1−k h SkD k=0

=

H 2 (D)

n−1 k=0

=

∗ T n−1−k h 2

D D

n−1

D

∗ T n−1−k h, T n−1−k hH D

k=0

= h, hH − T n h, T n hH . Then we see that sup UTn < ∞ if and only if sup T n < ∞.

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From [11] it follows that if sup T n < ∞ then limn→∞ UTn UT∗n exists (in the strong operator topology) and deﬁnes a projection with range ∞ RT := UTn KT . n=0 KT In particular, RT is a regular subspace of KT and PR = limn→∞ UTn UT∗n . T In this situation we get a Wold decomposition for UT so that RT reduces UT and UT |RT is unitary. KT Let ET be the closure of PR H. Since ET⊥ = (KT RT ) ⊕ (RT ∩ T

is regular, we get that ET is regular too. Besides, U ∗ ET ⊆ ET and H 2 (D)) T T := PEKTT UT |ET is a coisometric contraction on ET whose minimal isometric dilation is the unitary operator UT |RT . It can be seen that QT := (UT − T )ET is a Hilbert subspace of RT such that RT = ET ⊕ QT ⊕ UT QT ⊕ UT2 QT ⊕ · · · . Alternatively, PT := kernel (T ) is a Hilbert subspace of ET and RT = ET ⊕ UT PT ⊕ UT2 PT ⊕ UT3 PT ⊕ · · · . For more on Kre˘ın space isometries and their Wold decompositions we refer the reader to [10, 11, 12]. Let V be an isometry on a Kre˘ın space H whose defect subspaces N and M are Hilbert subspaces of H. Then there exist a Kre˘ın space F containing H as regular subspace and a unitary operator U ∈ L(F ) such that U |D = V , F H is a Hilbert space and F = n∈Z U n H. Under these conditions we say that U ∈ L(F ) is a minimal unitary Hilbert space extension of V . We identify two minimal unitary Hilbert space extensions of V , say U ∈ L(F ) and U ∈ L(F ), if there exists a unitary operator τ : F → F such that τ |H = 1 and τ U = U τ . Under the above identiﬁcation, U(V ) is the set of the minimal unitary Hilbert space extensions of V . If V is an isometry on a Kre˘ın space H whose defect subspaces N and M are Hilbert subspaces of H, then with each minimal unitary Hilbert space extension U of V acting on F we may associate the L(N , M)-valued function ϑU (z) given by F ϑU (z) := PM U (1 − zPFF H U )−1 |N . It turns out that ϑU ∈ S(N , M). The S(N , M)-functions are relevant in the analysis of the minimal unitary Hilbert space extensions of V as they provide the parameters of the labeling of U(V ). The parametric description of U(V ) is the main tool in our treatment of the abstract Hankel operators in the Kre˘ın space setting. To conclude this section we state the result encompassing such a description. The model is a Kre˘ın space extension of the one given by D.Z. Arov and L.Z. Grossman in the Hilbert space framework [2, 3]. A more general result for continuous isometric operators with regular domain and range is given in [7].

Theorem 2.1. Let V be an isometry on a Kre˘ın space H with domain D, range R and defect subspaces N and M. Assume that N and M are Hilbert subspaces of H. Given ϑ ∈ S(N , M), set Eϑ := H 2 (M) ⊕ ∆L2 (N ) ∩ {(ϑχ, ∆χ) : χ ∈ H 2 (N )}⊥ ,

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where 1

∆(ζ) := (1 − ϑ(ζ)∗ ϑ(ζ)) 2 ,

|ζ| = 1.

Deﬁne Fϑ := H ⊕ Eϑ and Uϑ : Fϑ → Fϑ by ⎤ ⎡ ⎡ ⎤ H h + φ(0) V PDH h + ϑ(0)PN h ⎥ ⎢ H h) S ∗ (φ + ϑPN Uϑ ⎣ φ ⎦ := ⎣ ⎦ ψ ∗ H S (ψ + ∆P h)

φ h ∈ H, ∈ Eϑ ψ

N

where S is the shift on either H 2 (M) or L2 (N ), depending on context. Then: (i) Uϑ ∈ L(Fϑ ) is a minimal unitary Hilbert space extension of V such that Fϑ Uϑ (1 − zPEFϑϑ Uϑ )−1 |N = ϑ(z) PM

for all z ∈ D. (ii) For any minimal unitary Hilbert space extension U of V on F , the function F U (1 − zPFF H U )−1 |N ϑU (z) := PM

belongs to S(N , M). (iii) Two minimal unitary Hilbert space extensions of V , say U ∈ L(F ) and U ∈ L(F ), are identiﬁed under a unitary operator τ : F → F such that τ |H = 1 and τ U = U τ , if and only if

F F PM U (1 − zPFF H U )−1 |N = PM U (1 − zPFF H U )−1 |N

for all z ∈ D. Therefore, the map ϑ −→ Uϑ ∈ L(Fϑ ) establishes a bijective correspondence between S(N , M) and U(V ) (up to isometric isomorphisms as far as U(V ) is concerned). We prove the theorem in the Appendix.

3. Hankel symbols for a given Hankel operator Let H1 and H2 be two Kre˘ın spaces, and let T1 ∈ L(H1 ) and T2 ∈ L(H2 ) be two contractions with minimal isometric dilations U1 ∈ L(K1 ) and U2 ∈ L(K2 ), respectively. We recall from the Introduction that X ∈ L(H1 , H2 ) is a Hankel operator for T1 and T2 if and only if XT1∗ = T2 X, and that a Hankel K2 Z|H2 = X. symbol for X is any Z ∈ L(K1 , K2 ) such that ZU1∗ = U2 Z and PH 2 The next propostion gives a necessary and suﬃcient condition for a given Hankel operator to have Hankel symbols.

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Proposition 3.1. Let T1 ∈ L(H1 ) and T2 ∈ L(H2 ) be Kre˘ın space contractions with corresponding minimal isometric dilations U1 ∈ L(K1 ) and U2 ∈ L(K2 ), and let X ∈ L(H1 , H2 ) be a Hankel operator for T1 and T2 . Assume that sup T1n < ∞ and sup T2n < ∞. For j = 1, 2 consider the regular subspace Ej of Kj given by K

Ej := PRjj Hj where Rj :=

∞

Ujn Kj

n=0

is the regular subspace of Kj which reduces Uj such that Uj |Rj is unitary and K PRjj = limn→∞ Ujn Uj∗n . Then the following statements are equivalent: i) There exists a Z ∈ L(K1 , K2 ) such that Z|R1 is a contraction, ZU1∗ = U2 Z

and

K2 PH Z|H1 = X. 2

∈ L(E1 , E2 ) such that ii) There exists a contraction X K2

K1 |H1 . | )∗ XP X = (PR R1 2 H2

j ∈ L(D

j , Hj ) is a defect operator for Proof. In what follows, for j = 1, 2, D Tj , so that Tj 0 Hj Hj Uj := ∗ :

j )

j ) → H 2 D Dj S H 2 (D is the minimal isometric dilation of Tj and

j )). Kj Ej = (Kj Rj ) ⊕ (Rj ∩ H 2 (D i) ⇒ ii): Let Z ∈ L(K1 , K2 ) be given such that Z|R1 is a contraction, ZU1∗ = K2 Z|H1 = X. U2 Z and PH 2 Since U1 and U2 are isometries we also have that ZU1 = U2∗ Z. Furthermore, for all n ∈ N, Z = U2∗n U2n Z = U2∗n ZU1∗n = ZU1n U1∗n and Z = ZU1∗n U1n = U2n ZU1n = U2n U2∗n Z. Therefore, K1 K2 K2 K1 Z = ZPR = PR Z = PR ZPR . 1 2 2 1

Thus, for all h1 ∈ H1 and h2 ∈ H2 , Xh1 , h2 H2

K2 K1 = Zh1 , h2 K2 = PR ZPR h , h2 K2 2 1 1 K1 K2 K1 K2 = ZPR h , PR h = PEK22 ZPR h , PR h 1 1 2 2 K2 1 1 2 2 K2 K2 K1 = (PR | )∗ PEK22 ZPR h , h2 H2 . 2 H2 1 1

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K1 |H1

:= P K2 Z|E1 then X

∈ L(E1 , E2 ), X = (P K2 |H2 )∗ XP Hence, if X E2 R2 R1 and, for all e1 ∈ E1 ,

1 K2 = P K2 Ze1 , P K2 Ze1 K2

1 , Xe Xe E2

E2

= Ze1 , Ze1 K2 − P K2

2) R2 ∩H 2 (D

Ze1 , P K2

2) R2 ∩H 2 (D

Ze1 K2

≤ Ze1 , Ze1 K2 ≤ e1 , e1 K1 ,

2 ) is a Hilbert subspace of K2 , and Z|R1 is contractive. since R2 ∩ H 2 (D

K1 |H1 .

∈ L(E1 , E2 ) be a contraction such that X= (P K2 |H2 )∗ XP ii) ⇒ i): Let X R2 R1 K

Set Tj := PEj j Uj |Ej for j = 1, 2. Then T1 ∈ L(E1 ) and T2 ∈ L(E2 ) are coisometric contractions whose minimal isometric dilations are the unitary

so

∗ = T X, operator U1 |R1 and U2 |R2 , respectively. It turns out that XT 1 2

∈ L(E1 , E2 ) is a Hankel operator for T and T . that X 1 2 In E1 × E2 deﬁne the hermitian sesquilinear form

1 , e2 K2 + e2 , Xe

1 K2 + e2 , e2 K2 . [(e1 , e2 ), (e1 , e2 )] := e1 , e1 K1 + Xe

∈ L(D

, E1 ) is a defect operator for X

(so that D

is a Hilbert If D X X X

=D

D

∗ ), set H := D

⊕ E2 with the standard inner

∗X space and 1 − X

X X X product (a, e2 ), (a , e2 )H = a, a D + e2 , e2 K2 . X

Then, for all (e1 , e2 ), (e1 , e2 ) ∈ E1 × E2 , [(e1 , e2 ), (e1 , e2 )]

1 , e K2 + e2 , Xe

K2 + e2 , e K2 = e1 , e1 K1 + Xe 2 1 2

∗ e1 , D

∗ e + Xe

1 + e2 , Xe

+ e K2 . = D 1 2

1 D X X X

Hence

∗ e1 , Xe

1 + e2 ) (e1 , e2 ) :−→ (D X

((e1 , e2 ) ∈ E1 × E2 )

gives an isometric map σ from (E1 × E2 , [·, ·]) into (H, ·, ·H ) such that σ(E1 × E2 ) is a dense subspace of H. Taking into account that Uj∗ |Rj is unitary and Uj∗ |Ej = Tj∗ , j = 1, 2,

we get that, for all

∗ = T X, and making use of the intertwining relation XT 1 2 (e1 , e2 ), (e1 , e2 ) ∈ E1 × E2 , σ(e1 , U2∗ e2 ), σ(e1 , U2∗ e2 )H = [(e1 , U2∗ e2 ), (e1 , U2∗ e2 )]

1 , U ∗ e K2 = e1 , e1 K1 + Xe 2 2

1 , K2 + U2∗ e2 , U2∗ e2 K2 + U2∗ e2 , Xe

1∗ e1 , e2 K2 = U1∗ e1 , U1∗ e1 K1 + XU

∗ e K2 + e2 , e K2 + e2 , XU 1 1 2 = σ(U1∗ e1 , e2 ), σ(U1∗ e1 , e2 )H . Hence, the operator V0 : σ(U1∗ E1 × E2 ) → σ(E1 × U2∗ E2 ) deﬁned by V0 σ(U1∗ e1 , e2 ) := σ(e1 , U2∗ e2 ) (e1 ∈ E1 , e2 ∈ E2 )

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is an isometry. Note that

∗ U ∗ E 1 ⊕ E2 σ(U1∗ H1 × E2 ) = D

1 X

is a regular subspace of H, thus a Kre˘ın space by itself. Viewing V0 as a weak

∗ U ∗ E1 ⊕ E2 to the Kre˘ın space H isomorphism from the Kre˘ın space D := D

1 X we can extend V0 to a continuous isometry V on H with domain D. Indeed, let E2 = E2+ ⊕ E2− be a fundamental decomposition of E2 . Then − E2 is a maximal uniformly negative subspace of E2 and, hence, of D. As E2 = σ ({0} × E2 ) is contained in the domain of V0 , so is E2− and the result follows. In addition, note that both E2− and V E2− = U2∗ E2− are maximal uniformly negative subspaces of H. From the above arguments we conclude that the defect subspaces of V are Hilbert subspaces of H. We can compute them to get

∗ U ∗ E1

D N =D

1 X X

and

a+X

∗ e2 = 0 . M = (a, e2 ) ∈ H : T2 e2 = 0 and D X

From the fundamental decomposition E2 = E2+ ⊕ E2− we get the fundamental decomposition

⊕ E +) ⊕ E − H = (D 2 2 X with corresponding norm · |H| , so that, for all e1 ∈ E1 ,

1 , P K−2 Xe

1 K2

σ(e1 , 0) 2|H| = σ(e1 , 0), σ(e1 , 0)H − 2PEK−2 Xe E 2

1 , P K−2 Xe

1 K2 . = e1 , e1 K1 − 2PEK−2 Xe E 2

2

2

Whence e1 −→ σ(e1 , 0) gives an isometry ϕ1 from E1 into H such that

1 , P K−2 Xe

1 K2

ϕ1 e1 2|H| = e1 , e1 K1 − 2PEK−2 Xe E 2

2

for all e1 ∈ E1 . In particular, ϕ1 ∈ L(E1 , H) and ϕ1 E1 is a regular subspace of H. Let U ∈ L(F ) be a minimal unitary Hilbert space extension of V . Notice that, for all e1 , e1 ∈ E1 and all n ∈ N ∪ {0}, U n σ(e1 , 0), σ(e1 , 0)F = σ(e1 , 0), U ∗n σ(e1 , 0)F = σ(e1 , 0), σ(U1∗n e1 , 0)H = e1 , U1∗n e1 K1 = U1n e1 , e1 K1 . Hence U n ϕ1 e1 , ϕ1 e1 F = U1n e1 , e1 K1 ,

e1 , e1 ∈ E1 , n ∈ N ∪ {0}.

(3.1)

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If T := PϕF1 E1 U |ϕ1 E1 then T is a contraction on ϕ1 E1 since, according with (3.1), for all e1 ∈ E1 , T ϕ1 e1 , T ϕ1 e1 F = U ϕ1 e1 , T ϕ1 e1 F = U1 e1 , ϕ∗1 T ϕ1 e1 K1 = ϕ1 PEK11 U1 e1 , U ϕ1 e1 F = PEK11 U1 e1 , U1 e1 K1 = T1 e1 , T1 e1 K1 and, therefore, T ϕ1 e1 , T ϕ1 e1 F ≤ e1 , e1 K1 = ϕ1 e1 , ϕ1 e1 F . Besides, for all e1 ∈ E1 , (U − T )ϕ1 e1 , (U − T )ϕ1 e1 F = U ϕ1 e1 , U ϕ1 e1 F − T ϕ1 e1 , T ϕ1 e1 F = e1 , e1 K1 − T1 e1 , T1 e1 K1 = U1 e1 , U1 e1 K1 − PEK11 U1 e1 , PEK11 U1 e1 K1 = P K1

K1

1 ) U1 e1 , PR1 ∩H 2 (D

1 ) U1 e1 K1 . R1 ∩H 2 (D

Thus (U − T )ϕ1 e1 , (U − T )ϕ1 e1 F ≥ 0 for all e1 ∈ E1 and the equality holds if and only if U1 e1 ∈ E1 . If this is the case then T ϕ1 e1 = ϕ1 U1 e1 by (3.1). So (U − T )ϕ1 e1 = U ϕ1 U1∗ e1 − ϕ1 e1 , where e1 := U1 e1 ∈ E1 . Then U ϕ1 U1∗ e1 = U σ(U1∗ e1 , 0) = σ(e1 , 0) = ϕ1 e1 and (U − T )ϕ1 e1 = 0. Then Q := (U − T )ϕ1 E1 is a Hilbert subspace of F . Next we show that U k Q ⊥ U j Q for all nonnegative integers k = j. Indeed, by (3.1), for all e1 , e1 ∈ E1 and all m ∈ N, T m ϕ1 e1 , ϕ1 e1 F = U T m−1 ϕ1 e1 , ϕ1 e1 F = U1 ϕ∗1 T m−1 ϕ1 e1 , e1 K1 = T m−1 ϕ1 e1 , ϕ1 U1∗ e1 F = U1 ϕ∗1 T m−2 ϕ1 e1 , U1∗ e1 K1 = T m−2 ϕ1 e1 , ϕ1 U1∗2 e1 F = · · · = U1 e1 , U1∗m−1 e1 K1 = U1m e1 , e1 K1 = U m ϕ1 e1 , ϕ1 e1 F , so that U m (U − T )ϕ1 e1 , (U − T )ϕ1 e1 F = U m ϕ1 e1 , ϕ1 e1 F − U m−1 T ϕ1 e1 , ϕ1 e1 F − U m+1 ϕ1 e1 , T ϕ1 e1 F + U m T ϕ1 e1 , T ϕ1 e1 F = 0. Now set F1 := ϕ1 E1 ⊕ Q ⊕ U Q ⊕ U 2 Q ⊕ · · · . Clearly F1 is a regular subspace of F , hence a Kre˘ın space. Recall that Q1 = (U1 − T1 )E1 is a Hilbert subspace of R1 such that R1 = E1 ⊕ Q1 ⊕ U1 Q1 ⊕ U12 Q1 · · · , and extend ϕ1 to a unitary operator Φ1 ∈ L(R1 , F1 ) by setting Φ1 |E1 := ϕ1 and Φ1 U1n (U1 − T1 )e1 := U n (U − T )ϕ1 e1 Φ1 U1∗ |R1

(n ∈ N ∪ {0}, e1 ∈ E1 ).

= U ∗ Φ1 . Clearly, Φ1 U1 |R1 = U Φ1 so that ∞ We canas well deﬁne an isometric isomorphism Φ2 from R2= n=0 U2n E2 ∞ onto F2 := n=0 U ∗n E2 , satisfying Φ2 U2 |R2 = U ∗ Φ2 and Φ2 U2∗ |R2 = U Φ2 .

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Now, view Φj as a linear operator from Rj into F ⊇ Fj , j = 1, 2, and K1 deﬁne Z : K1 → K2 by Z := Φ∗2 Φ1 PR . Then Z ∈ L(K1 , K2 ). 1 For all k1 ∈ K1 and k2 ∈ K2 , K1 ∗ K2 U1 k1 , Φ2 PR k ZU1∗ k1 , k2 K2 = Φ1 PR 1 2 2 F K1 K2 = Φ1 U1∗ PR k , Φ2 PR k 1 1 2 2 F K1 K2 = U ∗ Φ1 PR k1 , Φ2 PR k2 F 1 2 K1 K2 = Φ1 PR k1 , U Φ2 PR k2 F 1 2 K1 K2 ∗ = Φ1 PR k1 , Φ2 PR U2 k2 F 1 2

= U2 Zk1 , k2 K2 . ZU1∗

= U2 Z. Therefore, For all h1 ∈ H1 and h2 ∈ H2 , K1 K2 K1 K2 Zh1 , h2 K2 = Φ1 PR h , Φ2 PR h = σ(PR h1 , 0), σ(0, PR h )H 1 1 2 2 F 1 2 2 K1 K2

K1 h1 , P K2 h2 K2 = [(PR h , 0), (0, PR h )] = XP R1 R2 1 1 2 2 K2

K1 h1 , h2 K2 = Xh1 , h2 K2 . = (PR | )∗ XP R1 2 H2 K2 Thus PH Z|H1 = X. 2 The proof is complete if we show that Z|R1 is a contraction. To do that, note that E2− in any fundamental decomposition of E2 is maximal uniformly negative in R2 and in F as well. Since Φ2 E2− = E2− , it follows that Φ∗2 is a contraction. Therefore, for all r1 ∈ R1 , K1 K1 Zr1 , Zr1 K2 = Φ∗2 Φ1 PR r , Φ∗2 Φ1 PR r 1 1 1 1 F

= Φ∗2 Φ1 r1 , Φ∗2 Φ1 r1 F

≤ Φ1 r1 , Φ1 r1 F = r1 , r1 K1 .

As Propostion 3.1 clearly emphasizes, the relation XT1∗ = T2 X alone is not suﬃcient to grant the existence of symbols. In the Hilbert space case the diﬃculty is overtaken by assuming that: ii) There exists β ≥ 0 such that, for all h1 ∈ H1 and h2 ∈ H2 , K1 K2 |Xh1 , h2 H2 | ≤ β PR h PR h . 1 1 K1 2 2 K2

For a Hankel operator X for T1 and T2 satisfying the boundedness condition ii) , set X P V := inf β, where β runs over all nonnegative numbers satisfying

∈ L(E1 , E2 ) ii) . It turns out that ii) is equivalent to the the existence of an X K2 ∗ K1

such that X = (PR2 |H2 ) XPR1 |H1 and X ≤ β, and, in turn, is necessary and suﬃcient for the existence of a Z ∈ L(K1 , K2 ) such that ZU1∗ = U2 Z, K2 Z = X and Z = X P V (cfr. [14], [15], [16] and [5]). Therefore, when PR 2

X P V = 1, we can see that ii) is equivalent to i) and ii) in Proposition 3.1. The hypothesis X P V = 1 can be dropped provided we consider symbols Z of X such that Z = X P V . As a matter of fact, if X is a Hankel operator for T1 and T2 with X P V = b > 0, then X := 1b X is a Hankel operator for

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T1 and T2 with X P V = 1 and, moreover, Z is a Hankel symbol for X if and only if bZ is a Hankel symbol for X. In the Kre˘ın space setting a boundedness condition like (ii) and equivalent to (ii) in Proposition 3.1 is given next. Proposition 3.2. Let T1 ∈ L(H1 ) and T2 ∈ L(H2 ) be Kre˘ın space contractions with corresponding minimal isometric dilations U1 ∈ L(K1 ) and U2 ∈ L(K2 ), and let X ∈ L(H1 , H2 ) be a Hankel operator for T1 and T2 . Assume that sup T1n < ∞ and sup T2n < ∞. For j = 1, 2 consider the regular subspace Ej of Kj given by K

Ej := PRjj Hj where Rj :=

∞

Ujn Kj

n=0

is the regular subspace of Kj which reduces Uj such that Uj |Rj is unitary and K

∈ L(E1 , E2 ) such that PRjj = limn→∞ Ujn Uj∗n . Then there exists an X K2

K1 |H1 X = (PR |H2 )∗ XP R1 2

if and only if, for any ﬁxed norms · |K1 | and · |K2 | , there exists α ≥ 0 such that K1 K2 h1 |K1 | PR h2 |K2 | |Xh1 , h2 H2 | ≤ α PR 1 2

for all h1 ∈ H1 and h2 ∈ H2 . K2

K1 |H1 for some X

∈ L(E1 , E2 ). Let Proof. Assume that X = (PR | )∗ XP R1 2 H2

· |K1 | and · |K2 | be ﬁxed. Then, for all h1 ∈ H1 and h2 ∈ H2 , K2

K1 h1 , h2 K2 | = |(XP

K1 h1 , P K2 )h2 K2 | |Xh1 , h2 H2 | = |(PR |H2 )∗ XP R1 R1 R2 2

K1 h1 |K | P K2 h2 |K | ≤ XP 2 2 R1 R2 K1 K1

≤ X

P R1 h1 |K1 | PR2 h2 |K2 | ,

= supe =1 Xe

1 |K | . where X

2 1 |K1 | Conversely, suppose that, for ﬁxed norms · |K1 | and · |K2 | , there exists α ≥ 0 such that, for all h1 ∈ H1 and h2 ∈ H2 , K1 K2 |Xh1 , h2 H2 | ≤ α PR h

PR h

. 1 1 |K1 | 2 2 |K2 |

We can assume that, for j = 1, 2, · |Kj | is the norm associated with a fundamental decomposition of the form Kj = [(Kj Hj ) ⊕ Hj+ ] ⊕ Hj− and corresponding signature operator Jj , so that · |Kj | |Hj = · |Hj | and Jj |Hj = Jj with · |Hj | and Jj the norm and the signature operator arisen from the fundamental decomposition Hj = Hj+ ⊕ Hj− .

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K1 Deﬁne CPR h := Xh1 (h1 ∈ H1 ). Then, for all h1 ∈ H1 , 1 1 K1 h1 |H2 | =

CPR 1

sup h2 |H2 | =1

=

sup h2 |H2 | =1

=

sup h2 |H2 | =1

≤

sup h2 |H2 | =1

K1 |CPR h1 , h2 |H2 | | 1

|Xh1 , h2 |H2 | | |Xh1 , J2 h2 H2 | K1 K2 α PR h

PR J h

1 1 |K1 | 2 2 2 |K2 |

K2 K1 = α PR J |

PR h

. 2 2 H2 1 1 |K1 |

Hence C can be extended by continuity to all of E1 . K2 h2 := C ∗ h2 then, for all h2 ∈ H2 , In a similar fashion if DPR 2 K2

DPR h

= 2 2 |K1 |

≤

sup e1 |K1 | =1

sup e1 |K1 | =1

K2 |PR h2 , Ce1 |K2 | | 2 K2

PR h2 |K2 | Ce1 |H2 | 2

K2 = C

PR h2 |K2 | . 2

Whence D can be extended by continuity to get a continuous linear operator D : E2 → E1 .

:= D∗ . Then, for all h1 ∈ H1 and h2 ∈ H2 , Set X K2

K1 h1 , h2 H2 = Xh1 , h2 H2 . (PR |H2 )∗ XP R1 2

4. Labeling of all the Hankel symbols of a given Hankel operator We now turn our attention to the problem of describing the set of all Hankel symbols for a given Hankel operator. Theorem 4.1. Let T1 ∈ L(H1 ) and T2 ∈ L(H2 ) be Kre˘ın space contractions with corresponding minimal isometric dilations U1 ∈ L(K1 ) and U2 ∈ L(K2 ). Assume that sup T1n < ∞ and sup T2n < ∞. For j = 1, 2 consider the regular subspace Ej of Kj given by K

Ej := PRjj Hj where Rj :=

∞

Ujn Kj

n=0

is the regular subspace of Kj which reduces Uj such that Uj |Rj is unitary K and PRjj = limn→∞ Ujn Uj∗n . Let X ∈ L(H1 , H2 ) be a Hankel operator for

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K2

K1 |H1 for some contraction operator T1 and T2 such that X = (PR | )∗ XP R1 2 H2

∈ L(E1 , E2 ). Consider the Hilbert spaces X

∗ U ∗ E1

D N =D

1 X X and

X ⊕ E2 : P K2 U2 e2 = 0 and D

a+X

∗ e2 = 0 M = (a, e2 ) ∈ D E2 X

, E1 ) a defect operator for X.

Then there exists a bijection

∈ L(D with D X X between the Schur class S(N , M) and the set HS(X) of all Hankel symbols Z of X such that Z|R1 is a contraction. Before embarking in the proof of the theorem, let us mention that the map establishing the bijection between S(N , M) and HS(X) can be given by a close formula. Roughly speaking, there exist operator valued functions a, b, c, d, deﬁned and analytic on a disc around 0, such that a Z ∈ HS(X) is fully determined from a given ϑ ∈ S(N , M) by the Schur like formula a + bϑ(1 − cϑ)−1 d in such a way that the map ϑ → Z is one to one and onto. The reader is referred to formula (4.4) and the discussion that follows. Proof. We proceed stepwise. STEP 1. As we did to prove Propostion 3.1, we build up a Kre˘ın space H and an isometry V acting on H whose defect subspaces are the Hilbert spaces N and M in the statement of the theorem so that each minimal unitary Hilbert space extension U of V gives rise to a Hankel symbol Z of X. It turns out that any Hankel symbol Z of X can be obtained as before from a minimal unitary Hilbert space extension U of the isometry V . We show that in the next step. K1 K2 K2 K1 = PR Z = PR ZPR STEP 2. Let Z ∈ HS(X) be given. Then Z = ZPR 1 2 2 1

Z ∈ L(D

Z , R1 ) and Z|R1 ∈ L(R1 , R2 ) is a contraction. In the sequel, let D stand for a defect operator for Z|R1 . A hermitian sesquilinear form on R1 × R2 is deﬁned by setting

[(r1 , r2 ), (r1 , r2 )] := r1 , r1 K1 + Zr1 , r2 K2 + r2 , Zr1 R2 + r2 , r2 K2 . So, for all (r1 , r2 ), (r1 , r2 ) ∈ R1 × R2 , ∗ ∗

Z

Z r1 , D r1 D Z + Zr1 + r2 , Zr1 + r2 K2 . [(r1 , r2 ), (r1 , r2 )] = D

Z ⊕R2 , with the standard inner product, and τ is deﬁned Therefore, if F := D on R1 × R2 as ∗

Z τ (r1 , r2 ) := (D r1 , Zr1 + r2 ) (r1 ∈ R1 , r2 ∈ R2 ) then τ is an isometry from (R1 × R2 , [·, ·]) to (F , ·, ·F ) with dense range in F .

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K2 Since PH Z|H1 = X, it turns out that 2

PEK22 Z|E1 = X. Therefore, σ(e1 , e2 ), σ(e1 , e2 )H = τ (e1 , e2 ), τ (e1 , e2 ))F ,

e 1 ∈ E1 , e 2 ∈ E2 .

(4.1)

From (4.1) it follows that ρ0 σ := τ |E1 ×E2 is a densely deﬁned isometry from H to F . Note that a fundamental decomposition E2 = E2− ⊕ E2+ of the

⊕E + )⊕E − Kre˘ın space E2 gives rise to fundamental decompositions H = (D 2 2 X + − − 2

and F = [DZ ⊕ (R2 ∩ H (D2 )) ⊕ E2 ] ⊕ E2 so that E2 is maximal uniformly negative in both H and F . Since E2− happens to be contained in the domain of ρ0 (E2 = σ({0} × E2 )) ρ0 can be extended to an isometry ρ deﬁned on all of H. Therefore, via the isometry ρ, the Kre˘ın space H can be regarded as a regular subspace of the Kre˘ın space F . Moreover, as E2− is maximal uniformly negative in both H and F , and ρE2− = E2− , it turns out that F ρH is a Hilbert subspace of F . Set U0 τ (r1 , r2 ) := τ (U1 r1 , U2∗ r2 ),

r1 ∈ R1 , r2 ∈ R2 .

As ZU1∗ = U2 Z, the operator U0 is shown to be isometric. On the other hand, if r1 ∈ R1 and r2 ∈ R2 are given, then τ (r1 , r2 ) = τ (U1 U1∗ r1 , U2∗ U2 r2 ) = U0 τ (U1∗ r1 , U2 r2 ). It then turns out that U0 is a weak isomorphism on F . As in the proof of Proposition 3.1, when dealing with V0 , it can be seen that U0 can be extended to a unitary operator U on F . For all e1 ∈ E1 and e2 ∈ E2 , U ρσ(U1∗ e1 , e2 ) = U τ (U1∗ e1 , e2 ) = τ (U1 U1∗ e1 , U2∗ e2 ) = τ (e1 , U2∗ e2 ) = ρσ(e1 , U2∗ e2 ) = ρV σ(U1∗ e1 , e2 ).

Therefore, U ρ|D = ρV . In other words, by means of the isometry ρ, U can be interpreted as a unitary extension of V . If Φ1 : R1 → F and Φ2 : R2 → F are deﬁned as Φ1 r1 := τ (r1 , 0) (r1 ∈ R1 ) and Φ2 r2 := τ (0, r2 ) (r2 ∈ R2 ), then Φ1 and Φ2 are isometries such that (i) Φ1 R1 ∨ Φ2 R2 = F and, for all k1 ∈ K1 and k2 ∈ K2 , K1 K2 k , Φ2 PR k = Zk1 , k2 K2 . (ii) Φ1 PR 1 1 2 2 F

Furthermore, for all e1 ∈ E1 , e2 ∈ E2 and n ∈ N ∪ {0}, (iii) Φ1 U1n e1 = U n Φ1 e1 = U n ρσ(e1 , 0) and (iv) Φ2 U2n e2 = U ∗n Φ2 e2 = U ∗n ρσ(0, e2 ).

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From (iii) and (iv) we get that ∞

U n ρσ(E1 × {0}) =

n=0

and

∞

∞

U n Φ1 E 1 =

n=0

U ∗n ρσ({0} × E2 )

n=0

∞

∞

Φ1 U1n E1 = Φ1 R1 .

n=0

U ∗n Φ2 E2 =

n=0

∞

Φ2 U2n E2 = Φ2 R2 .

n=0

From the above relations and by applying (i), we conclude that F = n∈Z U n ρH, which shows that U is minimal. Finally, (ii) says that Z is given by U as in the correspondence we established in STEP 1. Therefore, we can conclude that the correspondence which associates to each minimal unitary extension U of V a Hankel symbol Z of X with Z|R1 contractive is surjective. STEP 3. Next we show that the correspondence is injective. Assume that U ∈ L(F ) and U ∈ L(F ) are two minimal unitary Hilbert space extensions of V and let Z and Z be the corresponding Hankel symbols. If Z = Z , then U n σ(e1 , 0), σ(0, e2 )F = ZU1n e1 , e2 K2

n

= Z U1n e1 , e2 K2 = U σ(e1 , 0), σ(0, e2 )F for all e1 ∈ E1 , e2 ∈ E2 and n ∈ N ∪ {0}. Besides, when n ∈ N ∪ {0}, U n σ(e1 , 0), σ(e1 , 0)F = U1n e1 , e1 K1 = U n σ(e1 , 0), σ(e1 , 0)F for all e1 , e1 ∈ E1 , and U n σ(0, e2 ) = σ(0, U2∗n e2 ) = U n σ(0, e2 ) for all e2 ∈ E2 . These relations, together with the minimality condition satisﬁed by both U and U and the fact that H = σ(E1 × E2 ), grants that U and U are weakly isomorphic under the map U n σ(e1 , e2 ) :→ U n σ(e1 , e2 ) ((e1 , e2 ) ∈ E1 × E2 , n ∈ Z) and also that

F n F PH U |σ(E1 ×E2 ) = PH U n |σ(E1 ×E2 ) ,

n ∈ N ∪ {0}.

As F H and F H are Hilbert subspaces of F and F , respectively, it turns out that, with respect to the Hilbert space inner products ·, ·|F | and ·, ·|F | determined by the fundamental decompositions F = [(F H) ⊕ H+ ] ⊕ H− and F = [(F H) ⊕ H+ ] ⊕ H− , U n σ(e1 , e2 ), U m σ(e1 , e2 )|F | = U n σ(e1 , e2 ), U m σ(e1 , e2 )F F n F m − 2PH σ(e1 , e2 )F − U σ(e1 , e2 ), PH− U

= U n σ(e1 , e2 ), U m σ(e1 , e2 )F

F n F m − 2PH σ(e1 , e2 ), PH σ(e1 , e2 )F −U −U

= U n σ(e1 , e2 ), U m σ(e1 , e2 )|F |

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for all e1 , e1 ∈ E1 , e2 , e2 ∈ E2 and n, m ∈ Z. Then U and U can be identiﬁed. The identiﬁcation is realized by the extension by continuity of the weak isomorphism, call it ω, which is a unitary operator from F onto F satisfying ωU = U ω and ω|H = 1. As for a summary of what we have achieved so far, let us point out that we have built up a Kre˘ın space H and an isometry V acting on H such that if U ∈ L(F ) is a minimal unitary Hilbert space extension of V , then there exist two isometries Φ1 : R1 → F and Φ2 : R2 → F such that the operator K1 is a symbol of X with Z|R1 a contraction, and that the Z := Φ∗2 Φ1 PR 1 mapping U → Z is a bijection between the family U(V ) of all minimal unitary Hilbert space extensions of V and the set HS(X) of all Hankel symbols Z of X such that Z|R1 is a contraction. At this point we are ready to consider the problem of labeling the set HS(X). STEP 4. Let Z ∈ HS(X) be given. Since ZU1 = U2∗ Z, then ZU1n e1 = U2∗n Ze1 for all e1 ∈ E1 and all n ∈ N. ∞ K2 K1 From this, as Z = PR ZPR and R1 = n=0 U1n E1 , it follows that Z is fully 2 1 K2 determined by PR Z|E1 . On the other hand, 2 R2 = E2 ⊕ U2 P2 ⊕ U22 P2 ⊕ U23 P2 ⊕ · · · , where P2 := kernel (T2 ) is a Hilbert subspace of E2 , and it turns out that K2

+ Z|E1 = X PR 2

∞ n=0

2

+ PUKn+1 Z|E1 = X P 2

2

∞ n=0

U2n+1 PPK22 U2∗n+1 Z|E1 .

Thus, Z is determined by the sequence of operators ∞ U2n+1 PPK22 U2∗n+1 Z|E1 . n=0

To Z ∈ HS(X) we associate an L(E1 , P2 )-valued function SZ (z) deﬁned around 0 by the power series ∞ K2 ∗n+1 z n S S Z|E1 (n ≥ 0). SZ (z) := Z (n), Z (n) := PP2 U2 n=0

We get that, for all e1 ∈ E1 and e2 ∈ P2 , ∞ ∞ z n S (n)e , e = z n Ze1 , U2n+1 e2 K2 SZ (z)e1 , e2 K2 = Z 1 2 K2 =

n=0 ∞

n=0 ∞

z n Φ1 e1 , U ∗n+1 e2 F =

n=0

n=0

∗ e1 , e2 F z n U n+1 D X

∗ e1 , e2 F . = U (1 − zU )−1 D

X View P2 as a regular subspace of F and let Ψ denote the isometric embedding of P2 into F (Ψ := σ|{0}×P2 ). Then, for all e1 ∈ E1 and e2 ∈ P2 ,

∗ e1 , e2 K2 . SZ (z)e1 , e2 K2 = Ψ∗ PPF2 U (1 − zU )−1 D X

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Therefore

∗

. SZ (z) = Ψ∗ PPF2 U (1 − zU )−1 |H D

X Choose the fundamental decomposition

⊕ P2 ⊕ U2∗ E + ) ⊕ U2∗ E − H = (D X

2

2

(4.2) (4.3)

and let · |H| be the corresponding quadratic norm. Observe that P2 , N , M are subspaces of H+ . Whence the restriction of ·, ·|H| to any of them equals the corresponding restriction of ·, ·H . In particular, we see that e2 |H| ≤ V e2 |H| for all e2 ∈ P2 . So V ≥ 1 when the operator norm is computed in terms of · |H| . The fundamental decomposition (4.3) yields the fundamental decomposition F = [(F H) ⊕ H+ ] ⊕ H− of F with norm · |F | . Since (F H) ⊕ N , (F H) ⊕ M ⊆ F + and U |(F H)⊕N is a unitary operator from (F H) ⊕ N onto(F H) ⊕ M, we get that, for all f ∈ F, F 2

U f 2|F | = V PDF f + U P(F H)⊕N f |F | F 2 = V PDF f 2|H| + U P(F H)⊕N f |F F 2 ≤ V 2 PDF f 2|H| + P(F H)⊕N f |F |

≤ V 2 f 2|F |. Therefore U = V when U is regarded as an operator on (F , · |F |). Set r := V PDH −1 . Then 1−zV PDH has a bounded inverse on (H, · |H|) for all |z| < r. Since P2 ⊆ H+ and U = V , we ﬁnd out that, for all |z| < r, 1 − z(1 − PPH2 )V PDH on (H, · |H| ) and 1 − zU on (F , · |F | ) have bounded inverses. All those inverse operators can be computed by means of Neumann series. Then, when P2 is endowed with the quadratic norm · P2 = · |H| |P2 , it comes that z ∈ D −→ PPF2 U (1 − rzU )−1 h is an H 2 (P2 )-function for each h ∈ H. F Now, let ϑU (z) := PM U (1 − zPFF H U )−1 |N be the S(N , M)-function associated to U via the Arov-Grossmann functional model (Theorem 2.1). Taking into account that V n P2 ⊆ D and V n+1 P2 ⊥ P2 for all n ∈ N ∪ {0}, and by a straightforward computation we omit in the present discussion, we get that, for all |z| < r, PPF2 U (1 − zU )−1 |H H H −1 = PPH2 (V PDH + ϑU (z)PN )[1 − z(V PDH + ϑU (z)PN )] |H

= PPH2 V PDH [1 − z(1 − PPH2 )V PDH ]−1 + PPH2 1 + zV PDH [1 − z(1 − PPH2 )V PDH ]−1 (1 − PPH2 ) −1 H [1 − z(1 − PPH2 )V PDH ]−1 ϑU (z) × ϑU (z) 1 − zPN H × PN [1 − z(1 − PPH2 )V PDH ]−1 .

From this expression and (4.2) it follows that, for all |z| < r, SZ (z) = a(z) + b(z)ϑ(z)(1 − c(z)ϑ(z))−1 d(z),

(4.4)

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where ϑ ≡ ϑU and

∗ ∈ L(E1 , P2 ), a(z) := Ψ∗ PPH2 V PDH [1 − z(1 − PPH2 )V PDH ]−1 D X ∗ H H H b(z):= Ψ PP2 1 + zV PD [1 − z(1 − PP2 )V PDH ]−1 (1 − PPH2 ) |M ∈ L(M, P2 ), H [1 − z(1 − PPH2 )V PDH ]−1 |M ∈ L(M, N ), c(z) := zPN H

∗ ∈ L(E1 , N ). [1 − z(1 − PPH2 )V PDH ]−1 D d(z) := PN

X

The Schur like formula (4.4) establishes the direct connection between S(N , M) and {SZ : Z ∈ HS(X)} while the map ϑ −→ SZ determined by (4.4) is a bijection between S(N , M) and HS(X) since the mappings Z ∈ HS(X) U ∈ U(V )

? ϑ ∈ S(N , M)

? {S Z (n)}

? SZ are all bijections (up to isomorphism as far as U ∈ U(V ) is concerned). Furthermore, we can realize the map ϑ → Z by a close formula as we show next. For each e1 ∈ E1 , z ∈ D → SZ (rz)e1 can be shown to be an H 2 (P2 )function when P2 is endowed with the quadratic norm · 2P2 = ·, ·K2 |P2 . Therefore, if Γ : H 2 (P2 ) → U2 P2 ⊕ U22 P2 ⊕ U23 P2 ⊕ · · · is the unitary operator given by ∞ ∞ ∞ n+1 n n 2 Γ z xn := U2 xn z xn ∈ H (P2 ) , x(z) = n=0

n=0

n=0

then ∞ n=0

−1 2 PUKn+1 Z| = Γ a(rz) + b(rz)ϑ(rz)(1 − c(rz)ϑ(rz)) d(rz) . E 1 P 2

2

K2 Consequently, PR Z|E1 : E1 → E2 ⊕ (R2 E2 ), 2 K2 Z|E1 PR 2

X

=

!

. Γ a(rz) + b(rz)ϑ(rz)(1 − c(rz)ϑ(rz))−1 d(rz)

K2 Since PR Z|E1 determines Z uniquely, the above representation gives a Schur 2 like formula for the map ϑ → Z. This completes the proof.

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Remark 4.2. In the proof of Theorem 4.1 one can ﬁnd the arguments to show that there exists a bijective correspondence between HS(X) and the set of all

If contractions Z ∈ L(R1 , R2 ) such that ZU1∗ |R1 = U2 Z and PER2 2 Z|E1 = X. the given contractions T1 and T2 are assumed to be coisometric contractions having Hilbert spaces for kernels, their corresponding minimal isometric dilations are unitary operators. Then, by assuming that X is contractive and considering

∗ U ∗ H1

X D N =D X

and

1

X ⊕ H2 : T2 h2 = 0 and D

X a + X

∗ h2 = 0 M = (a, h2 ) ∈ D

X , H1 ) a defect operator for X, no extra conditions on

X ∈ L(D with D {X, T1 , T2 } are required in Theorem 4.1 to grant that the set HS(X) is in bijective correspondence with S(N , M). The result reads: Let T1 ∈ L(H1 ) and T2 ∈ L(H2 ) be coisometric contractions such that kernel (T1 ) and kernel (T2 ) are Hilbert spaces, and let U1 ∈ L(K1 ) and U2 ∈ L(K2 ) be their corresponding minimal isometric dilations. If X ∈ L(H1 , H2 ) is a contractive Hankel operator for T1 and T2 , then there exists a bijective correspondence between HS(X) and S(N , M) with N and M as above. Remark 4.3. Since HS(X) is in bijective correspondence with the Schur class S(N , M), it is clear that HS(X) has a single element if and only if either N = {0} or M = {0}. In particular: For ﬁxed contractions T1 ∈ L(H1 ) and T2 ∈ L(H2 ) satisfying sup T1n < ∞ and sup T2n < ∞, any Hankel operator X for T1 K2

K1 |H1 , for some contraction operator and T2 such that X = (PR | )∗ XP R1 2 H2

∈ L(E1 , E2 ), has a unique Hankel symbol, say ZX , constrained to satisfy X that ZX |R1 is a contraction, if either PEK11 U1 |E1 is injective or PEK22 U2 |E2 is injective. Indeed, if PEK22 U2 |E2 is assumed to be injective then

∗ e2 = 0 = {0}. M := (a, e2 ) ∈ DX ⊕ E2 : PEK22 U2 e2 = 0 and DX a + X In a similar way one can show that if PEK11 U1 |E1 is injective then N = {0}. We last study the problem of establishing a necessary and suﬃcient condition for the set HS(X) to have a single element. We get the following criterion. Theorem 4.4. Let T1 ∈ L(H1 ) and T2 ∈ L(H2 ) be Kre˘ın space contractions with minimal isometric dilations U1 ∈ L(K1 ) and U2 ∈ L(K2 ), respectively. Assume that sup T1n < ∞ and sup T2n < ∞. For j = 1, 2 consider the regular subspace Ej of Kj given by K

Ej := PRjj Hj

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where ∞

Rj :=

Ujn Kj

n=0

is the regular subspace of Kj which reduces Uj such that Uj |Rj is unitary K and PRjj = limn→∞ Ujn Uj∗n . Let X ∈ L(H1 , H2 ) be a Hankel operator for

∈ L(E1 , E2 ) be a contraction operator such that X = T1 and T2 , and let X K2 ∗ K1 (PR2 |H2 ) XPR1 |H1 . On the Kre˘ın space E1 ⊕ E2 , with the standard inner product, consider the 2 × 2 block matrix operators K1 1 0 0 P U | T 1 := E1 1 E1 , T 2 := K2 0 PE2 U2 |E2 0 1 and

E :=

1

X

∗ X . 1

Then X has a unique Hankel symbol Z with Z|R1 contractive if and only if either (4.5) kernel (T 1 E) ⊆ kernel (E), or kernel (T 2 E) ⊆ kernel (E).

(4.6)

Proof. We apply Theorem 4.1 and use the notation introduced in its proof. In particular, if H is the Kre˘ın space built up in the proof of Theorem 4.1 and σ is the isometry from (E1 × E2 , [·, ·]) onto a dense subspace of H deﬁned therein, then, for all e1 , e1 ∈ E1 and e2 , e2 ∈ E2 , σ(e1 , e2 ), σ(e1 , e2 )H = [(e1 , e2 ), (e1 , e2 )] = E(e1 , e2 ), (e1 , e2 )E1 ⊕E2 . (4.7) It then follows that a set A of E1 × E2 is such that H if and only if, σA = ∗ regarding A as a set of E1 ⊕ E2 , it holds that kernel (E|A ) ⊆ kernel (E). In particular, by the identiﬁcations T 1∗ (E1 ⊕ E2 ) = U1∗ E1 × E2 and T 2∗ (E1 ⊕ E2 ) = E1 × U2∗ E2 , we get that either σ(U1∗ E1 × E2 ) = H or σ(E1 × U2∗ E2 ) = H if and only if kernel (T j E) ⊆ kernel (E) for j = 1, 2 (which corresponds with (4.5) and (4.6) respectively.) The result is now obtained just by recalling that HS(X) has a single element if and only if either of the Hilbert spaces N and M given in the statement of Theorem 4.1 is {0} and that N = H σ(U1∗ E1 × E2 ) and M = H σ(E1 × U2∗ E2 ).

5. Appendix We herein include a sketch of the proof of Theorem 2.1.

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∞ k (i) Given ϑ ∈ S(N , M), ϑ(z) = k=0 z ϑk , z ∈ D, let Uϑ be the linear operator deﬁned in Fϑ := H ⊕ Eϑ by ⎤ ⎡ ⎡ ⎤ H h + φ(0) V PDH h + ϑ(0)PN h φ ⎥ ⎢ ∗ H ⎣ ⎦ S (φ + ϑPN h) Uϑ φ := ⎣ h ∈ H, ∈ Eϑ . ⎦ ψ ψ ∗ H S (ψ + ∆P h) N

Here Eϑ := H 2 (M) ⊕ ∆L2 (N ) ∩ {(ϑχ, ∆χ) : χ ∈ H 2 (N )}⊥ , with 1

∆(ζ) := (1 − ϑ(ζ)∗ ϑ(ζ)) 2 , 2

|ζ| = 1,

2

and S is the shift on either H (M) or L (N ), depending on context. φ As Sϑ = ϑS|H 2 (N ) and S∆ = ∆S, given any h ∈ H and ∈ Eϑ , we ψ H H h) ∈ H 2 (M), S ∗ (ψ + ∆PN h) ∈ ∆L2 (N ) and that, for get that S ∗ (φ + ϑPN all u ∈ H 2 (N ), " ! # H S ∗ (φ + ϑPN h) ϑu H H H h, ϑSu+∆PN h, ∆Su = PN h, Su = 0. , = ϑPN H S ∗ (ψ + ∆PN h) ∆u Therefore, Uϑ ∈ L(Fϑ ).

φ ∈ Eϑ , It can be shown that Uϑ is unitary. In point of fact, if h ∈ H and ψ ⎡ ⎤ ⎡ ∗⎤ h h then ⎣ φ∗ ⎦ = Uϑ∗ ⎣ φ ⎦ is given by ψ∗ ψ H H ∗ h + PN (ϑ Sφ + ∆Sψ), h∗ = V ∗ PDH h + ϑ(0)∗ PM H H H h + Sφ − ϑ[ϑ(0)∗ PM h + PN ((ϑ∗ Sφ + ∆Sψ)], φ∗ = PM H H h + PN ((ϑ∗ Sφ + ∆Sψ). ψ ∗ = Sψ − ∆[ϑ(0)∗ PM

Also, it can be seen that Fϑ is the least Kre˘ın space containing Uϑn H for all n ∈ Z. From this and since Uϑ |D = V and Eϑ is a Hilbert space, it comes that Uϑ ∈ L(Fϑ ) is a minimal unitary Hilbert space extension of V . It remains to show that Fϑ Uϑ (1 − zPEFϑϑ Uϑ )−1 |N = ϑ(z) PM

for all z ∈ D. This follows from the relations ⎡ ⎤ 0 %k $ Fϑ PEϑ Uϑ h = ⎣ S ∗k ϑh ⎦ , h ∈ N , k ∈ N, S ∗k ∆h and

$ %k Fϑ Uϑ PEFϑϑ Uϑ h = ϑk h, PM

h ∈ N , k ∈ N ∪ {0}.

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(ii) Let U ∈ L(F ) be a minimal unitary Hilbert space extension of V . Then U11 0 U= : [(F H) ⊕ N ] ⊕ D → [(F H) ⊕ M] ⊕ R 0 V with U11 a unitary operator from the Hilbert space (F H) ⊕ N onto the Hilbert space (F H) ⊕ M. Write E F : (F H) ⊕ N → (F H) ⊕ M. U11 = G H Then {F H, N , M; E, F, G, H} is a unitary colligation of operators with state space F H, input space N and output space M, all of them Hilbert spaces. It is known that its characteristic function z −→ H + zG(1 − zE)−1 F is an S(N , M)-function. Just note that H + zG(1 − zE)−1 F ≡ ϑU (z). (iii) For a given U ∈ L(F ) belonging to U(V ), set Ω(U ) = {z ∈ D : (1 − zU )−1 ∈ L(F )}. If z ∈ Ω(U ) and h ∈ H then U (1 − zU )−1 h = f ∈ F if and only if U h = f − zU f . Notice that U h = f − zU f if and only if H F V PDH h + U11 PN h = f − zV PDF f − zU11 (PN f + PFF H f ).

Whence U (1 − zU )−1 h = f if and only if F H F F F U11 PN h = PH f − zV PDF f − zPM U11 (PN f + PFF H f ) (5.1) V PDH h + PM

and H F F PFF H U11 PN h = PFF H f − zV PDF f − zPM U11 (PN f + PFF H f ).

(5.2)

From the arguments in the proof of (ii) it follows that, for all z ∈ D, (1−zPFF HU )−1 |(F H)⊕N = (1−zPFF HU11 )−1 |(F H)⊕N ∈ L((F H)⊕N ). H (z ∈ D). From (5.1) and (5.2) it can be Deﬁne TU (z) := V PDH + ϑU (z)PN seen that, for all z ∈ Ω(U ), (1 − zTU (z))−1 ∈ L(H) and F U (1 − zU )−1 |H = TU (z)(1 − zTU (z))−1 = (1 − zTU (z))−1 TU (z). PH

In particular, if U ∈ L(F ) and U ∈ L(F ) are two minimal unitary Hilbert space extensions of V such that ϑU (z) ≡ ϑU (z) then, for all z ∈ Ω(U ) ∩ Ω(U ),

F F PH U (1 − zU )−1 |H = PH U (1 − zU )−1 |H .

Therefore, U n h, HF = U n h, HF , h ∈ H, n ∈ N, and the operator deﬁned on the linear span of {U n H}∞ n=−∞ and mapping U n h into U n h (h ∈ H, n ∈ Z) gives a weak isomorphism from F onto F .

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Since F H is a Hilbert space, the weak isomorphism can be extended to an isometric isomorphism τ : F → F such that τ |H = 1 and τ U = U τ . As for the converse, let U ∈ L(F ), U ∈ L(F ) belong to U(V ) and assume that there exists an isometric isomorphism τ : F → F such that τ |H = 1 and τ U = U τ . Then τ PFF H U = PFF H U τ . So, for all x ∈ N , y ∈ M, and k ∈ N ∪ {0}, ( ) ' & $ %k F k F F F PM U PF H U x, y = PM U PF H U x, y . H

H

Hence,

F F U (1 − zPFF HU )−1 |N = PM U (1 − zPFF H U )−1 |N = ϑU (z) ϑU (z) = PM

for all z ∈ D.

References [1] T. Andˆ o, Linear Operators in Kre˘ın Spaces, Hokkaido University, Research Institute of Applied Electricity, Division of Applied Mathematics, Sapporo, 1979. [2] D.Z. Arov and L.Z. Grossman, Scattering matrices in the theory of dilations of isometric operators, Soviet Math. Dokl., 27(1983), No. 3, 518–522. [3] , Scattering matrices in the theory of unitary extensions of isometric operators, Math. Nachr., 157(1992), 105–123. [4] T. Azizov and I.S. Iokhvidov, Foundations of the Theory of Linear Operators in Spaces with Indeﬁnite Metric, “Nauka”, Moscow, 1986; English transl. Linear Operators in Spaces with Indeﬁnite Metric, Wiley, New York, 1989. [5] S. Bermudo, S.A.M. Marcantognini and M.D. Mor´ an, Operators of Hankel type, Czechoslovak Mathematical Journal, 56(2006), No. 4, 1147–1163. [6] J. Bognar, Indeﬁnite Inner Product Spaces, Springer-Verlag, Berlin, 1974. [7] A. Dijksma, H. Langer and H.S.V. de Snoo, Generalized coresolvents of standard isometric operators and generalized resolvents of standard symmetric relations in Krein spaces, Operator Theory: Adv. Appl., Vol. 48(1990), Birkhuser, 261–274. [8] M.A. Dritschel and J. Rovnyak, Extension theorems for contraction operators on Kre˘ın spaces, Operator Theory: Adv. Appl., Vol. 47(1990), Birkhuser, 221– 305. , Operators on Indeﬁnite Product Spaces, Lectures presented at the [9] meeting held at the Fields Institute for Research in Mathematical Sciences, Waterloo, Ontario, September 1994, edited by Peter Lancaster, Fields Institute Monographs 3, American Mathematical Society, Providence, RI, 1996. [10] B.W. McEnnis, Shifts on indeﬁnite inner product spaces, Paciﬁc J. Math., 81(1979), No. 1, 113–130. , Shifts on indeﬁnite inner product spaces II, Paciﬁc J. Math., [11] 100(1982), No. 1, 177–183. , Shifts on Krein spaces, Proceedings of Symposia in Pure Mathematics [12] 51(1990), Part 2, American Mathematical Society, Providence, RI, 201–211.

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[13] I.S. Iokhvidov, M.G. Kre˘ın and H. Langer, Introduction to the Spectral Theory of Operators in Spaces with an Indeﬁnite Metric, Akademie-Verlag, Berlin, 1982. [14] V. Pt´ ak, Factorization of Toeplitz and Hankel operators, Math. Bohem., 122(1997), No. 2, 131–140. [15] V. Pt´ ak and P. Vrbov´ a, Operators of Toeplitz and Hankel type, Acta Sci. Math. (Szeged), 52(1988), No. 1-2, 117–140. , Lifting intertwining relations, Integral Equations Operator Theory, [16] 11(1988), No. 1, 128–147. [17] B. Sz.-Nagy and C. Foia¸s, Harmonic analysis of operators on Hilbert space, North-Holland Publishing Co., Amsterdam-London, 1970. S.A.M. Marcantognini Department of Mathematics Instituto Venezolano de Investigaciones Cient´ıﬁcas P.O. Box 21827 Caracas 1020A Venezuela e-mail: [email protected] M.D. Mor´ an Escuela de Matem´ aticas Facultad de Ciencias Universidad Central de Venezuela Apartado Postal 20513 Caracas 1020A Venezuela e-mail: [email protected] Submitted: October 19, 2009. Revised: November 25, 2009.

Integr. Equ. Oper. Theory 66 (2010), 425–440 DOI 10.1007/s00020-010-1751-5 Published online February 16, 2010 © Birkhäuser/Springer Basel AG 2010

Integral Equations and Operator Theory

Completely Bounded and Ideal Norms of Multiplication Operators and Schur Multipliers T. Oikhberg Abstract. We estimate the completely bounded norms, the completely p-nuclear norms, and the completely p-summing norms of certain multiplication operators and Schur multipliers. Mathematics Subject Classiﬁcation (2010). Primary: 46L07; Secondary: 47B10, 47B49, 47L20. Keywords. Operator space, Schatten space, Schur multiplier, multiplication operator, completely p-summing norm, completely p-nuclear norm.

1. Introduction The goal of this paper is to compute the c.b. norms, completely p-summing norms, and completely p-nuclear norms of multiplication operators and Schur multipliers. Throughout, we use standard operator space and Banach space terminology and results. The reader is referred to [5, 14, 17] for operator spaces, [12] for Banach spaces, and [4] for operator ideals. We use E ⊗ F to denote the minimal (or spatial) tensor product of operator spaces E and F . The space of n × n matrices (with its usual operator space structure) is referred to as Mn . The space of operators on 2 which belong to the Schatten p-class is denoted by S p , while Snp stands for its n × n version. The symbol S ∞ refers to the space of compact operators on 2 . We denote the norm of S p by · p , and write T p = ∞ if T ∈ / Sp. As common in the operator space literature, we denote the space of n × n matrices (that is, Sn∞ ) by Mn . For the sake of brevity, we often use Mn (E) instead of Mn ⊗ E (E is an operator space). The author wishes to thank the organizers of the Workshop on Operator Spaces and Quantum Groups at Fields Institute (December 2007), where part of this work was carried out.

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The pairing between a matrix space and its conjugate arises from the parallel duality: formatrices (ﬁnite or inﬁnite) T = (tij ) and S = (sij ), T, S = Tr T tS = i,j tij sij (tS is the transpose of S). p We ﬁrst investigate the multiplication operators Mp,q → Sq : A,B : S 2 T → AT B, with A, B ∈ B( )\{0}. If there is no possibility of confusion regarding p and q, we use the notation MA,B . In Section 2, we estimate the c.b. norms of these operators. We prove that Mp,q A,B = A2r B2r , with 1/r = |1/p − 1/q| (Theorem 2.1). In Section 3, we deal with the completely s-summing and s-nuclear norms of multiplication operators (see below for the p,q o deﬁnition). Theorem 3.1 shows that πso (Mp,q A,B ) = νs (MA,B ) = Ar Br for certain 4-tuples (p, q, s, r) (see below for the deﬁnition and basic properties of πso (·) and νso (·)).

Next we turn our attention to Schur multipliers. Recall that, for a matrix φ = (φij ), the Schur multiplier with symbol φ (denoted by Sφ ) is the map taking an operator T = (Tij ) to (φij Tij ) (the latter matrix is sometimes denoted by φ T ). Note that, if A = diag (ai ) and B = diag (bj ), then MA,B is the Schur multiplier with the symbol (ai bj )∞ i,j=1 . In Section 4, we complement the results of [15] by computing the c.b. norms of certain Schur multipliers into 1 (N × N). In Section 5, we deal with completely p-summing norms of Schur multipliers. Theorem 5.1 gives a factorization result for completely p -summing Schur multipliers from B(2 ) to S p (1 ≤ p ≤ 2). In the particular case of p = 2, this is a complete description of completely p-summing Schur multipliers. Furthermore, for a Schur multiplier from B(2 ) to S 2 , we have π2o (Sφ ) = Sφ cb . Finally, we show that πpo (Sφ ) = φp when Sφ is considered as a map from S p into S p , where 1/p + 1/p = 1 (Proposition 5.5). In the commutative case, the operator norms of the diagonal maps from p to q are easy to compute. The summing norms (and other ideal norms) of such diagonal operators were computed by Garling and Carl [2, 6]. For the convolution operators Tµ : f → f ∗ µ, acting on L1 (G) or C(G) (G is a metrizable compact Abelian group), criteria for being nuclear or p-summing were given in [19] (that paper also covers convolutions with vector-valued measures). Further information on p-summing norms of convolutors from Lq to Lr can be found in [1]. In the non-commutative setting, certain estimates on ideal norms of formal identities between Schatten spaces were obtained in [3]. A few more ideal norms were estimated in [13]. Recall that the completely p-summing norm (sometimes also referred to as the operator p-summing norm) πpo (u) (1 ≤ p ≤ ∞) of an operator u : E → F (E and F are operator spaces) is deﬁned as πpo (u) = IS p ⊗ u : S p ⊗ E → S p [F ]. This notion was introduced by G. Pisier in [16]. We refer the reader to that article for the necessary background, such as the deﬁnition and properties of the space S p [F ], the non-commutative Pietsch factorization, and a variety of other results. Here, we recall a few basic facts (having well-known classical counterparts).

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• The norm πpo (·) has is an ideal norm: πpo (v1 uv2 ) ≤ v1 cb πpo (u)v2 cb . Moreover, πpo (u) = supG πpo (u|G ), with the supremum taken over all ﬁnite dimensional subspaces of G of the domain of u. o (u) = ucb. • If q ≥ p, then πpo (u) ≥ πqo (u). In particular, πpo (u) ≥ π∞ p 2 p • For A, B ∈ S , πp (MA,B : B( ) → S ) ≤ A2p B2p (below, we see that actually we have equality). We also need a few easy facts concerning multiplication operators: • The composition of multiplication operators is again a multiplication operator: MA1 ,B1 MA2 ,B2 = MA1 A2 ,B2 B1 . • The adjoint of multiplication operators is again a multiplication operator: (MA,B )∗ = MtB,tA . One more ideal of operators will be investigated. Following Section 3.1.3 of [7], we say that an operator u : X → Y is completely p-nuclear (1 ≤ p ≤ ∞) if it has a factorization u = wMA,B v, with v : X → S ∞ , w : S p → Y , and MA,B : S ∞ → S p , with A, B ∈ S 2p . The completely p-nuclear norm νpo (u) is deﬁned as the inﬁmum of wcb A2p B2p vcb , running over all the factorizations as above. Denote the class of completely p-nuclear operators by Npo . The factorization results for completely p-summing operators show that, for any u, πpo (u) ≤ νpo (u). Furthermore, it was shown by M. Junge [7] that Πop (X, Y ) and Npo (Y, X) (1/p + 1/p ) are in trace duality, provided the spaces X and Y are ﬁnite dimensional. To ﬁnish this Introduction, recall that, by [16], Snp = Cnp ⊗h Rnp , where p Cn (respectively, Rnp ) is the column (row) subspace of Snp . Moreover, for any operator space X, Snp [X] = Cnp ⊗h X ⊗h Rnp . In the inﬁnite dimensional case, we have S p = C p ⊗h Rp , and S p [X] = C p ⊗h X ⊗h Rp , where C p and Rp are the column and row subspaces of S p . It is known that Cn∞ = Cn , and Rn∞ = Rn (the usual column and row spaces). Moreover, Cnp∗ = Rnp = Cnp (1/p + 1/p = 1), and (Cnp0 , Cnp1 )θ = Cnp for 1/p = (1 − θ)/p0 + θ/p1 . These results remain true for C p and Rp .

2. Completely bounded norms Here we compute the c.b. norms of certain multiplication operators MA,B , with A, B ∈ B(2 )\{0}. Our main result is Theorem 2.1. For 1 ≤ p, q ≤ ∞, set 1/r = |1/p − 1/q|. Then MA,B CB(S p ,S q ) = A2r B2r . If either A or B does not belong to S r , then MA,B ∈ / CB(S p , S q ). The following lemma will be used extensively throughout this paper, in order to reduce our problems to ﬁnite dimensional ones. Denote by pn the projection onto the linear span of the ﬁrst n elements of the canonical basis of 2 .

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Lemma 2.2. Suppose A, B ∈ B(2 ), and E and F are either S r (1 ≤ r ≤ ∞) or B(2 ). Then MA,B CB(E,F ) = supn Mpn A,Bpn CB(E,F ). Moreover, for any 1 ≤ p < ∞, πpo (MA,B : E → F) = supn πpo (Mpn A,Bpn : E → F). A further technical lemma is needed. Below, Pn is the truncation operator T → pn T pn (in other words, Pn = Mpn ,pn ). p Lemma 2.3. Suppose 1 ≤ p < ∞, N ∈ N, and y ∈ SN [F ], where F is either q 2 p S (1 ≤ q ≤ ∞) or B( ). Then y = limn (ISN ⊗ Pn )(y).

Proof. The case of F = S q (with 1 ≤ q ≤ ∞) is easy. Indeed, the map z → (ISNp ⊗ Pn )(z) is contractive. Furthermore, we can approximate y by eleM p ments of the form i=1 ai ⊗ui , with ai ∈ SN and ui ∈ F. As limn Pn ui = ui for each i, we are done. The case of F = B(2 ) is slightly more complicated. Suppose, for the p sake of contradiction, that y ∈ SN [B(2 )] is such that y > 1, yet there 3 exists δ > 0 s.t. yn < (1 − δ) for any n (here, yn = (ISNp ⊗ Pn )(y)). Viewing Mn as the upper left corner of B(2 ), we can consider yn as an p [Mn ]. By Theorem 1.5 of [16], for each n there exist nonelement of SN 2p negative Cn , Dn ∈ SN and vn ∈ MN (Mn ), such that Cn = Dn = vn < 1 − δ, and yn = (Cn ⊗ I2n )vn (Dn ⊗ I2n ). By compactness, there ˜ exists n1 < n2 < . . . s.t. the sequences (Cnk ) and (Dnk ) converge to C, ˜ ˜ ˜ respectively D. Let C = σ + C and D = σ + D, where σ > 0 is so small that 2p . Let C and D lie in the unit ball of SN uk = (C −1 Cnk ⊗ I2n )vnk (Dnk D−1 ⊗ I2n ). k

k

Note that uk ≤ 1 − δ for each k, and ynk = (C ⊗ I2n )uk (D ⊗ I2n ), with k k uk MN (Mnk ) < 1. We can view (uk ) as a sequence in MN (B(2 )). Passing to a further subsequence if necessary, we can assume that (uk ) converges weak∗ to u in the unit ball of MN (B(2 )). Then weak∗ −lim ynk = (C ⊗I2 )u(D⊗I2 ) p [B(2 )]. On the other hand, weak∗ − lim ynk = y, lies in the unit ball of SN which is assumed to have norm greater than 1. This yields a contradiction. Proof of Lemma 2.2. We work with the completely p-summing norm, as the case of the c.b. norm is similar. We show that, if πpo (MA,B ) > 1, then p πpo (Mpn A,Bpn ) ≥ 1 for n suﬃciently large. Find N ∈ N and x ∈ SN ⊗E s.t. x < 1, and (IMN ⊗ MA,B )xSNp [F ] = (I2N ⊗ A)x(I2N ⊗ B)SNp [F ] > 1. By Lemma 2.3, (I2N ⊗ A)x(I2N ⊗ B) = lim (ISNp ⊗ Pn )((I2N ⊗ A)x(I2N ⊗ B)) n

= lim (I2N ⊗ pn A)x(I2N ⊗ Bpn ), n

yielding the desired result.

The next lemma yields upper estimates for the c.b. norms of the multiplication operators.

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Lemma 2.4. For 1 ≤ p, q ≤ ∞ and 1/r = |1/p − 1/q|, MA,B CB(S p ,S q ) ≤ A2r B2r . The same estimate holds when S ∞ (when p or q equals ∞) is replaced by B(2 ). Proof. By Lemma 2.2, it suﬃces to show that MA,B CB(Snp ,Snq ) ≤ A2r B2r , for n × n matrices A and B. By [23], the natural identiﬁcations Φ : Sn2r → CB(Cnp , Cnq ) and Ψ : Sn2r → CB(Rnp , Rnq ) are isometries. In this notation, MA,B = Φ(A) ⊗ Ψ(B). Indeed, by polar decomposition we can assume that A and B are diagonal: write A = diag ((ai )ni=1 ), B = diag ((bi )ni=1 ). The matrix units in Snp are denoted by Eij (1 ≤ i, j ≤ n). Identify Cnp and Rnp with span[Ei1 : 1 ≤ i ≤ n] and span[E1j : 1 ≤ j ≤ n], respectively. The matrix unit Eij can be identiﬁed with Ei1 ⊗ E1j . For ξ = i ξi Ei1 , Aξ = i ai ξi Ei1 . The action of B can be described in a similar way. Thus MA,B Eij = Φ(A)Ei1 ⊗ Ψ(B)Ej1 . By the properties of tensor products, MA,B cb = Φ(A) ⊗ Ψ(B)CB(Cnp ⊗h Rpn ,Cnq ⊗h Rqn ) ≤ Φ(A)CB(Cnp ,Cnq ) Ψ(B)CB(Rpn ,Rqn ) ≤ A2r B2r ,

as desired. Lemma 2.5. For any A, B ∈ B(2 )\{0} MA,B CB(S 1 ,S ∞ ) = MA,B CB(S 1 ,B(2 )) = A2 B2 . Proof. We only need to consider the ﬁnite dimensional version: MA,B CB(Sn1 ,Sn∞ ) = A2 B2 .

By polar decomposition, we can assume that A = diag ((ai )ni=1 ) and B = diag ((bi )ni=1 ). Then MA,B Eij = ai bj Eij , where Eij are the matrix units. The canonical isometry between CB(E, F ) and E ∗ ⊗ F (for ﬁnite dimensional E and F ) allows us to identify the operator u = MA,B ∈ CB(Sn1 , Sn∞ ) with n u ˜ = i,j=1 ai bj Eij ⊗ Eij ∈ Mn ⊗ Mn . The space Mn ⊗ Mn can be identiﬁed with Mn2 , with Eij ⊗ Ek corresponding to Eik,j . Therefore, ucb

n n n n = ˜ u = ( ai Eii,1 )( bj E1,jj ) = ai Eii,1 · bj E1,jj i=1

=

n i=1

|ai |2

1/2

j=1 n

|bj |2

1/2

i=1

j=1

= A2 B2 .

j=1

Next we work with the “weighted transposition” operator. We denote by Θ the transposition operator A → tA on the space of n × n matrices. In other words, ΘEij = Eji , where (Eij )ni,j=1 are the matrix units. The same operator, acting from Snp to Snq , is denoted by Θp,q . Lemma 2.6. For 1 ≤ s ≤ 2, ΘCB(Sns ,Sns ) = 1. Together with Lemma 2.4, this lemma immediately implies

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Corollary 2.7. Suppose 1 ≤ q ≤ ∞, 1 ≤ r ≤ 2, and 1/t = |1 − 1/r − 1/q|. Suppose, furthermore, that the sequences α = (αi )ni=1 and β = (βj )nj=1 satisfy α2t = β2t = 1. Then the operator T : S r → S q : Eji → αi βj Eij is completely contractive. Proof of Lemma 2.6. Clearly, Θcb ≥ ΘE11 = 1. To prove the converse inequality, note that, for s = 2, ΘCB(Sn2 ) = ΘB(Sn2 ) = 1, as the operator 1 Hilbert space Sn2 is 1-homogeneous. For s = 1, we identify Θ ∈ CB(Sn , Mn ) with the element ij Eji ⊗ Eij ∈ Mn (Mn ), which has norm 1. We complete the proof by using complex interpolation. Lemma 2.8. For every A, B ∈ B(2 )\{0}, MA,B CB(S ∞ ,S 1 ) = A2 B2 . Proof. As before, we can only consider the n × n case. The upper estimate for MA,B CB(S ∞ ,S 1 ) follows from Lemma 2.4. To prove the lower estimate, assume A = diag ((ai )ni=1 ) and B = diag ((bj )nj=1 ) with ai , bj non-negative, and A2 = B2 = 1. 1 Identify n the transposition map Θ : Sn → Mn : Eij → Eji with E ⊗ E ∈ M ⊗ M , which has norm 1. Thus, it suﬃces u ˜ = ji n n i,j=1 ij to show that ˜ v ≥ 1, where v˜ = (IMn ⊗ MA,B )˜ u=

n

Eij ⊗ aj bi Eji ∈ Mn ⊗ Sn1 .

i,j=1

But v˜ corresponds to v : Mn → Mn : Eji → aj bi Eij . To show that v is n u = completely contractive, note that (IMn ⊗ v)˜ i,j=1 Eij ⊗ aj bi Eij can 2 be identiﬁed with ( i bi Eii,1 )( j aj E1,jj ) ∈ Mn , the latter having norm one. Proof of Theorem 2.1. The case of p = q is trivial. Suppose p < q, and show that MA,B CB(S p ,S q ) = A2r B2r , where 1/r = 1/p − 1/q. The upper estimate for MA,B cb was established in Lemma 2.4. It suﬃces to obtain the lower estimate when A and B are positive n × n matrices. Let 1/r1 = 1−1/p and 1/r2 = 1/q −1/∞. Find A1 , B1 ∈ Sn2r1 and A2 , B2 ∈ Sn2r2 so that A1 2r1 = B1 2r1 = A2 2r2 = B2 2r2 = 1, A0 2 = A2r , and B0 2 = B2r , where A0 = A2 AA1 and B0 = B1 BB2 . Then MA0 ,B0 = MA2 ,B2 MA,B MA1 ,B1 , hence MA0 ,B0 CB(Sn1 ,Sn∞ ) ≤ MA1 ,B1 CB(Sn1 ,Snp ) MA,B CB(Snp ,Snq ) MA2 ,B2 CB(Snq ,Sn∞ ) . By Lemma 2.5, MA0 ,B0 CB(Sn1 ,Sn∞ ) = A0 2 B0 2 = A2r B2r . By Lemma 2.4, MA1 ,B1 and MA2 ,B2 are complete contractions. Thus, the centered inequality implies MA,B cb ≥ A2r B2r . The case of p > q is handled similarly, except that one uses Lemma 2.8 instead of Lemma 2.5.

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3. Completely p-summing norms Throughout this section, we use the notation w = w/(w − 1), for w ∈ [1, ∞] (in other words, w satisﬁes 1/w+1/w = 1). The main result of this section is Theorem 3.1. Consider MA,B as an operator from S p1 to S p2 . In each of the following situations, πpo (MA,B ) = νpo (MA,B ) = A2r B2r . 1. 2. 3. 4. 5.

p1 ≤ p ≤ p2 , 1/r = 1/p1 + 2/p − 1/p2 . p2 ≤ p ≤ p1 , 1/r = 1/p1 + 1/p2 . p2 ≤ min{2, p} ≤ max{2, p} ≤ p1 , 1/r = 1/p1 + 1/p2 . p1 ≤ p ≤ p2 ≤ 2, 1/r = 1/p2 + 2/p − 1/p1 . p1 = ∞, 1/r = 1/p + |1/p − 1/p2 |. In particular, if p1 = ∞ ≥ p ≥ p2 , then νpo (MA,B ) = πpo (MA,B ) = MA,B cb = A2p B2p .

Note that part (5) a strengthening of the Maurey Factorization Theorem for the class of multiplication operators (see Chapter 10 of [4] for the classical case, and [9, 10] for the recent non-commutative results). Observing that MI,I is the identity operator on Snq , we conclude: Corollary 3.2. For n ∈ N, q ∈ [2, ∞], and q/(q − 1) ≤ p ≤ q, πpo (ISnq ) = n2/p . More generally, for such p and q, πpo (ISnq 1 n2 ) = (n1 n2 )1/p . Remark 3.3. For p = 2,√this result follows from Chapter 6 of [16], where it was shown that π2o (IE ) = dim E for any ﬁnite dimensional operator space E. For p = 2, the completely p-summing norms of operators on OH, and some other homogeneous Hilbertian spaces, were recently computed by M. Junge, Q. Xu, and K.-L. Yew [11, 24]. In the classical case, the ideal norms (including the p-summing norms) of Iqn can be found in e.g. [2, 6]. Furthermore, by Lemma 5.2 of [3], π2 (id : Snp1 n2 → Snq 1 n2 ) =

√ max{1, min{n1 , n2 }1/q−1/2 } . n1 n2 max{1, min{n1 , n2 }1/p−1/2 }

By Lemma 2.2, it suﬃces to establish the ﬁnite dimensional case of Theorem 3.1. Accordingly, we restrict ourselves to the ﬁnite dimensional situation for the rest of this section. Furthermore, we assume that A = (diag (ai )ni=1 ) and B = (diag (bj )nj=1 ), with (ai ) and (bj ) positive. 1 ,p2 Lemma 3.4. In each of the following cases, νpo (MpA,B ) ≤ A2r B2r :

1.

1 1 1 1 1 1/p1 + 1/p2 , p ≥ p2 = + + − = 2/p + 1/p − 1/p , p ≤ p2 . r p1 p p p2 1 2

2. p1 ≤ 2 ≤ p2 ,

1 1 1 1 1 1/p1 + 1/p2 , p ≥ p2 = + + − = 2/p + 1/p1 − 1/p2 , p ≤ p2 . r p1 p p p2

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1 ,p2 Lemma 3.5. In each of the following cases, πpo (MpA,B ) ≥ A2r B2r : 1. 1 1 1 1 1 1/p1 − 1/p2 + 2/p, p ≥ p1 = + − − = 1/p1 + 1/p2 , p ≤ p1 . r p2 p p p1

2.

1 1 1 1 1 p1 ≥ p 1/p1 + 1/p2 , = + − − = 2/p + 1/p2 − 1/p1 , p1 ≤ p. r p2 p p p1

3. p1 = ∞, 1/r = 1/p + |1/p − 1/p2 |. Proof of Lemma 3.4. (1) It suﬃces to consider the case of A2r = B2r = 1. Write A = A3 A2 A1 and B = B1 B2 B3 , where A1 2p1 = B1 2p1 = 1, A2 2p = B2 2p = 1, and A3 2q = B3 2q = 1, where 1/q = |1/p − 1/p2 |. We regard MA1 ,B1 , MA2 ,B2 , and MA3 ,B3 as acting from S p1 to S ∞ , from S ∞ to S p , and from S p to S p2 , respectively. Then νpo (MA,B ) ≤ MA1 ,B1 cb νpo (MA2 ,B2 )MA3 ,B3 cb ≤ 1.

p ,p

p1 ,p1 1 ,p2 1 2 = Θp2 ,p2 ◦ MtB, , Lemma 2.6, (2) The factorization MpA,B tA ◦ Θ and part (1) yield the result.

Proof of Lemma 3.5. By Lemma 2.2, it suﬃces to estimate πpo (MA,B ) from below when A and B are n × n matrices. By scaling and approximation, we can assume that A, B ∈ Mn , A2r = B2r = 1, A = diag ((ai )ni=1 ), and B = diag ((bj )nj=1 ), with ai , bj positive. (1) Let 1/s = 1/p2 + 1/p + |1/p1 − 1/p |, and note that 1/r + 1/s = 2. (1) (1) Find positive operators A1 = (diag (ai )ni=1 ) and B1 = (diag (bj )nj=1 ) s.t. A1 A1 = A1 2s = A2r = 1 = BB1 1 = B1 2s = B2r . (1)

(1)

Then MA1 B1 MAB Eij = ai ai bj bj Eij for any matrix unit Eij , hence the trace of MA1 B1 MAB (acting on Snp1 ) equals (1) (1) ai ai bj bj = A1 A1 BB1 1 = 1. i,j

By the trace duality, 1 = Tr (MA1 B1 MAB ) ≤ πpo (MAB )νpo (MA1 B1 ). By Lemma 3.4, νpo (MA1 B1 ) ≤ A1 2s B1 2s = 1, which yields the desired estimate for πpo (MAB ). (2) Suppose, for the sake of contradiction, that πpo (MA,B ) < 1. Let 1/s = |1/p2 − 1/p|, 1/t = |1/p − 1/p1 |, and q = min{p, p2 }. Find sequences (1) (1) (2) (2) a(1) = (ai )ni=1 , b(1) = (bj )nj=1 , a(2) = (ai )ni=1 , and b(2) = (bj )nj=1 in such a way that (1)

(2)

(1) (2)

(ai )2t = (ai )2s = (ai ai ai )p2 (1)

(2)

(1)

(2)

= (bj )2t = (bj )2s = (bj bj bj )q = 1

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(this is possible, since (ai )2r = (bj )2r = 1, and 1/r + 1/s + 1/t = 2/q). (1) (1) Consider u = ij ai bj Eji ⊗ Eij ∈ Snp ⊗ Snp1 . By Corollary 2.7, u ≤ 1. Let (1) (1) v = (ISnp ⊗ MA,B )(u) = ai ai bj bj Eji ⊗ Eij . ij

πpo (MA,B )

If < 1, then vSnp [Snp2 ] < 1. We achieve the desired contradiction by showing that vSnp [Snp2 ] ≥ 1. Suppose ﬁrst p ≥ p2 . By [22], vSnp [Snp2 ] =

sup C2s ,D2s ≤1

(MC,D ⊗ ISnp2 )vSnp2 [Snp2 ] .

(2)

(3.1)

(2)

Taking C = (diag (bj )) and D = (diag (ai )), we have: vSnp [Snp2 ] ≥ =

n i,j=1

n

i,j=1

(1) (2)

(1) (2)

Eji ⊗ ai ai ai bj bj bj Eij Snp2 [Snp2 ] |ai ai ai bj bj bj |p2 (1) (2)

(1) (2)

(1) (2)

1/p2

(1) (2)

= (ai ai ai )p2 (bj bj bj )p2 = 1, which is the estimate we need. For p ≤ p2 , the proof proceeds along similar lines, with one exception: instead of (3.1), we use the inequality vSnp [Snp2 ] ≥ (I2n ⊗ C)v(I2n ⊗ D)Snp [Snp ] whenever C and D are in the unit ball of Sn2s (this follows from the fact that MC,D : Snp2 → Snp is completely contractive, and Corollary 1.2 of [16]). In particular, vSnp [Snp2 ] ≥ (I2n ⊗ A2 )v(I2n ⊗ B2 )Snp [Snp ] (1) (2) (1) (2) = ai ai ai bj bj bj Eji ⊗ Eij Snp [Snp ] (1) (2)

(1) (2)

= (ai ai ai )p (bj bj bj )p = 1. (3) In the case of p2 ≤ p, the inequality πpo (T ) ≥ T cb (valid for any operator T ), and the equality MA,B CB(S ∞ ,S p2 ) = A2p2 B2p2 (Theorem 2.1), yield the proof. For p2 ≥ p, Remark 5.10 of [16] yields a factorization MA,B = wMA1 ,B1 via Snp , with A1 2p = B2p = 1, and wcb = πpo (MA,B ). As the operators A and B are invertible, so are A1 and B1 , and o therefore, w = MA,B M−1 A1 ,B1 = MAA−1 ,B −1 B . Then wcb = πp (MA,B ) = 1

1

−1 AA−1 1 2s B1 B2s , with 1/s = 1/p−1/p2 . As 1/r = 2/p−1/p2 = 1/s+1/p, ∞,p2 −1 o A2r B2r ≤ AA−1 1 2s A1 2p B1 2p B1 B2s ≤ πp (MA,B ).

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Proof of Theorem 3.1. For i, j ∈ {0, 1}, case (2i+j+1) of our theorem follows from Lemma 3.4(i) and Lemma 3.5(j). Case (5) follows from Lemma 3.4(1), Lemma 3.5(3), and Theorem 2.1.

4. Toeplitz Schur multipliers into 1 Here, we consider the c.b. norms of Toeplitz Schur multipliers from a matrix space into 1 (N × N). For a sequence m = (ms )s∈Z , we deﬁne the Toeplitz Schur multiplier Tm : (aij )i,j∈N → mj−i aij . The result below (essentially contained in [15]) describes the c.b. norm of Tm as acting from S 1 to 1 (N×N). Proposition 4.1. In the above notation, Tm cb = Tm = m1 . Proof. It is shown in [15] that Tm = m1 . It remains to show that Tm cb ≤ m1 . By an extreme point argument, it suﬃces to consider the case of ms = 1 for s = p, ms = 0 otherwise. For such an m, we have to show that IS 1 ⊗ Tm : S 1 [S 1 ] → S 1 [1 (N × N) ≤ 1. Note that S 1 [S 1 ] is isometric to S 1 (N × N). The extreme points of the closed unit ball of the latter space are of the form x ⊗ y, where x and y are unit vectors in 2 (N × N). Write 2 x = (xj ) and y = (yk ), where xj , yk ∈ , and j xj 2 = k yk 2 = 1. Then, xj ⊗ yk , j − k = p IS 1 ⊗ Tm (x ⊗ y) jk = 0, otherwise. If p ≥ 0, we have: ∞ xj yk = xk+p yk Tm (x ⊗ y) = ≤

j−k=p ∞

xk+p 2

k=1 ∞

1/2

k=1

yk 2

1/2

≤ 1.

k=1

The case of p ≤ −1 is handled similarly.

Next we consider a Schur multiplier from the upper triangular trace class matrices to 1 . More precisely, we consider the subspace HS 1 of S 1 , consisting of all trace class matrices (aij ) s.t. aij = 0 for i > j. A sequence (ms )s≥0 deﬁnes a multiplier Tm : HS 1 → 1 (N × N) : (aij ) → (mj−a aij ). Following [15], we deﬁne r(q+1) ∞ 2 1/2 ρ(m) = |m0 |2 + |m1 |2 + sup |ms |2 . r≥1 q=1

s=rq+1

Proposition 4.2. In the above notation, there exists a constant c such that ρ(m) cρ(m) ≥ Tm cb ≥ Tm ≥ √ . 3

√ Sketch of proof. By Theorem 3.3 of [15], cρ(m) ≥ Tm ≥ ρ(m)/ 3. In fact, that proof can be easily modiﬁed to yield cρ(m) ≥ Tm cb .

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Finally, we consider the Schur multipliers from B(2 ) (or S ∞ ) to (N × N). The space of bounded (completely bounded) Schur multipliers between matrix spaces E and F shall be denoted by B S (E, F ) (resp. CB S(E, F )). The space 1 (2 ) is viewed as the space of matrices a = (aij ), equipped with the norm a1 (2 ) = i ( j |aij |2 )1/2 . The space t1 (2 ) is equipped with the norm at1 (2 ) = ta1 (2 ) . Finally, we deﬁne the space 1

X = 1 (2 ) + t1 (2 ). That is, aX = inf b1 (2 ) + ct1 (2 ) . a=b+c

In this notation, we have: Proposition 4.3. The following spaces are isomorphic (via the natural identity): (1) X , (2) B S (B(2 ), 1 (N × N)), (3) CB S (B(2 ), 1 (N × N)), (4) B S (S ∞ , 1 (N × N)), (5) CB S (S ∞ , 1 (N × N)). Proof. It is easy to see that the spaces (2) and (4) are isometric, as are (3) and (5). It is shown in [15] that (1) and (3) are isomorphic. Moreover, id : CB S (B(2 ), 1 (N × N)) → B S (B(2 ), 1 (N × N)) is a contraction. By duality, it suﬃces to show that id : X → CB S (c0 (N×N), S 1 ) is a contraction. By symmetry, we need to prove that id : 1 (2 ) → CB S (c0 (N × N), S 1 ) is a contraction. By an extreme point argument, it suﬃces to show that Sa CB S (c0 (N×N),S 1 ) ≤ 1 whenever the matrix a = (aij ) is such that aij = 0 for i > 1, and j |a1j |2 = 1. This, in turn, is equivalent to proving that, for any family (xij ) of contractive n × n matrices, j a1j E1j ⊗ x1j S 1 ⊗Mn ≤ 1. The matrix units E1jspan a copy of C (the column space) in S 1 . Thus, we need to show that j a1j Ej1 ⊗ x1j S ∞ ⊗Mn ≤ 1. However, a1j Ej1 ⊗ x1j 2S ∞ ⊗Mn = |a1j |2 x∗1j x1j ≤ |a1j |2 x∗1j x1j ≤ 1, j

j

j

as desired.

5. Schur multipliers into Schatten spaces In this section we consider Schur multipliers from S p (or B(2 )) to S q . Theorem 5.1. Consider 1 ≤ p ≤ 2, a matrix φ = (φij ), and the following statements: 1. We have πpo (Sφ ) ≤ 1, where Sφ is viewed as acting from B(2 ) (or S ∞ ) to S p . 2. There exist sequences a = (ai )i∈N and b = (bj )j∈N in the unit ball of 2p , such that |φij | ≤ |ai bj | for any i, j ∈ N.

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3. There exists a matrix ψ, the sequences a = (ai )i∈N and b = (bj )j∈N in the unit ball of 4 , and the sequences a = (ai )i∈N and b = (bj )j∈N in the unit ball of 2r (1/r = 1/p − 1/2), such that |ψij | ≤ |ai bj |, and Sφ = Mdiag (ai ),diag (bi ) Sψ . Here, π2o (Sψ ) ≤ 1 (Sψ is viewed as a map from B(2 ) to S 2 ), and furthermore, Mdiag (ai ),diag (bi ) CB(S 2 ,S p ) ≤ 1.

Then (1) ⇒ (2) ⇔ (3). For p = 2, (2) ⇔ (1).

By Theorem 5.5 and Remark 5.6, the implication (2) ⇒ (1) fails for p = 1. We do not know whether it is true for p ∈ (1, 2). The next result shows that, for p = 2, the c.b. norms and the completely 2-summing norms of Schur multipliers from B(2 ) to S 2 are equivalent. Theorem 5.2. There is a constant C such that π2o (Sφ ) ≤ CSφ cb for any Schur multiplier Sφ from B(2 ) (or S ∞ ) to S 2 . Remark 5.3. The classical Grothendieck Theorem tells us that any bounded operator from C(K) to a Hilbert space is 2-summing. This is no longer true in the non-commutative case. Indeed, by Corollary 5 of of [8], for every N there exists a projection P from B(2 ) to OHN s.t. P cb ≤ c N/(1 + log √N ). On o o the other hand, by Theorem 6.13 of [16], π2 (P ) ≥ π2 (idOHN ) = N . See [9, 10] for some positive results in this direction. Considering multiplication operators from L∞ (µ) to Lp (µ), instead of Schur multipliers, we complement the known results by computing the completely p-summing norm precisely. Theorem 5.4. Suppose µ is a σ-ﬁnite measure, and φ is a µ-measurable function. Denote by Mφ the operator of multiplication by φ, acting from L∞ (µ) to Lp (µ). Then, for 1 ≤ p ≤ ∞, πp (Mφ ) = πpo (Mφ ) = Mφ = Mφ cb = φp . For the sake of completeness, we include a formula for the completely p-summing norms of Schur multipliers from S p to S p . Proposition 5.5. Suppose φ is a matrix, determining the Schur multiplier Sφ from S p to S p (1 ≤ p ≤ ∞, 1/p + 1/p = 1). Then πpo (Sφ ) = φp . For p = 1, the result is also true for Schur multipliers from B(2 ) to S 1 . Note that, by applying this theorem to MA,B , we recover a particular case of Theorem 3.1 (that of p = p2 = p1 /(p1 − 1)). Remark 5.6. In [18, 22], Sφ cb was computed for 1 ≤ p ≤ 2. A counterpart of the previous proposition for sequence spaces was established in [6], where the following was proved: suppose 1 ≤ p ≤ ∞, and a = (ai ) is a bounded sequence, and Da : p → p maps δi to ai δi (here, (δi ) and (δi ) are the canonical bases in p and p , respectively). Then πp (Da ) = ap . In particular, for p = 1, Da = π1 (Da ). In contrast to this, the c.b. norm and the completely 1-summing norm of Schur multipliers from S ∞ or B(2 ) into S 1 are not equivalent. Indeed, by [20], there exist matrices φ and ψ s.t. |ψij | ≤ φij for any pair (i, j), φ ∈ S 1 , while ψ ∈ / S 1 . By Proposition 5.5, Sψ

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is not completely 1-summing. However, by [18], Sψ is a completely bounded map from S ∞ (or B(2 )) to S 1 . To prove the results of this section, we state an immediate consequence of [16], Theorem 5.9. Lemma 5.7. Consider an operator u : Mn → Snp (1 ≤ p ≤ ∞). Then πpo (u) ≤ 1 if and only if there exist non-negative a and b in the unit ball of Sn2p , such that (u(x(ij) ))SNp [Snp ] ≤ (ax(ij) b)SNp [Snp ] whenever x(ij) (1 ≤ i, j ≤ N ) are elements of Mn . Next we prove Theorem 5.1. The implication (3) ⇒ (2) follows immediately from Theorem 2.1. Furthermore, (3) ⇒ (1) for p = 2 is immediate. Proof of Theorem 5.1, (2) ⇒ (3), p = 2. It suﬃces to show the ﬁnite dimensional version. That is, consider an n × n matrix φ = (φk ), such that 4 |φ | ≤ a b , where (a ), (b ) are non-negative numbers, with k k k k ak = 4 b = 1. By Lemma 5.7, we have to show that (Sφ (x(ij) ))SN2 [Sn2 ] ≤ (ax(ij) b)SN2 [Sn2 ] , where a = diag ((ai )) and b = diag ((bj )). However, Sφ (x(ij) )2Sn2 (Sφ (x(ij) ))2S 2 [S 2 ] = N

n

and (ax(ij) b)2S 2 [S 2 ] = N

n

i,j

i,j

ax(ij) b2Sn2 ,

hence it suﬃces to show that Sφ xSn2 ≤ axbSn2 for any n × n matrix x. Writing x = (xk ) and φ = (φk ), we see that Sφ x2Sn2 =

n k,=1

|φk |2 |xk |2 ≤

n k,=1

a2k b2 |xk |2 = axb2Sn2 .

Proof of Theorem 5.1, (1) ⇒ (2). We handle the ﬁnite dimensional case ﬁrst. Suppose an n × n matrix φ is such that πpo (Sφ ) ≤ 1 (Sφ is viewed as a map from Mn to Snp ). We shall show that there exist a = (ai )ni=1 and b = (bj )nj=1 in the unit ball of 2p n , with non-negative entries, such that |φij | ≤ ai bj . Suppose Sφ : Mn → Snp satisﬁes πpo (Sφ ) ≤ 1. By Lemma 5.7, there exist c and d in the unit ball of Sn2p s.t. Sφ xp ≤ cxdp for any n × n matrix x. If U and V are n × n matrices with ±1 on the main diagonal, and zeroes elsewhere, we have Sφ xp = Sφ (U xV )p ≤ cU xV dp , and p/2 p/2 = Tr d∗ V x∗ U c∗ cU xV d . φ xpp ≤ Tr (cU xV d)(d∗ V x∗ U c∗ ) (5.1) The last inequality follows from the fact that any continuous function on a compact interval can be approximated by a polynomial arbitrarily well, hence Tr (F (AB)) = Tr (F (BA)) whenever AB and BA are non-negative matrices, and F is continuous on [0, ∞) (in our case, F (t) = tp/2 ). By a special case of

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Lieb’s concavity (see e.g. Theorem 8.10 of [21]), the function T → Tr T α is concave on the set of non-negative matrices for 0 < α ≤ 1. Note that, for any matrix y, Ave(U yU ) is a matrix with the same entries as y on the main diagonal, and zeroes elsewhere (here, we average over all possible choices of signs in U ). Averaging with respect to U in (5.1), we obtain a diagonal operator a in the unit ball of Sn2p with non-negative entries, such that a2 = AveU (U c∗ cU ), and p/2 p/2 φ xpp ≤ Tr d∗ V x∗ a2 xV d = Tr x∗ a2 xV dd∗ V for any V as above. There exists a diagonal operator b in the unit ball of Sn2p with non-negative entries, such that b2 = AveV (V d∗ dV ). Then p/2 p/2 = Tr (bx∗ a)(axb) = axbpp φ xpp ≤ Tr x∗ a2 xb2 for any matrix x. Now consider the inﬁnite dimensional case. By truncating to the upper left corner of size n × n, and applying the ﬁnite dimensional result prove (n) (n) above, we conclude that, for every n, there exist a(n) = (a1 , . . . , an , 0, . . .) (n) (n) and b(n) = (b1 , . . . , bn , 0, . . .) in the unit ball of 2p , with non-negative (n) (n) entries, such that |φij | ≤ ai bj for 1 ≤ i, j ≤ n. Then there exist a = (ai ), (n )

(n )

b = (bj ), and n1 < n2 < . . . s.t. limk ai k = ai for every i, and limk bj k = bj for every j. Clearly, a and b are still in the unit ball of 2p , and |φij | ≤ ai bj for all i and j. Proof of Theorem 5.1, (2) ⇒ (3). Suppose an n × n matrix φ = (φij ) satis ﬁes |φ | ≤ ai bj , where ai and bj are non-negative numbers, and i a2p i = 2pij 1−p/2 b ≤ 1. Write φ = ω |φ |, with |ω | = 1. Let ψ = ω |φ | , ij ij ij ij ij ij ij j j 4 p/2 p/2 (2−p)/2 (2−p)/2 ai = a i , b i = b i , a i = ai , and bi = bi . Then ai = 4 2r 2r a2p = 1, and, similarly, b = 1, and a = b = 1. Moreover, i i i i |ψij | ≤ ai bj . As proved before, π2o (Sψ ) ≤ 1 (Sψ is viewed as a map from S ∞ or B(2 ) to S 2 ). Moreover, by Theorem 2.1, Mdiag (ai ),diag (bi ) is a complete contraction from S 2 to S p . Proof of Theorem 5.2. Suppose Sφ : S ∞ → S 2 is a complete contraction, and prove that π2o (Sφ ) ≤ C, for some universal constant C. Note that S∗φ : S 2 → S 1 is a completely contractive Schur multiplier, given by the same matrix φ. Therefore, Sψ = S∗φ Sφ is a complete contraction (here, ψ = φ φ, or in other words, ψij = φ2ij for i, j ∈ N). By [18], there exist sequences (αi ) 2 C 2 (C is and (βj ) of non-negative numbers, such that i α2i = j βj ≤ √ an absolute constant), such that |ψij√| ≤ αi βj for all i, j. Let ai = αi , and bj = βj . Then (ai )4 , (bj )4 ≤ C, and |φij | ≤ ai bj . By Theorem 5.1, π2o (Sφ ) ≤ C. Proof of Theorem 5.4. Clearly, Mφ = φp ≤ Mφ cb ≤ πpo (Mφ ), and Mφ ≤ πp (Mφ ). It remains to show that φp ≥ max{πp (Mφ ), πpo (Mφ )}.

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Suppose φp = 1, and, without loss of generality, φ > 0 everywhere. Consider the probability measure ν = φp µ, and the complete isometries T∞ : L∞ (µ) → L∞ (ν) : f → f and Tp : Lp (ν) → Lp (µ) : g → φg. Let J be the formal identity map from L∞ (ν) to Lp (ν). Then Mφ = Tp JT∞ . By Pietsch Factorization Theorem, πp (Mφ ) ≤ 1. Finally, by Proposition 5.12 of [16], πpo (J) ≤ 1, hence πpo (Mφ ) ≤ 1 Proof of Proposition 5.5. As usual, it suﬃces to consider the ﬁnite dimensional version of the problem. That is, suppose φ is an n × n matrix, deter mining a Schur multiplier Sφ : Snp → Snp (1/p + 1/p = 1). We have to show that πpo (Sφ ) = φp . n We can identify Sφ with an element u = i,j=1 φij Eij ⊗ Eij ∈ Snp [Snp ], n or with u = i,j=1 φij Eii,jj ∈ Snp2 . By Lemma 5.14 of [16], πpo (Sφ ) = u = u = φp .

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[15] A. Pelczy´ nski and F. Sukochev, Some remarks on Toeplitz multipliers and Hankel matrices. Studia Math. 175 (2006), 175–204. [16] G. Pisier, Lp spaces. Asterisque 247 (1998). [17] G. Pisier, An introduction to the theory of operator spaces, Cambridge University Press, Cambridge, 2003. [18] G. Pisier and D. Shlyakhtenko, Grothendieck’s theorem for operator spaces. Invent. Math. 150 (2002), 185–217. [19] M. Robdera and P. Saab, Convolution operators associated with vector measures. Glasgow Math. J. 40 (1998), 367–384. [20] B. Simon, Pointwise domination of matrices and comparison of Sp norms. Paciﬁc J. Math. 97 (1981), 471–475. [21] B. Simon, Trace ideals and their applications. Second Edition, Amer. Math. Soc., Providence RI, 2005. [22] Q. Xu, Operator-space Grothendieck inequalities for noncommutative Lp spaces. Duke Math. J. 131 (2006), 525–574. [23] Q. Xu, Embedding of Cq and Rq into non-commutative Lp spaces, 1 ≤ p < q ≤ 2. Math. Ann. 335 (2006), 109–131. [24] K.-L. Yew, Completely p-summing maps on the operator Hilbert space OH. Preprint. T. Oikhberg Department of Mathematics University of California at Irvine Irvine CA 92697 USA and Department of Mathematics University of Illinois at Urbana-Champaign Urbana IL 61801 USA e-mail: [email protected] Submitted: May 03, 2009. Revised: October 26, 2009.

Integr. Equ. Oper. Theory 66 (2010), 441–461 DOI 10.1007/s00020-010-1755-1 Published online February 23, 2010 © Birkhäuser/Springer Basel AG 2010

Integral Equations and Operator Theory

Pseudodiﬀerential Operators with Rough Negative Deﬁnite Symbols Alexander Potrykus Abstract. We consider state-space dependent continuous negative definite functions and use their associated pseudodiﬀerential operators to construct Feller semigroups. Our method works with “rough” symbols p(x, ξ), i.e. ξ → p(x, ξ) only needs to be continuous. The main part of this work concerns the development of an asymptotic expansion formula for the composition of two pseudodiﬀerential operators with rough negative deﬁnite symbols. This presents an improvement over other symbolic calculi that typically require the symbols to be smooth. As an application we show how to adapt existing techniques to construct and approximate Feller semigroups to the case of rough symbols. Mathematics Subject Classiﬁcation (2010). Primary 47G30; Secondary 35S05, 47D06, 60J35. Keywords. Pseudodiﬀerential operators, rough symbols, symbolic calculus, Feller semigroups, asymptotic expansions.

1. Introduction A pseudodiﬀerential operator with negative deﬁnite symbol is an operator of the form n eix·ξ p(x, ξ) u(ξ) dξ, p(x, D)u(x) = (2π)− 2 Rn

n

n

n

u ∈ S(R ), where p : R × R → C is the symbol of the operator and ξ → p(x, ξ) is a continuous negative deﬁnite function for all x ∈ Rn . Negative deﬁnite functions are perhaps best characterized by their L´evy-Khinchin representation, i.e. for ﬁxed x ∈ Rn we have p(x, ξ) = a(x) − il(x) · ξ + ξ · Q(x)ξ iy · ξ iy·ξ + 1−e N (x, dy), + 1 + |y|2 Rn \{0} The author is supported by an RCUK-Fellowship.

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where for each x ∈ Rn , a(x) ∈ R+ , l(x) ∈ Rn , Q(x) is a positive semideﬁnite matrix and N (x, ·) is a Radon measure on Rn \{0} such that |y|2 N (x, dy) < ∞. 2 Rn \{0} 1 + |y| Pseudodiﬀerential operators of this type have frequently been investigated as generators of Feller semigroups. Their connection to the theory of stochastic processes can be seen by considering the constant coeﬃcient case. Let ψ : Rn → C be a continuous negative deﬁnite function. Then it is well known that ψ is the characteristic exponent of a L´evy process (Xt )t≥0 , E x eiξ·(Xt −x) = E 0 (eiξ·Xt ) = e−tψ(ξ) , as standard reference we give the monograph of Sato [18]. Two classical examples are given by ψ(ξ) = |ξ|2 , which corresponds to Brownian motion, and ψ(ξ) = |ξ| which corresponds to the Cauchy process. It can be shown that every L´evy process is generated by a pseudodiﬀerential operator with a constant coeﬃcient negative deﬁnite symbol ψ(ξ). Conversely, every such pseudodiﬀerential operator generates a L´evy process. In this paper a construction of Feller processes is discussed. Feller processes appear for example as solutions to stochastic diﬀerential equations with Lipschitz continuous coeﬃcients. As such their construction and approximation is of immediate interest for applications. The starting point is to consider the variable coeﬃcient case, i.e. for a symbol p(x, ξ), we are interested in showing under which additional conditions the corresponding pseudodiﬀerential operators generate Feller processes. It is no longer true that all such pseudodiﬀerential operators lead to Feller processes. Some initial solutions to this problem go back to Jacob [6], and further details can be found in the survey article [12] and the monographs [7, 8, 9]. A stochastic approach via the martingale problem can be found in Hoh [3, 4]. In order to see how our result ﬁts in with the existing ones, we now take a look at some of the conditions imposed on the symbol p(x, ξ) by the diﬀerent approaches. We will not give a complete list but rather concentrate on some cases that are of particular interest to us. In the following we always assume that p : Rn × Rn → C is a negative deﬁnite symbol. We also often need a negative deﬁnite “reference” function ψ : Rn → R that satisﬁes ψ(ξ) ≥ c|ξ|r0 for some c > 0, r0 > 0 and |ξ| ≥ 1. The approach by Jacob [8] uses the same decomposition of the symbol p as we will do later on. Assume that p has the decomposition p(x, ξ) = p1 (ξ) + p2 (x, ξ), where p1 is continuous negative deﬁnite function that satisﬁes for γ0 > 0, γ1 , γ2 > 0, |ξ| ≥ 1 γ0 ψ(ξ) ≤ Re p1 (ξ) ≤ γ1 ψ(ξ),

(1.1)

|Im p1 (ξ)| ≤ γ2 Re p1 (ξ),

(1.2)

and

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n

for all ξ ∈ Rn . Further assume that x → p2 (x, ξ) belongs to C r0 +n+3 (Rn ) for all ξ ∈ Rn and that for β ∈ Nn0 with |β| ≤ rn0 + n + 3 the estimate (1.3) |∂xβ p2 (x, ξ)| ≤ ϕβ (x) 1 + ψ(ξ) holds where ϕα ∈ L1 (Rn ). The ﬁnal assumption is that ϕα L1 ≤ cn,r0 ,ψ

(1.4)

|α|≤ rn +n+3 0

where the precise conditions such that cn,r0 ,ψ > 0 are given in [8]. Conditions (1.1) and (1.2) allow us to take p1 as the reference function ψ. Assumptions (1.3) and (1.4) require p2 to have a high diﬀerentiability order with respect to x ∈ Rn and that the symbol p2 is a small perturbation of p1 . These conditions are needed to construct a solution to the pseudodiﬀerential equation λu + p(x, D)u = f on some anisotropic Sobolev spaces, for further details we refer to Theorem 2.6.4 in [8]. The estimates required to prove this result are not straightforward and one often has to resort to using commutators. Note that Hoh’s proof of the uniqueness of the martingale problem essentially uses the ideas just outlined combined with some stopping-time techniques [3]. Hoh introduced in [5] a symbolic calculus for pseudodiﬀerential operators with negative deﬁnite symbols. In particular he develops an asymptotic expansion formula for the composition of two pseudodiﬀerential operators which allows us to employ techniques well-established in the theory of classical pseudodiﬀerential operators. For Hoh’s approach we need that the reference function is smooth and satisﬁes 2−ρ(|α|) 2 (1.5) |∂ξα 1 + ψ(ξ) | ≤ c 1 + ψ(ξ) for all α ∈ Nn0 where ρ(k) = min(k, 2). Then we can introduce the symbol classes Sρm,ψ which consist of all smooth functions p : Rn × Rn → C such that for all α, β ∈ Nn0 , m−ρ(|α|) 2 |∂ξα ∂xβ p(x, ξ)| ≤ c 1 + ψ(ξ) . For the construction of Feller semigroups we need p(x, ξ) to be a negative deﬁnite symbol of class Sρ2,ψ with p(x, ξ) ≥ δ 1 + ψ(ξ) for some δ > 0 and all suﬃciently large |ξ|, ξ ∈ Rn . In comparison to the approach discussed above, we have restricted our choice of reference function ψ by introducing (1.5) and also now require smoothness of the symbol p(x, ξ) in both variables. Jacob and the author followed in [10] an idea from Roth [17] to construct Feller semigroups. Here, ξ → p(x, ξ) needs to be twice diﬀerentiable, p(x, ·) ∈ C 2 (Rn ), and the symbol needs to satisfy for α ∈ Nn0 , |α| ≤ 2, |∂ξα p(x, ξ)| ≤ cα

(1.6)

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where cα > 0 is a constant independent of x ∈ Rn . The main idea is to freeze the coeﬃcients of the symbol, i.e. to temporarily ﬁx some x0 ∈ Rn and to consider the symbol ξ → p(x0 , ξ) and the associated Feller semigroup. Using condition (1.6) we show uniform estimates that hold independently of the choice of x0 ∈ Rn , and in this way obtain the same estimates also for the variable coeﬃcient case. An outline of this approach can also be found in [9]. Overall, the crucial restriction is condition (1.6), which means we consider pseudodiﬀerential operators bounded on C∞ (Rn ). In [15] the author showed how to use non-smooth negative deﬁnite symbols within Hoh’s symbol classes to construct Feller semigroups. Essentially we stay within Hoh’s symbol classes, but now consider carefully how much diﬀerentiability is needed: the composition of two pseudodiﬀerential operators is again a pseudodiﬀerential operator where the symbol is an oscillatory integral of type one. These oscillatory integrals exist only if the involved amplitudes, i.e. symbols, are smooth enough. We need the reference function to satisfy (1.4) and that p ∈ Sρ2,ψ (Mx , Mξ ), i.e. that for all α, β ∈ Nn0 with |α| ≤ Mξ , |β| ≤ Mx , 2−ρ(|α|) 2 |∂ξα ∂xβ p(x, ξ)| ≤ c 1 + ψ(ξ) . Here,

n Mx ≥ − 1 + 4n + 12 r0

and Mξ ≥ 2n + 9. As an additional restriction we need x → ∂xβ p(x, ξ) to vanish at inﬁnity for |β| = 1. This means we consider only slowly varying, or asymptotically constant, symbols. Furthermore, we need that p(x, ξ) ≥ δ 1 + ψ(ξ) . This result shows that it is possible to use Hoh’s ideas to construct Feller semigroups starting with non-smooth negative deﬁnite symbols. However, the conditions on Mx , Mξ are still very strong, i.e. a high order of diﬀerentiability is needed. We also brieﬂy refer to a general result obtained by Baldus [1], Theorem 4.12. Since we do not want to introduce several new notions with all the required esimates, we just mention that Baldus’ result is based on choosing a metric γ on Rn × Rn and an appropriate weight function M : Rn × Rn → R+ . The symbols under consideration are smooth and need to satisfy estimates based on the choice of γ and M . Hoh’s symbolic calculus is included in Baldus’ considerations if one chooses n (dξj )2 (dxj )2 + γ(x,ξ) (dx, dξ) = 1 . (1 + ψ(ξ)) 2 j=1 All of these methods either require high diﬀerentiability or smoothness of the negative deﬁnite symbol p(x, ξ), or restrict the choice of reference function ψ(ξ) and require a nice decay behavior for x → p(x, ξ). Our construction of Feller semigroups is best compared to the one we described ﬁrst since we

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will use a similar decomposition. However, our approach will be less technical and shorter. Since we achieve the construction of the semigroup using a symbolic calculus we can extend the approximation result from [11] to the case involving rough negative deﬁnite symbols. This paper is structered as follows: in Section 2 we formally introduce rough negative deﬁnite symbols and develop an asymptotic expansion formula. In Section 3 we then construct Feller semigroups and give an approximation procedure. Finally we will comment on how other results obtained using Hoh’s symbolic calculus for smooth symbols can be transferred to the case of rough symbols.

2. Rough Negative Deﬁnite Symbols In the introduction we gave a characterization of continuous negative deﬁnite functions using the L´evy-Khintchin formula. We give an alternative deﬁnition which was originally used by Schoenberg. Deﬁnition 2.1. A function ψ : Rn → C is called negative deﬁnite if for any m ∈ N and all ξ1 , . . . , ξm ∈ Rn the m × m matrix ψ(ξj ) + ψ(ξk ) − ψ(ξj − ξk ) j,k=1,...,m is positive Hermitian, i.e. if for all λ1 , . . . , λm ∈ C, m ψ(ξj ) + ψ(ξk ) − ψ(ξj − ξk ) λj λk ≥ 0. j,k=1

Negative deﬁnite functions also satisfy the following estimate. Lemma 2.2. For any locally bounded negative deﬁnite function ψ : Rn → C there exists a constant cψ > 0 such that for all ξ ∈ Rn |ψ(ξ)| ≤ cψ (1 + |ξ|2 ). Let us now deﬁne our symbol classes. When discussing Feller semigroups later on we will have to work with negative deﬁnite functions. Hence it is clear, in light of Lemma 2.2, that the symbols under considerations are of order at most 2. Deﬁnition 2.3. Let j ∈ {0, 1, 2}, and let p : Rn × Rn → C be a function that may be represented in the form p(x, ξ) = p1 (ξ) + p2 (x, ξ) p Mξ 1

p2

p2

where p1 ∈ C (Rn ), p2 ∈ C Mx ,Mξ (R2n ) for Mξp1 , Mxp2 , Mξp2 ∈ N0 ∪{∞}. In addition, let ψ : Rn → R be a function such that for all α ∈ Nn0 , |α| ≤ Mξp1 , m−ρ2j (|α|) |∂ξα p1 (ξ)| ≤ cα,m 1 + ψ(ξ)

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where m ∈ R and ρj (|α|) := min(j, |α|). Assume also that for α, β ∈ Nn0 , |α| ≤ Mξp2 , |β| ≤ Mxp2 , m−ρ2j (|α|) |∂ξα ∂xβ p2 (x, ξ)| ≤ cα,β,m 1 + ψ(ξ) and that x → p2 (x, ξ) has for all ξ ∈ Rn compact support K ⊂ Rn , i.e. K is independent of ξ ∈ Rn . Then p is called a rough symbol of order m ∈ R m,ψ (Mξp1 , Mxp2 , Mξp2 ). If p1 ≡ 0 this is simpliﬁed to and is denoted by p ∈ SK,ρ j m,ψ (Mxp2 , Mξp2 ), and if p2 ≡ 0 this is also written as Sρm,ψ (Mξp1 ). SK,ρ j j

The next Lemma is standard and is easily proved using Schwarz’ inequality, [14]. Lemma 2.4. Let Rn |A(ζ)| dζ < ∞ and a, b ∈ L2 (Rn ). Then n n A(ξ − η)a(ξ)b(η) dξdη ≤ n |A(ζ)| dζ a L2 b L2 . R

R

R

We also need Peetre’s inequality for negative deﬁnite functions on several occasions, [7]. Lemma 2.5. Let ψ : Rn → C be a negative deﬁnite function. Then for s ∈ R, s s |s| 1 + |ψ(ξ)| ≤ 2|s| 1 + |ψ(η)| 1 + |ψ(ξ − η)| . The next lemma is fundamental for the proof of Theorem 2.7. m,ψ Lemma 2.6. Let p ∈ SK,ρ (Mx , Mξ ), i.e. p1 ≡ 0. In addition, let α ∈ Nn0 be j such that |α| ≤ Mξ and N ∈ N0 be such that N ≤ Mx . Then the inequality − N m−ρ2j (|α|) 1 + |η|2 2 . |∂ξα p(η, ξ)| ≤ cN · diam K · 1 + ψ(ξ)

holds for all ξ, η ∈ Rn . Proof. Let γ ∈ Nn0 such that |γ| = N and denote the compact support of x → p(x, ξ) for every ξ ∈ Rn by K ⊂ Rn . Then γ α γ α −n −ix·η 2 e p(x, ξ) dx |η ∂ξ p(η, ξ)| = η ∂ξ (2π) Rn ∂xγ (e−ix·η )∂ξα p(x, ξ) dx ≤ |∂ξα ∂xγ p(x, ξ)| dx ≤ Rn

K

m−ρ2j (|α|) . ≤cN · diam K · 1 + ψ(ξ) Since |γ| = N the result follows.

Note that Lemma 2.6 gives us a precise control on the decay behavior of ∂ξα ∂xβ p(η, ξ) by choosing K ⊂ Rn accordingly This control will be needed later on for the construction of Feller semigroups. The next result allows us to consider asymptotic expansions of the composition of two pseudodiﬀerential operators. It is important to point out that Theorem 2.7 does not give us a symbolic calculus in the usual sense. In our case we will not be able to say that the remainder term is again a pseudodiﬀerential operator of lower order,

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but instead we simply ﬁnd that the remainder term is “some” operator of lower order. In order to illustrate the diﬀerence, consider two symbol p ∈ Sρm,ψ and q ∈ Sρl,ψ , where Sρm,ψ denotes Hoh’s symbol classes. We ﬁnd that the symbol of the composition of the two corresponding pseudodiﬀerential operators p(x, D) ◦ q(x, D) can be written as (p ◦ q)(x, ξ) = p(x, ξ)q(x, ξ) + r(x, ξ) where r ∈ Sρm+l−1,ψ . It way that r ∈ Sρm+l−2,ψ .

is possible to expand the above formula in such a This is rarely needed, since ﬁrst order expansions suﬃce to construct parametrices and to prove G˚ arding-type lower estimates for the operators. However, in many situation it is not necessary to know that r ∈ Sρm+l−1,ψ , but it suﬃces to know that the corresponding operator r(x, D) is a continuous linear operator mapping the space H s+m+l−1,ψ (Rn ) into H s,ψ (Rn ) for s ∈ R. These spaces, which are deﬁned as

s H s,ψ (Rn ) := u ∈ S (Rn ) : 1 + ψ(·) 2 u (·) L2 < ∞ , have the nice property that for large s they can be continuously embedded into the space C∞ (Rn ); for more details see [8]. We follow ideas from Oleinik and Radkevic [14] and state m,ψ l,ψ (Mξp1 , Mxp2 , Mξp2 ) and q ∈ SL,ρ (Mξq1 , Mxq2 , Mξq2 ). Theorem 2.7. Let p ∈ SK,ρ j j Also, choose k ∈ {−1, . . . , j − 1} and assume that

Mxp2 ≥ |s| + n + 1,

Mxq2 ≥ |m − k − 1| + |s| + 2n + 3 + k,

Mξp1 ≥ k ∨ 0, Mξp2 ≥ k ∨ 0.

Deﬁne

1 ∂ α p(x, ξ)Dxα q(x, ξ). pq k (x, ξ) := α! ξ |α|≤k

Dxα

(−i)|α| ∂xα

where := and the sum on the right-hand side is deﬁned to be 0 if k = −1. Then for all u ∈ S(Rn ) it holds that p(x, D) ◦ q(x, D)u = pq k (x, D)u + Ru, where R is an operator of order m + l − k − 1, i.e. for all s ∈ R Ru H s,ψ ≤ C u H s+m+l−k−1),ψ .

(2.1)

Proof. It is clear that (2.1) follows once we have shown that n Ru(x)v(x) dx ≤ c u H m+l−k−1+s,ψ v H −s,ψ . R

We begin by observing that p(x, D) ◦ q(x, D) = p1 (D) ◦ q1 (D) + p2 (x, D) ◦ q1 (D) + p1 (D) ◦ q2 (x, D) + p2 (x, D) ◦ q2 (x, D).

(2.2)

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The symbol (pq)k of the operator (pq)k (x, D) can be written as p1 (ξ)q1 (ξ) + p2 (x, ξ)q1 (ξ) +

1 ∂ α p1 (ξ)Dxα q2 (x, ξ) α! ξ

|α|≤k

1 + ∂ α p2 (x, ξ)Dxα q2 (x, ξ). α! ξ

(2.3)

|α|≤k

Now we set Ru := p(x, D) ◦ q(x, D)u − (pq)k (x, D)u as well as (p1 q2 )k (x, ξ) :=

1 ∂ α p1 (ξ)Dxα q2 (x, ξ), α! ξ

|α|≤k

and (p2 q2 )k (x, ξ) :=

1 ∂ α p2 (x, ξ)Dxα q2 (x, ξ). α! ξ

|α|≤k

Since the symbols of p1 (D) ◦ q1 (D) and p2 (x, D) ◦ q1 (D) are p1 (ξ)q1 (ξ) and p2 (x, ξ)q1 (ξ) respectively, it is clear when comparing (2.2) and (2.3) that R = p1 (D) ◦ q2 (x, D) − (p1 q2 )k (x, D) + p2 (x, D) ◦ q2 (x, D) − (p2 q2 )k (x, D).

(2.4)

Let us deﬁne R1 := p1 (D) ◦ q2 (x, D) − (p1 q2 )k (x, D) and R2 := p2 (x, D) ◦ q2 (x, D) − (p2 q2 )k (x, D). Hence, in order to arrive at (2.1), it suﬃces to show ≤ C u H s+m+l−(k+1),ψ v H −s,ψ R u(x)v(x) dx n 1 R

and

Rn

R2 u(x)v(x) dx ≤ C u H s+m+l−(k+1),ψ v H −s,ψ

for u, v ∈ S(Rn ). We start by noting that with Parseval’s equation we get R R1 u(x)v(x) dx = u(η) dη. 1 u(η) Rn

Rn

Hence we are now interested in the Fourier transform R 1 u(η). First of all note that p1 (D)u(ξ) = p1 (ξ) u(ξ). Furthermore, using the well-known fact

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that u · v = (2π)− 2 u ∗ v, we conclude −n

2 R1 (η) = (2π) p1 (η) q2 (η − ξ, ξ) u(ξ) dξ (2.5) n R n 1 α ∂ξ p1 (ξ)q2 (η − ξ, ξ)(η − ξ)α u − (2π)− 2 (ξ) dξ α! n R |α|≤k ⎞ ⎛ n 1 α ∂ξ p1 (ξ)(η − ξ)α ⎠ u = (2π)− 2 q2 (η − ξ, ξ) ⎝p1 (η) − (ξ) dξ. α! n R |α|≤k

2 (η). We have Next, we do a similar calculation for R F [p2 (x, D) ◦ q2 (x, D)u](η) −n = (2π) p2 (η − τ, τ )q2 (τ − ξ, ξ) u(ξ) dξdτ Rn

Rn

and

F [(p2 q2 )k (x, D)u](η) =

Rn

u(ξ) dξ. (p 2 q2 )k (η − ξ, ξ)

But (p 2 q2 )k (ζ, ξ) =

1 Fx→ζ [∂ξα p2 (x, ξ)Dxα q2 (x, ξ)] α!

|α|≤k

1 Fx→ζ [∂ξα p2 (x, ξ)] ∗ Fx→ζ [Dxα q2 (x, ξ)] α! |α|≤k 1 = ∂ α p2 (ζ − ρ, ξ)ρα q2 (ρ, ξ) dρ. α! Rn ξ =

|α|≤k

Substituting ρ = τ − ξ we ﬁnally get F [(p2 q2 )k (x, D)u](η) 1 −n ∂ α p2 (η − τ, ξ)(τ − ξ)α q2 (τ − ξ, ξ) = (2π) u(ξ) dτ dξ. α! ξ Rn Rn |α|≤k

We conclude that

2 (η) = (2π)−n R ⎛

Rn

q2 (τ − ξ, ξ)

Rn

⎞ 1 ∂ α p2 (η − τ, ξ)(τ − ξ)α ⎠ u (ξ) dξdτ. × ⎝p2 (η − τ, τ ) − α! ξ |α|≤k

Recall that by (2.4) Ru(x)v(x) dx ≤ Rn

Rn

R v (η)| dη + 1 u(η)| |

Rn

|R v (η)| dη. 2 u(η)| |

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We will now estimate R 2 u(η) and the estimate for R1 u(η) works along the same lines. From Lemma 2.6 we ﬁnd l N |q2 (τ − ξ, ξ)| ≤ 1 + ψ(ξ) 2 (1 + |τ − ξ|2 )− 2 where N > 0. According to Taylor’s formula we also ﬁnd 1 ∂ α p2 (η − τ, ξ)(τ − ξ)α p2 (η − τ, τ ) − α! ξ |α|≤k α = ∂ξ p2 η − τ, ξ + θ(τ − ξ) (τ − ξ)α |α|=k+1

where 0 ≤ θ ≤ 1. A further application of Lemma 2.6 gives α α η − τ, ξ + θ(τ − ξ) (τ − ξ) ∂ p ξ 2 |α|=k (2.6) m−k−1 k+1 N 2 (1 + |τ − ξ|2 ) 2 ≤ C(1 + |η − τ |2 )− 2 1 + ψ(ξ + θ(τ − ξ)) for some N > 0. Note that by Peetre’s inequality for negative deﬁnite functions we have − s − s |s| |s| 1 + ψ(ξ) 2 ≤ 2 2 1 + ψ(η) 2 1 + ψ(η − ξ) 2 . Then for s ∈ R m−k−1 2 1 + ψ(ξ + θ(τ − ξ)) m−k−1 |m−k−1| 2 2 ≤ c 1 + ψ(ξ) 1 + ψ(θ(τ − ξ)) |m−k−1| m−k−1 2 2 1 + θ2 |τ − ξ|2 ≤ c 1 + ψ(ξ) m−k−1 |m−k−1| 2 2 1 + |τ − ξ|2 ≤ c 1 + ψ(ξ) m−k−1+s − s |m−k−1| 2 = c 1 + ψ(ξ) 1 + ψ(ξ) 2 (1 + |τ − ξ|2 ) 2 m−k−1+s − s |s| |m−k−1| 2 1 + ψ(η) 2 1 + ψ(η − ξ) 2 (1 + |τ − ξ|2 ) 2 ≤ c 1 + ψ(ξ) m−k−1+s − s |s| |m−k−1| 2 ≤ c 1 + ψ(ξ) 1 + ψ(η) 2 1 + |η − ξ|2 2 (1 + |τ − ξ|2 ) 2 . Again by Peetre’s inequality we also ﬁnd N N N N (1 + |η − τ |2 )− 2 ≤ 2 2 1 + |η − ξ|2 )− 2 1 + |τ − ξ|2 2 . Hence it follows from (2.6) that α α ∂ξ p2 η − τ, ξ + θ(τ − ξ) (τ − ξ) |α|=k +k+1 − s |m−k−1|+N m−k−1+s 2 2 1 + ψ(η) 2 1 + |τ − ξ|2 ≤ c 1 + ψ(ξ) |s|−N × 1 + |η − ξ|2 ) 2 .

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We conclude that 1 α α q2 (τ − ξ, ξ) p2 (η − τ, τ ) − ∂ξ p2 (η − τ, ξ)(τ − ξ) α! |α|≤k (2.7) m+l−k−1+s − s |m−k−1|+N +k+1−N 2 2 ≤ C 1 + ψ(ξ) 1 + ψ(η) 2 1 + |τ − ξ|2 ) |s|−N 2 × 1 + |η − ξ|2 . Hence ≤ R u(x)v(x) dx |R v (η)| dη 2 u(η)| | n 2 R Rn m+l−k−1+s − s 2 ≤C 1 + ψ(ξ) 1 + ψ(η) 2 Rn

Rn

(2.8)

Rn

|s|−N |m−k−1|+N +k+1−N 2 2 × 1 + |τ − ξ|2 ) | u(ξ)| | v (η)| dξdηdτ. 1 + |η − ξ|2

Fixing N such that |s| − N ≤ −(n + 1) we then choose N in such a way that |m − k − 1| + N + k + 1 − N ≤ −(n + 1). Now we integrate in (2.8) with respect to τ and ﬁnd n R2 u(x)v(x) dx R m+l−k−1+s − s 2 1 + ψ(ξ) 1 + ψ(η) 2 ≤ c Rn

Rn

|s|−N × 1 + |η − ξ|2 ) 2 | u(ξ)| | v (η)| dξdη ≤ c u H m+l−k−1+s,ψ v H −s,ψ . Hence R2 u H s,ψ ≤ c u H m+l−k−1+s,ψ .

1 u can be achieved in As mentioned before, the corresponding estimate for R a similar manner. Instead of (2.7) we now obtain, compare also (2.5), 1 |q2 (η − ξ, ξ)| p1 (η) − ∂ξα p1 (ξ) (η − ξ)α α! |α|≤k 1 α α ∂ξ p1 ξ + θ(η − ξ) (η − ξ) = |q2 (η − ξ, ξ)| |α|=k α! − s m+l−k−1+s |m−k−1|+|s|+k+1−N 2 2 1 + ψ(η) 2 (1 + |η − ξ|2 ) . ≤ 1 + ψ(ξ)

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for 0 ≤ θ ≤ 1, s ∈ R and some N ≥ 0. Now we choose N so that |m − k − 1| + |s| + k + 1 − N ≤ −(n + 1) and ﬁnd

Rn

R1 u(x)v(x) dx

|R v (η)| dη 1 u(η)| | m+l−k−1+s − s 2 1 + ψ(ξ) 1 + ψ(η) 2 ≤ c

≤

Rn

Rn

Rn

× (1 + |η − ξ|2 )

|m−k−1|+|s|+1−N 2

| u(ξ)| | v (η)| dξdη

≤ c u H m+l−k−1+s,ψ v H −s,ψ . It follows that R1 u H s,ψ ≤ c u H m+l−k−1+s,ψ . We conclude that Ru H s,ψ ≤ R1 u H s,ψ + R2 u H s,ψ ≤ c u H m+l−k−1+s,ψ which proves the result. Let us now look at the diﬀerentiability conditions. First of all we observe that our applications of Lemma 2.6 only give us additional restrictions on Mxq2 and Mxp2 . It is important to note that the diﬀerentiability with respect to ξ is is only restricted by how many terms of the asymptotic expansion we want to use. Going through the proof we ﬁnd that N ≥ |s| + n + 1

and

N ≥ |m − k − 1| + N + k + 1 + n + 1

where we need by Lemma 2.6 that Mxq2 ≥ N as well as Mxp2 ≥ N . Since we want to minimize the diﬀerentiability conditions we choose N and N as small as possible. Hence Mxp2 ≥ |s| + n + 1 and Mxq2 ≥ |m − k − 1| + |s| + 2n + k + 3. Obviously, this result deviates from the usual symbolic calculus where one tries to identify operations on the level of operators with operations on the level of symbols. On the other hand, the technical machinery involved in the classical approach is quite involved, i.e. oscillatory integrals and their estimates. The approach used by Oleinik and Radkevich in [14] and by us is somewhat simpler: the main tools to achieve our goals are the deﬁnition of our symbol classes as in Deﬁnition 2.3, Peetre’s inequality for negative deﬁnite functions, and the function spaces H s,ψ (Rn ) to achieve our goals. As a direct corollary of Theorem 2.7 we have m,ψ (Mξp1 , Mxp2 , Mξp2 ) and assume that for s ∈ R Corollary 2.8. Let p ∈ SK,ρ j

Mxp2 ≥ |s| + n + 1.

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Then the following inequality holds for all u ∈ H s+m,ψ (Rn ): p(x, D)u H s,ψ ≤ c u H s+m,ψ . Proof. This follows by applying Theorem 2.7 for p(x, D) ◦ Id and k = −1 where Id denotes the identity operator. Let us point out that we have not included a discussion about the symbol of the adjoint of a pseudodiﬀerential operator with rough negative deﬁnite symbols. Such a result is important for proving lower estimates, e.g. G˚ ardingtype inequalities. We could in principle use the same techniques as before to prove results concerning the symbol of the adjoint operator. However, since we are working with non-smooth symbols we run into problems straightaway. Usually, any such discussion starts by showing that p(x, D) : S(Rn ) → S(Rn ) for which we need the estimate (m)

|p(x, D)u|k,S ≤ ck |p|k |u|m+2(n+1)+k,S , for all k ∈ N, where | · |k,S denotes the Schwartz space seminorm. Clearly, (m) |p|k exists only for k ≤ min{Mxp , Mξp }, i.e. the above estimate only works for smooth symbols p. W Hoh’s construction in [5] for smooth negative definite symbols follows this path to construct Feller semigroups. On the other hand, N Jacob in [6] uses a decomposition similar to ours and constructs Feller semigroups without using a symbolic calculus. We want to combine both approaches, i.e. we will follow [6] to derive lower estimates for our pseudodiﬀerential operators with rough negative deﬁnite symbols. Then we use our symbolic calculus, as in [5], to make the regularity argument with the Friedrichs molliﬁer easier.

3. Feller semigroups In this section we want to use the Hille-Yosida theorem to construct Feller semigroups (Tt )t≥0 that are generated by the closure of pseudodiﬀerential operators −p(x, D) with rough negative deﬁnite symbols. In the following we use the notion of Feller semigroups in the Feller-Dynkin sense as introduced e.g. in Rogers and Williams [16]. We start by considering symbols 2,ψ (Mξp1 , Mxp2 , Mξp2 ) and want to ﬁnd conditions on the negative defp ∈ SK,ρ 2 inite function ψ ∈ C ∞ (Rn ), K ⊂ Rn , and Mξp1 , Mxp2 , Mξp2 ∈ N0 ∪ {∞} such that the corresponding pseudodiﬀerential operator extends to the generator of a Feller semigroup. Note that p is of order 2 since ξ → p(x, ξ) needs to be negative deﬁnite for all x ∈ Rn . Recall that a Feller semigroup (Tt )t≥0 is a strongly continuous contraction semigroup on C∞ (Rn ) which is positivity preserving. In order to use the Hille-Yosida theorem we need to check that the operator −p(x, D) with symbol p satisﬁes three conditions. We need that the domain of −p(x, D) is dense in C∞ (Rn ). If ψ(ξ) ≥ c|ξ|r0 for some c, r0 > 0 and suﬃciently large |ξ|, then for s > rn0 we have the embedding H s,ψ → C∞ (Rn ). Thus, under these assumptions, we choose D − p(x, D) = H s+2,ψ (Rn ). The second condition

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is that −p(x, D) is a dissipative operator on H s+2,ψ (Rn ) which follows from the fact that it satisﬁes the positive maximum principle, see [2]. It remains to show that there exists a unique solution to the equation λu + p(x, D)u = f

(3.1)

for λ > 0 and f ∈ H s,ψ (Rn ). Note that this setup only makes sense if p(x, D) is a mapping from H s+2,ψ (Rn ) to H s,ψ (Rn ) which by Corollay 2.8 is the case provided that Mxp2 ≥ |s| + n + 1. Since s > rn0 it follows that we need to have n p2 Mx ≥ + n + 1. r0 We are now looking for a unique solution u ∈ H s+2,ψ (Rn ) that solves (3.1). We proceed as follows: ﬁrst we construct a weak solution in H 1,ψ (Rn ) using the Lax-Milgram theorem, see [19], then we improve the regularity of this weak solution using the Friedrichs molliﬁer to ﬁnd that it is indeed a strong solution. We deﬁne for u, v ∈ S(Rn ) the sesquilinear forms B(u, v) := p(x, D)u, v L2 and Bλ (u, v) := B(u, v) + λ(u, v)L2 , for some λ > 0. Then the weak formulation of (3.1) can be written as Bλ (u, ϕ) = (ϕ, f )L2

(3.2)

for all ϕ ∈ C0∞ (Rn ). First of all, we note that |(ϕ, f )L2 | ≤ f L2 ϕ H 1,ψ , s+2,ψ

n

(R ) deﬁnes a continuous linear functional on H 1,ψ (Rn ). i.e. every f ∈ H It remains to check the other conditions of the Lax-Milgram theorem. 2,ψ Lemma 3.1. If p ∈ SK,ρ (Mξp1 , Mxp2 , Mξp2 ) and 2

Mxp2 ≥ 2n + 4, then the sesquilinear forms B and Bλ have continuous extensions onto H 1,ψ (Rn ). Proof. Using Theorem 2.7 and Theorem 2.8 we have |B(u, v)| = p(x, D)u, v L2 1 1 2 = (1 + ψ(D))− 2 p(x, D)u, (1 + ψ(D)) 2 v L ≤ c u H 1,ψ v H 1,ψ .

The estimate for Bλ follows immediately and we ﬁnd |Bλ (u, v)| ≤ (λ + c) u H 1,ψ v H 1,ψ .

(3.3)

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We work from now on with the continuous extensions of B and Bλ as given by Lemma 3.1 without changing notation. 2,ψ Theorem 3.2. Let p ∈ SK,ρ (Mξp1 , Mxp2 , Mξp2 ) such that 2

Mxp2 ≥ 2n + 4, p1 (ξ) ≥ δ 1 + ψ(ξ) for |ξ| ≥ 1 and some δ > 0. It follows for u ∈ H 1,ψ (Rn ) that 1 1 + ψ(D) − 2 p2 (x, D)u 2 ≤ c0 · diam(K) · u H 1,ψ L

(3.4)

holds for some c0 > 0. If furthermore diam(K) <

δ c0

then for u ∈ H 1,ψ (Rn ) and λ > 0 suﬃciently large the following inequality is valid: (3.5) |Bλ (u, u)| ≥ δ − c0 · diam(K) u 2H 1,ψ . Proof. Estimate (3.4) follows immediately from Theorem 2.7 and Corollary 2.8. In order to prove (3.5) we ﬁrst of all note that 2 p1 (D)u, u L2 = δ 1 + ψ(ξ) − λ0 | u(ξ)|2 dξ p1 (ξ)| u(ξ)| dξ ≥ Rn

=

δ u 2H 1,ψ

−

λ0 u 2L2 .

Rn

It follows that for λ > λ0 |Bλ (u, u)| ≥ p1 (D)u, u L2 − p2 (x, D)u, u L2 + λ u 2L2 1 1 = p1 (D)u, u L2 − (1 + ψ(D))− 2 p2 (x, D)u, (1 + ψ(D)) 2 u L2 + λ u 2L2 ≥ p1 (D)u, u L2 − c0 · diam(K) · u 2H 1,ψ + λ u 2L2 ≥ δ u 2H 1,ψ + (λ − λ0 ) u 2L2 − c0 · diam(K) · u 2H 1,ψ ≥ δ − c0 · diam(K) u 2H 1,ψ .

We have now shown (3.3) and (3.5) which enables us to use the LaxMilgram theorem to ﬁnd the existence of a unique element u ∈ H 1,ψ (Rn ) that solves (3.2). Let us now introduce the Friedrichs molliﬁer which is given for u ∈ L2 (Rn ) and ε ∈ (0, 1] by Jε u := (jε ∗ u)(x) = jε (x − y)u(y) dy, Rn

where jε (x) := ε

−n

j( xε )

and

j(x) :=

c0 e(|x| 0

2

−1)−1

, |x| < 1, , |x| ≥ 1.

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The Friedrichs molliﬁer has many nice properties, see e.g. Proposition 2.3.15 in N. Jacob [8]. In particular, once we have shown for some u ∈ L2 (Rn ) Jε (u) H s,ψ ≤ c with c independent of ε, then we already have u ∈ H s,ψ (Rn ). Using Lemma 2.5.9 in [8] we ﬁnd that for any ε > 0 the function ξ → j(εξ) belongs to the (∞), see Deﬁnition 2.3. symbol class Sρ0,ψ 2 2,ψ Theorem 3.3. For p ∈ SK,ρ (Mξp1 , Mxp2 , Mξp2 ), where 2

Mxp2 ≥ |s| + 2n + 4, and s ∈ R there exists a constant c > 0 independent of ε, 0 < ε ≤ 1, such that [p(x, D), Jε ]u H s,ψ ≤ c u H s+1,ψ for all u ∈ H s+1,ψ (Rn ). Proof. By deﬁnition, [p(x, D), Jε ] = p(x, D)Jε − Jε p(x, D). Clearly, −n 2

p(x, D)Jε u(x) = (2π)

Rn

eix·ξ p(x, ξ) j(εξ) u(ξ) dξ.

By Theorem 2.7 we also have if Mxp2 ≥ |s| + 2n + 4 that Jε p(x, D) = p(x, D)Jε + R where R is an operator of order 1. We conclude [p(x, D), Jε ]u H s,ψ = R H s,ψ ≤ c u H s+1,ψ and the independence of c from ε follows from our asymptotic expansion formula developed in Theorem 2.7. The following lower estimate is crucial to make our argumentation using the Friedrichs molliﬁer work. 2,ψ Theorem 3.4. Let p ∈ SK,ρ (Mξp1 , Mxp2 , Mξp2 ) such that 2 p1 (ξ) ≥ δ 1 + ψ(ξ)

for |ξ| ≥ 1 and some δ > 0. Then for all u ∈ H s+2,ψ (Rn ) the following inequality holds p2 (x, D)u H s,ψ ≤ c1 · diam(K) · u H s+2,ψ . If furthermore diam(K) < then

p(x, D)u H s,ψ ≥

δ 2c1

δ − c1 · diam(K) u H s+2,ψ − γ u L2 . 2

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Proof. First of all we note that for all ε > 0 and s1 < s2 < s3 we have the estimate s3 s1 s2 1 + ψ(ξ) 2 ≤ ε2 1 + ψ(ξ) 2 + c(ε)2 1 + ψ(ξ) 2 which implies that u H s2 ,ψ ≤ ε u H s3 ,ψ + c(ε) u H s1 ,ψ . Now, for some suitable λ1 ≥ 0 s p1 (D)u 2H s,ψ = u(ξ)|2 dξ 1 + ψ(ξ) p1 (ξ)2 | Rn s 2 u(ξ)| dξ 1 + ψ(ξ) δ 2 1 + ψ(ξ) − λ1 | ≥ Rn

= δ 2 u 2H s+2,ψ − λ1 u 2H s,ψ 3 ≥ δ 2 u 2H s+2,ψ − δ 2 u 2H s+2,ψ − γ 2 u 2L2 4 δ2 = u 2H s+2,ψ − γ 2 u 2L2 4 for some γ > 0. Moreover, by Corollary 2.8, we know that p2 (x, D)u H s,ψ ≤ c1 · diam(K) · u H s+2,ψ We conclude that p(x, D)u H s,ψ ≥ p1 (D)u H s,ψ − p2 (x, D)u H s,ψ δ u H s+2,ψ − γ u L2 − c1 · diam(K) · u H s+2,ψ 2 δ − c1 · diam(K) u H s+2,ψ − γ u L2 . = 2 ≥

Given f ∈ H s,ψ (Rn ) we have so far constructed a unique weak solution u ∈ H 1,ψ (Rn ) that solves Bλ (u, ϕ) = (ϕ, f )L2 for all ϕ ∈

C0∞ (Rn ).

We now have for u ∈ H 1,ψ (Rn ) Bλ (Jε u, ϕ) = Bλ (u, Jε ϕ) − [Jε , p(x, D) + λ]u, ϕ L2 = (Jε ϕ, f )L2 − [Jε , p(x, D) + λ]u, ϕ L2 = (ϕ, Jε f )L2 − [Jε , p(x, D) + λ]u, ϕ L2

In other words, we ﬁnd for suitable t ≥ 0, (λ + p(x, D))Jε u H t,ψ ≤ Jε f H t,ψ + [Jε , p(x, D) + λ]u H t,ψ . Then by Theorem 3.4 we have Jε u H 2,ψ ≤ γ0 Jε u L2 + cK (p(x, D) + λ)Jε u L2 ≤ c ( Jε u L2 + Jε f L2 + [Jε , p(x, D) + λ]u L2 ) ≤ c0 ( u L2 + f L2 + u H 1,ψ )

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Using properties of the Friedrichs molliﬁer we ﬁnd that u ∈ H 2,ψ (Rn ). Clearly we can now repeat this procedure until we arrive at the estimate Jε u H s+2,ψ ≤ c0 ( u H s,ψ + f H s,ψ + u H s+1,ψ ) . Again we conclude that u ∈ H s+2,ψ (Rn ). We are now able to construct a Feller semigroup. Let us summarize all the conditions on the rough symbol mentioned so far. 2,ψ 2,ψ (0, Mxp2 , 0), i.e. p ∈ SK,ρ (Mξp1 , Mxp2 , Mξp2 ) with Theorem 3.5. Let p ∈ SK,ρ 2 2 Mξp1 = Mξp2 = 0 such that

ψ(ξ) ≥ c|ξ|r0 , c, r0 > 0, n p2 +n+1 Mx ≥ max 2n + 4, r0 p1 (ξ) ≥ δ 1 + ψ(ξ) , δ δ diam(K) < min , c0 2c1 where c0 , c1 > 0 are constants taken from Theorem 3.4 and Theorem 3.2. for all |ξ| ≥ 1 and some δ > 0. Then the corresponding pseudodiﬀerential operator −p(x, D) extends to the generator of a Feller semigroup. Let us now remark how this result compares to the existing results mentioned in the introduction. In all approaches a smallness assumption similar to the one on diam(K) in Theorem 3.5 is needed. Compared to the result in [15], we still need high diﬀerentiability of the symbol with respect to x ∈ Rn , but we do not need diﬀerentiability conditions with respect to ξ ∈ Rn anymore. Our result also slightly improves the diﬀerentiability conditions given in [6] where it is needed that Mxp2 ≥ rn0 +n+3. On the other hand we restrict ourselves to symbols p that can be decomposed as p(x, ξ) = p1 (ξ) + p2 (x, ξ) where x → p2 (x, ξ) has compact support. In [3, 4] it was shown that the associated martingale problem is well-posed. Furthermore, a localization procedure is used to extend this result to the case where no smallness assumption is required. The diﬀerentiability condition is Mxp2 ≥ 2 rn0 + n + 1. One of the main advantages of having a symbolic calculus for rough symbols is that many results already proven for symbols out of Hoh’s symbol classes also hold for rough symbols. The reason for this is that many arguments and proofs use mapping properties of our pseudodiﬀerential operators on the Sobolev spaces H s,ψ (Rn ). As already seen in our construction of Feller semigroups, many times a combination of our symbolic calculus and techniques developed orginially by Jacob [6], allows us to draw conclusions about symbols of pseudodiﬀerential operators which generate Feller semigroups. In order to emphasize this point, let us consider the case of approximating a Feller semigroup by using the Yosida approximation of the symbol of the generator, compare [11] for details. We begin by considering a negative

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deﬁnite symbol that satisﬁes the conditions of Theorem 3.5. Let pν1 (ξ) =

νp1 (ξ) ν + p1 (ξ)

and

pν2 (x, ξ) =

νp2 (x, ξ) , ν + p2 (x, ξ)

and note that x → pν2 (x, ξ) still has compact support, i.e. pν (x, ξ) := pν1 (ξ) + pν2 (x, ξ) satisﬁes Assumption 2.3. We have p(x, ξ) − pν (x, ξ) = p1 (ξ) − pν1 (ξ) + p2 (x, ξ) − pν2 (x, ξ) and hence for s ∈ R p(x, D)u − pν (x, D)u H s,ψ ≤ p1 (D)u − pν1 (D)u H s,ψ + p2 (x, D)u − pν2 (x, D)u H s,ψ . Now p1 (ξ) − pν1 (ξ) =

p1 (ξ)2 ν + p1 (ξ)

and

p2 (x, ξ) − pν2 (x, ξ) =

p2 (x, ξ)2 ν + p2 (x, ξ)

where again we note that p2 (x, ξ) − pν2 (x, ξ) satisﬁes Assumption 2.3. Since |p1 (ξ) − pν1 (ξ)| ≤ and

2m 1 1 + ψ(ξ) 2 ν

2m 1 1 + ψ(ξ) 2 ν ≥ |s| + n + 1 that

|p2 (x, ξ) − pν2 (x, ξ)| ≤ it follows from Theorem 2.8 if Mxp2

p1 (D)u − pν1 (D)u H s,ψ ≤

1 u H s+2m,ψ ν

and p2 (x, D)u − pν2 (x, D)u H s,ψ ≤

1 u H s+2m,ψ . ν

Since the approximation procedure carried out in [11] does not depend on the diﬀerentiability conditions on the symbol p we skip the details here and give the ﬁnal result. Theorem 3.6. Assume p is a rough negative deﬁnite symbol as in Theorem 3.5 and denote the Feller semigroup generated by the corresponding pseudodiﬀerential operator by (Tt )t≥0 . Then ν lim e−tp (x,D) u − Tt u = 0. ν→∞

∞

It might be worthwhile to investigate whether other results, such as on subordination, or the parameter-dependent symbolic calculus [13], hold when replacing smooth symbols with rough symbols.

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4. Conclusion An asymptotic expansion formula for the composition of pseudodiﬀerential operators with non-smooth, “rough”, negative deﬁnite symbols was developed. This asymptotic expansion formula was then used to construct Feller semigroups generated by the closure of pseudodiﬀerential operators with rough negative deﬁnite symbols p(x, ξ). Precise diﬀerentiability conditions on p(x, ξ) were given and compared to existing approaches by Jacob [6], Hoh [4] and Potrykus [15]. In particular it was found that the diﬀerentiability condition with respect to the x-variable of the symbol p(x, ξ) can be slightly weakened when compared to [4, 6]. This was achieved by using the simpler approach by Oleinik and Radkevich [14] to construct Feller semigroups without needing technical tools such as G˚ arding-type inequalities or the Friedrichs symmetrization. On the other hand, having a symbolic calculus means that certain results from [6] are easier to attain. Another advantage is that we were able to use techniques so far reserved for smooth negative deﬁnite symbols also for rough symbols and therefore many results from the smooth symbolic calculus could be carried over to our situation. As an example we extended an approximation method for Feller semigroups using the Yosida approximation of the symbol of the generator [11] to the case of pseudodiﬀerential operators with rough negative deﬁnite symbols. Acknowledgment The author would like to thank the anonymous referee for several helpful suggestions and corrections.

References [1] F. Baldus. Application of the Weyl-H¨ ormander calculus to generators of Feller semigroups. Math. Nachr., 252:3–23, 2003. [2] S. N. Ethier and T. G. Kurtz. Markov processes. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. John Wiley & Sons Inc., New York, 1986. Characterization and convergence. [3] W. Hoh. The martingale problem for a class of pseudo-diﬀerential operators. Math. Ann., 300(1):121–147, 1994. [4] W. Hoh. Pseudodiﬀerential operators with negative deﬁnite symbols and the martingale problem. Stochastics Stochastics Rep., 55(3-4):225–252, 1995. [5] W. Hoh. A symbolic calculus for pseudo-diﬀerential operators generating Feller semigroups. Osaka J. Math., 35(4):789–820, 1998. [6] N. Jacob. Feller semigroups, Dirichlet forms, and pseudodiﬀerential operators. Forum Math., 4(5):433–446, 1992. [7] N. Jacob. Pseudo diﬀerential operators and Markov processes. Vol. I. Imperial College Press, London, 2001. Fourier analysis and semigroups. [8] N. Jacob. Pseudo diﬀerential operators & Markov processes. Vol. II. Imperial College Press, London, 2002. Generators and their potential theory.

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[9] N. Jacob. Pseudo diﬀerential operators and Markov processes. Vol. III. Imperial College Press, London, 2005. Markov processes and applications. [10] N. Jacob and A. Potrykus. Roth’s method applied to some pseudo-diﬀerential operators with bounded symbols. A case study. Rend. Circ. Mat. Palermo (2) Suppl., (76):45–57, 2005. [11] N. Jacob and A. Potrykus. Approximating a Feller semigroup by using the Yosida approximation of the symbol of its generator. Positivity, 11(1):1–13, 2007. [12] N. Jacob and R. L. Schilling. L´evy-type processes and pseudodiﬀerential operators. In L´evy processes, pages 139–168. Birkh¨ auser Boston, Boston, MA, 2001. [13] N. Jacob and A. G. Tokarev. A parameter-dependent symbolic calculus for pseudo-diﬀerential operators with negative-deﬁnite symbols. J. London Math. Soc. (2), 70(3):780–796, 2004. [14] O. A. Ole˘ınik and E. V. Radkeviˇc. Second order equations with nonnegative characteristic form. Plenum Press, New York, 1973. Translated from the Russian by Paul C. Fife. [15] A. Potrykus. A symbolic calculus and a parametrix construction for pseudodiﬀerential operators with non-smooth negative deﬁnite symbols. Rev. Mat. Complut., 22(1):187–207, 2009. [16] L. C. G. Rogers and D. Williams. Diﬀusions, Markov processes, and martingales. Vol. 2. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. Itˆ o calculus, Reprint of the second (1994) edition. [17] J.-P. Roth. Les op´erateurs elliptiques comme g´en´erateurs inﬁnit´esimaux de semi-groupes de Feller. In S´eminaire de Th´ eorie du Potentiel, No. 3 (Paris, 1976/1977), volume 681 of Lecture Notes in Math., pages 234–251. SpringerVerlag, Berlin, 1978. [18] K.-i. Sato. L´evy processes and inﬁnitely divisible distributions, volume 68 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 1999. Translated from the 1990 Japanese original, Revised by the author. [19] K. Yosida. Functional analysis. Classics in Mathematics. Springer-Verlag, Berlin, 1995. Reprint of the sixth (1980) edition. Alexander Potrykus Department of Mathematics Swansea University Singleton Park Swansea SA2 8PP UK e-mail: [email protected] Submitted: June 12, 2009. Revised: October 22, 2009.

Integr. Equ. Oper. Theory 66 (2010), 463–494 DOI 10.1007/s00020-010-1757-z Published online March 20, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Odd BMO(R) Functions and Carleson Measures in the Bessel Setting J. J. Betancor, A. Chicco Ruiz, J. C. Fari˜ na and L. Rodr´ıguez-Mesa Abstract. In this paper, we characterize the odd functions in BMO(R) by using Carleson measures associated with Poisson and heat semigroups for Bessel operators. Mathematics Subject Classification (2000). Primary 42C05; Secondary 42C15. Keywords. BMO, Carleson measure, Bessel operator, Poisson and heat semigroups.

1. Introduction As it is well-known (see [11, Chapter 4 Sect. 1]), a measurable function f belongs to the classical space of functions of bounded mean oscillation, BMO(R), when 1 ||f ||BMO(R) = sup |f (x) − fI |dx < ∞, I⊂R |I| I

where the supremum is taken over all bounded intervals I in R, 1 fI = |I| f (x)dx and |I| denotes the length of I. I In this paper, we deal with the space BMOo (R) of all the odd functions in BMO(R). This space was considered in [5]. A useful property is the following. Let 1 ≤ p < ∞. A measurable function f is in BMOo (R) if and only if there exists C > 0 such that ⎫1/p ⎧ ⎬ ⎨ 1 |f (x) − fI |p dx ≤ C, (1.1) ⎭ ⎩ |I| I

This paper is partially supported by MTM2007/65609.

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for every interval I = (a, b), 0 < a 0 : (1.1) and (1.2) hold} is equivalent to ||f ||BMO(R) . (See [11, Corollary p. 144]). It is well-known that the classical space BMO(R) can be characterized by means of Carleson measures. We say that a positive measure µ on (0, ∞) × (0, ∞) is Carleson, when there exists C > 0 such that, for every interval I ⊂ (0, ∞) µ(I × (0, |I|)) ≤ C. |I| , when µ is a Carleson We denote by ||µ||C = supI⊂(0,∞), I interval µ(I×(0,|I|)) |I| measure on (0, ∞) × (0, ∞). The aim of this paper is to characterize the space BMOo (R) in terms of Carleson measures involving the heat and Poisson semigroups associated with the Bessel operator ∆λ = −x−λ Dx2λ Dx−λ , λ > 0. Assume that λ > 0. For each y ∈ (0, ∞), the function ϕy (x) = √ xyJλ−1/2 (xy), x ∈ (0, ∞), is an eigenfunction of ∆λ . According to [14, Sect. 3.2 (5) and (6)] we have that √ √ ∆λ ( xyJλ− 12 (xy)) = y 2 xyJλ− 12 (xy), x, y ∈ (0, ∞). (1.3) Here Jν represents the Bessel function of the ﬁrst kind and order ν. The heat semigroup {Wtλ }t>0 associated with the Bessel operator ∆λ is deﬁned by ∞ Wtλ (f )(x)

Wtλ (x, y)f (y)dy,

=

t, x ∈ (0, ∞),

0

where the heat kernel is given by ∞ Wtλ (x, y)

=

2

e−tz ϕx (z)ϕy (z)dz,

t, x, y ∈ (0, ∞).

0

By [14, Sect. 13.31 (1)] we have √ xy x2 +y2 xy Iλ− 12 e− 4t , t, x, y ∈ (0, ∞), Wtλ (x, y) = 2t 2t where Iν denotes the modiﬁed Bessel function of the ﬁrst kind and order ν. The Poisson semigroup {Ptλ }t>0 for ∆λ is deﬁned by ∞ Ptλ (f )(x)

Ptλ (x, y)f (y)dy,

= 0

t, x ∈ (0, ∞),

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where ∞ Ptλ (x, y)

=

e−tz ϕx (z)ϕy (z)dz,

t, x, y ∈ (0, ∞).

0

By taking into account [10, (16.4)] (see also [15]) we can write Ptλ (x, y)

2λ(xy)λ t = π

π 0

(sin θ)2λ−1 dθ, ((x − y)2 + t2 + 2xy(1 − cos θ))λ+1

t, x, y ∈ (0, ∞).

(1.4) L1loc [0, ∞)

such Now by BMO+ we denote the space of all those f ∈ that the odd extension fo of f to R is in BMO(R). We consider the natural norm on BMO+ . The following theorem is the main result of the paper. Theorem 1.1. Let λ > 0. Assume that f ∈ L1loc [0, ∞). The following assertions are equivalent. (i) f ∈ BMO+ . (ii) (1 + x2 )−1 f ∈ L1 (0, ∞) and the measure 2 dxdt 2 ∂ λ W (f )(x) dµf (x, t) = t ∂s s t s=t2 (iii)

is Carleson on (0, ∞) × (0, ∞). (1 + x2 )−1 f ∈ L1 (0, ∞) and the measure 2 ∂ λ dxdt dγf (x, t) = t Pt (f )(x) ∂t t is Carleson on (0, ∞) × (0, ∞). Moreover, the quantities ||f ||2BMO+ , ||µf ||C and ||γf ||C are equivalent.

On the way of the proof we obtain new characterizations of the subspace Ho1 (R) of H 1 (R) constituted by the odd functions which belong to the Hardy space H 1 (R). The space Ho1 (R) was considered by Fridli [6], who described it by using what he called Telyakovskii transform, a local Hilbert transform studied by Andersen and Muckenhoupt [1]. Fridli also obtained a description of the space Ho1 (R) in terms of (odd)-atoms (see [6, Theorem 2.1]). A measurable function a on (0, ∞) is an (odd)-atom when it satisﬁes one of the following properties: (a) (b)

a = 1δ χ(0,δ) , for some δ > 0, where χ(0,δ) denotes as usual the characteristic function on the interval (0, δ); I ⊂ (0, ∞) such that supp a ⊂ I, there exists a bounded interval a(x)dx = 0, and ||a||∞ ≤ |I|−1 . I

In Proposition 4.1 we characterize the Hardy space Ho1 (R) using nontangential Littlewood–Paley g-functions associated to the Poisson semigroup for the Bessel operators ∆λ .

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Assume that f ∈ BMOo (R). If a represents the odd extension to R of an (odd)-atom, we denote by +∞ f (y)a(y)dy. Φf (a) = −∞

By arguing in a standard way (see, for instance [11, pp. 142 and 143]) we can prove that Φf admits a unique extension to Ho1 (R) deﬁning an element of the dual space (Ho1 (R)) of Ho1 (R) and being ||Φf ||(Ho1 (R)) ≤ C||f ||BMOo (R) . Moreover, according to [4, Corollary 1.6], [2, Proposition 32(b)] and [5, Proposition 3.1(ii)], the mapping f −→ Φf is an isomorphism between BMOo (R) and (Ho1 (R)) and ||f ||BMOo (R) is equivalent to ||Φf ||(Ho1 (R)) , for every f ∈ BMOo (R). The Hankel transform deﬁned by ∞ √ xyJλ− 12 (xy)f (y)dy, x ∈ (0, ∞), hλ (f )(x) = 0

will be useful in the sequel. hλ is an isometry in L2 (0, ∞) ([13, p. 473 (1)]). In [7] and [8] a convolution operation for hλ is deﬁned. The main properties of the Hankel transform hλ was established in [16]. Throughout this paper we denote by C a suitable positive constant that is not necessarily the same in each ocurrence.

2. Proof of (i) =⇒ (ii) in Theorem 1.1 For every t > 0, set Qλt (f ) = t2

∂ λ W (f ) . ∂s s s=t2

We can see that ∞ Qλt (f )(x)

= 0

t2

∂ λ W (x, y) f (y)dy, ∂s s s=t2

t, x ∈ (0, ∞).

By [9, (5.7.9)], for all x, y, s ∈ (0, ∞), xy x2 +y2 ∂ λ (xy)λ xy −λ+ 12 ∂ 1 Ws (x, y) = e− 4s I λ− 2 ∂s ∂s (2s)λ+ 12 2s 2s xy x2 +y2 λ + 12 x2 + y 2 xy −λ+ 12 (xy)λ 1 − e− 4s =− I 1 λ− 2 s 4s2 2s 2s (2s)λ+ 2 xy x2 +y2 (xy)λ (xy)2 xy −λ− 12 1 e− 4s . − I (2.1) 1 λ+ 3 λ+ 2 4s 2s 2s 2 (2s) Hence, by [9, (5.11.10)] we get 2 − (x−y) ∂ λ 4s e x2 + y 2 Ws (x, y) ≤ C 1 + , 3 ∂s s s2

x, y, s ∈ (0, ∞).

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λ the square function Denote by SQ ⎞ 12 ⎛∞ 2 dt λ ⎠ , SQ (f )(x) = ⎝ Qλt (f )(x) t

467

x ∈ (0, ∞).

0

The following result concerns to the boundedness of this square function. Lemma 2.1. For all λ > 0 one has 1 λ ||SQ (f )||L2 (0,∞) = √ ||f ||L2 (0,∞) , 2 2

f ∈ L2 (0, ∞).

Lemma 2.1 follows from the fact that hλ is an isometry in L2 (0, ∞) and the equality 2 2

Qλt (f ) = −t2 hλ (e−t

y

y 2 hλ (f )),

t ∈ (0, ∞) and f ∈ L2 (0, ∞).

(See [12, Lemma 1]). Assume now that f ∈ BMO+ . Since f ∈ L1loc [0, ∞), [11, p. 141] leads to (1 + x2 )−1 f ∈ L1 (0, ∞). To see that the measure 2 dxdt ∂ dxdt = |Qλt (f )(x)|2 dµf (x, t) = t2 Wsλ (f )(x) ∂s t t 2 s=t is Carleson on (0, ∞) × (0, ∞), and ||µf ||C ≤ C||f ||2BMO+ , we will establish that, for a certain C > 0, 1 |I|

|I| 0

|Qλt (f )(x)|2

dxdt ≤ C||f ||2BMO+ , t

(2.2)

I

for every I = (a, b), 0 ≤ a < b < ∞. Assume that I = (a, b) with 0 ≤ a < b < ∞. As in the classical case ([11, p. 160]) we split f as follows f = (f − f2I )χ2I + (f − f2I )χ(0,∞)\2I + f2I = f1 + f2 + f3 , where 2I denotes the double of the interval I in (0, ∞), that is, if xI = then 2I = (xI − |I|, xI + |I|) ∩ (0, ∞). By Lemma 2.1 we get 1 |I|

|I| 0

dxdt |Qλt (f1 )(x)|2 t

I

1 ≤ |I| C ≤ |I|

λ |SQ (f1 )(x)|2 dx ≤

I

a+b 2 ,

1 ||S λ (f1 )||2L2 (0,∞) |I| Q

|f (x) − f2I |2 dx ≤ C||f ||2BMO+ .

(2.3)

2I

On the other hand, by using [12, Lemma 1] we can write Wsλ (x, y)

(xy)λ 1 = 2λ 1 1 2 Γ(λ + 2 ) sλ+ 2

∞ 0

z2

e− 4s Dλ (x, y, z)z 2λ dz,

s, x, y ∈ (0, ∞).

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Here Dλ (x, y, z) =

22−2λ Γ(λ + 12 ) [(z 2 − (x − y)2 )((x + y)2 − z 2 )]λ−1 √ , (xyz)2λ−1 πΓ(λ)

when |x − y|< z < x + y, and Dλ (x, y, z) = 0, otherwise. It has, for x, y ∈ ∞ (0, ∞), that 0 Dλ (x, y, z)z 2λ dz = 1 [8]. Hence, if s, x, y ∈ (0, ∞), ∂ λ (xy)λ Ws (x, y) = 3 ∂s 22λ Γ(λ + 12 )sλ+ 2

x+y

|x−y|

z2 1 z2 −λ − + e− 4s Dλ (x, y, z)z 2λ dz. 2 4s

We conclude that, x+y λ ∂ λ z2 (xy) Ws (x, y) ≤ C e− 8s Dλ (x, y, z)z 2λ dz 3 ∂s λ+ s 2 |x−y|

≤C

(xy)λ 3 sλ+ 2

(xy)λ ≤C s

x+y

1

|x−y| x+y

|x−y|

1+

z 2 λ+1 8s

√ (s +

s

z 2 )λ+1

Dλ (x, y, z)z 2λ dz

Dλ (x, y, z)z 2λ dz

s, x, y ∈ (0, ∞).

By [12, (4.1) and (6.7)] and [10, p. 86 (b)] one obtains, ∂ λ C Ws (x, y) ≤ √ , s, x, y ∈ (0, ∞). ∂s s(s + |x − y|2 ) Then, 2 ∂ λ t ∂s Ws (x, y)

t t ≤C ≤C , 2 2 t + |x − y| (t + |x − y|)2 2

t, x, y ∈ (0, ∞).

s=t

It is not difﬁcult to note that t λ |f (y) − f2I |dy |Qt (f2 )(x)| ≤ C (t + |x − y|)2 ≤C (0,∞)\2I

≤C

∞

(0,∞)\2I

t |f (y) − f2I |dy (t + |xI − y|)2

k=1 {y∈(0,∞):2k−1 |I|≤|xI −y|<2k |I|}

t |f (y) − f2I |dy (t + |xI − y|)2

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Odd BMO(R) Functions and Carleson Measures ⎛

≤C

∞ k=1

469

t ⎜ ⎝ 2k 2 |I|2

|f (y) − f2k+1 I |dy

{y∈(0,∞):|xI −y|<2k |I|}

⎞

⎟ +2k |I||f2k+1 I − f2I |⎠ ⎛ ∞ 1 ⎝ 1 t ≤C |I| 2k 2k |I| k=1

⎞ |f (y) − f2k+1 I |dy + |f2k+1 I − f2I |⎠

2k+1 I

t ≤ C ||f ||BMO+ , |I|

x ∈ I, t > 0.

Indeed if k ∈ N\{0} and 2k |I| > xI then 2k+1 I = (0, xI +2k |I|) ⊂ (0, 2k+1 |I|) and 2k+1 |I|

|f (y) − f2k+1 I |dy ≤

(|f (y)| + |f2k+1 I |)dy 0

2k+1 I

⎛

⎜ ≤ 2k+1 |I| ⎝||f ||BMO+ + ≤2

k+3

1 2k |I|

2k+1 |I|

⎞

⎟ |f (y)|dy ⎠

0

|I|||f ||BMO+ .

Hence 1 |I|

|I| 0

dxdt |Qλt (f2 )(x)|2 t

≤C

||f ||2BMO+ |I|3

I

|I|

tdt

0

dx ≤ C||f ||2BMO+ . (2.4)

I

Finally we have to analyze 1 |I|

|I| 0

dxdt |Qλt (f3 )(x)|2 t

I

|f2I |2 = |I|

|I| 0

|Qλt (1)(x)|2

dxdt . t

I

In the classical case this term does not appear ([11, p. 160]). We ﬁrst observe that, since 2I = (xI − |I|, xI + |I|), xI+|I|

1 |f2I | ≤ |I|

|f (y)|dy ≤ 0

xI + |I| ||f ||BMO+ . |I|

In order to estimate 1 |I|

|I| 0

I

|Qλt (1)(x)|2

dxdt t

(2.5)

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we write, for t, x ∈ (0, ∞), ⎛

t2

x ⎜ λ 2⎜ Qt (1)(x) = t ⎝ + 0

max

x

2

+

t2

x

, 3x 2

⎞

∞ +

2 2 2 min tx , x max tx , x max tx , 3x 2 2 2

⎟ ∂ λ ⎟ Ws (x, y) s=t2 dy. ⎠ ∂s

By (2.1) it has λ 2 +y 2 ∂ λ Ws (x, y) ≤ C (xy)3 e− x 8s , ∂s λ+ s 2 provided that s, x, y ∈ (0, ∞) and xy ≤ s. Then 2 t x x2 ∞ y2 xλ e− 8t2 ∂ − 8t λ 2 λ ≤ C W (x, y) dy e y dy s 2 ∂s s=t t2λ+3 0 0 x2

xλ e− 8t2 ≤ C λ+2 , t

t, x ∈ (0, ∞).

Hence 2 t2 |I| x dxdt 1 ∂ λ t 2 Ws (x, y) 2 dy s=t |I| ∂s t 0 I 0 C ≤ |I|

∞ I

0

x2

x2λ e− 4t2 dtdx ≤ C. t2λ+1

(2.6)

Also, if xI > |I| we can write 2 t2 |I| x |I| 2 dxdt C ∂ x2λ − x22 1 λ t W ≤ (x, y) dy e 4t dxdt s 2 s=t |I| ∂s t |I| t2λ+1 0 I 0 0 I C ≤ |I| C ≤ |I|

|I|

t2λ+1

0

t2 dt

0

≤ C|I|

t2 x2

λ+ 32 dxdt

I

|I|

3

x2λ

1 dx x3

I

(4x2I

xI . − |I|2 )2

(2.7)

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Then according to (2.5), (2.6) and (2.7) we obtain 2 2 |I| tx 2 dxdt |f2I |2 ∂ λ t Ws (x, y) 2 dy s=t |I| ∂s t 0 I 0 ⎧ 2 when xI < |I|, ⎪ ⎨ ||f ||BMO+ , ≤C ⎪ ⎩ xI |I| ||f ||2 BMO+ , when xI > |I|. (2xI −|I|)2 Thus we conclude that 2 2 |I| tx 2 dxdt ∂ λ |f2I | t 2 Ws (x, y) 2 dy ≤ C||f ||2BMO+ . s=t |I| ∂s t 0 I 0

(2.8)

On the other hand, by [9, (5.7.9)] one has xy

1 − (x−y)2 xy − xy 4s 2s √ e Iλ− 12 e 2s 2s 2s xy

2 λ xy xy −λ+ 12 1 − (x−y) ∂ − xy 4s 2s Iλ− 12 +√ e e ∂s 2s 2s 2s 2s

(x−y)2 xy xy 1 xy ∂ √ e− 4s Iλ− 12 e− 2s = ∂s 2s 2s 2s xy

(x−y)2 xy xy xy I 1 e− 2s + √ 5 e− 4s 2s λ− 2 2s 2 2s 2 xy

xy 2λs xy Iλ− 12 −e− 2s xy 2s 2s

xy xy xy Iλ+ 12 − e− 2s , t, x, y ∈ (0, ∞). 2s 2s

∂ λ ∂ W (x, y) = ∂s s ∂s

Then by [9, 5.11.10] we get ∂ ∂ λ W (x, y) = ∂s s ∂s

(x−y)2 1 √ e− 4s 2 πs

+ e−

(x−y)2 8s

O

1 √ xy s

provided that s, x, y ∈ (0, ∞) and xy ≥ s. Hence we deduce that 2 − (x−y) ∂ λ 8s e Ws (x, y) ≤ C , 3 ∂s s2

s, x, y ∈ (0, ∞) and xy ≥ s.

,

(2.9)

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Thus, if t, x ∈ (0, ∞), ∞ ∞ y2 C ∂ λ ≤ Ws (x, y) 2 dy e− 72t2 dy 3 s=t ∂s t 2 max 3x , t2 max 3x , t 2

x

2

≤ C

x

∞

x2 − 64t 2

e

t3 max x2

e− 64t2 ≤C t3

∞

y2

e− 144t2 dy

3x t2 2 , x

y2

e− 144t2 dy

0

2

≤ C

x − 64t 2

e

t2

,

(2.10)

and, x x 2 2 (x−y)2 C ∂ λ Ws (x, y) 2 dy ≤ 3 e− 8t2 dy s=t t 2 ∂s 0 min t , x x

2

x2

x2

e− 64t2 xe− 32t2 ≤ C . ≤C t3 t2

(2.11)

By combining (2.10) and (2.11) we obtain ⎛ 2 ⎞ x |I| ∞ 2 2⎜ dxdt ⎟ ∂ λ 1 t ⎜ ⎟ + ⎝ ⎠ ∂s Ws (x, y)s=t2 dy |I| t 2 2 0 I max t , 3x min t , x x

C ≤ |I|

|I|

2

x − 32t 2

e 0

I

2

x

C dxdt ≤ t |I|

2

|I| ∞ x2 1 e− 32t2 dxdt ≤ C, t 0

0

and also, when xI > |I|, we can write 2 ⎛ ⎞ x |I| ∞ 2 dxdt 2⎜ ⎟ ∂ λ 1 t ⎜ ⎟ Ws (x, y)|s=t2 dy + ⎝ ⎠ |I| ∂s t 2 2 0 I max t , 3x min t , x x

C ≤ |I|

|I|

2

x − 32t 2

e 0

I

2

C dxdt ≤ t |I|

x

I

2

1 dx x3

|I| 0

t2 dt ≤ C|I|3

xI . (4x2I − |I|2 )2

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By (2.5) and proceeding as above we conclude that 2 ⎛ ⎞ x |I| ∞ 2 dxdt 2⎜ ⎟ ∂ λ |f2I |2 t ⎜ ⎟ + ⎝ ⎠ ∂s Ws (x, y)s=t2 dy |I| t 2 2 0 I max t , 3x min t , x x

≤

2

x

2

C||f ||2BMO+ .

(2.12)

We now prove that

|f2I |2 |I|

|I| 0

I

2 max tx2 , 3x 2 2 dxdt ∂ λ t Ws (x, y) 2 dy ≤ C||f ||2BMO+ . s=t ∂s t max t2 , x x

2

(2.13) Equality (2.9) suggests us to write, for each t, x ∈ (0, ∞), max

max

t2

x

t2 x

, 3x 2

,x 2

∂ λ W (x, y) 2 dy s=t ∂s s

=

( x2 , 3x 2 )∩ ⎛ +

where Ws (x, y) = e We note that

,∞

⎞

⎟ ∂ ⎜ ⎜ Ws (x, y)dy ⎟ ⎠ ⎝ ∂s

x 3x t2 ( 2 , 2 )∩ x ,∞

−

3x

(x−y)2 4s

√ 2 πs

3x

2

2 Ws (x, y)dy =

x 2

t2 x

∂ λ Ws (x, y) − Ws (x, y) 2 dy s=t ∂s

x 2

x

(x−y)2

∞ 0

s=t2

, s, x, y ∈ (0, ∞).

e− 4s 1 √ dy = √ 2 πs 2 π

1 = √ π

,

2 −u 4s

e 1 √ du − √ s π

1 = 1− √ π

∞ x 2

2 −u 4s

e √

s

du,

2 −x 2

∞ x 2

u2

e− 4s 1 √ du = √ s π 2 −u 4s

e √ du s

s, x ∈ (0, ∞).

x

2 0

u2

e− 4s √ du s

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Then 3x

∂ ∂s

2 x 2

1 Ws (x, y)dy = − √ π

∞ x 2

∂ ∂s

u2

e− 4s √ s

s, x ∈ (0, ∞),

du,

and, for all t, x ∈ (0, ∞), ⎛ ⎞ 3x 2 ∞ − u22 ∂ ⎜ e 8t ⎟ ⎝ Ws (x, y)dy ⎠ ≤C du ∂s t3 x x s=t2 2 2 ∞

x2

e− 64t2 ≤ C t3

2

u − 16t 2

e 0

x2

e− 64t2 du ≤ C . t2

Moreover, for each t, x ∈ (0, ∞), t2

2

2 x − (x−y) ∂ e 8t2 Ws (x, y) dy ≤ C dy s=t2 ∂s t3

0

3x t (x 2 , 2 )∩(0, x )

t2

x ≤C

e−

x2 +y 2 8t2

t3

0

x2

e− 8t2 dy ≤ C 2 . t

By proceeding as in the proof of (2.12) we obtain 2 ⎛ ⎞ |I| dxdt 2 ⎟ |f2I | 2 ∂ ⎜ ⎜ ⎟ ≤ C||f ||2BMO+ . W (x, y)dy t s ⎝ ⎠ |I| ∂s t 0 I 3x t2 (x s=t2 2 , 2 )∩( x ,∞) (2.14) On the other hand, equation (2.9) leads to 3x 2 λ (x−y)2 dy C ∂ Ws (x, y) − Ws (x, y) 2 dy ≤ e− 8t2 s=t ∂s xt y x t2 ( x2 , 3x 2 2 )∩( x ,∞) C ≤ 2 x t for each t, x ∈ (0, ∞).

+∞ (x−y)2 C e− 8t2 dy ≤ 2 , x

−∞

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Hence, 2 |I| dxdt ∂ λ 1 t 2 Ws (x, y) − Ws (x, y) 2 dy s=t |I| ∂s t

0 I t2 ∩ ,∞ ( x2 , 3x ) 2 x C ≤ |I|

|I| t 0

3

∞

√2

1 dxdt ≤ C, x4

3t

and if xI > |I|, we can also write 2 |I| 2 dxdt λ 1 ∂ t Ws (x, y) − Ws (x, y) 2 dy s=t |I| ∂s t

0 I x 3x t2 , ∩ ,∞ (2 2 ) x C ≤ |I|

|I| 0

t3 dt

I

1 (2xI + |I|)3 − (2xI − |I|)3 dx ≤ C|I|3 . 4 x (4x2I − |I|2 )3

These estimates and (2.5) allow us to obtain, when xI < |I|, 2 |I| 2 dxdt |f2I | ∂ λ t 2 Ws (x, y) − Ws (x, y) 2 dy s=t |I| ∂s t

0 I t2 ∩ ,∞ ( x2 , 3x ) 2 x ≤ C||f ||2BMO+ , and, in the case that xI > |I|, 2 |I| 2 dxdt λ |f2I |2 ∂ t Ws (x, y) − Ws (x, y) 2 dy s=t |I| ∂s t

0 I t2 ( x2 , 3x 2 )∩ x ,∞ ≤C

|I| (2xI + |I|)3 − (2xI − |I|)3 ||f ||2BMO+ ≤ C||f ||2BMO+ . xI + |I| (2xI − |I|)3

Thus

2 |I| 2 dxdt ∂ |f2I | λ t 2 W (x, y) − W (x, y) dy s s 2 s=t |I| ∂s t

0 I x 3x t2 ( 2 , 2 )∩ x ,∞ ≤ C||f ||2BMO+ , that, jointly (2.14), leads to (2.13).

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By (2.8), (2.12) and (2.13) we obtain |I|

1 |I|

0

|Qλt (f3 )(x)|2

dxdt ≤ C||f ||2BMO+ . t

(2.15)

I

Hence, by (2.3), (2.4) and (2.15) we establish (2.2) and the proof is thus ﬁnished.

3. Proof of (ii) =⇒ (iii) in Theorem 1.1 Let λ > 0 and f a measurable function on (0, ∞) such that (1 + x2 )−1 f ∈ L1 (0, ∞). Assume that dµf is a Carleson measure. By using subordination formula we have that ∞ −u e 1 λ √ W λt2 (f )(x)du, t, x ∈ (0, ∞). Pt (f )(x) = √ π u 4u 0

Then, ∂ 1 t Ptλ (f )(x) = √ ∂t π

∞ 0

t2 e−u ∂ λ √ W (f )(x) t2 du, s= 4u 2u u ∂s s

t, x ∈ (0, ∞).

The Minkowski inequality leads to ⎧ ⎫ 12 2 ⎪ ⎪ ⎨ 1 |I| ∂ ⎬ t Ptλ (f )(x) dxdt ⎪ ∂t t ⎪ ⎩ |I| ⎭ 0 I ⎧ ⎫ 12 2 ⎪ ∞ −u ⎪ |I| 2 ⎨ t ∂ λ dxdt ⎬ 1 e 1 √ ≤√ du t2 2u ∂s Ws (f )(x)s= 4u t ⎪ π u⎪ ⎩ |I| ⎭ 0 0 I ⎫ 12 ⎧ |I| √ ⎪ ⎪ 2 u ∞ ⎪ ⎪ −u ⎨ 2 ∂ λ 2 dxdv ⎬ 1 1 e √ =√ du 2v ∂s Ws (f )(x)|s=v2 |I| v ⎪ π u⎪ ⎪ ⎪ ⎭ ⎩ 0 0 I ⎛ ⎧ ⎫ 12 ⎪ 2 ⎪ |I| ⎨ −u 2 ∂ λ dxdv ⎬ ⎜ 1 e ⎜ √ ≤C⎝ du 2v ∂s Ws (f )(x)|s=v2 ⎪ |I| ⎪ v ⎭ u⎩ √ 0

2 u≥1

e−u

+

3

√ 2 u<1 1

u4 ∞

≤ C(||µf ||C ) 2

0

−u 2

e

3

0

I

⎧ ⎪ ⎨ 1 |J| ⎪ |J| ⎩

u4

J

⎫ 12 ⎞ 2 ⎪ 2 ∂ λ dxdv ⎬ ⎟ 2v 2 W (f )(x) du⎟ |s=v s ⎠ ⎪ ∂s v ⎭ 1

du ≤ C(||µf ||C ) 2 ,

for every I ⊂ (0, ∞). Hence dγf is a Carleson measure and ||γf ||C ≤ C||µf ||C .

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4. Proof of (iii) =⇒ (i) in Theorem 1.1 To prove that (iii ) ⇒ (i ) in Theorem 1.1 we ﬁrst have to show several results. We establish new characterizations of the space Ho1 (R) by using nontangential g-functions associated with the Poisson semigroup for the Bessel operators ∆λ . We denote by Pt (x) the classical Poisson kernel, that is, Pt (x) =

t 1 , 2 π t + x2

t ∈ (0, ∞), x ∈ R.

The classical Poisson integral of f is deﬁned by Pt (f )(x) = Pt (y − x)f (y)dy, t ∈ (0, ∞), x ∈ R. R

We consider the sets Γ(x) = {(y, t) ∈ R × (0, ∞) : |x − y| < t} ,

x ∈ R,

and Γ+ (x) = {(y, t) ∈ (0, ∞) × (0, ∞) : |x − y| < t} ,

x ∈ (0, ∞).

In the next proposition we use the nontangential g-functions deﬁned as follows: ⎧ ⎫ 12 ⎪ 2 ⎪ ⎨ ∂ dtdy ⎬ g(f )(x) = t ∂t Pt (f )(y) t2 ⎪ , x ∈ R, ⎪ ⎩ ⎭ Γ(x)

⎧ ⎪ ⎨ g+ (f )(x) =

⎪ ⎩

Γ+ (x)

and

⎧ ⎪ ⎨ gλ (f )(x) =

⎪ ⎩

Γ+ (x)

⎫ 12 2 ⎪ ⎬ ∂ t Pt (f )(y) dtdy ∂t t2 ⎪ , ⎭ ⎫ 12 2 ⎪ ⎬ ∂ λ t Pt (f )(y) dtdy ∂t t2 ⎪ , ⎭

x ∈ (0, ∞),

x ∈ (0, ∞).

Proposition 4.1. Let λ > 0. Suppose that f ∈ L1 (R) and f is odd. Then the following assertions are equivalent: (a) (b) (c) (d)

f ∈ H 1 (R). g(f ) ∈ L1 (R). g+ (f ) ∈ L1 (0, ∞). gλ (f ) ∈ L1 (0, ∞).

Moreover, the quantities ||f ||L1 (R) +||g(f )||L1 (R) , ||f ||L1 (0,∞) +||g+ (f )||L1 (0,∞) and ||f ||L1 (0,∞) + ||gλ (f )||L1 (0,∞) are equivalent to ||f ||H 1 (R) . Proof. (a) ⇔ (b). It is a well-known result ([11, Proposition 4, p. 124]).

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(b) ⇔ (c). It is clear that (b) ⇒ (c). On the other hand, since f is odd, we can write, for every y ∈ R and t > 0, +∞ ∞ Pt (z − y)f (z)dz = (Pt (z − y) − Pt (z + y))f (z)dz. Pt (f )(y) = −∞

0

Hence Pt (f ), t > 0, is an odd function. Then, for every t, x ∈ (0, ∞), we can write {y∈R:(y,t)∈Γ(−x)}

−x+t 2 2 ∂ ∂ t Pt (f )(y) dy = t Pt (f )(y) dy ∂t ∂t −x−t

x+t

2 ∂ t Pt (f )(y) dy = ∂t

= x−t

{y∈R:(y,t)∈Γ(x)}

2 ∂ t Pt (f )(y) dy, ∂t

and also, if 0 < x < t, {y∈R:(y,t)∈Γ(x)}

⎞ ⎛ 0 x+t 2 2 ∂ ∂ t Pt (f )(y) dy = ⎝ ⎠ + ∂t t ∂t Pt (f )(y) dy x−t

0

x+t 2 ∂ ≤2 t ∂t Pt (f )(y) dy 0

≤2 {y∈(0,∞):(y,t)∈Γ+ (x)}

2 ∂ t Pt (f )(y) dy. ∂t

√ Thus, we get that g(f ) is an even function verifying that g(f )(x) ≤ 2g+ (f )(|x|), x ∈ R, and we conclude that (c) ⇒ (b). Let us prove (c) ⇔ (d). First we show that (c) is equivalent to the following property: ⎫ 12 ⎧ 2 2x ⎪ ∂ dtdy ⎪ ⎬ ⎨ (c ) g+,loc (f )(x) = Pt (z − y)f (z)dz ∈ L1 (0, ∞). t ∂t t2 ⎪ ⎪ ⎭ ⎩ x Γ+ (x) 2 Note that, for every x ∈ (0, ∞), |g+ (f )(x) − g+,loc (f )(x)| ⎧ ⎫ 12 ⎤ 2 ⎡ ⎪ ⎪ 2x ⎨ ⎬ ∂ ⎢ ⎥ dtdy ≤ t ⎣Pt (f )(y) − Pt (z −y)f (z)dz ⎦ 2 t ⎪ ∂t ⎪ ⎩ ⎭ x Γ+ (x) 2

Vol. 66 (2010) ⎧ ⎪ ⎨ =

⎪ ⎩

Γ+ (x)

⎧ ⎪ ⎨ =

⎪ ⎩

Γ+ (x)

Odd BMO(R) Functions and Carleson Measures

479

⎫ 12 ⎤ 2 ⎡ ⎪ ∞ 2x ⎬ ∂⎢ ⎥ dtdy t ⎣ (Pt (z−y) − Pt (z +y))f (z)dz− Pt (z−y)f (z)dz ⎦ t2 ⎪ ∂t ⎭ x 0 2

2 ⎫ 12 ⎪ ∞ ⎬ ∂ dtdy ∂ Pt (z −y)f (z)dz − t Pt (z+y)f (z)dz . t t2 ⎪ ∂t ∂t ⎭ 0 (0,∞)\( x ,2x) 2

By using the Minkowski inequality, it gets, for every x ∈ (0, ∞), |g+ (f )(x) − g+,loc (f )(x)| ≤ H1 (f )(x) + H2 (f )(x), where H1 (f )(x) =

|f (z)|

(0,∞)\( x 2 ,2x)

⎧ ⎪ ⎨ ⎪ ⎩

Γ+ (x)

and 2x H2 (f )(x) =

|f (z)| x 2

⎫ 12 2 ⎪ ⎬ ∂ t [Pt (z − y) − Pt (z + y)] dtdy ∂t t2 ⎪ dz, ⎭

⎧ ⎪ ⎨ ⎪ ⎩

Γ+ (x)

⎫ 12 2 ⎪ ∂ dtdy ⎬ t Pt (z + y) ∂t t2 ⎪ dz. ⎭

Then the equivalence between (c) and (c ) will be established when we see that Hi (f ), i = 1, 2, belongs to L1 (0, ∞). We begin analyzing H2 . We can write, for every t, z, y ∈ (0, ∞), ∂ t t Pt (z + y) = |(1 − 2πtPt (z+y))Pt (z+y)| ≤ CPt (z + y) ≤ C . ∂t (z + y + t)2 Hence, for each x, z ∈ (0, ∞), 2 ∂ t Pt (z + y) dtdy ≤ C ∂t t2 Γ+ (x)

Γ+ (x)

∞ ≤C 0

dtdy ≤C (z + y + t)4

∞ 0

dy (z + y + |x − y|)3

1 C dy ≤ 2 , 3 (z + y) z

and then, ∞

∞ 2x |H2 (f )(x)|dx ≤ C

0

0

x 2

|f (z)| dzdx ≤ C||f ||L1 (0,∞) . z

On the other hand, it has ∂ 4zy [Pt (z − y) − Pt (z + y)] = 2 ∂t π((z − y) + t2 )((z + y)2 + t2 ) 16zyt2 (z 2 + y 2 + t2 ) − , π((z − y)2 + t2 )2 ((z + y)2 + t2 )2 for each z, y, t ∈ (0, ∞).

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Then, ∂ Czy [Pt (z − y) − Pt (z + y)] ≤ ∂t ((z − y)2 + t2 )((z + y)2 + t2 ) √ C z √ ≤ (|z − y| + t)2 z + y + t √ C z ≤ z, y, t ∈ (0, ∞). 5 , (|z − y| + t) 2 Hence, for z, x ∈ (0, ∞), 2 ∂ t [Pt (z − y) − Pt (z + y)] dtdy ∂t t2 Γ+ (x)

≤C Γ+ (x)

⎡

⎢ ≤C⎣

z dtdy ≤ C (|z − y| + t)5

max{x,z}

min{x,z}

⎛ z ⎜ dy + ⎝ (x − z)4

∞ 0

z dy (|z − y| + |x − y|)4

min{x,z}

∞

+ 0

max{x,z}

z , ≤C |x − z|3

⎞

⎤

⎟ ⎠

z ⎥ dy ⎦ (x + z − 2y)4 (4.1)

and we get ∞

∞ |H1 (f )(x)|dx ≤ C

0

⎛ √ ⎜ |f (z)| z ⎝

0

z

2 0

⎞ ∞ ⎟ + ⎠ 2z

1 3

|x − z| 2

dxdz ≤ C||f ||L1 (0,∞) .

Thus we have proved that ||g+ (f ) − g+,loc (f )||L1 (0,∞) ≤ ||f ||L1 (0,∞) which implies that (c) is equivalent to (c ). We now show that (c ) ⇔ (d). The Poisson kernel Ptλ given by (1.4) is splited into two parts as follows ⎞ ⎛ π 2 π λ (sin θ)2λ−1 2λ(yz) t ⎜ ⎟ Ptλ (y, z) = dθ ⎝ + ⎠ π ((z − y)2 + t2 + 2zy(1 − cos θ))λ+1 0

π 2

λ λ (y, z) + Pt,2 (y, z), = Pt,1

t, y, z ∈ (0, ∞).

We observe that, for each x ∈ (0, ∞), ⎧ ⎫ 12 ⎡ ⎤ 2 ⎪ ⎪ 2x ⎨ ⎬ ∂ ⎢ λ ⎥ dtdy |gλ (f )(x) − g+,loc (f )(x)| ≤ . t ⎣Pt (f )(y)− Pt (z −y)f (z)dz ⎦ 2 ∂t t ⎪ ⎪ ⎩ ⎭ x Γ+ (x) 2

Then, the Minkowski inequality leads to

Vol. 66 (2010)

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481

|gλ (f )(x) − g+,loc (f )(x)| ⎧ ⎫ 12 ⎪ 2 ⎪ ∞ ⎨ ∂ dtdy ⎬ λ t Pt,2 ≤ |f (z)| (y, z) ∂t t2 ⎪ dz ⎪ ⎩ ⎭ 0

Γ+ (x)

|f (z)|

+ (0,∞)\( x 2 ,2x)

2x |f (z)|

+ x 2

=

⎪ ⎩

Kj (f )(x),

⎪ ⎩

Γ+ (x)

⎧ ⎪ ⎨ Γ+ (x)

3

⎧ ⎪ ⎨

⎫ 12 2 ⎪ ∂ λ dtdy ⎬ t Pt,1 (y, z) ∂t t2 ⎪ dz ⎭

⎫ 12 2 ⎪ ∂ λ dtdy ⎬ t ∂t Pt,1 (y, z) − Pt (z − y) t2 ⎪ dz ⎭

x ∈ (0, ∞).

j=1

To see that (c ) is equivalent to (d) it is sufﬁcient to show that Kj (f ) ∈ L1 (0, ∞), j = 1, 2, 3. Note ﬁrstly that, since λ > 0, ⎡ π ∂ λ (sin θ)2λ−1 Pt,2 (y, z) ≤ C(yz)λ ⎢ dθ ⎣ ∂t (z 2 + y 2 + t2 − 2zy cos θ)λ+1 π 2

⎤

π +

(z 2

π 2

≤C

2

(z 2

+

y2

2λ−1

t (sin θ) ⎥ dθ⎦ + t2 − 2zy cos θ)λ+2

zλ (yz)λ ≤C , 2 2 λ+1 +y +t ) (z + y + t)λ+2

t, y, z ∈ (0, ∞).

Hence, for every z, x ∈ (0, ∞), Γ+ (x)

2 ∞ ∞ ∂ λ z 2λ t Pt,2 (y, z) dtdy ≤ C dtdy ∂t t2 (z + y + t)2λ+4 0 |x−y|

∞

z 2λ dy (z + y + |x − y|)2λ+3 0 ⎛ x ⎞ ∞ 1 1 dy + dy ⎠ ≤ Cz 2λ ⎝ (z + x)2λ+3 (z − x + 2y)2λ+3

≤C

≤C

0 2λ

z . (z + x)2λ+2

x

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Then, ∞

∞ |K1 (f )(x)|dx ≤ C

0

∞ |f (z)|

0

0

zλ dxdz ≤ C||f ||L1 (0,∞) . (z + x)λ+1

Let us now see that K2 (f ) ∈ L1 (0, ∞). Since sin θ ∼ θ and 2(1−cos θ) ∼ θ , θ ∈ [0, π2 ], by considering separately z > y2 and 0 < z ≤ y2 , we have that 2

π

2 ∂ λ (yz)λ (sin θ)2λ−1 Pt,1 (y, z) ≤ C dθ ∂t (z 2 + y 2 + t2 − 2zy cos θ)λ+1 0

π

2 ≤C 0

(yz)λ θ2λ−1 dθ ((z − y)2 + t2 + zyθ2 )λ+1 π

2 ≤ Cz λ

λ

0

≤C

θλ−1 ((z − y)2 + t2 + yzθ2 ) 2 +1

zλ , (|z − y| + t)λ+2

dθ

t, y, z ∈ (0, ∞).

Hence, by proceeding as in (4.1), it follows that, for each x, z ∈ (0, ∞), Γ+ (x)

2 ∞ ∂ λ z 2λ z 2λ t Pt,1 (y, z) dtdy ≤ C dy ≤ C , ∂t t2 (|z − y| + |x − y|)2λ+3 |x − z|2λ+2 0

and ⎛z ⎞ ∞ ∞ 2 ∞ ⎜ ⎟ |K2 (f )(x)|dx ≤ C |f (z)|z λ ⎝ + ⎠ 0

0

0

2z

1 dxdz ≤ C||f ||L1 (0,∞) . |x − z|λ+1

Finally, we are going to see that K3 (f ) ∈ L1 (0, ∞). The proof of this fact is divided in several parts. In a ﬁrst step we will prove that, if 0 < x2 < z < 2x, then ⎧ ⎪ ⎨

2 ⎫ 12 π 2 dtdy ⎪ ∂ ⎬ (zy)λ t((sin θ)2λ−1 − θ2λ−1 ) dθ t ∂t ⎪ ((z − y)2 + t2 + 2zy(1 − cos θ))λ+1 t2 ⎪ ⎩ ⎭ 0 Γ+ (x) z2 C ≤ . (4.2) 1 + log+ z |x − z|2

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Since sin θ ∼ θ, and 2(1−cos θ) ∼ θ2 , as θ ∈ [0, π2 ], the mean value theorem leads to π2 ∂ 2λ−1 2λ−1 t((sin θ) −θ ) dθ ∂t ((z − y)2 + t2 + 2zy(1 − cos θ))λ+1 0 π

2 ≤C 0

|(sin θ)2λ−1 −θ2λ−1 | dθ ((z − y)2 +t2 +2zy(1−cos θ))λ+1

π

2 ≤C 0

θ2λ+1 dθ, ((z − y)2 +t2 +zyθ2 )λ+1

(4.3)

for every t, z, y'∈ (0, ∞). According to [10, p. 61] and making the change of zy variables u = θ (z−y) 2 +t2 , we get π

2

((z −

0

y)2

θ2λ+1 dθ + t2 + zyθ2 )λ+1 π 2

≤

C (zy)λ+1

'

zy (z−y)2 +t2

0

u2λ+1 du (1 + u2 )λ+1

(

2λ+2 zy/((z − y)2 + t2 ) zy ( 1 + log+ (z − y)2 + t2 1 + zy/((z − y)2 + t2 ) zy C ≤ 1 + log , + (zy + (z − y)2 + t2 )λ+1 (z − y)2 + t2 ⎧ 1 z ⎪ , 0 < y < or y > 2z > 0, t > 0, ⎨ (zy+(z − y)2 +t2 )λ+1 2 ≤C zy 1 z ⎪ ⎩ 1+log+ , < y < 2z, t > 0. (zy+(z − y)2 +t2 )λ+1 (z − y)2 + t2 2

≤

C (zy)λ+1

In the last inequality we have taken into account that, when 0 < 2z < y, it has that |z − y|2 + t2 ≥ |z − y|2 ≥ zy 2 . Thus we can write 2 π 2 λ 2λ−1 2λ−1 (zy) t((sin θ) −θ ) dtdy ∂ dθ t 2 2 λ+1 t2 ∂t ((z − y) + t + 2zy(1 − cos θ)) 0 Γ+ (x) 2 ⎞ ⎛ π 2z ∞ 2 λ 2λ−1 2λ−1 (zy) t((sin θ) −θ ) ∂ ⎟ ⎜ =⎝ + ⎠ dθ dtdy 2 2 λ+1 ∂t ((z − y) + t + 2zy(1 − cos θ)) z 0 |x−y| (0,∞)\( z ,2z)

2

=

2 j=1

Ij (x, z),

2

x, z ∈ (0, ∞).

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x 2

IEOT

< z < 2x we have that

⎞ ⎛ z 2 ∞ ∞ ⎟ ⎜ I1 (x, z) ≤ C ⎝ + ⎠ 0

2z

|x−y|

⎞ ⎛ z 2 ∞ ∞ ⎟ ⎜ ≤ C⎝ + ⎠ 0

2z

|x−y|

⎞ ⎛ z 2 ∞ ⎟ ⎜ = C⎝ + ⎠ 0

(zy)2λ dtdy (zy + (z − y)2 + t2 )2λ+2

2z

⎞ ⎛z 2 ∞ dtdy 1 ⎟ ⎜ ≤C⎝ + ⎠ dy 4 (|z − y|+t) (|z − y|+|y − x|)3 0

2z

1 C dy ≤ 2 . |z + x − 2y|3 z

On the other hand, if |x − y| < t and z ∈ (0, ∞), then log+ log+

2

z |z−y|2 +|x−y|2

≤ log+

2z ∞ I2 (x, z) ≤ C z 2

|x−y|

2

2z |x−z|2 .

2z 2 ≤ C 1 + log+ |x − z|2 ≤ C 1 + log+

z2 |x − z|2

≤

Hence, we get, for x, z ∈ (0, ∞),

z 4λ (z 2 + (z − y)2 + t2 )2λ+2

z2 |z−y|2 +t2

2 2z ∞ z 2

|x−y|

1 + log+

z2 |z − y|2 + t2

2 dtdy

1 dtdy (z + |z − y| + t)4

2 2z z 2

1 dy (z+|z − y|+|y − x|)3

2 C z2 ≤ 2 1+log+ , z |x − z|2

and the proof of (4.2) is thus ﬁnished. In the second step, our objective is to establish that ⎧ ⎪ ⎨

π ∂ 2 (zy)λ tθ2λ−1 t 2 ∂t ⎪ ((z − y) + t2 + 2zy(1 − cos θ))λ+1 ⎩ 0 Γ+ (x) )1

2 dtdy 2 (zy)λ tθ2λ−1 − dθ ((z − y)2 + t2 + zyθ2 )λ+1 t2 z2 C x ≤ , 0 < < z < 2x. 1 + log+ 2 z (x − z) 2

(4.4)

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Odd BMO(R) Functions and Carleson Measures

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By using that 2(1 − cos θ) ∼ θ2 , as θ ∈ [0, π2 ], and the mean value theorem, we can write J(z, t, y) π2

∂ 2λ−1 2λ−1 tθ tθ − = dθ ∂t ((z − y)2 + t2 + 2zy(1 − cos θ))λ+1 ((z − y)2 + t2 + zyθ2 )λ+1 0 π

2 ≤

θ2λ−1

0

1 1 dθ − ((z − y)2 + t2 + 2zy(1 − cos θ))λ+1 ((z − y)2 + t2 + zyθ2 )λ+1

π

2 + t2 θ2λ−1 0

2(λ+1) 2(λ + 1) dθ − ((z − y)2 +t2 +2zy(1 − cos θ))λ+2 ((z − y)2 + t2 + zyθ2 )λ+2

π

2 ≤C 0

zyθ2λ+3 dθ ((z − y)2 + t2 + zyθ2 )λ+2

π

2 ≤C 0

((z −

y)2

θ2λ+1 dθ, + t2 + zyθ2 )λ+1

z, t, y ∈ (0, ∞).

The last term is the same one that appears in the last term in (4.3). Then (4.4) is established. In the third step we will prove that, for x, z ∈ (0, ∞), ⎫ 12 ⎧ ⎡ ⎤2 π 2 ⎪ ⎪ ∂ 2λ ⎬ C ⎨ (zy)λ tθ2λ−1 ⎢ ⎥ dtdy dθ − P (z − y) ≤ . t ⎣ ⎦ t ∂t π t2 ⎪ ⎪ ((z − y)2 + t2 + yzθ2 )λ+1 z ⎭ ⎩ 0 Γ+ (x) (4.5) We ﬁrst write the following equality: π

2 0

tθ2λ−1 dθ ((z − y)2 + t2 + zyθ2 )λ+1 ⎞ ⎛ ∞ ∞ tθ2λ−1 ⎟ ⎜ =⎝ − ⎠ dθ, t, z, y ∈ (0, ∞). 2 ((z − y) + t2 + zyθ2 )λ+1 0

π 2

Moreover, by making the change of variable u = θ ∞ 0

'

tθ2λ−1 t 1 dθ = 2 2 2 λ+1 λ ((z − y) + t + zyθ ) (zy) (z − y)2 + t2

zy (z−y)2 +t2

∞ 0

we obtain

u2λ−1 du (1 + u2 )λ+1

π 1 = Pt (z − y), t, z, y ∈ (0, ∞). 2λ (zy)λ

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Then, it has, for every t, z, y ∈ (0, ∞), π

2λ π

2

(zy)λ tθ2λ−1 dθ − Pt (z − y) ((z − y)2 + t2 + zyθ2 )λ+1

0

2λ =− π

∞ π 2

(zy)λ tθ2λ−1 dθ. ((z − y)2 + t2 + zyθ2 )λ+1

(4.6)

We now analyze 2 ⎫ 12 ∞ dtdy ⎪ ∂ ⎬ (zy)λ tθ2λ−1 dθ , x, z ∈ (0, ∞). J (x, z) = t ∂t ((z − y)2 + t2 + zyθ2 )λ+1 t2 ⎪ ⎪ ⎭ ⎩ π Γ+ (x) ⎧ ⎪ ⎨

2

The Minkowski inequality leads to 2 ⎫ 12 ∞ ⎪ ⎬ (zy)λ θ2λ−1 dθ dtdy J (x, z) ≤ C ((z − y)2 + t2 + zyθ2 )λ+1 ⎪ ⎪ ⎭ ⎩ Γ+ (x) π 2 ⎫ 12 ⎧ ⎪ ∞ ⎪ ⎬ ⎨ (zy)2λ θ4λ−2 ≤C dtdy dθ ⎪ ⎪ ((z − y)2 + t2 + zyθ2 )2λ+2 ⎭ ⎩ π ⎧ ⎪ ⎨

2

Γ+ (x)

⎫ 12 ⎪ ⎬

⎧ ∞ ⎪ 1⎨ ≤C θ⎪ ⎩ π 2

∞ ≤C π 2

Γ+ (x)

⎧∞ ⎨

1 θ⎩

0

1 dtdy dθ √ ⎪ (|z − y| + t + θ zy)4 ⎭

⎫ 12 ⎬ 1 dθ √ 3 dy ⎭ (|z − y| + |y − x| + θ zy)

⎫ 12 ⎧⎛ z ⎞ ⎪ ∞ ⎪ 2 ∞ ⎬ ⎨ 1 1 ⎜ ⎟ ≤C + dy dθ, x, z ∈ (0, ∞). √ ⎠ ⎝ θ⎪ (|z − y| + θ zy)3 ⎪ ⎭ ⎩ π z 2

0

2

Moreover, by using the H¨ older inequality we get, for x, z ∈ (0, ∞) and θ ∈ ( π2 , ∞), z

2 0

z

1 dy ≤ C √ (|z − y| + θ zy)3

2

1 √ 5 1 dy |z − y| 2 (θ zy) 2 0 ⎫ 12 ⎧ z ⎫ 12 ⎧ z ⎪ ⎪ ⎪ ⎬ ⎪ ⎨2 ⎨2 1 dy ⎬ C ≤√ . ≤C √ dy 5 ⎪ θz 2 ⎭ ⎪ ⎩ |z − y| ⎪ ⎭ ⎩ θ zy ⎪ 0

0

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On the other hand, ∞ z 2

1 dy ≤ C √ (|z − y| + θ zy)3

∞ z 2

1 C dy ≤ 3 2 . √ (θ zy)3 θ z

Then, we obtain that C J (x, z) ≤ z

∞ π 2

dθ θ

5 4

≤

C , z

x, z ∈ (0, ∞).

From (4.6) we get then (4.5). By combining (4.2), (4.4) and (4.5) it follows that, when 0 < x2 < z < 2x, ⎧ ⎪ ⎨ ⎪ ⎩

Γ+ (x)

⎫ 12 ⎪ ∂ λ 2 dtdy ⎬ z2 C t P (z, y) − P (z − y) ≤ . 1 + log t + ∂t t,1 t2 ⎪ z |z − x|2 ⎭

Hence ∞

∞ 2x |K3 (f )(x)|dx ≤ C 1 + log+

0

0

x 2

∞ ≤C

2z |f (z)| z 2

0

∞

2 |f (z)|

≤C 0

1 2

1 x 1 u

z2 |z − x|2

|f (z)| dzdx z

z2 1 + log+ |z − x|2

1 + log+

1 |1 − u|

dxdz

dudz ≤ C||f ||L1 (0,∞) .

Then we conclude that ||gλ (f ) − g+,loc (f )||L1 (0,∞) ≤ ||f ||L1 (0,∞) , which leads to (c ) ⇔ (d). Finally, from the estimations established above and the fact that ||f ||H 1 (R) ∼ ||f ||L1 (R) + ||g(f )||L1 (R) , we deduce that ||f ||H 1 (R) ∼ ||f ||L1 ((0,∞) + ||g+ (f )||L1 (0,∞) ∼ ||f ||L1 ((0,∞) + ||gλ (f )||L1 (0,∞) . Thus the proof of this proposition is ﬁnished.

Remark 4.2. Since the operator f −→ g(f ) is bounded from Lp (0, ∞) into itself, when 1 0, the operator f −→ gλ (f ) is also bounded from Lp (0, ∞) into itself, when 1 < p < ∞, and from L1 (0, ∞) into L1,∞ (0, ∞). As far as we know, this result is new. We now present other useful results in order to establish (iii) ⇒ (i).

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If F is a measurable function on (0, ∞) × (0, ∞), we deﬁne F*, Φ(F ) and Ψ(F ) as follows: + F (y, t), (y, t) ∈ (0, ∞) × (0, ∞), F*(y, t) = F (−y, t) (y, t) ∈ (−∞, 0) × (0, ∞), ⎛ ⎞ 12 |I| dydt ⎟ ⎜ 1 |F (y, t)|2 Φ(F )(x) = sup ⎝ ⎠ , x ∈ (0, ∞), |I| t I⊂(0,∞),I x 0

and

⎛ ⎜ Ψ(F )(x) = ⎝

I

⎞ 12 |F (y, t)|2

Γ+ (x)

dydt ⎟ ⎠ , t2

x ∈ (0, ∞).

From a well-known result about tent spaces (see [11, p. 162]) we can deduce the following. Proposition 4.3. Let F and G be measurable functions on (0, ∞) × (0, ∞). Suppose that Φ(F ) ∈ L∞ (0, ∞) and Ψ(G) ∈ L1 (0, ∞). Then ∞ ∞ ∞ dydt ≤ C Φ(F )(x)Ψ(G)(x)dx. |F (y, t)G(y, t)| t 0

0

0

Proof. According to [11, p. 162] we have ∞ ∞ ∞ ∞ dydt * t)| dydt = 2 |F (y, t)G(y, t)| |F*(y, t)G(y, t t 0

0 −∞

0

+∞ * ≤C Υ(F*)(x)Ω(G)(x)dx,

* where Ω(G)(x) =

Γ(x)

* t)|2 dydt |G(y, t2 ⎛

Υ(F*)(x) =

⎜ 1 ⎝ |I| I⊂R,I x

|I|

sup

0

12

(4.7)

−∞

, x ∈ R, and ⎞ 12

|F*(y, t)|2

dydt ⎟ ⎠ , t

x ∈ R.

I

Let I = (a, b) with −∞ < a < b < +∞. We deﬁne ⎧ a ≥ 0, ⎨ I, I = (−b, −a), b ≤ 0, ⎩ (0, |I|), a < 0 < b. Then, since F*(y, t) = F*(−y, t), t ∈ (0, ∞) and y ∈ R, it gets, for each interval I in R, 1 |I|

|I| 0

I

1 dydt ≤ |F*(y, t)|2 t |I|

|I| 0

I

|F (y, t)|2

dydt . t

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Hence, Υ(F*)(x) = Φ(F )(x), x ∈ (0, ∞), and Υ(F*)(x) = Φ(F )(−x), x ∈ (−∞, 0). * t) = G(−y, * On the other hand, since G(y, t), t ∈ (0, ∞) and y ∈ R, we have dydt * t)|2 dydt = * t)|2 dydt ≤ 2 |G(y, | G(y, |G(y, t)|2 2 , x ∈ (0, ∞). 2 2 t t t

Γ(x)

Γ(−x)

Γ+ (x)

√ * * * Then Ω(G)(x) = Ω(G)(−x), x ∈ R, and Ω(G)(x) ≤ 2Ψ(G)(x), x ∈ (0, ∞). From (4.7) it deduces that ∞ ∞ ∞ dydt ≤ C Φ(F )(x)Ψ(G)(x)dx. |F (y, t)G(y, t)| t 0

0

0

Proposition 4.4. Suppose that f is a measurable function on (0, ∞) such that (1 + x2 )−1 f ∈ L1 (0, ∞) and dγf is a Carleson measure on (0, ∞). Then 1 4

∞

∞ ∞ f (x)a(x)dx =

0

t 0

0

∂ λ ∂ dydt Pt (f )(y)t Ptλ (a)(y) , ∂t ∂t t

provided that a is an (odd)-atom. Proof. According to Propositions 4.1 and 4.3 we have that ∞ ∞ ∂ λ t Pt (f )(y)t ∂ Ptλ (a)(y) dydt < ∞. ∂t t ∂t 0

0

Then ∞ ∞ t 0

0

∂ λ ∂ dydt P (f )(y)t Ptλ (a)(y) = lim H(ε, N ), ε→0,N →∞ ∂t t ∂t t

where, for 0 < ε < N < ∞, N ∞ t

H(ε, N ) = ε

0

∂ λ ∂ P (f )(y) Ptλ (a)(y)dydt. ∂t t ∂t

Our next objective is to prove that ∞ ∞ ∞ ∂ ∂ ∂ ∂ t Ptλ (f )(y) Ptλ (a)(y)dy = f (z) t Ptλ (y, z) Ptλ (a)(y)dydz, t > 0. ∂t ∂t ∂t ∂t 0

0

0

(4.8) To see (4.8) it is sufﬁcient to show that ∞ ∂ λ t Pt (y, z) ∂ Ptλ (a)(y) dy ≤ C , ∂t ∂t 1 + z2 0

t, z ∈ (0, ∞).

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Since a ∈ L2 (0, ∞), according to [3, proof of Proposition 6.1] it has that ∂ λ ∂ Pt (a)(y) = [hλ (e−tu hλ (a)(u))](y) = −hλ (ue−tu hλ (a)(u))(y), t, y ∈ (0, ∞). ∂t ∂t (4.9)

By using (1.3) and well-known properties of Bessel functions ([9, Sect. 5.3]) we get ∞ 1 ∂ λ √ Pt (a)(y) = − (1 + ∆λ,u )( yuJλ− 12 (yu))ue−tu hλ (a)(u)du 2 ∂t 1+y 0

1 =− hλ [(1 + ∆λ,u )(ue−tu hλ (a)(u))](y), 1 + y2

t, y ∈ (0, ∞).

Moreover, we can write ∆λ,u (ue−tu hλ (a)(u)) = −ue−tu hλ+2 (x2 a(x))(u) +(2λ + 3−2tu)e−tu hλ+1 (xa(x))(u) 2λ −tu − ut2 −(2λ+2)t+ e hλ (a)(u), u

t, u ∈ (0, ∞).

Note also that, since a is an (odd)-atom and z −ν Jν (z) is a bounded function on (0, ∞), when ν > −1/2, it has |hν (xn a(x))(u)| ≤ Cuν ,

u ∈ (0, ∞), n ∈ N,

provided that ν > 0. Then we deduce that |(1 + ∆λ,u )(ue−tu hλ (a)(u))| ≤ Ce−tu p(t)q(u)uλ−1 , t, u ∈ (0, ∞), √ where p and q are polynomials. Hence, since zJλ− 12 (z) is a bounded function on (0, ∞), and λ > 0, it follows that ∂ λ Pt (a)(y) ≤ A(t) , t, y ∈ (0, ∞). (4.10) ∂t 1 + y2 Here A denotes a continuous function on (0, ∞). On the other hand, according to [10, (b) p. 86] it has that ∂ λ t Pt (y, z) ≤ C , t, z, y ∈ (0, ∞). ∂t (z − y)2 + t2

(4.11)

Estimations (4.10) and (4.11) lead to ⎛ z ⎞ ∞ 2 ∞ ∂ λ t ⎟ t Pt (y, z) ∂ Ptλ (a)(y) dy ≤ A(t)⎜ dy ⎝ + ⎠ ∂t ∂t ((z − y)2 + t2 )(1 + y 2 ) 0

⎛ A(t) ⎝ ≤ 1 + z2

0

∞ 0

1 dy + 1 + y2

∞ 0

z 2

⎞ t A(t) dy ⎠ ≤ , 2 2 (y − z) + t 1 + z2

t, z ∈ (0, ∞),

and (4.8) is thus proved. In the last chain of inequalities A represents a continuous function on (0, ∞) not necessarily the same in each ocurrence.

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Odd BMO(R) Functions and Carleson Measures

Also we have that ∞ N ∞ ∂ ∂ H(ε, N ) = f (z) t Ptλ (z, y) Ptλ (a)(y)dydtdz, ∂t ∂t 0

ε

491

0 < ε < N < ∞.

0

Moreover, by taking into account that hλ is an isometry in L2 (0, ∞) and using (4.9) it follows that ⎛∞ ⎞ ∞ ∂ ∂ ∂ ∂ t Ptλ (z, y) Ptλ (a)(y)dy = t ⎝ Ptλ (z, y) Ptλ (a)(y)dy ⎠ ∂t ∂t ∂t ∂t 0

0

∞ −

tPtλ (z, y) 0

2

∂ P λ (a)(y)dy ∂t2 t

∂ hλ (ue−2tu hλ (a))(z) − thλ (u2 e−2tu hλ (a))(z) ∂t = hλ (tu2 e−2tu hλ (a))(z), t, z ∈ (0, ∞). √ Since the function zJλ− 12 (z) is bounded in (0, ∞), Fubini theorem allows us to write ⎛N ⎞ N ∞ ∂ λ ∂ λ −2tu 2 t Pt (z, y) Pt (a)(y)dydt = hλ ⎝ te dtu hλ (a)(u)⎠ (z) ∂t ∂t = −t

ε

0

ε

1 = − hλ (2N u + 1)e−2N u hλ (a)(u) (z) 4 1 + hλ (2εu + 1)e−2εu hλ (a)(u) (z) 4 λ = MN (a)(z) − Mελ (a)(z), z ∈ (0, ∞),

where Mtλ (a) =

1 4

t

(4.12)

∂ λ λ P2v (a)|v=t − P2t (a) , ∂v

t > 0.

A straightforward manipulation leads to ∂ λ λ t P2v (z, y)|v=t − P2t (z, y) ∂v π t2 (sin θ)2λ−1 32λ(λ + 1) λ (zy) =− dθ, π ((z − y)2 + 4t2 + 2zy(1 − cos θ))λ+2 0

for every t, z, y ∈ (0, ∞). Then it has, λ ∂ λ λ ≤ C (zy) t P2v (z, y)|v=t − P2t (z, y) , ∂v |z − y|2λ+2

t, z, y ∈ (0, ∞).

Also, by [10, (b) p. 86], we get ∂ λ t λ t P2v (z, y)|v=t − P2t ≤C (z, y) , ∂v (z − y)2 + t2

t, z, y ∈ (0, ∞).

(4.13)

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Let α > 0 such that supp a ⊂ [0, α]. It follows that, when 0 < z ≤ 2α and t > 0, ∞ |Mtλ (a)(z)|

≤ C||a||L∞ (0,∞) 0

t C dy ≤ ||a||L∞ (0,∞) ≤ . (z − y)2 + t2 1 + z2

Moreover (4.13) implies that, if z ≥ 2α and t > 0, α |Mtλ (a)(z)|

≤C 0

(zy)λ C |a(y)| dy ≤ λ+2 |z − y|2λ+2 z

α |a(y)|y λ dy ≤ 0

C . 1 + z2

Hence we conclude that sup |Mtλ (a)(z)| ≤ t>0

C , 1 + z2

z ∈ (0, ∞).

From here it deduces that N ∞ C ∂ ∂ sup t Ptλ (z, y) Ptλ (a)(y)dydt ≤ , ∂t ∂t 1 + z2 0<ε
z ∈ (0, ∞).

0

Moreover, by taking into account this estimate and that (1 + x2 )−1 f ∈ L (0, ∞), the proof will be completed if we show that 1

Nk ∞ lim

k→∞

t 1 Nk

0

∂ λ ∂ a(z) P (z, y) Ptλ (a)(y)dydt = , ∂t t ∂t 4

a.e. z ∈ (0, ∞),

(4.14)

for some increasing sequence {Nk }k∈N of nonnegative integers. To see this, it is sufﬁcient to prove that N ∞ t

lim

ε→0,N →∞ ε

0

∂ λ ∂ a(z) P (z, y) Ptλ (a)(y)dydt = ∂t t ∂t 4

in L2 (0, ∞).

(4.15)

Note that (4.14) and (4.15) can be seen as Calderon reproducing formulas in our setting. Since hλ is an isometry in L2 (0, ∞) ([13, p. 473 (1)]), by (4.12) we have that (4.15) is equivalent to lim

ε→0,N →∞

(2N u + 1)e−2N u hλ (a)(u) + (2εu + 1)e−2εu hλ (a)(u) = hλ (a)(u), (4.16)

in L2 (0, ∞). We obtain (4.16) as a consequence of the dominated convergence theorem because hλ (a) ∈ L2 (0, ∞). Thus the proof of this proposition is ﬁnished.

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Proof of (iii) ⇒ (i). Suppose that a is a ﬁnite combination of (odd)-atoms. Since γf is a Carleson measure, from Propositions 4.1, 4.3 and 4.4 we deduce ∞ ∞ ∞ ∂ λ f (x)a(x)dx ≤ 4 t Pt (f )(y)t ∂ Ptλ (a)(y) dydt ∂t t ∂t 0 0 0 ∂ Ψ t ∂ Ptλ (a) ≤ C Φ t Ptλ (f ) 1 ∂t ∂t L∞ (0,∞) L (0,∞) 1

≤ C(||γf ||C ) 2 ||ao ||H 1 (R) . Here ao denotes the odd extension of a to R. By remembering that (Ho1 (R)) = BMOo (R), we conclude that f ∈ BMO+ and ||f ||2BMO+ ≤ C||γf ||C .

References [1] Andersen, K.F., Muckenhoupt, B.: Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions. Stud. Math. 72(1), 9–26 (1982) [2] Auscher, P., Russ, E., Tchamitchian, P.: Hardy Sobolev spaces on strongly Lipschitz domains of Rn . J. Funct. Anal. 218(1), 54–109 (2005) [3] Betancor, J.J., Fari˜ na, J.C., Mart´ınez, T., Torrea, J.L.: Riesz transform and g-functions associated with Bessel operators and their appropriate Banach spaces. Israel J. Math. 157, 259–282 (2007) [4] Chang, D.-C., Krantz, S.G., Stein, E.M.: H p theory on a smooth domain in RN and elliptic boundary value problems. J. Funct. Anal. 114(2), 286–347 (1993) [5] Deng, D., Duong, X.T., Sikora, A., Yan, L.: Comparison of the classical BMO with the BMO spaces associated with operators and applications. Rev. Mat. Iberoam. 24(1), 267–296 (2008) [6] Fridli, S.: Hardy spaces generated by an integrability condition. J. Approx. Theory 113(1), 91–109 (2001) [7] Haimo, D.T.: Integral equations associated with Hankel convolutions. Trans. Am. Math. Soc. 116, 330–375 (1965) [8] Hirschman, I.I. Jr.: Variation diminishing Hankel transforms. J. Anal. Math. 8, 307–336 (1960) [9] Lebedev, N.N.: Special functions and their applications. Dover, New York (1972) [10] Muckenhoupt, B., Stein, E.M.: Classical expansions and their relation to conjugate harmonic functions. Trans. Am. Math. Soc. 118, 17–92 (1965) [11] Stein, E.M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. In: Princeton Mathematical Series, vol. 43. Princeton University Press, Princeton (1993) [12] Stempak, K.: The Littlewood–Paley theory for the Fourier-Bessel transfom. Preprint n. 45, Math. Inst. Univ. Wroclaw, Poland (1985) [13] Titchmarsh, E.C.: Hankel transforms. Proc. Camb. Phil. Soc. 21, 463–473 (1923)

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[14] Watson, G.N.: A treatise on the theory of Bessel functions. Cambridge Mathematical Library. Cambridge University Press, Cambridge (1995) [15] Weinstein, A.: Discontinuous integrals and generalized potential theory. Trans. Am. Math. Soc. 63, 342–354 (1948) [16] Zemanian, A.H.: Generalized integral transformations. Dover, New York (1987) J. J. Betancor, J. C. Fari˜ na and L. Rodr´ıguez-Mesa Departamento de An´ alisis Matem´ atico Universidad de la Laguna Campus de Anchieta, Avda. Astrof´ısico Francisco S´ anchez, s/n 38271 La Laguna (Sta. Cruz de Tenerife), Spain e-mail: [email protected]; [email protected]; [email protected] A. Chicco Ruiz Instituto de Matem´ atica Aplicada del Litoral G¨ uemes 3450 Santa Fe, Argentina e-mail: [email protected] Received: January 23, 2009. Revised: August 11, 2009.

Integr. Equ. Oper. Theory 66 (2010), 495–528 DOI 10.1007/s00020-010-1760-4 Published online March 19, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

On Weighted Toeplitz, Big Hankel Operators and Carleson Measures Carme Cascante, Joan F`abrega and Joaqu´ın M. Ortega Abstract. We compute the norm of pointwise multiplication operators, Toeplitz and Big Hankel operators with antiholomorphic symbols, defined on Besov spaces. These norms will be given in terms of Carleson measures for Besov spaces related to the symbol. Mathematics Subject Classification (2000). Primary 47B35; Secondary 32A37. Keywords. Toeplitz operators, Hankel operators, Carleson measures.

1. Introduction ¯ its closure, S the unit sphere Let B be the open unit ball in Cn , n ≥ 1, B and let ν (resp. σ) be the normalized Lebesgue measure on B (resp. on S). ¯ denote the space of holomorphic functions on B Let H(B) (resp. H(B)) ¯ (resp. on B). A classical problem in complex analysis, which has been extensively treated by many authors (see for instance the books [3,18,20,24] and the references therein), is the study of the boundedness of the Toeplitz and Hankel operators in Hardy or weighted Bergman spaces. If n = 1 and P denotes the orthogonal projection from L2 (S) to the Hardy space H 2 , then the Toeplitz operator Tϕ with symbol ϕ ∈ L1 (S) deﬁned on the polynomials by Tϕ (f ) = P(ϕf ) extends to a bounded operator from H 2 to itself if and only if ϕ ∈ L∞ (S). If the symbol ϕ is holomorphic then Tϕ (f ) = ϕf and therefore the above result gives a characterization of the pointwise multipliers from H 2 to itself. It is also well-known that the Big Hankel operator Hϕ deﬁned by Hϕ (f ) = ϕf − P(ϕf ), extends to a continuous operator from H 2 to L2 if and only if P (ϕ) = Hϕ (1) ∈ BM O. In this case, if we denote by g¯ the antiholomorphic function P (ϕ), we can write the above condition in terms of Partially supported by DGICYT Grant MTM2008-05561-C02-01, Grant MTM200730904-E and DURSI Grant 2009SGR 1303.

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Carleson measures, that is Hϕ is bounded if and only if the measure dµg (z) = |g (z)|2 (1 − |z|2 )dν(z) satisﬁes ¯ < r} ≤ Cµ r, sup µg {z ∈ B; |1 − ζz| g

ζ∈S

which is in turn equivalent to the fact that the embedding H 2 ⊂ L2 (µg ) is bounded. Since Mϕ (f ) = ϕf = Tϕ (f ) + Hϕ (f ), it is easy to deduce from the above results and the fact that L∞ ⊂ BM O that Mϕ is continuous from H 2 to L2 if and only if ϕ ∈ L∞ . In this paper we study the boundedness of multiplication, weighted Toeplitz and Big Hankel operators, from holomorphic Besov spaces on the unit ball of Cn to some related Sobolev spaces. These Toeplitz and Big Hankel operators are associated to the orthogonal projection from L2 ((1 − |z|2 )N −1 dν) to H(B) ∩ L2 ((1 − |z|2 )N −1 dν). In order to state the main results, we introduce some notations. If α = (α1 , . . . , αn ) is a multiindex with αj ≥ 0, j = 1, . . . , n, we denote by n |α| = j=1 αj . If ∂j = ∂z∂ j , ∂α = ∂1α1 · · · ∂nαn is the differential operator of order |α| and ψ ∈ C k (B), let |∂ k ψ| = |α|=k |∂α ψ| and |dk ψ| = |α|+|β|=k |∂¯β ∂α ψ|. We will write the identity operator by I and the radial derivative n by R = j=1 zj ∂j . If 1 ≤ p < ∞, k a non-negative integer and δ > 0, the weighted Sobolev space Lpk,δp = Lpk ((1 − |z|2 )δp−1 dν) consists of measurable functions ψ on B such that ψpLp = |dj ψ(z)|p (1 − |z|2 )δp−1 dν(z) < ∞. k,δp

0≤j≤k B

The basic properties of these real spaces can be found for instance in [1] or [23]. For 1 ≤ p ≤ ∞ and s ∈ R, the holomorphic Besov space Bsp consists of holomorphic functions f on B, such that (1 − |z|2 )k−s (I + R)k f (z) ∈ dv(z) Lp ( 1−|z| 2 ) for some nonnegative k > s. It is well-known that if p < ∞, p Bs = H(B) ∩ Lpk,(k−s)p and that different values of k give equivalent norms ¯ is the space H(B) ¯ endowed by the norm in B p and on Bsp (see [9]). If Bsp (B) s p p < ∞, this space is dense in Bs . For N > 0, let P N denote the integral operator deﬁned on L10,N by (1 − |w|2 )N −1 Γ(n + N ) N . (1.1) dν(w), cN = P (ϕ)(z) = cN ϕ(w) (1 − wz) ¯ n+N n!Γ(N ) B

¯ by, Related to P , the Toeplitz operator TψN , ψ ∈ L10,N , is deﬁned on H(B) N

TψN (f ) = P N (ψf ). If g ∈ H(B), we also introduce the following operators: (i)

the holomorphic pointwise multiplication operator Mg : Bsp (B) → H(B), deﬁned by Mg (f ) = gf ,

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the antiholomorphic pointwise multiplication operator Mg¯ : Bsp (B) → C ∞ (B) deﬁned by Mg¯ (f ) = g¯f , 1 ¯ → H(B), deﬁned by , the Toeplitz operator Tg¯N : Bsp (B) if g ∈ B−N N N N g f )(z) = P (Mg¯ (f )), and Tg¯ (f ) = P (¯ 1 p ¯ ∞ if g ∈ B−N , the Big Hankel operator HgN ¯ : Bs (B) → C (B), deﬁned N N by HgN ¯ − Tg ¯ = Mg ¯ = (I − P )(Mg ).

In order to simplify, in what follows we will just say Hankel instead of Big Hankel operator. Observe that Mg = TgN , and that Mg¯ = Tg¯N +HgN ¯ . As usual, T L(X→Y ) denotes the norm of the operator T : X → Y . In the particular case X = Y , we will just write T L(X) . There is an extense literature in the study of the above operators in the p classical case of Bergman spaces B−1/p where the corresponding Toeplitz or Hankel operators are given by the Bergman projection (N = 1 which corre2 = H ∩ L2 (dν)). sponds to the orthogonal projection from L2 (dν) to B−1/2 p The same occurs for weighted Bergman spaces Bs , 1 1, some references can be obtained in [10,24] and more recently [27]. The object of this paper is to extend to any s ∈ R and N > 0, the study of the necessary and sufﬁcient conditions on the symbol g to assure the boundedness from Bsp to Lpk,(k−s)p of the above operators, and also to give estimates of the corresponding norms. In many cases, the estimates will be given in terms of the norms of certain Carleson measures on Besov spaces which depend on g, p, s and N . We recall that a positive measure µ on B is a Bsp -Carleson measure if the space Bsp is continuously embedded in Lp (µ), i.e. if there exists C > 0 such that for any f ∈ Bsp , f Lp (µ) ≤ Cf Bsp . We will denote by CMsp the set of these measures, and by µCMsp the norm of the embedding Bsp ⊂ Lp (µ). The Carleson measures for Besov and Hardy-Sobolev spaces have been studied by many authors ([2,4–6,11,13,14,16,22,25] among others). See Sect. 2.3 for a more complete description of these measures. For 1 s a positive integer and g ∈ Bsp , we consider the measure µg given by dµg (z) = |Rk g(z)|p (1 − |z|2 )(k−s)p−1 dν(z). We also consider the space CBsp = {g ∈ Bsp ; µg ∈ CMsp } , normed by gCBsp = |g(0)| + µg CMsp . Observe that in order to be precise we should add an index k in the definition of the space CBsp . We will prove that as it happens in the Besov case, the space just deﬁned is independent of the integer k > s selected and that it can be deﬁned replacing the radial derivatives by other differential operators.

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As it is well known, these spaces appear in a natural way when we consider some problems on Toeplitz or Hankel operators on Besov spaces even using other projections (see for instance [7,28]). They have applications to interpolating sequences [5], and to corona-type theorems (see [7,15,19] and the references therein). In some particular cases it is possible to give an explicit description of CBsp . For instance, since the space B02 is the Hardy space H 2 , µ is a Carleson ¯ < r} ≤ Cµ rn , and measures for B02 if and only if supζ∈S µ{z ∈ B; |1 − ζz| 2 consequently CB0 = BM OA. The fact that for s > n/p the Bsp -Carleson measures are just the ﬁnite ones gives that CBsp = Bsp . The characterizations of the Bsp -Carleson measures given in Sect. 2.3 in other particular cases, can be used to obtain descriptions of the spaces CBsp . In addition to these classical situations, we will show that: Proposition 1.1. If 1 < p < ∞ and s ∈ R, then, CBsp ⊂ B0∞ . If s < 0, then CBsp = B0∞ . We also obtain the following characterization of the space CBsp in terms of a pointwise multiplication operator. Theorem 1.2. Let 1 < p < ∞ and s ∈ R. Then, g ∈ CBsp if and only if the p . operator MRg (f ) = f Rg is continuous from Bsp to Bs−1 p Moreover MRg L(Bsp →Bs−1 ) ≈ g − g(0)CBsp . The next theorem gives an estimate of the norm of the operators Mg and Mg¯ . Theorem 1.3. Let 1 s a non-negative integer and g ∈ H(B). The following assertions are equivalent: (i) The operator Mg : Bsp → Bsp is bounded. (ii) The operator Mg¯ : Bsp → Lpk,(k−s)p is bounded. (iii) g ∈ H ∞ ∩ CBsp . Moreover, Mg L(Bsp ) ≈ Mg¯ L(Bsp →Lpk,(k−s)p ) ≈ g∞ + gCBsp . In one complex variable, the equivalence between (i) and (iii) has been proved in [21,26]. The general case has been proved in [19]. As an immediate consequence of Theorem 1.3 and the duality (Bsp ) = p B−s−N with respect to the pairing

f, hN = lim cN f (rz)h(rz)(1 − |z|2 )N −1 dν(z), (1.2) r1

B

we obtain the following estimate of the norm of the Toeplitz operator. 1 Theorem 1.4. Let 1 0, s ∈ R and g ∈ B−N . Then the operator N p Tg¯ extends to a bounded operator from Bs to itself, if and only if the operator

p to itself. Moreover, Tg¯N Bsp ≈ Mg B p Mg is bounded from B−s−N

−s−N

.

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Observe that Theorem 1.3 together with Proposition 1.1 and Theorem 1.4 give that if s > −N , then Tg¯N Bsp ≈ g∞ . The characterization of the Hankel operator also depends on the sign of −s − N , as it is stated in the following theorem. Theorem 1.5. Let 1 0, s ∈ R, k > s a positive integer and 1 . g ∈ B−N (i) If s > −N , then the Hankel operator Hg¯ extends to a bounded operator from Bsp to Lpk,(k−s)p , if and only if, g ∈ CBsp . p p p Moreover, HgN ¯ L(Bs →Lk,(k−s)p ) ≈ g − g(0)CBs . (ii) If s ≤ −N , then the Hankel operator Hg¯ extends to a bounded operp ator from Bsp to Lpk,(k−s)p , if and only if, g ∈ CB−s−N . Moreover, p p HgN ≈ g − g(0) . ¯ L(Bs →Lk,(k−s)p ) CB p −s−N

Proposition 1.1 gives that in any situation, at least one of the spaces p CBsp or CB−s−N coincides with the Bloch space B0∞ . Hence, Theorem 1.5 can be rewritten as follows: The operator Hg¯ : Bsp → Lpk,(k−s)p is bounded if

p . and only if g ∈ CBsp ∩ CB−s−N The following table summarizes the results stated above.

Boundedness of multiplication, Toeplitz and Hankel operators. Necessary and sufficient conditions. 1 s

Mg : Bsp → Bsp

Mg¯ : Bsp → Lpk,(k−s)p

Tg¯N : Bsp → Bsp

p p HgN ¯ : Bs → Lk,(k−s)p

g ∈ H∞ g ∈ H∞

g ∈ Bsp

−N < s < 0

g ∈ Bsp g ∈ H∞ g ∈ CBsp g ∈ H∞

−n/p − N

g ∈ H∞

g ∈ H∞

s > n/p 0 ≤ s ≤ n/p

≤ s ≤ −N s< −n/p − N

g ∈ H∞

g∈

H∞

g ∈ CBsp g ∈ B0∞

p g ∈ CB−s−N

p g ∈ CB−s−N

p g ∈ B−s−N

p g ∈ B−s−N

The paper is organized as follows: in Sect. 2 we recall some representation and integral formulas as well as the properties of the Besov spaces and Carleson measures needed for the proof of the main results. In Sect. 3, we study the different characterizations of the space CBsp and prove Theorem 1.2. In Sect. 4, we give the proof of the Theorems 1.3 and 1.4, concerning the boundedness of the multiplication and Toeplitz operators. Finally, in Sect. 5 we give the proof of the Theorem 1.5, characterizing the boundedness of the Hankel operator.

2. Preliminaries In this section we will recall the results on spaces of functions and integral operators needed in the following sections.

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2.1. Notations In order to facilitate the reading of this paper, in this section we include most of the definitions of operators, spaces of functions and measures that we will use throughout the paper and that we have not already introduced. Throughout the paper, the letter C may denote various non-negative numerical constants, possibly different in different places. The notation f (z) g(z) means that there exists C > 0, which does not depend on z, f and g, such that f (z) ≤ Cg(z). We denote by dνδ (z) = cδ (1 − |z|2 )δ−1 dν(z), where cδ is a normalizing constant so that νδ (B) = 1. If 1 ≤ i < j ≤ n, let Di,j be the complex-tangential differential operator deﬁned by Di,j = z¯i ∂j − z¯j ∂i . Let |∂T f | = 1≤i<j≤n |Di,j f |. For l > 0 and k a positive integer, let Rkl =

Γ(l) ((l + k − 1)I + R) · · · (lI + R). Γ(l + k)

p Although both operators (I + R)k , Rkl give bijections from Bsp to Bs−k , the use of the second one in some of the forthcoming sections will simplify the technicalities. Let 1 s a positive integer. If g ∈ Bsp , let

dµg,p,s,k (z) = |∂ k g(z)|p (1 − |z|2 )(k−s)p−1 dν(z), k p 2 (k−s)p−1 dν(z). dµrad g,p,s,k (z) = |R g(z)| (1 − |z| )

We introduce the following subspaces of Bsp . p = {g ∈ Bsp ; µg,p,s,k ∈ CMsp } . CBs,k p p C rad Bs,k = g ∈ Bsp ; µrad g,p,s,k ∈ CMs .

We will prove that these two spaces coincide and will denoted them simply by CBsp . We recall that CMsp denotes the set of Bsp -Carleson measures. 2.2. Representation Formulas and Estimates In this section we recall some well-known representation formulas, which generalize the classical Cauchy–Pompeiu formula in the unit disk. Among the different representation formulas, we will use the ones obtained in [12]. Theorem 2.1. For each N > 0 there exists an integral operator KN , satisfying ¯ and z ∈ B, that for any ψ ∈ C 1 (B) ¯ ψ(z) = P N (ψ)(z) + KN (∂ψ)(z). and

¯ |K (∂ψ)(z)| N

Ψ(w) B

(1 − |w|2 )N −1 dν(w). |1 − wz| ¯ N −n+1 D(w, z)n−1/2

¯ + (1 − |w|2 )1/2 |∂¯T ψ(w)|, and D(w, z) = where Ψ(w) = (1 − |w|2 )|∂ψ(w)| 2 2 2 |1 − wz| ¯ − (1 − |w| )(1 − |z| ).

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In particular if ψ is a holomorphic function on B we have that ψ = P N (ψ)(z). In order to state estimates of the kernels deﬁned in the above formulas, we introduce the following nonnegative integral kernels and operators. Definition 2.2. Let N, M, L be real numbers satisfying N > 0 and L < n. For z, w ∈ B, let N (w, z) = KM,L

(1 − |w|2 )N −1 . |1 − wz| ¯ M D(w, z)L

N will also denote the corresponding integral operator given by KM,L N N (ψ)(z) = ψ(w)KM,L (w, z) dν(w). ψ(z) → KM,L B

Observe that |P (w, z)| ≈ The proof of the following result can be found in Lemma I.1 in [12]. N

N Kn+N,0 (w, z).

Lemma 2.3. Let N, M, L be real numbers satisfying N > 0 and L < n. If n + N − M − 2L = 0, then N KM,L (w, z) dν(w) 1 + (1 − |z|2 )n+N −M −2L , z ∈ B. B N Definition 2.4. We deﬁne the type of the kernel KM,L by N type KM,L = n + N − M − 2L.

Roughly speaking, an operator whose associated kernel is of positive type improves the regularity of the function. H¨ older’s inequality easily gives a pointwise estimate of the operators N KM,L , which will be frequently used in the next sections. N Lemma 2.5. Let 1 0, L < n, KM,L a p kernel of type 0. If 0 < ε < N and ψ ∈ L0,(N −ε)p , then N p KM,L (ψ)(z)p (1 − |z|2 )−εp K(N −ε)p M −N +(N −2ε)p,L (|ψ| )(z).

In particular,

N p (N −ε)p (ψ)(z) (1 − |z|2 )−εp K(N −2ε)p,0 (|ψ|p )(z). |P N (ψ)(z)|p KN,0 Proof. Assume ψ ≥ 0. By H¨ older’s Inequality,

p/p p N (N −ε)p εp (1) . KM,L (ψ) ≤ KM −N +(N −2ε)p,L (ψ p ) KM −N +2εp ,L

εp N Since KM,L is of type 0, n + N − M − 2L = 0 and consequently, KM −N +2εp ,L is of type −εp . Therefore, Lemma 2.3 gives that

p/p εp KM (1) (1 − |z|2 )−εp , −N +2εp ,B

which ends the proof.

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As a consequence of the above lemma, Fubini’s Theorem and Lemma 2.3 with ε < min(δ, N − δ), we have: Proposition 2.6. Let 1 ≤ p < ∞, M, N, L ∈ R such that N > 0, L < n and N N is of type 0, then the operator KM,L is bounded from N > δ > 0. If KM,L p L0,δp to itself. The next formulas state the relation with the differentiation of the operators P N and KN . ¯ We then have Lemma 2.7. Let N > 0, |α| = k and ψ ∈ C k+1 (B). ¯ = KN +|α| (∂α ∂ψ). ¯ ∂α P N (ψ) = P N +|α| (∂α ψ) and ∂α KN (∂ψ) Proof. Let us check ﬁrst that if 1 ≤ j ≤ n, ∂j P N (ψ) = P N +1 (∂j ψ). Indeed, w ¯j (1 − |w|2 )N −1 dν(w) ∂j P N (ψ)(z) = (n + N )cN ψ(w) (1 − wz) ¯ n+N +1 B ∂j (1 − |w|2 )N (n + N )cN ψ(w) dν(w) =− N (1 − wz) ¯ n+N +1 B (1 − |w|2 )N dν(w) = cN +1 ∂j ψ(w) (1 − wz) ¯ n+N +1 B

= P N +1 (∂j ψ)(z). Iterating the above formula we obtain that ∂α P N (ψ) = P N +|α| (∂α ψ). The second equality is a consequence of the representation formula. Indeed, ¯ = ∂α (ψ − P N (ψ)) = ∂α ψ − P N +|α| (∂α ψ)) = KN +|α| (∂α ∂ψ), ¯ ∂α KN (∂ψ)

and that concludes the proof.

We will give other type of integral formulas that also arise from integration by parts. ¯ Then, Lemma 2.8. Let M > 0 and let Ψ ∈ C 1 (B). M Ψ(z)(1 − |z|2 )M −1 dν(z) = ((n + M )I + R)Ψ(z)(1 − |z|2 )M dν(z). B

B

Replacing R by R we obtain an analogous formula. Proof. Since M (1 − |z|2 )M −1 = (M I − R)(1 − |z|2 )M , integration by parts gives that the left hand side integral above equals to n M Ψ(z)(1 − |z|2 )M dν(z) + ∂j (zj Ψ(z))(1 − |z|2 )M dν(z). B

j=1 B

And that gives the lemma. Iterating Lemma 2.8 and using the fact that cM = we obtain:

Γ(n+M ) n!Γ(M )

[see (1.1)],

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¯ Then, Lemma 2.9. Let M > 0, k a positive integer and Ψ ∈ C k (B). cM Ψ(z)(1 − |z|2 )M −1 dν(z) = cM +k Rkn+M Ψ(z)(1 − |z|2 )M +k−1 dν(z). B

B

The following results state some different integral formulas obtained by choosing particular functions Ψ in Lemma 2.9. ¯ Then, Corollary 2.10. Let N > 0, k a positive integer and f, h ∈ H(B).

f, hN = Rkn+N f, hN +k = f, Rkn+N hN +k . ¯ in Lemma 2.9. Proof. Take Ψ = f h

¯ then Corollary 2.11. Let N > 0, k a positive integer and ψ ∈ C k (B), (1 − |w|2 )N +k−1 P N (ψ)(z) = cN +k Rkn+N ψ(w) dν(w). (1 − wz) ¯ n+N B

Proof. Take Ψ(w) =

ψ(w) (1−wz) ¯ n+N

in Lemma 2.9.

As a consequence of Corollary 2.11 and Proposition 2.6 we have: Proposition 2.12. Let 1 ≤ p < ∞, s a real number, N > max{0, −s} and k > s a non negative integer. Then, the operator P N maps Lpk,(k−s)p to Bsp . In the next two lemmas we apply Lemma 2.9 to a function Ψ = ϕψ in order to obtain weighted Sobolev-type estimates of products of functions. ¯ and ψ ∈ C k (B), ¯ Lemma 2.13. Let M > 0, and 1 ≤ l ≤ k integers. If ϕ ∈ C l (B) then ϕ(z)ψ(z)(1 − |z|2 )M −1 dν(z) B

=

j k−1

ai,j

j=0 i=0

+

j l−1

B

+

j=0 i=0

bi,j

j=0 i=0 l−1 k−1

Rl ϕ(z)Ri ψ(z)(1 − |z|2 )M +l+j−1 dν(z) Ri ϕ(z)Rk ψ(z)(1 − |z|2 )M +k+j−1 dν(z)

B

ci,j B

Rj ϕ(z)Ri ψ(z)(1 − |z|2 )M +k+l−1 dν(z).

(2.1)

Remark 2.14. Observe that by (2.1), B ϕ(z)ψ(z)(1 − |z|2 )M −1 dν(z) can be decomposed as a sum of dominating terms of type k−1 Rl ϕ(z)Rj ψ(z)(1 − |z|2 )M +l+j−1 dν(z) j=0 B

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and of type l−1

Rj ϕ(z)Rk ψ(z)(1 − |z|2 )M +k+j−1 dν(z),

j=0 B

and other terms with better estimates. Proof. Before we prove the general case, we begin by sketching the proof in ¯ By Lemma 2.8, particular case ϕ, ψ ∈ C 1 (B). M

ϕ(z)ψ(z)(1 − |z|2 )M −1 dν(z) =

B

Rϕ(z)ψ(z)(1 − |z|2 )M dν(z)

B

+

ϕ(z)Rψ(z)(1 − |z|2 )M dν(z)

B

+(n+M )

ϕ(z)ψ(z)(1−|z|2 )M dν(z).

B

Applying Lemma 2.9 to the last term we obtain the formula in this case. We next sketch brieﬂy the proof of the general case. By Lemma 2.9 applied to Ψ = ϕψ, we have ϕψ dνM =

j+i≤l

B

Rj ϕRi ψ dνM +l .

ci,j B

If j = l, then B Rl ϕψdνM +l is one of the terms included in the ﬁrst summand. If i = k, we have that k = l and j = 0, and consequently the integral ϕRk ψ dνM +l is one of the terms considered in the second summand. If B i, j < l ≤ k, then we apply Lemma 2.8. In this case

R ϕR ψ dνM +l = c0 j

B

R ϕR ψ dνM +l+1 + c1

i

j

B

B

R ϕR j

+c2

Rj+1 ϕRi ψ dνM +l+1

i

i+1

ψ dνM +l+1 .

B

As it happens in the previous case, if j + 1 = l, the corresponding integral is one of the terms considered in the ﬁrst summand of (2.1), and if i+1 = k then the integral is one of the terms included the second summand. Otherwise, we apply again Lemma 2.8. Repeating this argument k times, we obtain the desired formula. Similar arguments and the fact that Rj =

|α|≤j cα z

α

∂α give:

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¯ and ψ ∈ C k (B), ¯ Lemma 2.15. Let M > 0, and 1 ≤ l ≤ k integers. If ϕ ∈ C l (B) there exist holomorphic polynomials pγ,κ,i,j (z) such that ϕ(z)ψ(z)(1 − |z|2 )M −1 dν(z) B

=

k−1

pγ,κ,l,j (z)∂γ ϕ(z)∂κ ψ(z)(1 − |z|2 )M +l+j−1 dν(z)

|γ|=l j=0 |κ|≤j B

+

k−1

pγ,κ,j,k (z)∂γ ϕ(z)∂κ ψ(z)(1 − |z|2 )M +k+j−1 dν(z)

|κ|=k j=0 |γ|≤j B

+

pγ,κ,i,j (z)∂γ ϕ(z)∂κ ψ(z)(1 − |z|2 )M +l+k−1 dν(z)

|γ|=l−1 |κ|≤k−1 B

Remark 2.16. Formula (2.13) states that | B ϕ(z)ψ(z)(1 − |z|2 )M −1 dν(z)| is bounded by sums of dominating terms of type k−1 ∂ l ϕ(z)||∂ j ψ(z) (1 − |z|2 )M +l+j−1 dν(z) j=0 B

and of type l−1

|∂ j ϕ(z)||∂ k ψ(z)|(1 − |z|2 )M +k+j−1 dν(z),

j=0 B

and other terms with better estimates. 2.3. Holomorphic Besov Spaces We summarize here the main properties on Besov spaces that will be used in the next sections, whose proofs can be found in Section 5 in [9] (see also [29]). Proposition 2.17. Let 1 ≤ p ≤ ∞, s ∈ R and let k > s a non-negative integer. (i) If (1 − |z|2 )k−s |∂ k f (z)| ∈ Lp (1 − |z|2 )−1 dν(z) , then f ∈ Bsp . p , then f ∈ Bsp . (ii) If Rk f ∈ Bs−k p p (iii) If β ≥ α, then ∂β f Bs−|β| ∂α f Bs−|α| . (iv) If f ∈ B0∞ , then (1 − |z|2 )1/2 |∂T f |(z)∞ (1 − |z|2 )|∂f |(z)∞ . Proof. (i) is a consequence of the fact that |∂ j f (0)| + (1 − |z|2 )k−s |∂ k f (z)| Lp ((1−|z|2 )−1 dν(z)) 0≤j≤k−1

is an equivalent norm in Bsp . The proof of this result arises from the Barrow’s 1 d h(tz)dt applied to both f and its derivatives (see formula h(z) = h(0) + 0 dt for instance Theorems 2.16, 2.17 and 6.2 in [29]). The proof of (ii) is analogous. Assertion (iii) follows form the definition of the spaces Bsp , and the last one can be found for instance in Sect. 7 in [29].

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Proposition 2.18. Let 1 ≤ p ≤ q ≤ ∞ and let s, t be real numbers. We have: (i) (ii) (iii) (iv)

If p < ∞, the space H(B) is dense in Bsp . If s > t, then Bsp ⊂ Btp . q 1 p ∞ If 1 n/p, then Bs is a subspace of the Lipschitz space Bs−n/p , and Bsp is a multiplicative algebra.

Proof. Assertion (i) can be found in Lemma 5.2 in [9]. In fact we have that if f ∈ Bsp , 0 < r < 1 and fr (z) = f (rz), then f − fr Bsp → 0 if r 1. The other assertions can be found in Theorems 5.13 and 5.14 in [9]. The next result states that the family of Besov spaces Bsp are stable under the Calderon complex interpolation method. Proposition 2.19. Let 1 ≤ p0 , p1 < ∞ and let s0 , s1 real numbers and 0 < θ < 1. Assume that p and s satisfy 1/p = (1 − θ)/p0 + θ/p1 and s = (1 − θ)s0 + θs1 . Then, p0 p1 1−θ 1 θ = Bs0 , Bs1 [θ] = Bsp , + , s = (1 − θ)s0 + θs1 . p p0 p1

Proof. See for instance, Theorem 1.3 in [8].

The dual of the space Bsp can be identiﬁed with another Besov space using an adequate pairing. More precisely, for N > 0, and f, h ∈ H(B), we consider the pairing given by (2.2)

f, hN = lim cN f (rz)h(rz)(1 − |z|2 )N −1 dν(z). r1

B

Proposition 2.20. Let 1 0 and let s be a real number. Then, the p dual of Bsp with respect to the pairing given by (2.2) is B−s−N . In particular,

p if Λ ∈ (Bsp ) then Λ(f ) = f, hN for some h ∈ B−s−N and Λ ≈ hB p

−s−N

Proof. See for instance Proposition 5.7 in [9].

.

2.4. Bsp -Carleson Measures As an immediate consequence of Lemma 2.3 we obtain a necessary condition for a measure to be Carleson for Bsp . Namely, Lemma 2.21. Let 1 ≤ p < ∞, s ∈ R and µ a Bsp -Carleson measure. If M > n/p − s, then dµ(w) µpCMsp (1 − |z|2 )n−sp−M p , z ∈ B. (2.3) |1 − w¯ z |M p B

Proof. The fact that µ is a Bsp -Carleson measure gives that dµ(w) ≤ µpCMsp (1 − w¯ z )−M pBsp . |1 − w¯ z |M p B

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Next, Lemma 2.3 gives that provided M > n/p − s, (1 − |w|2 )(k−s)p−1 dν(w) (1 − w¯ z )−M pBsp ≈ ≈ (1 − |z|2 )n−sp−M p , |1 − w¯ z |kp+M p B

which proves the result.

If s < n/p, decomposing the above integral into a sum of integrals over tents, it can be given a reformulation of the necessary condition in Lemma 2.21, in terms of the measure of tents (see for instance Theorem 5.4 in [29]). Namely, ¯ < r} µp p sup rsp−n µ{z ∈ B; |1 − z ζ| CMs

(2.4)

ζ∈S 0
Observe that, if s = 0, then the positive measures satisfying (2.4) are the usual Carleson measures, that is the H p -Carleson measures. We recall that in some particular cases, there exist characterizations of the Bsp -Carleson measures. For instance, if p = 1 and s ∈ R, if 1 < p < ∞ and s < 0, or if 1 < p ≤ 2, s = 0, we have that µ ∈ CMsp if and only if µ satisﬁes (2.3). The proof of this result for the case p = 1 follows easily from Corollary 2.11 and Fubini’s Theorem. The case 1 −s, M > 0 (1 − wz) ¯ n+N (1 − z¯w)M B

and H¨ older’s Inequality. In the case 1 < p ≤ 2 and s = 0, we can proceed as in [17, Section 3.11, p. 178]. Since for this range of p s, B0p ⊂ H p (see [9, Theorem 5.13]), we have f Lp (µ) f H p f B0p . Another situation where we can give an easy characterization is the reg∞ ular case. If s > n/p, then Bsp ⊂ Bs−n/p ⊂ H ∞ , and consequently in this p case the Bs -Carleson measures are just the ﬁnite ones. Near the regular case, there are also characterizations of the Bsp -Carleson measures. By Corollary 2.11, is clear that if µ satisﬁes ⎛ ⎞p |ϕ(w)|dν(w) ⎝ ⎠ dµ(z) ϕp p (2.5) L (dν) , |1 − wz| ¯ n+1−(s−1/p) B

B

for all the functions ϕ ∈ Lp (dν), then µ ∈ CMsp . In [2,13,14], it is proved that if (n−1)/p < s ≤ n/p, i.e. if 0 ≤ n−sp < 1, then µ ∈ CMsp if and only if µ satisﬁes condition (2.5). This last condition can be expressed in terms of nonisotropic Riesz capacities of open sets in S. In the extreme case p = 2, and n − 2s = 1, which corresponds to the Drury–Arveson space [6,22] have obtained characterizations of the corresponding Carleson measures. Recently [25] (see also [16]) have characterized the Carleson measures for any Hs2 in terms of a T 1-type condition. To conclude this section we prove the following proposition.

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Proposition 2.22. Let 1 0. If |α| = j, M > j + s and k > s, Corollary 2.11 gives that (1 − |w|2 )M +k−1 dν(w). ∂α f (z) = cM +k Rkn+M (∂α f )(w) (1 − wz) ¯ n+M B

Next, if ε > 0, by Lemma 2.5 we have that |∂α f (z)|p (1 − |z|2 )−εp

B

2 (M +k−ε)p−1 k Rn+M (∂α f )(w)p (1 − |w| ) dν(w). |1 − wz| ¯ n+(M −2ε)p

Therefore, applying Fubini’s Theorem we obtain (1 − |z|2 )j |∂ j f (z)|pLp (dµ) p (1 − |z|2 )(j−ε)p dµ(z) Rkn+M (∂α f )(w) dν(M +k−ε)p (w). |1 − wz| ¯ n+(M −2ε)p B

B

Choosing ε < 1 ≤ j = |α|, Lemma 2.21 gives (1 − |z|2 )j |∂ j f (z)|pLp (dµ) p µCMsp |Rkn+M (∂α f )(w)|p (1 − |w|2 )(k+j−s)p−1 dν(w)

B p µCMsp f pBsp ,

which concludes the proof.

3. The Space CBsp The main goal of this section is to show that for any k > s the two spaces p p and C rad Bs,k already introduced coincide and moreover, that these CBs,k spaces are, in fact, independent of k. In consequence, we will drop the index k and the superindex rad and just write CBsp . The techniques used to prove the results are based on representation formulas and linear operator theory. There could have also been used analogous alternative proofs using similar techniques like the ones considered in [5]. We begin by stating some technical results needed in the proof of the main result of this section and that will also used in the forthcoming sections. These results provide both pointwise and Lp -norm estimates of the derivap p and C rad Bs,k and will be used in the tives of functions in the spaces CBs,k proof of Proposition 1.1 and Theorem 1.2.

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Lemma 3.1. Let 1 min{0, −s} and h ∈ H(B). If the measure dµ(z) = |h(z)|p (1 − |z|2 )δp−1 dν(z) is a Bsp -Carleson measure, then (1 − |z|2 )s+δ h(z)∞ µCMsp . p ∞ ∩ B−δ . In particular, if µCMsp is finite, then h ∈ B−s−δ 2 s+δ Moreover, if s < 0, then (1 − |z| ) h(z)∞ ≈ µCMsp . p and Proof. Since dµ is a Bsp -Carleson measure, then it is clear that h ∈ B−δ therefore for N > δ and M > 0, we have (1 − |w|2 )N −1 (1 − |z|2 )M dν(w). h(z) = cN h(w) (1 − wz) ¯ n+N (1 − z¯w)M B

Choosing N > δp and M > −(s + δ)p, H¨ older’s Inequality and Lemma 2.3, give then that (1 − |w|2 )N −1 |h(z)|p (1 − |z|2 )M |h(w)|p dν(w) |1 − wz| ¯ n+N +M B (1 − |w|2 )δp−1 (1 − |z|2 )M |h(w)|p dν(w) |1 − wz| ¯ n+δp+M B

µpCMsp (1 µpCMsp (1

− |z|2 )M (1 − z¯w)−n/p−δ−M/p pBsp − |z|2 )−(s+δ)p .

Therefore, (1 − |z|2 )s+δ h(z) ∞ µCMsp . If s < 0, then f pBsp ≈ B |f (w)|p (1 − |z|2 )−sp−1 dν(z), and therefore the measure (1 − |z|2 )−sp−1 dν(z) is in CMsp . Thus µCMsp (1 − |z|2 )s+δ h(z)∞ (1 − |z|2 )−sp−1 dν(z)CMsp ∞ (1 − |z|2 )s+δ h(z)∞ ≈ hB−s−δ ,

and the result is proved.

p p and C rad Bs,k are The next two corollaries show that the spaces CBs,k ∞ subspaces of the Bloch space B0 .

Corollary 3.2. Let 1 s a positive integer and g ∈ H(B). Then, (1 − |z|2 )k |∂ k g(z)|∞ µg,p,s,k CMsp . In particular, if µg,p,s,k CMsp is finite, then g is in B0∞ ∩ Bsp . Moreover, if s < 0, then (1 − |z|2 )k |∂ k g(z)|∞ ≈ µg,p,s,k CMsp . Proof. Apply the above lemma to h = ∂β g, |β| = k and δ = k − s.

Corollary 3.3. Let 1 s a positive integer and g ∈ H(B). p Then, (1 − |z|2 )Rg(z)∞ µrad g,p,s,k CMs . rad In particular, if µg,p,s,k CMsp is finite, then g is in B0∞ ∩ Bsp . p Moreover, if s < 0, then (1 − |z|2 )Rg(z)∞ ≈ µrad g,p,s,k CMs .

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Proof. Lemma 3.1 with h = Rk g and δ = k − s proves that (1 − |z|2 )k Rk 2 2 k k p g(z)∞ µrad g,p,s,k CMs . The estimate (1 − |z| )Rg(z)∞ (1 − |z| ) R g(z)∞ is an easy consequence of 1 1 · · · dt dt 1 k−1 |Rg(z)| = · · · (Rk g)(t1 · · · tk−1 z) t1 · · · tk−1 0

0

2 k

1

(1 − |z| ) R g(z)∞

1 ···

k

0

0

dt1 · · · dtk−1 (1 − |t1 · · · tk−1 z|)k

(1 − |z|2 )k Rk g(z)∞ (1 − |z|2 )−1 .

And that ends the proof.

In the following two propositions we give Sobolev-norm estimates of p p derivatives of products of functions f ∈ Bsp and g ∈ CBs,k or g ∈ C rad Bs,k , p p rad which we will use to prove that CBs,k = C Bs,k . Proposition 3.4. Let 1 t a positive integer and δ ≥ m − t. Let h be a holomorphic function on B, and assume that the measure dµ(z) = |∂ m h(z)|p (1 − |z|2 )δp−1 dν(z) is in CMtp . We then have that for any |α| + |β| ≤ m, ∂α f ∂β hLp0,δp ((1 − |z|2 )δ+t−m+|β| |∂β h(z)|∞ + µCMtp )f Btp . Remark 3.5. If δ < m − t, then (1 − |z|2 )δ+t−m |h(z)|∞ is ﬁnite if and only if h = 0. We include the condition δ ≥ m − t in the hypothesis in order to consider only non-trivial cases. Proof. The fact that µ ∈ CMtp gives that if |β| = m, then f ∂β hLp0,δp µCMtp f Btp . Therefore, it remains to consider the case |β| < m. The representation formula obtained in Theorem 2.1, with M large enough that we will choose later, gives then that (1 − |w|2 )M −1 dν(w). (∂β h∂α f )(z) = cM (∂β h∂α f )(w) (1 − wz) ¯ n+M B

Lemma 2.15, with l = m − |β|, a positive integer k > m, ϕ(w) = ∂β h(w) ∂α f (w) , gives that we can write the last integral as a ﬁnite and ψ(w) = (1 − wz) ¯ n+M sum of terms of type (1 − |w|2 )M +l+j−1 dν(w), Il,j (z) = p(w)∂γ ∂β h(w)∂κ ∂α f (w) (1 − wz) ¯ n+M B

with |γ| = l and |κ| ≤ j < k, (1 − |w|2 )M +k+j−1 Jj,m (z) = p(w)∂γ ∂β h(w)∂κ ∂α f (w) dν(w), (1 − wz) ¯ n+M B

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with |γ| ≤ j ≤ l − 1 and |κ| = k, and terms of type (1 − |w|2 )M +l+k−1 dν(w), Li,j (z) = p(w)∂γ ∂β h(w)∂κ ∂α f (w) (1 − wz) ¯ n+M B

with |γ| ≤ j ≤ l − 1 and |κ| ≤ k − 1. In all the cases considered, p(w) denotes a holomorphic polynomial. We will obtain Lp0,δp -norm estimates of each of this type of integrals. In order to estimate the terms of type Il,j , observe that (1 − |w|2 )M +l+j−1 Il,j (z) |∂ m h(w)||∂κ+α f (w)| dν(w). |1 − wz| ¯ n+M B

By Proposition 2.6, p Il,j Lp |∂ m h(w)|p |∂κ+α f (w)|p (1 − |w|2 )(δ+l+j)p−1 dν(w) 0,δp

B

(1 − |w|2 )(l+j)p |∂κ+α f (w)|p dµ(w).

B

Since |κ| ≤ j and |α| ≤ m − |β| = l, Proposition 2.22 proves that the last term is bounded by µCMsp f pB p . t The estimates of the Lp0,δp -norm of terms of type Jj,m and Li,j are easier, because it is enough to use pointwise estimates of the function h and its derivatives. Since |γ| ≤ j we have (1 − |z|2 )δ+t−m+j+|β| ∂γ+β h(z)∞ (1 − |z|2 )δ+t−m+|β| ∂β h(z)∞ , and |Jj,m (z)| (1 − |z|2 )δ+t−m+|β| ∂β h(z)∞

B

|(∂κ+α f )(w)| dνM −δ+k−t+m−|β| (w). |1 − wz| ¯ n+M

Choosing M > δ, and using the fact that the kernel of type 0, Proposition 2.6 gives

(1 − |w|2 )M −1 is |1 − wz| ¯ n+M

Jj,m (z)Lp0,δp (1−|z|2 )δ+t−m+|β| ∂β h(z)∞ (1−|w|2 )k−t−δ+m−|β| (∂κ+α f )(w)Lp0,δp ≈ (1 − |z|2 )δ+t−m+|β| ∂β h(z)∞ ∂κ+α f Lp0,(k−t+m−|β|)p p (1 − |z|2 )δ+t−m+|β| ∂β h(z)∞ f Bt−k+|κ|−m+|α|+|β| .

Since |κ| = k and |α| + |β| ≤ m, we have t − k + |κ| − m + |α| + |β| ≤ t, and therefore the last term in the above inequalities is bounded by (1 − |z|2 )δ+t−m+|β| ∂β h(z)∞ f Btp .

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Similar arguments prove that Li,j (z)Lp0,δp (1 − |z|2 )δ+t−m+|β| ∂β h(z)∞ f Btp , which concludes the proof.

In the same lines of the proof of proposition 3.4 we obtain: Proposition 3.6. Let 1 t a positive integer and δ ≥ m − t. Let h ∈ H(B) and dµ(z) = |Rm h(z)|p (1 − |z|2 )δp−1 dν(z) be a Btp −Carleson measure. If i + j ≤ m, and j ≥ 1, then Ri f Rj hLp0,δp µCMtp f Btp . Proof. Replacing ∂α f by Ri f and ∂β h by Rj f in the proof of the above proposition, and using Lemma 2.13 instead of Lemma 2.15, we obtain Ri f Rj hLp0,δp ((1 − |z|2 )δ+t−m+j Rj h(z)∞ + µCMtp )f Btp . To conclude the proof of the Proposition, we show that ((1 − |z|2 )δ+t−m+j Rj h(z)∞ µCMtp )f Btp . By Lemma 3.1 we have that (1 − |z|2 )t+δ Rm h(z)∞ µCMtp . Since j ≥ 1 and t + δ ≥ m, the argument used in the proof of Corollary 3.3 shows that (1 − |z|2 )δ+t−m+j Rj h(z)∞ (1 − |z|2 )δ+t Rm h(z)∞ , and that ends the proof.

Corollary 3.7. Let 1 s a positive integer. Let g ∈ H(B). Then, ∂α f ∂β gLp0,(k−s)p (g∞ + µg,p,s,k CMsp )f Bsp . |α|+|β|≤k

p p Ri f Rj gLp0,(k−s)p (g∞ + µrad g,p,s,k CMs )f Bs .

i+j≤k

Proof. Take m = k, t = s and δ = k − s in the above propositions.

The next result is a consequence of Proposition 3.6, and it will we used to estimate the norm of the Hankel operators. Corollary 3.8. Let 1 t a positive integer. Let h be a holomorphic function on B, and dµ(z) = |Rm h(z)|p (1 − |z|2 )(m−t)p−1 dν(z) and let l ≤ m be a positive integer. Then, p Ri f Rj hBt−l (h∞ + µCMtp )f Btp . i+j≤l

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Proof. Observe that p p Ri f Rj hBt−l ≈ (I + R)m−l (Ri f Rj h)Bt−m i+j≤l

i+j≤l

Ri f Rj hLp0,(m−t)p .

i+j≤m

Therefore, the result is a consequence of the second estimate in Corollary 3.7. Proposition 3.9. Let 1 s a positive integer. We then have p p (i) CBs,k = C rad Bs,k . rad p (ii) µg,p,s,k CMs ≈ (1 − |z|2 )|∂g(z)|∞ + µg,p,s,k CMsp . Proof. We deﬁne (a)

Qg,p,s,k =

sup

¯ 0=f ∈H(B) |α|+|β|≤k

∂α f ∂β gLp0,(k−s)p f Bsp

.

|β|≥1

(b)

Qrad g,p,s,k =

sup

Ri f Rj gLp0,(k−s)p

¯ 0=f ∈H(B) i+j≤k j≥1

f Bsp

.

We will show that

rad Qg,p,s,k ≈ Qrad g,p,s,k ≈ µg,p,s,k CM p s

≈ (1 − |z|2 )|∂g(z)|∞ + µg,p,s,k CMsp

(3.1)

Since Rl = |γ|≤l cγ z γ ∂γ , it is clear that Qrad g,p,s,k Qg,p,s,k . By takrad p ing i = 0 and j = k in (b), it is also clear that µrad g,p,s,k CMs Qg,p,s,k . rad Proposition 3.6, gives the converse inequality and therefore, µg,p,s,k CMsp ≈ p Qrad g,p,s,k . Next, taking |α| = 0 and |β| = k in (a), we have µg,p,s,k CMs 2 Qg,p,s,k . Propositions 3.4 and 2.17 gives that Qg,p,s,k (1−|z| )|∂g(z)|∞ + µg,p,s,k CMsp . Since (1 − |z|2 )|∂g(z)|∞ ≈ (1 − |z|2 )Rg(z)∞ and p (1 − |z|2 )Rg(z)∞ µrad g,p,s,k CMs (see Corollary 3.3), we have (1 − 2 p |z| )|∂g(z)|∞ Qg,p,s,k , and Qg,p,s,k ≈ (1−|z|2 )|∂g(z)|∞ +µrad g,p,s,k CMs . rad Therefore, it remains to prove that Qg,p,s,k Qg,p,s,k . Decomposing the differential operator ∂j as ∂j = z¯j R + (1 − |z|2 )∂j + (|z|2 ∂j − z¯j R), we obtain that if i = |α| ≥ 1, then ∂α = z¯α Ri + Dα . Here Dα is a differential operator given by a ﬁnite sum of compositions of i differential operators of type z¯j R, (1 − |z|2 )∂j or (|z|2 ∂j − z¯j R). Any of these summands includes at most i − 1 times the differential operators z¯j R. The differential operator (1 − |z|2 )∂j maps Bsp to Lpk,(k−s)p , and the operator |z|2 ∂j − z¯j R is a complex tangential operator that maps Bsp to Lpk,(k−s+1/2)p . Hence, the operator Dα is a differential operator that maps Bsp to Lpk,(k−s+i−1/2)p and consequently the operator Dα f is more regular than ∂α f or Ri f . And this regularity is essential to prove the statement.

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Let us demonstrate these assertions. If α = 0, then we write Dα = 0. If |α| = i and |β| = j, then ∂α f (z)∂β g(z) = z¯α+β Ri f (z)Rj g(z) + z¯α Ri f (z)Dβ g(z) +¯ z β Dα f (z)Rj g(z) + Dα f (z)Dβ g(z). ∞

Since β ≥ 1, we can assume g(0) = 0, and in particular, (1−|z|2 )|∂g(z)| ≈ gB0∞ , and

2 |g(z)| (1−|z|2 )|∂g(z)|∞ (1+| log(1−|z|2 )) Qrad g,p,s,k (1+| log(1−|z| )).

Let us estimate Ri f Dβ gLp0,(k−s)p , |β| = j > 0. We ﬁrst observe that

¯ −n−N | |1 − wz| ¯ −n−N −|β|+1/2 . This last inequality is a conse|Dβ (1 − wz) quence of the definition of Dβ and the following facts: ¯ 1 − |w|2 + |z − w|2 |1 − wz| ¯ = |w ¯j |(1 − |z|2 ) ≤ 2|1 − wz| ¯ (1 − |z|2 )|∂j (1 − wz)| |(|z|2 ∂j − z¯j R)(1 − wz)| ¯ = |¯ zj − w ¯j + w ¯j (1 − |z|2 ) − z¯j (1 − wz)| ¯ |1 − wz| ¯ 1/2 . If N > 0 is large enough and ε > 0 is small enough, that we choose later, the above estimate together with Corollary 2.11, and Lemma 2.5 give p k Rn+N f (w) (1 − |w|2 )(N +k−ε)p−1 i p 2 −εp dν(w). |R f (z)| (1 − |z| ) |1 − wz| ¯ n+(N +i−2ε)p B p k Rn+N g(u) (1 − |u|2 )(N +k−ε)p−1 p 2 −εp |Dβ g(z)| (1 − |z| ) dν(u). |1 − u ¯z|n+(N +j−1/2−2ε)p B

Next we observe that if N > k + |s| and ε < 1/2, then (see for instance Lemma 3.6 in [19]) (1 − |z|2 )(k−s−2ε)p−1 dν(z) n+(N +i−2ε)p |1 − u |1 − wz| ¯ ¯z|n+(N +j−1/2−2ε)p B

(1 − |w|2 )(k−s−2ε)p−(N +i−2ε)p (1 − |u|2 )(k−s−2ε)p−(N +j−1/2−2ε)p + . |1 − u ¯w|n+(N +j−1/2−2ε)p |1 − wu| ¯ n+(N +i−2ε)p

Fubini’s Theorem gives that Ri f Dβ gpLp

0,(k−s)p

by B B

is bounded, up to a constant,

|Rkn+N f (w)|p |Rkn+N g(u)|p dν(N +k−ε)p (u) dν(k−s+k−i−ε)p (w) |1 − u ¯w|n+(N +j−1/2−2ε)p

+ B B

|Rkn+N f (w)|p |Rkn+N g(u)|p dν(k−s+k−j+1/2−ε)p (u) dν(N +k−ε)p (w). |1 − u ¯w|n+(N +i−2ε)p

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In order to estimate the above terms observe that k k Rn+N g(u) |g(u)| + |Rl g(u)| l=1 2 Qrad g,p,s,k (1 + | log(1 − |u| )|) +

k

|Rl g(u)|.

l=1

Therefore, B

|Rkn+N g(u)|p dν(N +k−ε)p (u) |1 − u ¯w|n+(N +j−1/2−2ε)p

Qrad g,p,s,k (1

+

k l=1 B

2

− |z| )|∂g(z)|∞ B

(1 + | log(1 − |u|2 )|) dν(N +k−ε)p (u) |1 − u ¯w|n+(N +j−2ε)p

|R g(u)| dν(N +k−ε)p (u) |1 − u ¯w|n+(N +j−1/2−2ε)p l

For δ > 0, (1 + | log(1 − |u|2 )|) (1 − |u|2 )−δ and since k − j ≥ 0, Lemma 2.3 gives that if δ < ε, the ﬁrst integral on the right hand side of the above inequality is bounded. Since N + k − ε ≥ k − s, we have |Rl g(u)|p dν(N +k−ε)p (u) |Rl g(u)|p dν(k−s)p (u) |1 − u ¯w|n+(N +j−1/2−2ε)p |1 − u ¯w|n+(j−1/2−s−ε)p B B rad p ¯w)−n/p−(j−1/2−s−ε) Bsp Qg,p,s,k (1 − u rad p Qg,p,s,k (1 − |z|2 )−(j−1/2−ε)p . Therefore, these estimates together with the fact that (1 − |z|2 )|∂g(z)|∞ Qrad g,p,s,k give p p k Rn+N f (w) Rkn+N g(u) dν(N +k−ε)p (u)dν(k−s+k−i−ε)p (w) |1 − u ¯w|n+(N +j−1/2−2ε)p B B p k Rn+N f (w)p (1 − |w|2 )(k−s+k−i−j+1/2−ε)p−1 dν(w) Qrad g,p,s,k B

p f pBsp . Qrad g,p,s,k In the last step we have used that 0 ≤ i+j ≤ k and 0 < ε < 1/2. Analogously, using that k − j + 1/2 − ε > 0, we obtain p k Rn+N g(u) dν(k−s+k−j+1/2−ε)p (u) |1 − u ¯w|n+(N +i−2ε)p B

p 2 (k−s−N +1/2−i−j+ε)p (Qrad , g,p,s,k ) (1 − |z| )

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p p k Rn+N f (w) Rkn+N g(u) dν(k−s+k−j+1/2−ε)p (u) dν(N +k−ε)p (w) |1 − u ¯w|n+(N +i−2ε)p B B p k Rn+N f (w)p (1 − |w|2 )(k−s+k−i−j+1/2−ε)p−1 dν(w) Qrad g,p,s,k

p Qrad g,p,s,k

B

f pBsp .

p Therefore, Rj f Dβ gLp0,(k−s)p Qrad g,p,s,k f Bs . Arguing similarly we get p Dα f Rj gLp0,(k−s)p + Dα f Dβ gLp0,(k−s)p Qrad g,p,s,k f Bs ,

which concludes the proof.

p The following theorem states a relation between the spaces C rad Bs,k and the multiplication operator MRg (f ) = f Rg.

Theorem 3.10. Let 1 s a positive integer. Let g be a holomorphic function on B, and let MRg be the multiplication operator defined by MRg (f ) = f Rg. Then the following assertions are equivalent: (i) (ii)

p . MRg is bounded from Bsp to Bs−1 rad p µg,p,s,k ∈ CMs . rad p Moreover, MRg L(Bsp →Bs−1 ) ≈ µg,p,s,k CMsp .

Proof. If k > s is a non-negative integer, then p MRg L(Bsp →Bs−1 ) ≈

sup

p (I + R)k−1 (f Rg)Bs−k

¯ 0=f ∈H(B)

f Bsp

.

Therefore, by (3.1) in Proposition 3.9, we have rad p MRg L(Bsp →Bs−1 ) µg,p,s,k CMsp .

In order to prove the converse, let us show that p (1 − |z|2 )Rg(z)∞ MRg L(Bsp →Bs−1 ).

Observe that, for any N, M > 0, the representation formula in Theorem 2.1 gives (1 − |w|2 )N −1 (1 − |z|2 )M Rg(z) = cN Rg(w) dν(w). (1 − wz) ¯ n+N (1 − z¯w)M B

Let l be a positive integer. By Corollary 2.11, Rg(w) (1 − |w|2 )N +l−1 2 M l Rn+N dν(w). Rg(z) = cN +l (1 − |z| ) (1 − z¯w)M (1 − wz) ¯ n+N B

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Therefore, since N > −s + 1, H¨ older’s Inequality gives ⎛ ⎞1/p 2 (N +s−1)p −1 (1 − |w| ) |Rg(z)| (1 − |z|2 )M ⎝ dν(w)⎠ |1 − wz| ¯ (n+N )p B

⎛

⎞1/p p l Rg(w) × ⎝ Rn+N (1 − |w|2 )(l−s+1)p−1 dν(w)⎠ (1 − z¯w)M B

Choosing M and l satisfying M > n/p − s and l > s, Lemma 2.3 gives p ¯w)−M Bsp |Rg(z)| (1 − |z|2 )M +s−1−n/p MRg L(Bsp →Bs−1 ) (1 − z p (1 − |z|2 )−1 MRg L(Bsp →Bs−1 ), p which proves that (1 − |z|2 )Rg(z)∞ MRg L(Bsp →Bs−1 ) . Observe that if s < 0, Corollary 3.3 gives that rad µg,p,s,k ≈ (1 − |z|2 )Rg(z)∞ ,

which proves the desired result for this range of s. To conclude we prove that rad µg,p,s,k p MRg L(B p →B p ) . CMs

s

s−1

By Leibnitz’s formula we have f Rk g =

k−1

cj Rk−1−j (RgRj f ).

j=0

Therefore, p ≈ f Rk gBs−k f Lp (dµrad g,p,s,k )

k−1

p Rk−1−j (RgRj f )Bs−k

j=0

k−1

p MRg (Rj f )Bs−j−1 .

(3.2)

j=0

If s0 < 0, Corollary 3.3 gives that, MRg L(Bsp0 →Bsp

0 −1

)

p ≈ (1 − |z|2 )Rg(z)∞ MRg L(Bsp →Bs−1 ).

(3.3)

Since the Besov spaces Btp , s0 < t < s, are interpolation spaces between p p and Bsp0 , we have that MRg L(Btp →Bt−1 ) MRg L(Bsp →Bs−1 ) for any t < s. Therefore, for any 0 ≤ j ≤ k − 1,

Bsp

j p p p MRg (Rj f )Bs−j−1 MRg L(Bsp →Bs−1 ) R f Bs−j p ≈ MRg L(Bsp →Bs−1 ) f Bsp ,

and plugging these estimates in (3.2), we obtain p MRg L(Bsp →Bs−1 f Lp (dµrad ) f Bsp . g,p,s,k ) p p p Therefore, µrad g,p,s,k CMs MRg L(Bs →Bs−1 ) , and the theorem is proved.

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Corollary 3.11. Let 1 n/p, then MRg L(Bsp →Bs−1 ) ≈ g − g(0)Bsp . 2 2 MRg L(B0 →B−1 ) ≈ g − g(0)BM OA . p p If s > t, then MRg L(Btp →Bt−1 ) MRg L(Bsp →Bs−1 ).

Proof. Assertion (i) is just (3.3). If s > n/p, the Bsp -Carleson measures are 1 rad 2 p p just the ﬁnite ones and µrad g,p,s,k ≈ µg,p,s,k (B) ≈ g − g(0)Bs . Since B0 = 2 2 2 2 H , |Rg(z)| (1 − |z| )dν(z) ∈ CM0 iﬀ g ∈ BM OA. Using again (3.2) and the complex interpolation method we obtain (iv). As a corollary of Proposition 3.9 and Theorem 3.10 we also obtain the following result. Corollary 3.12. Let 1 s positive integers. Then, p p p p = CBs,m = C rad Bs,k = C rad Bs,m . Therefore, we can omit the index CBs,k p k in the notation and use CBs to denote all these spaces. To conclude we rewrite some of the results obtained in this section in terms of the spaces CBsp . Let 1 s a positive integer. p We consider the space CBsp normed by gCBsp = |g(0)| + µrad g,p,s,k CMs . Proposition 3.13. Let 1 n/p, then CBsp = Bsp . (iii) CB02 = BM OA. Theorem 3.14. Let 1 < p < ∞, s ∈ R. The following assertions are equivalent: p (i) MRg : Bsp → Bs−1 is bounded. p (ii) g ∈ CBs . p Moreover, MRg L(Bsp →Bs−1 ) ≈ g − g(0)CBsp .

4. Multiplication and Toeplitz Operators In this section we give the proof of Theorems 1.3 and 1.4 stated in the Introduction. Theorem 4.1. Let 1 s a positive integer and g ∈ H(B). Then the following assertions are equivalent: (i) (ii) (iii)

Mg : Bsp → Bsp is bounded. Mg¯ : Bsp → Lpk,(k−s)p is bounded. g ∈ H ∞ ∩ CBsp .

Moreover, Mg L(Bsp ) ≈ Mg¯ L(Bsp →Lpk,(k−s)p ) ≈ g∞ + gCBsp .

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Proof. It is clear that Mg L(Bsp ) ≈

p ∂γ (f g)Bs−k

sup

f Bsp

¯ 0=f ∈H(B) |γ|≤k

∂α f ∂β gLp0,(k−s)p

sup

(4.1)

f Bsp

¯ 0=f ∈H(B) |α|+|β|≤k

and that Mg¯ L(Bsp →Lpk,(k−s)p ) ≈ =

sup

∂α f ∂β gLp0,(k−s)p

f Bsp

¯ 0=f ∈H(B) |α|+|β|≤k

sup

∂α f ∂β gLp0,(k−s)p

¯ 0=f ∈H(B) |α|+|β|≤k

f Bsp

.

(4.2)

Clearly Mg L(Bsp ) Mg¯ L(Bsp →Lpk,(k−s)p ) and by Corollary 3.7, Mg¯ L(Bsp →Lpk,(k−s)p ) g∞ + gCBsp . We next show that g∞ +gCBsp Mg L(Bsp ) . We ﬁrst observe that, g∞ =

1/m

lim g m Lp

m→+∞

0,(k−s)p

1/m

≤ Mg L(Bsp ) lim 1Lp m→+∞

0,(k−s)p

.

Therefore g∞ ≤ Mg L(Bsp ) . This boundedness could be also be deduced using appropriate test functions. Now, if s0 < 0, then gf Bsp0 g∞ f Bsp0 , and consequently Mg L(Bsp0 ) g∞ Mg L(Bsp ) .

(4.3)

Btp ,

Since the spaces s0 < t < s, are interpolation spaces between Bsp0 p and Bs (see Proposition 2.19), the operator Mg is also bounded from Btp to itself for any s0 < t < s and by (4.3), Mg L(Btp ) Mg L(Bsp ) .

(4.4)

For |α| + |β| ≤ k, Leibnitz formula gives ∂β g∂α f Lp0,(k−s)p = c ∂ (g∂ f ) γ,κ γ κ+α |γ|+|κ|=|β| p Bs−k g∂κ+α f B p s−k+|γ|

|γ|+|κ|=|β|

p p Mg L(Bs−k+|γ| ) f Bs−k+|γ|+|κ|+|α|

|γ|+|κ|=|β|

Mg L(Bsp ) f Bsp . In the two last inequalities we have used that |γ|+|κ|+|α| = |β|+|α| ≤ k. Observe that the above inequality gives in particular that f |∂ k g|Lp0,(k−s)p Mg L(Bsp ) f Bsp , which implies that gCBsp Mg L(Bsp ) .

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Corollary 4.2. Let g ∈ H(B), 1 n/p, then Mg L(Bsp ) ≈ gBsp . If t < s, then Mg L(Btp ) Mg L(Bsp ) .

Proof. (i) is a consequence of (4.3). If s > n/p, Proposition 3.13, gives that CBsp = Bsp . The last assertion is just (4.4). The next theorem gives precise estimates of the norm of the Toeplitz operator Tg¯N . 1 Theorem 4.3. Let N > 0, 1 < p < ∞ and s ∈ R. Let g ∈ B−N . Then, N p Tg¯ extends to a bounded operator from Bs to itself, if and only if, g ∈

p . Moreover, Tg¯N L(Bsp ) ≈ Mg L(B p H ∞ ∩ CB−s−N

N T p ≈ g∞ , g¯N L(Bs ) Tg¯ ≈ g∞ + gCB p L(B p )

−s−N

s

−s−N )

,

, that is

if N > −s, and if N ≤ −s.

¯ then T N (f ), hN = f, Mg (h)N and Proof. Observe that if f, h ∈ H(B), g ¯ by Proposition 2.20 we have N Tg¯ ≈ Mg L(B p ) . L(B p ) s

−s−N

Therefore the result is a consequence of Theorem 4.1.

1 Corollary 4.4. Let 1 0 and g ∈ B−N . Then,

(i) If s < −n/p − N , then Tg¯N L(Bsp ) ≈ gB p (ii)

If s > t, then Tg¯N L(Bsp ) Tg¯N L(Btp ) .

−s−N

.

5. Hankel Operators The ﬁrst result in this section states necessary conditions derived from the p p continuity of the operator HgN ¯ : Bs → Lk,(k−s)p . 1 Proposition 5.1. Let 1 0, g ∈ B−N , s a real number and N p k > s a positive integer. If the operator Hg¯ : Bs → Lpk,(k−s)p is bounded, then g ∈ B0∞ ∩ Bsp and p p (1 − |z|2 )|∂g(z)|∞ + g(z) − g(0)Bsp HgN ¯ L(Bs →Lk,(k−s)p ) .

N Proof. The continuity of the operator HgN ¯ and g(z) − g(0) = Hg ¯ (1)(z), give N immediately that g(z) − g(0)Bsp Hg¯ L(Bsp →Lpk,(k−s)p ) . In order to prove p p that (1 − |z|2 )|∂g(z)|∞ HgN ¯ L(Bs →Lk,(k−s)p ) , let M > max{0, −s} (to ensure that the integral formula in Corollary 2.11 makes sense) and ε > 0

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small enough. Applying Corollary 2.11, and Lemma 2.5, we have (1 − |w|2 )M +k−1 k−1 R |∂j g(z)| dν(w) n+M +1 ∂j g(w) |1 − wz| ¯ n+M +1 B

⎛ ⎞1/p 2 (k−s)p−1 (1−|w| ) p (1−|z|2 )−ε ⎝ |Rk−1 dν(w)⎠ . n+M +1 ∂j g(w)| |1− wz| ¯ n−sp−εp+p B

Since p k−1

p R k−1 n+M +1 ∂j g(w) (1 − u¯ z )−n/p+s+ε−1 (w) , = Rn+M +1 ∂¯j HgN ¯ n−sp−εp+p |1 − w¯ z| we have that if we choose 0 < ε < 1/p , then

2 −ε −n/p+s+ε−1 p p (1 − u¯ z ) |∂j g(z)| HgN (1 − |z| ) ¯ L(Bs →Lk,(k−s)p )

Bsp

2 −1 p p HgN , ¯ L(Bs →Lk,(k−s)p ) (1 − |z| )

which concludes the proof.

We next prove that for some values of s the condition g ∈ B0∞ is a sufﬁcient condition for the boundedness of the operator HgN ¯ . Proposition 5.2. Let 1 < p < ∞. If −N < s < 0, and g ∈ B0∞ , then 2 p p HgN ¯ L(Bs →L0,−sp ) ≈ (1 − |z| )|∂g(z)|∞ .

Proof. Proposition 5.1 gives that in order to prove the Proposition it is enough to show that for any f ∈ Bsp , N Hg¯ (f ) p (1 − |z|2 )|∂g(z)|∞ f Bsp . L 0,−sp

Theorem 2.1 applied to the function ψ = g¯f , gives that 2 N |HgN ¯ (f )(z)| (1 − |z| )|∂g(z)|∞ KN +1−n,n−1/2 (|f |)(z) N +(1 − |w|2 )1/2 |∂T g(w)|∞ KN +1−n,n−1/2 (|f |)(z). N Since KN +1−n,n−1/2 is a kernel of type 0 and N > −s, Proposition 2.6 N p shows that the Lp0,−sp -norm of KN +1−n,n−1/2 (|f |)(z) is bounded by f L0,−sp , which is in turn equivalent to f Bsp . Since (1 − |w|2 )1/2 |∂T g(w)|∞ (1 − |w|2 )|∂g(w)|∞ (see Proposition 2.17) the proof is ﬁnished.

Theorem 5.3. Let 1 max{0, −s} and let 1 . Then, the Hankel operator HgN g ∈ B−N ¯ extends to a bounded operator from p p Bs to Lk,(k−s)p , if and only if, g ∈ CBsp . p p p Moreover, HgN ¯ L(Bs →Lk,(k−s)p ) ≈ g − g(0)CBs .

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p p Proof. Assume that HgN ¯ : Bs → Lk,(k−s)p is bounded. Proposition 5.1 gives 2 N (1 − |z| )|∂g(z)|∞ Hg¯ L(Bsp →Lpk,(k−s)p ) . In order to prove that µ = µg,p,s,k is a Bsp -Carleson measure, observe that for any f ∈ Bsp , f Lp (dµ) = |∂¯k HN (f )|Lp ≤ HN (f )Lp g ¯

g ¯

0,(k−s)p

≤

k,(k−s)p

p p p HgN ¯ L(Bs →Lk,(k−s)p ) f Bs .

p p Therefore, µCMsp ≤ HgN ¯ L(Bs →Lk,(k−s)p ) . Observe that in the proof of the above estimates we do not use any additional condition on N , and in fact, we have proved that for all N > 0 we p p have g − g(0)CBsp HgN ¯ L(Bs →Lk,(k−s)p ) . Let us prove the converse. We want to show that if |α| + |β| ≤ k then ∂¯β ∂α HN (f )Lp ,(k−s)p gCB p f B p .

g ¯

0

s

s

We consider the following cases: N +|α|

(∂α f ). 1. |α| ≤ s, |β| = 0: Lemma 2.7 gives ∂α HgN ¯ ¯ (f ) = Hg By Proposition 5.2 with 0 < δ < min{N, k − s}, we have N +|α| ∂α HgN Hg¯ (∂α f ) p ¯ (f ) Lp 0,(k−s)p

L0,δp

2

p . (1 − |z| )|∂g(z)|∞ ∂α f B−δ p p Since −δ < 0 ≤ s−|α|, we have |α|−δ < s and ∂α f B−δ f B|α|−δ f , which proves 1.

Bsp

2. s < |α| ≤ k, |β| = 0: This case is analogous to the previous one. Since N > −s we have N + |α| > |α| − s and Proposition 5.2 gives that N +|α| ∂α HgN Hg¯ (∂α f ) p ¯ (f ) Lp 0,(k−s)p

L0,(|α|−s)p

2

(1 − |z| )|∂g(z)|∞ ∂α f Lp0,(|α|−s)p (1 − |z|2 )|∂g(z)|∞ f Bsp . k 3. |α| = 0, |β| = k: Since |∂¯β HgN ¯ (f )| = |f ∂β g| ≤ |f ||∂ g|, we have that ∂¯β HgN gCBsp f Bsp . ¯ (f ) Lp 0,(k−s)p

4. |α|,|β| = 0: Since ∂¯β ∂α HgN ¯ (f ) = ∂β g∂α f , we need to show that ∂β g∂α f Lp0,(k−s)p gCMsp f Bsp , which is a consequence of (3.1) in Proposition 3.9. And that concludes the proof. We ﬁnally deal with the case 0 < N ≤ −s. The proof will be based on duality, as it has been used in the characterization of the Toeplitz operators. The main difﬁculty here is that we now deal with duals of the “real” spaces Lpk,(k−s)p , which are less manageable to work with. In order to avoid

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this handicap, we obtain the characterization by studying the continuity of differences of Hankel operators, which in turn coincide with difference of Toeplitz operators. And these differences of Toeplitz operators map Besov spaces to itselves under some conditions on the symbol. 1 ¯ then and f, h ∈ H(B), Lemma 5.4. If N > 0, g ∈ B−N

(i) Tg¯N +l (f ), hN = f, gRln+N hN +l . (ii) (Tg¯N +l − Tg¯N )(f ), hN = f, gRln+N h − Rln+N (gh)N +l . Proof. Observe that N +l Tg¯ (f ), h N (1 − |w|2 )N +l−1 ¯ = cN cN +l g¯(w)f (w) dν(w)h(z)(1 − |z|2 )N −1 dν(z) (1 − wz) ¯ n+N +l B B (1 − |z|2 )N −1 ¯ dν(z)(w)dνN +l−1 (w) = cN +l g¯(w)f (w)cN h(z) (1 − wz) ¯ n+N +l B B = cN +l f (w)g(w)Rln+N h(w)dνN +l−1 (w) B

= f, gRln+N h N +l . And that proves (i). By Corollary 2.10,

Tg¯N (f ), h

N

= cN B

f (w)g(w)h(w)(1 − |w|2 )N −1 dν(w)

= cN +l

f (w)Rln+N (gh)(w)(1 − |w|2 )N +l−1 dν(w)

B

= f, Rln+N (gh) N +l ,

which, together with assertion (i) concludes the proof of (ii). In particular if l = 1 we have: 1 Corollary 5.5. If N > 0, g ∈ B−N and f, h ∈ H(B), then

N +1 (Tg¯ − Tg¯N )(f ), h N = f, hRgN +1 = f, MRg hN +1 . Corollary 5.6. Let N > 0, 1 < p < ∞ and s ∈ R. Then, Tg¯N +1 − Tg¯N is

p to bounded from Bsp to itself, if and only if, MRg is bounded from B−s−N

p B−s−N −1 , with equivalent norms.

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Proof. This result is a consequence of the above corollary and the fact that p (see Proposition 2.20). the dual of Bsp respect the pairing ·, ·M is B−s−M N +1 N T (f ), h − T N +1 g ¯ g ¯ N Tg¯ − Tg¯N L(B p ) ≈ sup s f Bsp hB p ¯ 0=f,h∈H(B) −s−N

| f, MRg hN +1 | = sup ¯ f Bsp hB p 0=f,h∈H(B) −s−N

≈ MRg L(B p

p −s−N →B−s−N −1 )

.

Proposition 5.7. Let N > 0, 1 < p < ∞, s ∈ R and l a positive integer. Then, Tg¯N +l − Tg¯N L(Bsp ) MRg L(B p

p −s−N →B−s−N −1 )

.

In particular, if −s − N < 0, then N +l Tg¯ − Tg¯N L(B p ) (1 − |z|2 )|∂g(z)|∞ . s

Proof. By Corollary 5.6 we have l−1 N +l N +j+1 Tg¯ Tg¯ − Tg¯N L(B p ) ≤ − Tg¯N +j s

L(Bsp )

j=0

≈

l−1

MRg L(B p

p −s−N −j →B−s−N −j−1 )

j=0

.

By Corollary 3.11, (iv), MRg L(B p

p −s−N −j →B−s−N −j−1 )

MRg L(B p

p −s−N →B−s−N −1 )

,

which concludes the proof.

Theorem 5.8. Let 1 < p < ∞, s < 0, k a positive integer, 0 < N ≤ −s and p 1 p . Then, HgN g ∈ B−N ¯ extends to a bounded operator from Bs to Lk,(k−s)p , if

p . and only if, g ∈ CB−s−N Moreover, N Hg¯ ≈ MRg L(B p L(B p →Lp ) s

p −s−N →B−s−N −1 )

k,(k−s)p

≈ g − g(0)CB p

−s−N

.

Proof. The last equivalence is a consequence of Theorem 3.14. Let us prove the ﬁrst one. Assume that MRg L(B p →B p is ﬁnite. By Theorem 3.10 ) −s−N

2

−s−N −1

(1 − |z| )Rg(z)∞ MRg L(B p

p −s−N →B−s−N −1 )

.

Let l > −s − N a positive integer. Theorem 5.3 gives that N +l Hg¯ ≈ (1 − |z|2 )∂g(z)∞ . L(B p →Lp ) s

k,(k−s)p

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N +l +Tg¯N +l − Using the above estimates, together with the fact that HgN ¯ ¯ = Hg N Tg¯ , and Proposition 5.7, we obtain

N Hg¯ L(B p →Lp s

k,(k−s)p )

(1 − |z|2 )|∂g(z)|∞ + Tg¯N +l − Tg¯N L(B p ) s

MRg L(B p

p −s−N →B−s−N −1 )

.

In order to prove the converse, assume that HgN ¯ is bounded. By p p Proposition 5.1 we have (1 − |z|2 )|∂g(z)|∞ HgN ¯ L(Bs →Lk,(k−s)p ) , and

+l L(Bsp →Lpk,(k−s)p ) ≈ by Theorem 5.3, we have that if l > −s − N , then HgN ¯ 2 (1 − |z| )∂g(z)∞ . Therefore, for any l > −s − N we have N +l N +l Tg¯ Hg¯ − Tg¯N L(B p ) HgN + p p ¯ L(Bs →Lk,(k−s)p ) L(Bsp →Lp s k,(k−s)p ) N Hg¯ L(B p →Lp . (5.1) ) s

k,(k−s)p

Moreover, if s0 > 0, by Propositions 5.7 and 5.1, then N +l Tg¯ − Tg¯N L(B p ) (1 − |z|2 )|∂g(z)|∞ HgN ¯ L(B p →Lp s0

k,(k−s)p )

s

.

Since the spaces Btp , s0 > t > s, are interpolation spaces between Bsp0 and Bsp , we have that N +l Tg¯ − Tg¯N L(B p ) HgN ¯ L(B p →Lp ) s

t

for all t ≥ s. In particular N +l Tg¯ − Tg¯N L(B p

s+1 )

k,(k−s)p

HgN ¯ L(B p →Lp

k,(k−s)p )

s

We will use this estimate to prove that N +l+1 Tg¯ − Tg¯N +1 L(B p ) HgN ¯ L(B p →Lp s

Indeed R1n+N

and R1n+M Tg¯M

s

.

k,(k−s)p )

(5.2)

.

N −M n+M R R = I+ =I+ I+ n+N n+N n+N n+M n+M 1 N −M I+ R , = n+N n + N n+M = Tg¯M +1 R1n+M (see Corollary 2.11), we have

R1n+N Tg¯N +l − Tg¯N = Tg¯N +l+1 − Tg¯N +1 R1n+N N +l+1 l T − Tg¯N +l . + n + N g¯

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p Using this formula and the fact that R1n+N is a bijective operator from Bs+1 p to Bs , we get N +l+1 Tg¯ − Tg¯N +1 L(B p ) s N +l+1 1 N +1 Rn+N L(B p →B p ) ≈ Tg¯ − Tg¯ s s+1 1 N +l N Rn+N Tg¯ − Tg¯ + Tg¯N +l+1 − Tg¯N +l L B p →B p p L(Bs+1 →Bsp ) ( s+1 s ) N +l+1 N +l N N +l − Tg¯ L(B p ) + Tg¯ − Tg¯ Tg¯ L(Bsp ) s+1 N Hg¯ L(B p →Lp + (1 − |z|2 )Rg(z) ∞ s k,(k−s)p ) HgN ¯ L(B p →Lp ) s

k,(k−s)p

where in the last two inequalities we have used (5.2) and Propositions (5.7) and (5.1). Therefore, N +1 Tg¯ − Tg¯N L(B p ) s ≤ Tg¯N +l+1 − Tg¯N +1 L(B p ) + Tg¯N +l+1 − Tg¯N L(B p ) s s N Hg¯ L(B p →Lp ). s

k,(k−s)p

In the estimate of the second summand we have used the ones obtained above and (5.1). Since Corollary 5.6 gives that N Hg¯ MRg ≈ Tg¯N +1 − Tg¯N , p p p p p L(B−s−N →B−s−N −1 )

we have ﬁnished the proof of the theorem.

L(Bs )

L(Bs →Lk,(k−s)p )

References [1] Adams, D.R., Hedberg, L.I.: Function Spaces and Potential Theory. Springer, Berlin (1996) [2] Ahern, P., Cohn, W.S.: Exceptional sets for Hardy–Sobolev spaces. Indiana Math. J. 39, 417–451 (1989) [3] Aleksandrov, A.B.: Function theory in the ball. Several complex variables. II. Function theory in classical domains. Complex potential theory. Encyclopaedia of Mathematical Sciences, vol. 8, pp. 107–178. Springer, Berlin (1994) [4] Arcozzi, N., Rochberg, R., Sawyer, E.: Carleson measures for analytic Besov spaces. Rev. Mat. Iberoamericana 18, 443–510 (2002) [5] Arcozzi, N., Rochberg, R., Sawyer, E.: Carleson measures and interpolation sequences for Besov spaces on complex balls. Mem. Am. Math. Soc. 182(859), vi+163 (2006) [6] Arcozzi, N., Rochberg, R., Sawyer, E.: Carleson measures for Drury–Arvesson Hardy space and other Besov–Sobolev spaces on complex balls. Adv. Math. 218(4), 1107–1180 (2008) [7] Arcozzi, N., Rochberg, R., Sawyer, E., Wick, B.D.: Bilinear forms on the Dirichlet space (25 Nov 2008). arXiv:0811.4107v1

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[8] Beatrous, F.: Estimates for derivatives of holmorphic functions in pseudoconvex domains. Math. Z. 191, 91–116 (1986) [9] Beatrous, F., Burbea, J.: Sobolev spaces of holomorphic functions in the ball. Dissertationes Math. 276 (1989) [10] B´ekoll´e, D., Berger, C.A., Coburn, L.A., Zhu, K.H.: BMO in the Bergman metric on bounded simmetric domains. J. Funct. Anal. 93, 310–350 (1990) [11] Cascante, C., Ortega, J.M.: Carleson measures on spaces of Hardy–Sobolev type. Can. J. Math. 47, 1177–1200 (1995) [12] Charpentier, Ph.: Formules explicites pour les solutions minimales de l’equation ∂u = f dans la boule et dans le polydisque de Cn . Ann. Inst. Fourier Grenoble 30, 121–154 (1980) [13] Cohn, W.S., Verbitsky, I.E.: Trace inequalities for Hardy–Sobolev functions in the unit ball of Cn . Indiana Univ. Math. J. 43, 1079–1097 (1994) [14] Cohn, W.S., Verbitsky, I.E.: Non-linear potential theory on the ball, with applications to exceptional and boundary interpolation sets. Michigan Math. J. 42, 79–97 (1995) [15] Costea, S., Sawyer, E.T., Wick, B.D.: The corona theorem for the DruryArveson Hardy space and other holomorphic Besov-Sobolev spaces on the unit ball in Cn (8 Apr 2009). arXiv:0811.0627v7 [16] Hyt¨ onen, T., Martikainen, H.: Non-homogeneous Tb theorem and random dyadic cubes on metric measure spaces (23 Nov 2009). arXiv:0911.4387v1 [17] Maz’ya, V.G., Shaposhnikova, T.O.: Theory of multipliers in spaces of differentiable functions. Pitmann (1985) [18] Nikolski, N.K.: Operators, functions, and systems: an easy reading, vol. 1. Hardy, Hankel, and Toeplitz. Mathematical Surveys and Monographs, 92. American Mathematical Society, Providence (2002) [19] Ortega, J.M., F` abrega, J.: Pointwise multipliers and decomposition theorems in analytic Besov spaces. Math. Z. 235, 53–81 (2000) [20] Peller, V.V.: Hankel Operators and their Applications. Springer Monographs in Mathematics. Springer, New York (2003) [21] Stegenga, D.A.: Multipliers of the Dirichlet space. Illinois J. Math. 24, 113–139 (1980) [22] Tchoundja, E.: Carleson measures for the generalized Bergman spaces via a T (1)-type theorem. Ark. Mat. 46, 377–406 (2008) [23] Triebel, H.: Theory of function spaces. Monographs in Mathematics, vol. 78. Birkh¨ auser, Basel (1993) [24] Upmeier, H.: Toeplitz operators and index theory in several complex variables. Operator Theory: Advances and Applications, vol. 81. Birkh¨ auser, Basel (1996) [25] Volberg, A., Wick, B.D.: Bergman-type singular operator and the characterization of Carleson measures for Besov–Sobolev spaces on the complex ball (8 Oct 2009). arXiv:0910.1142v2 [26] Verbitsky, I.E.: Multipliers in spaces of fractional norms, and inner functions. Sibirsk. Mat. Zh. 26, 51–72 (1985, Russian) [27] Xia, J.: Boundedness and compactness of Hankel operators on the sphere. J. Funct. Anal. 255(1), 25–45 (2008)

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[28] Wu, Z., Zhao, R., Zorboska, N.: Toeplitz operators on analytic Besov spaces. Int. Equ. Oper. Theory 60, 435–449 (2008) [29] Zhu, K.: Spaces of holomorphic functions in the unit ball. Graduate Texts in Mathematics, vol. 226. Springer, New York (2005) Carme Cascante, Joan F` abrega and Joaqu´ın M. Ortega Department Matem` atica Aplicada i An` alisi Universitat de Barcelona Gran Via 585 08071 Barcelona Spain e-mail: [email protected]; joan [email protected]; [email protected] Received: March 9, 2009. Revised: December 10, 2009.

Integr. Equ. Oper. Theory 66 (2010), 529–538 DOI 10.1007/s00020-010-1761-3 Published online March 19, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Curvature Condition for Non-contractions Does not Imply Similarity to the Backward Shift Hyun-Kyoung Kwon and Sergei Treil Abstract. We give an example of an operator that satisfies the curvature condition as defined in Kwon and Treil (Publ Mat 53(2):417– 438, 2009), but is not similar to the backward shift S ∗ on the Hardy class H 2 . We conclude therefore that the contraction assumption in the similarity characterization given in Kwon and Treil (Publ Mat 53(2): 417–438, 2009) is a necessary requirement. Mathematics Subject Classification (2000). Primary 47A99; Secondary 47B32, 30D55, 53C55. Keywords. Similarity, backward shift, eigenvector bundle, curvature condition, Carleson measure.

Contents Notation 1. Introduction and Result 2. The Construction 3. Main Estimates References

529 530 532 534 538

Notation := C D

equal by definition; the complex plane; the unit disk, D := {z ∈ C : |z| < 1};

The work of S. Treil was supported by the National Science Foundation under Grant DMS-0501065.

530 ∂ ∂ ∂z , ∂z

∆ · H 2, H ∞

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∂ ∂ ∂ ∂ ∂ ∂ ∂ and ∂ derivatives: ∂z := ( ∂x − i ∂y )/2, ∂z := ( ∂x + i ∂y )/2, the symbols ∂ and ∂ are sometimes used; ∂2 ∂2 normalized Laplacian, ∆ = ∂∂ = ∂∂ = 14 ( ∂x 2 + ∂y 2 ); norm; Hardy classes of analytic functions, ⎫ ⎧ ⎬ ⎨ |dz| = 0 for k < 0 , H p := f ∈ Lp (T) : fˆ(k) := f (z)z −k ⎭ ⎩ 2π T

Hardy classes can be identiﬁed with the spaces of functions that are analytic in the unit disk D: in particular, H ∞ is the space of all functions bounded and analytic in D.

1. Introduction and Result In this paper we consider operators with a complete analytic family of eigenvectors (the backward shift in the Hardy space H 2 , or, more generally in some reproducing kernel Hilbert space being a typical example). We are interested in the classiﬁcation of such operators in terms of the geometry of their eigenvector bundles. The problem of unitary classiﬁcation was completely solved by Cowen and Douglas [1]. One of the ﬁrst results in [1] was a version of Calabi’s Rigidity Theorem, which stated that if the eigenvector bundles of operators T1 and T2 are isomorphic (as Hermitian vector bundles), then the operators are unitarily equivalent. In the case of dim ker(T1,2 − λI) = 1, the fact that the eigenvector bundles are isomorphic simply means that the curvatures coincide. Note, that the general case of dim ker(T1,2 − λI) = n < ∞ was also treated in [1], and numerous local criteria for determining unitary equivalence were obtained there. The problem of classiﬁcation of operators up to similarity, as some examples in [1] did show, is significantly more complicated. For example, an obvious conjecture that uniform equivalence of curvatures is sufﬁcient for similarity (it is obviously necessary) is trivially wrong; the curvatures of the (eigenvector bundles of the) backward shifts in H 2 and in the Bergman space A2 differ by a factor of 2, but these operators are very far from being similar. In [2] a particular case of the similarity problem, namely the problem of similarity to the backward shift in the Hardy space H 2 was considered. The structure of the backward shift is very well understood, so there was hope that it will be possible to get a solution in this special case, and that this solution may lead to a better understanding of the general case. This program was partially realized by the authors: in [2] a complete description (in terms of the geometry of the eigenvector bundles) of operators similar to backward shifts of ﬁnite multiplicity was obtained, but under an additional assumption that the operator T is a contraction, T ≤ 1. More precisely, it was assumed in [2] that the operator T in a Hilbert space H satisﬁes the following four conditions:

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(i) T is contractive, i.e., T ≤ 1; (ii) dim ker(T − λI) = n < ∞ for all λ ∈ D; (iii) span{ker(T − λI) : λ ∈ D} = H; (iv) The subspaces E(λ) = ker(T − λI) depend analytically on the spectral parameter λ ∈ D. Remark 1.1. Condition (iv) means that for each ω ∈ D, there exist a neighborhood Uω of ω and an analytic operator-valued function Fω deﬁned on Uω that is left invertible in L∞ such that ran Fω (λ) = E(λ), λ ∈ Uω . Since the columns of Fω (λ) form a holomorphic basis in E(λ), the disjoint union λ∈D E(λ) = {(λ, vλ ) : λ ∈ D, vλ ∈ E(λ)} is a holomorphic vector bundle over D (subbundle of the trivial bundle D × H) with the natural projection π, π(λ, vλ ) = λ. Note, that thesubspaces E(λ) inherit the metric from the Hilbert space H, so our bundle λ∈D E(λ) is a Hermitian holomorphic vector bundle. The main result of [2] was a necessary and sufﬁcient condition for the operator T to be similar to the backward shift of multiplicity n, i.e., to the direct sum of n backward shifts S ∗ in H 2 , S ∗ f (z) = (f (z) − f (0))/z, f ∈ H 2 . For the case n = 1, which we are considering in this paper, this result can be stated as follows: Theorem 1.2 [2, Theorem 0.1]. The following statements are equivalent: 1. T is similar to the backward shift operator S ∗ on H 2 . 2. The eigenvector bundles of T and S ∗ are “uniformly equivalent”, i.e., there exists a holomorphic bundle map bijection Ψ from the eigenvector bundle of S ∗ to that of T such that for some constant c > 0, 1 vλ H 2 ≤ Ψ(vλ )H ≤ cvλ H 2 c for all vλ ∈ ker(S ∗ − λI) and for all λ ∈ D. 3. There exists a bounded subharmonic solution ϕ to the Poisson equation ∆ϕ(z) = |κS ∗ (z) − κT (z)|

4.

for all z ∈ D, where κS ∗ and κT denote the curvatures of the eigenvector bundles of the operators S ∗ and T , respectively. We say that the “curvature condition” is satisfied in this case. The measure |κS ∗ (z) − κT (z)|(1 − |z|)dx dy, where κS ∗ and κT are the corresponding curvatures, is Carleson and the estimate 1 C |κS ∗ (z) − κT (z)| 2 ≤ 1 − |z| holds.

Remark 1.3. Without stating the exact definition, let us recall that the curvature κT of the eigenvector bundle of T (in fact, the curvature of any Hermitian line bundle over a planar domain) can be calculated as

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κT (z) = −∆ ln γ(z)2 , where γ is a section of the bundle and ∆ is the normalized Laplacian, i.e., ∆ = ∂∂ = ∂∂ [1]. 2 It can also be expressed as κT (z) = − ∂Π(z) ∂z , where Π : D → B(H) denotes the projection-valued function that assigns to each λ ∈ D, the orthogonal projection onto ker(T − λI), Π(λ) := Pker(T −λI) (this formula is in fact true for any line subbundle of the trivial bundle Ω × H → Ω, where H is a Hilbert space and Ω ⊂ C). In [2], the latter formula for the curvature was used, but it is well known and not hard to show that both formulas give the same result. Remark 1.4. For the backward shift S ∗ , the curvature κS ∗ can be easily computed: κS ∗ (z) = −(1 − |z|2 )−2 . Remark 1.5. It was shown in [2] that if T is a contraction (with dim(T −λI) = 1 for all λ ∈ D), then κS ∗ − κT ≥ 0. Thus, in the statement of the main theorem (Theorem 0.1) in [2], we used the term κS ∗ − κT instead of |κS ∗ − κT |. This nonnegativity is consistent with a more general result in [3], but we can no longer take it for granted when it comes to non-contractions. A natural question is whether it is possible to get rid of the assumption that T is contractive in Theorem 1.2. This question was partially answered in [2], where a non-contractive operator satisfying statement 2, but which is not similar to S ∗ was constructed. That means the assumption T ≤ 1 is needed to prove Theorem 1.2 in full generality. However, there was hope, that maybe statement 3 and/or 4 is sufﬁcient for similarity even without assuming that T is a contraction. In the present paper, we destroy this hope by giving an example of an operator that satisﬁes statements 3 and 4, but is not similar to S ∗ . This example is a slight modiﬁcation of the one given in [2]. Now we state the main result: Theorem 1.6. Given α > 0, there exists an operator T that is “almost” a coisometry in the sense that (1 − α)x ≤ T ∗ x ≤ (1 + α)x, and such that it is not similar to S ∗ on H 2 , but satisfies statements 2–4 of Theorem 0.1.

2. The Construction The operator T will be constructed as the backward shift on the weighted

2 n 2 2 Hardy class Hw := f = n≥0 an z : f w := n≥0 |an | wn < ∞ , where w = {wn }∞ 0 , wn > 0. Let us remind the reader that the backward shift in Hw2 is the adjoint of the forward shift f → zf , and it can be easily shown that ∞ ∞

wn+1 n = an z an+1 z n . T w n n=0 n=0 Let us also remind the reader that the eigenvector bundle of T is the family of the reproducing kernel kλw of Hw2 , T kλw = λkλw ,

λ ∈ D,

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Figure 1. The function ln wn : two “spikes” are shown and that the reproducing kernel kλw can be computed by the following formula: n ∞

λ zn . (2.1) kλw (z) = wn k=0

The weight sequence w = {wn }∞ 0 (wn > 0) will be of the form (see Fig. 1) ⎧ ⎨ 2j ln(1 + α) n = Nk + j, 0 ≤ j ≤ k, ln wn = 2j ln(1 + α) n = Nk + 2k − j, 0 ≤ j ≤ k, ⎩ 0 otherwise, where the numbers Nk , k ∈ N, satisfying Nk + 2k < Nk+1 will be chosen later. Using the fact that the weight sequence w = {wn }∞ 0 is unbounded, one can show that the operator T is not similar to S ∗ . To see this, let us assume operator A so that the contrary, i.e., AT = S ∗ A for some bounded, invertible

∞ T ∗ n A∗ = A∗ S n . If we take an f ∈ H 2 such that A∗ f = 0 an z n = 0 and an m such that am = 0, then T ∗ n A∗ f 2 =

∞

|aj |2 wj+n ≥ |am |2 wm+n ,

j=0 ∗n

∗

so supn T A f = ∞. But A∗ S n f Hw2 ≤ A∗ f H 2 , and we have a contradiction. Also, it is easy to see that the adjoint T ∗ of the operator is “almost” an isometry, i.e., (1 − α)x ≤ T ∗ x ≤ (1 + α)x for all x ∈ Hw2 . This is due to the slope condition (1 + α)−2 ≤ wn+1 /wn ≤ (1 + α)2 , for all n ≥ 0.

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In the next section, we will prove that it is possible to choose the numbers Nk so that statements 2–4 of Theorem 1.2 are true. Since statement 3 follows from (with estimates) 4 (see [2,5]), it sufﬁces to check only 2 and 4.

3. Main Estimates In fact, we will prove even more than what is stated in Theorem 1.6. Namely we will show that given ε > 0, one can pick the numbers Nk ’s in such a way that 1.

2.

(1 + ε)−1 ≤ |kλ1 (λ)/kλ2 (λ)| ≤ 1 + ε for all λ ∈ D, where kλ1 and kλ2 are the reproducing kernels for the Hardy space H 2 and for the weighted Hardy space Hw2 , respectively.1 This means that statement 2 of Theorem 1.2 holds with c = 1 + ε (the bundle map between the eigenvector bundles is kλ1¯ → kλ2¯ , λ ∈ D). The difference of the curvatures satisﬁes the estimate

|κT (z) − κS ∗ (z)| ≤ , (1 − |z|)2 and the measure dµ = |κT (z) − κS ∗ (z)|(1 − |z|)dx dy is Carleson with Carleson norm µCarl ≤ .

Let us ﬁrst recall the definition of a Carleson measure. Denote by I ⊂ T an arc of the unit circle with length |I|. Then a positive measure µ in the closed unit disk is called a Carleson measure if µ(QI ) ≤ c

|I| , 2π

for some constant c > 0, where |I| z QI := z ∈ C : ∈ I, 1 − |z| ≤ |z| 2π is the Carleson window [4, Chap. 6.3]. The infimum of such c > 0 is called the Carleson norm, denoted µCarl . The maps z → kz1¯ and z → kz2¯ are sections of the eigenvector bundles of z ) = kzi (z), the operators S ∗ and T , respectively. Since kzi¯2 = (kzi¯, kzi¯) = kzi¯(¯ we get by recalling the formula for the curvature (see Remark 1.3) that the corresponding curvatures equal −∆ ln kzi (z), i = 1, 2. 1 We know that kz1 (λ) = 1/(1−zλ), so kz1 (z) = 1−|z| 2 . On the other hand, (2.1) implies that kz2 (z) =

∞

1 1 |z|2n = + gk (z), wn 1 − |z|2

n≥0 1

k=1

2 instead of k w for the reproducing kernel of H 2 . We will use the symbol kλ w λ

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⎛ ⎞ k k−1

gk (z) = |z|2Nk ⎝ cj |z|2j + cj |z|2(2k−j) ⎠ , j=1

j=1

and cj = Thus, |κS ∗

1 − 1. (1 + )2j

∞

kz2 (z) 2 − κT | = ∆ ln 1 = ∆ ln 1 + gk (z)(1 − |z| ) . k (z) z

k=1

Since |f ∆f − |∂f |2 | , |f |2 to prove that conditions 1 and 2 hold, we need to show that given δ > 0, the k2 (z) Nk ’s can be chosen so that the following conditions hold for f (z) = kz1 (z) = z

∞ 1 + k=1 gk (z)(1 − |z|2 ): 1. 1 − δ ≤ |f | ≤ 1 + δ; δ 2. (a) |∆f | ≤ (1−|z|) 2; |∆ ln f | =

√

δ ; (b) |∂f | ≤ 1−|z| 3. dµ = |∆f |(1 − |z|)dx dy is a Carleson measure with µCarl ≤ δ; 4. dµ = |∂f |2 (1 − |z|)dx dy is a Carleson measure with µCarl ≤ δ.

Remark 3.1. Since f is a quotient of the reproducing kernels, the above condition 1 implies condition 2 of Theorem 1.2 with c = 1/(1 − δ). But note that these conditions follow from the following four conditions where we consider the functions hk (z) := gk (z)(1 − |z|2 ) instead of f (z): 1. |hk | ≤ 2δk ; δ 2. (a) |∆hk | ≤ 2k (1−|z|) 2; (b) |∂hk | ≤

δ ; 2k (1−|z|)

dµ = |∆hk |(1 − |z|)dx dy is a Carleson measure with µCarl ≤ 2δk ; dµ = |∂hk |2 (1 − |z|)dx dy is a Carleson measure with µCarl ≤ 2δ2k . It is easy to see that the ﬁrst three conditions for hk imply those for f because they are just linear (in hk ) estimates. As for the fourth condition, let us notice that condition 4 for hk simply means that for every Carleson window QI , √ δ 1 ∂hk L2 (QI ,(1−|z|)dx dy) ≤ k |I| 2 . 2

Since ∂f = k≥1 ∂hk , we conclude that ∞ √

δ 1/2 √ ∂f L2 (QI ,(1−|z|)dx dy) ≤ |I| = δ|I|1/2 , k 2

3. 4.

k=1

which is exactly statement 4 for f .

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Next, note that since the hk can be written down as hk = z

2Nk

2

(1 − |z| )

2k−1

akj |z|2j ,

j=1

it sufﬁces to show that for the elementary functions ψN = |z|2N (1 − |z|2 ) = z N z¯N − z N +1 z¯N +1 , the analogues of the above conditions 1–4 hold, and that by taking sufﬁciently large N we can make the right sides of the inequalities there as small as we want. Hence Theorem 1.6 will follow from the Lemma given below: Lemma 3.2. The following statements are true for the function ψN = |z|2N (1 − |z|2 ): 1. ψN (z) −→ 0 uniformly on D as N → ∞; 2. |∆ψN (z)|(1 − |z|)2 −→ 0 uniformly on D as N → ∞; N (z) 2 3. ∂ψ∂z (1 − |z|)2 −→ 0 uniformly on D as N → ∞; 4. The measure dµ = dµN = |∆ψN (z)|(1 − |z|)dx dy is a Carleson measure with µCarl → 0 as N → ∞; N (z) 2 5. The measure dµ = dµN = ∂ψ∂z (1 − |z|)dx dy is a Carelson measure with µCarl → 0 as N → ∞. Proof. Direct calculations show that ∂ψN = N z N −1 z¯N − (N + 1)z N z¯N +1 , and ∂z ∂ψN = N z N z¯N −1 − (N + 1)z N +1 z¯N . ∂z We then have that ∆ψN = N 2 |z|2(N −1) − (N + 1)2 |z|2N , and ∂ψN 2 2(2N −1) (N − (N + 1)|z|2 )2 . ∂z = |z| N N N 1 Since the maximum of ψN attained at |z| = is N +1 N +1 N +1 , statement 1 is obvious. Let us prove 2. Letting r = |z|, we have |∆ψN (z)|(1 − |z|)2 = N 2 r2N −2 − (N + 1)2 r2N (1 − r)2 = (N + 1)2 r2N −2 (1 − r2 ) − (2N + 1)r2N −2 (1 − r)2 ≤ 2(N + 1)2 r2N −2 (1 − r)3 + (2N + 1)r2N −2 (1 − r)2 . We can see that both terms on the right side of the inequality can be estimated by functions of the same type: 3 (N + 1)2 r2N −2 (1 − r)3 ≤ C nk rkn (1 − r) , k = 2/3, n = N − 1, 2 (2N + 1)r2N −2 (1 − r)2 ≤ C nk rkn (1 − r) , k = 1/2, n = 2(N − 1).

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The function f (r) = nk rkn (1 − r) attains its maximum value kn kn nk kn + 1 kn + 1 at r = kn/(kn + 1). Since k < 1, the maximum value clearly goes to 0 as n → ∞. Estimating the square root of the expression in statement 3, we get ∂ψN (z) 2N −1 |N − (N + 1)r2 |(1 − r) ∂z (1 − |z|) = r ≤ r2N −1 |(N + 1)(1 − r2 )|(1 − r) + r2N −1 (1 − r) ≤ 2r2N −1 (N + 1)(1 − r)2 + r2N −1 (1 − r). Since the maxima for the two terms on the right side of the last inequality are 2N −1 2 2 2N − 1 , 2(N + 1) 2N + 1 2N + 1 and

2N −1 1 1 , 1− 2N 2N N (z) respectively, we see that ∂ψ∂z (1 − |z|) approaches 0 as N → ∞. z For statement 4, let SI = {z ∈ D : |z| ∈ I} be the smallest sector of D with vertex at 0 containing QI ∩ D. Since 1 1 µ(QI ) ≤ µ(SI ) |I| |I| 1 N 2 |z|2(N −1) − (N + 1)2 |z|2N dx dy, = |I| SI

we get the following by integrating in polar coordinates: 1 1 µ(QI ) ≤ |I| |I| |I| 1 =

1

N 2 r2(N −1) − (N + 1)2 r2N rdr

0

|N 2 − (N + 1)2 r2 |r2N −1 (1 − r)dr

0 2

1

≤ 2(N + 1)

r 0

2N −1

1

2

(1 − r) dr + (2N + 1)

r2N −1 (1 − r)dr

0

1 2 1 − + +(2N +1) 2N 2N + 1 2N + 2 1 1 = 2(N + 1)2 O + 2(N + 1)O . N3 N2

= 2(N + 1)2

1 1 − 2N 2N + 1

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Finally, as with statement 4, replacing QI with the sector SI and integrating in polar coordinates, we reduce statement 5 to the following estimate: 1 r4N −1 (N − (N + 1)r2 )2 (1 − r)dr 0

1 ≤8

r

4N −1

0

2

3

1

(N + 1) (1 − r) dr + 2

r4N −1 (1 − r)dr

0

1 3 3 1 1 1 − + − − = 8(N + 1) +2 . 4N 4N + 1 4N + 2 4N + 3 4N 4N + 1 1 1 + 2O . = 8(N + 1)2 O N4 N2 The lemma follows by letting N → ∞. 2

References [1] Cowen, M.J., Douglas, R.G.: Complex geometry and operator theory. Acta Math. 141, 187–261 (1978) [2] Kwon, H., Treil, S.: Similarity of operators and geometry of eigenvector bundles. Publ. Mat. 53(2), 417–438 (2009) [3] Misra, G.: Curvature inequalities and extremal properties of bundle shifts. J. Oper. Theory 11, 305–317 (1984) [4] Nikolski, N.K.: Operators, Functions, and Systems: An Easy Reading, vol. 1: Hardy, Hankel, and Toeplitz, Mathematical Surveys and Monographs, vol. 92. American Mathematical Society, Providence, RI (2002) (translated from the French by Andreas Hartmann) [5] Nikolski, N.K.: Treatise on the Shift Operator, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 273. Springer-Verlag, Berlin (1986) (Spectral function theory, With an appendix by S. V. Hruˇsˇcev [S. V. Khrushch¨ev] and V. V. Peller, Translated from the Russian by Jaak Peetre) Hyun-Kyoung Kwon and Sergei Treil PDE and Functional Analysis Research Center, School of Mathematical Sciences Seoul National University San 56-1, Shinrim-dong, Kwanak-gu, Seoul, 151-747, Republic of Korea e-mail: [email protected] Received: April 1, 2009. Revised: October 12, 2009.

Integr. Equ. Oper. Theory 66 (2010), 539–551 DOI 10.1007/s00020-010-1763-1 Published online March 16, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Reconstruction of the Dirac Operator From Nodal Data Chuan-Fu Yang and Zhen-You Huang Abstract. Inverse nodal problems consist in constructing operators from the given nodes (zeros) of their eigenfunctions. In this work, we deal with the inverse nodal problem of reconstructing the Dirac operator on a finite interval. We prove that a dense subset of nodal points uniquely determine the parameters of the boundary conditions, the mass of a particle and the potential function of the Dirac system. We also provide a constructive procedure for the solution of the inverse nodal problem. Mathematics Subject Classification (2000). Primary 34A55; Secondary 34L05, 34L40, 45C05. Keywords. Dirac operator, inverse nodal problem, potential function, reconstruction formula.

1. Introduction Inverse spectral problems consist in recovering operators from their spectral characteristics. Such problems play an important role in mathematics and have many applications in natural sciences (see, for example, [6,12,14,16,22, 23]). In 1988, the inverse nodal problem was posed and solved for Sturm–Liouville problems by McLaughlin [15], who showed that knowledge of a dense subset of nodal points of the eigenfunctions alone can determine the potential function of the Sturm–Liouville problem up to a constant. Some numerical schemes were provided by Hald and McLaughlin [7] for the reconstruction of the potential. This is the so-called inverse nodal problem. From the physical point of view this corresponds to finding, e.g., the density of a string or a beam from the zero-amplitude positions of their eigenvibrations. Recently, some authors have reconstructed the potential function for generalizations of the Sturm–Liouville problem from the nodal points (for example, refer to [1–5,7–11,15,17–19,21,24]). Inverse spectral problems for weighted Dirac systems are considered in [20], but for the Dirac systems inverse nodal problems have not been studied yet. In this work we concern ourselves with the reconstruction of the one

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dimensional Dirac operator from nodal data. The novelty of this work lies in the use of a dense subset of nodal points for a component ui (x, λn ) of two component vector eigenfunctions u(x, λn ) = (u1 (x, λn ), u2 (x, λn ))T as the given spectral data for the reconstruction of the Dirac system.

2. Main Results We consider the eigenvalue problem corresponding to the Dirac operator u2 + (V (x) + m)u1 = λu1 in (0, π)

−u1 + (V (x) − m)u2= λu2

(2.1)

with the boundary conditions u1 (0) cos α + u2 (0) sin α = 0, u1 (π) cos β + u2 (π) sin β = 0,

(2.2) 0 ≤ α, β < π.

(2.3)

Here λ is a spectral parameter. We suppose that the real-valued potential V (x) ∈ AC[0, π] (absolutely continuous function) and the mass m is positive. The Dirac operator is the relativistic Schr¨ odinger operator in quantum physics. It is well known that the spectrum of the problem (2.1) with the boundary conditions (2.2) and (2.3) consists of the eigenvalues λn , n ∈ Z, which are all real and simple, and the sequence {λn , n ∈ Z} satisfies the classical asymptotic form [13] c1 υ +O λn = n + + π n c1 =

1 n2

π ,

υ =β−α+

V (x)dx, 0

m(sin 2α − sin 2β) + m2 π π . 2π cos2 ( 0 V (x)dx − α + β)

(2.4)

Let (u1 (x, λ), u2 (x, λ))T be the solution of (2.1) with the initial conditions u1 (0, λ) = sin α,

u2 (0, λ) = − cos α,

then by the method of successive approximations [13], there hold ⎛ ⎞ |λ|x x e U u1 (x, λ) = sin ⎝λx − V (x)dx + α⎠ − + O , λ λ2 ⎛

0

u2 (x, λ) = − cos ⎝λx −

x 0

⎞ V (x)dx + α⎠ −

V +O λ

|λ|x

e

λ2

(2.5)

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⎛ ⎞ x 2 m U = −m sin ⎝λx − V (x)dx⎠ cos α + x cos ⎝λx − V (x)dx + α⎠ , 2 0 0 ⎛ ⎛ ⎞ ⎞ x x 2 m V = m sin ⎝λx − V (x)dx⎠ sin α + x sin ⎝λx − V (x)dx + α⎠ . 2 x

⎞

0

0

(2.6) T

Let u(x, λn ) = (u1 (x, λn ), u2 (x, λn )) be the eigenfunction corresponding to the eigenvalue λn of the Dirac operator (2.1). Using an analog of the Sturm oscillation theorem, for sufficiently large |n| we get ui (x, λn ) has exactly |n| + 1 − i nodal points in (0, π), i = 1, 2. Suppose xn,i k are the nodal points of the ith component of the eigenfunction u(x, λn ). In other words, n,i n,i n,i n,i ui (xn,i k , λn ) = 0. Let Ik = (xk , xk+1 ), and the nodal length lk by n,i lkn,i = xn,i k+1 − xk . n,i For convenience we agree that in what follows xn,i 0 = 0 and x|n|+1−i = π. We

also define the function jn (x) to be the largest index j such that 0 ≤ xn,i j ≤x for n > 0, and the function jn (x) to be the largest index j such that 0 ≤ x ≤ n,i xn,i for n < 0. Thus, j = jn (x) if and only if x ∈ [xn,i j j , xj+1 ) for n > 0, and n,i n,i x ∈ [xj+1 , xj ) for n < 0. 2 1 X is called the set of nodal Denote X i {xn,i k }, i = 1, 2. X = X points of the Dirac operator. We consider the following inverse problem.

Problem. Given set X of nodal points or a subset of X which is dense in (0, π), find the parameters α and β of the boundary conditions, the mass m of a particle and the potential V (x). The main theorems are the following. i Theorem 2.1. Fix i = 1, 2 and x ∈ [0, π]. Let {xn,i j } ⊂ X be chosen such that n,i lim|n|→∞ xj = x. Then the following finite limits exist and the corresponding equalities hold:

(−1)i + 1 n,i lim (2.7) nxj − j − π g i (x), 4 |n|→∞ π lim (−1)j+i n2 ljn,i − nπ − π[V (x) − π1 0 V (x)dx] + α − β |n|→∞ (2.8) hi (x)

and g i (x) =

V (x)dx − 0

i

⎡

x

x⎣ β−α+ π 2

h (x) = m sin(2α) + m x.

π 0

⎤ V (x)dx⎦ − α, (2.9)

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Let us now formulate a uniqueness theorem and provide a constructive procedure for the solution of the inverse nodal problem. Theorem 2.2. Fix i = 1, 2. Let X0 ⊂ X i be a subset of nodal points which is dense in (0, ππ). Then, the specification of X0 uniquely determines the potential V (x) − π1 0 V (x)dx in (0, π), the mass m and the coefficients α and β of the π boundary conditions. The potential V (x) − π1 0 V (x)dx, the mass m and the numbers α and β can be constructed via the following algorithm: n,i (1) for each x ∈ [0, π] choose a sequence {xn,i j } ⊂ X0 such that xj → x as |n| → ∞; (2) find the function g i (x) via (2.7) and in turn, calculate

α = −g i (0),

V (x) −

1 π

π

β = −g i (π), V (x)dx =

0

d i β−α g (x) + ; dx π

(2.10)

i

(3) calculate h (x) by formula (2.8) and determine ⎧ i ⎨ h (0) (α = 0), sin(2α) m= i ⎩ h (π) (α = 0). π

(2.11)

i Theorem 2.3. Given i = 1, 2 and x ∈ [0, π]. Let {xn,i j } ⊂ X be chosen such n,i that lim|n|→∞ xj = x. Define

n2 n,i π (−1)j+i m sin 2α (−1)j+i (2j + 2 − i)m2 π υ − lj − + 2 − , π n n n2 2n3 n υ (−1)j+i m sin 2α (−1)j+i (2j + 2 − i)m2 π nljn,i − π + − − Gin (x) = π n n 2n2 |n|+1−i υ (−1)j+i m sin 2α (−1)j+i (2j + 2 − i)m2 π 1 − − nljn,i −π+ − . π j=1 n n 2n2 Fni (x) =

Then Fni (x) converges to V (x) for almost all x ∈ [0, π] and in L1 (0, π)-norm, π and Gin (x) converges to V (x) − π1 0 V (x)dx for almost all x ∈ [0, π]. π Using only the nodal data and the constant, namely π1 0 V (x)dx, we can reconstruct the potential. Our reconstruction formulae are direct and automatically implies the uniqueness of this inverse problem.

3. Proofs First we prove that, being numbered in a natural way, the i-th component ui (x, λn ) of the eigenfunction u(x, λn ) of the Dirac operator (here n ∈ Z) has exactly |n|+ 1− i nodes in the interval (0, π) for sufficiently large |n|, which is an analog of the classical Sturm’s oscillation theorem for the Sturm–Liouville operator.

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Lemma 3.1. For sufficiently large |n|, the ith component ui (x, λn ) of the eigenfunction u(x, λn ) of the Dirac operator has exactly |n| + 1 − i nodes in the interval (0, π): < · · · < xn,1 0 < xn,1 n < π for n > 0, 1 < · · · < xn,2 0 < xn,2 n < π for n > 0, 1

n,1 < xn,1 0 < xn,1 0 −1 < · · · < xn+1 < π for n < 0, n,2 < xn,2 0 < xn,2 0 −1 < · · · < xn+2 < π for n < 0.

Moreover, xn,1 j

=

jπ +

xn,1 j 0

V (x)dx − α υ(jπ + − n

xn,1 j 0

V (x)dx − α)

n2 π

(−1)j (m sin 2α + m2 xn,1 (jπ + j ) + + 2 2n 1 +O , |n| → ∞ n3

xn,1 j 0

V (x)dx − α)(υ 2 − c1 π 2 ) n3 π 2 (3.1)

and xn,2 j

=

(j − 12 )π +

xn,2 j 0

V (x)dx − α

−

υ[(j − 12 )π +

n (−1)j+1 (m sin 2α + m2 xn,2 j ) + 2 2n xn,2 1 [(j − 2 )π + 0 j V (x)dx − α](υ 2 − c1 π 2 ) + n3 π 2 1 +O , |n| → ∞, n3

xn,2 j 0 n2 π

V (x)dx − α]

(3.2)

uniformly with respect to j ∈ Z. Proof. We note that the eigenfunctions (u1 (x, λn ), u2 (x, λn ))T of the Dirac operator (2.1) are real-valued functions. From (2.5) and (2.6) the eigenfunctions have the following asymptotic formulae for |n| → ∞ uniformly in x: ⎛ u1 (x, λn ) = sin ⎝λn x − ⎛

x

⎞ V (x)dx + α⎠ −

0

u2 (x, λn ) = − cos ⎝λn x −

x 0

U +O λn

1 λ2n

⎞ V (x)dx + α⎠ −

V +O λn

1 λ2n

,

(3.3)

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for large |λn |, where ⎛

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⎛ ⎞ x 2 m x cos ⎝λn x − V (x)dx + α⎠, U = −m sin ⎝λn x− V (x)dx⎠ cos α+ 2 0 0 ⎛ ⎞ ⎞ ⎛ x x 2 m x sin ⎝λn x− V (x)dx + α⎠. V = m sin ⎝λn x− V (x)dx⎠ sin α+ 2 x

⎞

0

0

(3.4) Taking (2.4), (3.3) and (3.4) into account and using Taylor’s expansions for the arccosx and arcsinx, we obtain the following asymptotic formulae for nodal points as |n| → ∞ uniformly in j ∈ Z: xn,1 n,1 jπ + 0 j V (x)dx − α (−1)j (m sin 2α + m2 xj ) 1 n,1 xj = + +O 2 λn 2λn λ3n and xn,2 j =

(j − 12 )π+

xn,2 j 0

λn

V (x)dx−α

(−1)j+1 (m sin 2α + m2 xn,2 j ) + +O 2λ2n

Moreover, using the asymptotic formulae 1 c1 1 υ2 υ −1 − 2 + 2 2 +O , λn = 1− n nπ n n π n3 1 2c1 1 3υ 2 2υ − = + + O 1 − , λ−2 n n2 nπ n2 n2 π 2 n3

1 λ3n

.

(3.5)

we obtain that the equalities (3.1) and (3.2) hold. The equalities (3.1) and (3.2) give 1 π n,1 n,1 xj+1 − xj = + O , |n| → ∞, n n2 < xn,1 uniformly with respect to j. Consequently, for large |n| we have xn,1 j j+1 n,1 n,1 for positive n and xj > xj+1 for negative n. For j = 0, ±1, n, n + 1 formula (3.1) gives 1 1 −π − α α n,1 xn,1 + O + O = = − , x , −1 0 n n2 n n2 1 1 π−α α n,1 + O + O = = π − , . . . , x xn,1 , n 1 n n2 n n 1 π−α + O = π + . xn,1 n+1 n n Thus, according to the order of xn,1 j , for large |n|, the 1th component u1 (x, λn ) of the eigenfunction u(x, λn ) of the Dirac operator has exactly |n| nodes in n,1 the interval (0, π), i.e., xn,1 j , j = 1, n, for positive n and xj , j = n + 1, 0, for negative n. Similarly, we get the 2nd component u2 (x, λn ) of the eigenfunc tion u(x, λn ) has exactly |n| − 1 nodes in the interval (0, π).

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Lemma 3.2. The asymptotic formulae for nodal lengths ljn,i are the following: xn,1 xn,1 j+1 j+1 υ π + V (x)dx n,1 π + xn,1 V (x)dx xj j − ljn,1 = n n2 π j+1 j+1 1 (−1) m sin 2α (−1) (2j + 1)m2 π + + +O , |n| → ∞ 2 3 n 2n n3 (3.6) and ljn,2 =

π+

xn,2 j+1 xn,2 j

υ[π +

V (x)dx

xn,2 j+1 xn,2 j

−

V (x)dx]

n n2 π 1 (−1)j m sin 2α (−1)j jm2 π + + +O , 2 3 n n n3

|n| → ∞. (3.7)

Proof. By the asymptotic formulae (3.1) and (3.2) for nodal points we can obtain the asymptotic expansions of nodal lengths as follows. xn,1 xn,1 j+1 j+1 π + xn,1 V (x)dx υ[π + xn,1 V (x)dx] j j n,1 − lj = n n2 π j+1 2 n,1 (−1) [2m sin 2α + m (xj + xn,1 1 j+1 )] +O + , |n| → ∞ 2 2n n3 and ljn,2 =

π+

xn,2 j+1 xn,2 j

υ[π +

V (x)dx

xn,2 j+1

−

xn,2 j

V (x)dx]

n n2 π 2 n,2 (−1) [2m sin 2α + m (xj + xn,2 1 j+1 )] + O + , 2n2 n3 j

Note that xn,1 j

+

xn,1 j+1

(2j + 1)π +O = n

then it yields (3.6) and (3.7).

1 , n

xn,2 j

+

xn,2 j+1

|n| → ∞.

2jπ +O = n

1 , n

In the above results, the order estimate is independent of j. As a result, 1 π n,i lj = + o , i = 1, 2. (3.8) n n Corollary 3.3. From Lemmas 3.1 and 3.2 it follows that the sets X i = {xn,i k }, i = 1, 2, are dense in [0, π]. According to the classical theorem that, for f ∈ L1 (0, π), the mapping A → A |f (t)|dt defines a measure that is absolutely continuous with respect to the Lebesgue one, we have Lemma 3.4.

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Lemma 3.4. Suppose that V ∈ L1 (0, π). Then for x ∈ (0, π), with j = jn (x), xn,i j+1

lim

|n|→∞

V (t)dt = 0,

i = 1, 2.

(3.9)

xn,i j

Lemma 3.5. Suppose that V ∈ L1 (0, π). Then, with j = jn (x), xn,i j+1

n lim |n|→∞ π and

V (t)dt = V (x)

for almost all x ∈ [0, π]

(3.10)

as |n| → ∞, i = 1, 2.

(3.11)

xn,i j

xn,i j+1 n V (t)dt − V (x) π →0 n,i xj 1

Proof. Note that xn,i j+1

n π

V (t)dt = xn,i j

n − λn π

xn,i j+1

xn,i j+1

V (t)dt + xn,i j

xn,i j+1

= O(1) xn,i j

λn π

V (t)dt xn,i j

xn,i j+1

λn V (t)dt + π

V (t)dt. xn,i j

Applying the result in [3,5]: Suppose that f ∈ L1 (0, π), with j = jn (x), there hold xn,i j+1

λn lim |n|→∞ π and

for almost all x ∈ [0, π]

V (t)dt = V (x) xn,i j

xn,i j+1 λn V (t)dt − V (x) π →0 n,i xj

as |n| → ∞,

1

we conclude that (3.10) and (3.11) hold. The proof is complete. Now we can give the proofs of the theorems in this work.

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Proof of Theorem 2.1. Using the asymptotic expansions (3.1) and (3.2) for nodal points, for i = 1, 2 we get n,i

xj 1 (−1)i + 1 (−1)i + 1 υ n,i +O V (t)dt−α− j − nxj − j − π= 4 4 n n

0

(3.12) and

⎫ ⎧ ⎡ ⎤ π ⎬ ⎨ 1 V (x)dx⎦ + α − β (−1)j+i n2 ljn,i − nπ − π ⎣V (x) − ⎭ ⎩ π 0

xn,i j+1

= m sin(2α) + m

2 (2j

−(−1)j+i πV (x) −

+ 1)π + (−1)j+i n 2n j+i

(−1) π

V (x)dx xn,i j

xn,i j+1

υ

V (x)dx + O xn,i j

1 . n

(3.13) i

(−1) +1 π ]n = x Also, the fact that lim|n|→∞ xn,i j = x implies that lim|n|→∞ [j − 4

and lim|n|→∞

π(2j+1) 2n

= x , and n,i

xj

V (x)dx =

lim

|n|→∞

x

0

V (x)dx. 0

From this it follows that as |n| → ∞ the limits of left-hand sides in (3.12) and (3.13) exist and

(−1)i + 1 − j − nxn,i g i (x) = lim π j 4 |n|→∞ x υx V (x)dx − α − = π 0 (3.14) ⎡ ⎤ x π x⎣ = β − α + V (x)dx⎦ − α, V (x)dx − π 0

0

hi (x) = m sin(2α) + m2 x. This proves the theorem.

Proof of Theorem 2.2. Now given a nodal subset X0 , by Theorem 2.1 we can build up the reconstruction formulae. Formulae (2.10) and (2.11) can be derived from (3.14) stepwise. We obtain the following procedure. Step 1: Take x = 0, it follows g i (0) = −α. Hence, α = −g i (0).

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Step 2: Take x = π, then it yields g i (π) = −β, which implies β = −g i (π). Step 3: After α and β are reconstructed, from ⎡ ⎤ x π x g i (x) = V (x)dx − ⎣β − α + V (x)dx⎦ − α, π 0

0

by taking derivatives we obtain ⎛ ⎞ π 1⎝ d i V (x) − β − α + V (x)dx⎠ = g (x), π dx 0

i.e., 1 V (x) − π

π V (x)dx = 0

Step 4: After α, β and V (x) −

1 π

π 0

d i β−α g (x) + . dx π

V (x)dx are reconstructed, by

hi (x) = m sin(2α) + m2 x, we obtain

⎧ ⎨ m=

⎩

hi (0) sin(2α)

hi (π) π

(α = 0),

(3.15)

(α = 0).

Thus these formulae are constructed. Since each nodal data only determine a set of reconstruction formulae which only depend on nodal data, the uniqueness holds obviously. Proof of Theorem 2.3. Using the asymptotic expansions (3.6) and (3.7) for nodal lengths, we get xn,i j+1

Fni (x)

n = π

xn,i j

υ V (t)dt − 2 π

xn,i j+1

xn,i j

1 V (t)dt + O . n

From this we get

xn,i xn,i j+1 j+1 n 1 υ i V (t)dt − 2 V (t)dt + O − V (x) |Fn (x) − V (x)| ≤ π n π n,i xj xn,i j xn,i xn,i j+1 j+1 n υ 1 ≤ V (t)dt − V (x) + 2 V (t)dt + O . π π n n,i xj xn,i j

Combining Lemmas 3.4 and 3.5, then it yields lim |Fni (x) − V (x)| = 0

|n|→∞

for almost all x ∈ [0, π].

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Moreover, π 0

n,i π xj+1 1 n i |Fn (x) − V (x)|dx ≤ V (t)dt − V (x) dx + O . π n 0 xn,i j

Lemma 3.5 tells us π

|Fni (x) − V (x)|dx = 0.

lim

|n|→∞

0

Thus, Fn (x) converges to V (x) for almost all x ∈ [0, π] and in L1 (0, π)-norm. Using the asymptotic expansions (3.6) and (3.7) for nodal lengths, we get xn,i j+1

n Gin (x) = π

xn,i j

xn,i j+1

υ V (t)dt − 2 π

xn,i j+1

=

n π

V (t)dt − xn,i j

1 π

xn,i j

1 V (t)dt − π

j=1

xn,i j+1

xn,i j

1 V (t)dt + O n

xn,i j+1

π V (t)dt − 0

|n|+1−i

υ π2

V (t)dt + O xn,i j

1 . n

Combining Lemmas 3.4 and 3.5, then it yields Gin (x) converges to V (x) − 1 π π 0 V (x)dx for almost all x ∈ [0, π]. The proof is complete. Acknowledgements The authors acknowledge helpful comments from the referees. This work was supported by a Grant-in-Aid for Star of Purple and Gold from Nanjing University of Science and Technology, and a Grant-in-Aid for Scientific Research from Nanjing University of Science and Technology (XKF 09041).

References [1] Browne, P.J., Sleeman, B.D.: Inverse nodal problem for Sturm–Liouville equation with eigenparameter dependent boundary conditions. Inverse Probl. 12, 377–381 (1996) [2] Buterin, S.A., Shieh, C.T.: Inverse nodal problem for differential pencils. Appl. Math. Lett. 22, 1240–1247 (2009) [3] Chen, Y.T., Cheng, Y.H., Law, C.K., Tsay, J.: L1 convergence of the reconstruction formula for the potential function. Proc. Am. Math. Soc. 130, 2319– 2324 (2002) [4] Cheng, Y.H., Law, C.K., Tsay, J.: Remarks on a new inverse nodal problem. J. Math. Anal. Appl. 248, 145–155 (2000) [5] Currie, S., Watson, B.A.: Inverse nodal problems for Sturm–Liouville equations on graphs. Inverse Probl. 23, 2029–2040 (2007)

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[6] Freiling, G., Yurko, V.A.: Inverse Sturm–Liouville Problems and Their Applications. NOVA Science Publishers, New York (2001) [7] Hald, O.H., McLaughlin, J.R.: Solutions of inverse nodal problems. Inverse Probl. 5, 307–347 (1989) [8] Koyunbakan, H.: A new inverse problem for the diffusion operator. Appl. Math. Lett. 19, 995–999 (2006) [9] Law, C.K., Shen, C.L., Yang, C.F.: The inverse nodal problem on the smoothness of the potential function. Inverse Probl. 15, 253–263 (1999); Errata, 17 (2001), 361–364 [10] Law, C.K., Tsay, J.: On the well-posedness of the inverse nodal problem. Inverse Probl. 17, 1493–1512 (2001) [11] Law, C.K., Yang, C.F.: Reconstructing the potential function and its derivatives using nodal data. Inverse Probl. 14(2), 299–312 (1998) [12] Levitan, B.M.: Inverse Sturm–Liouville Problems. VNU Science Press, Utrecht (1987) [13] Levitan, B.M., Sargsjan, I.S.: Sturm–Liouville and Dirac Operators (Russian), Nauka, Moscow 1988: English transl. Kluwer, Dordrecht (1991) [14] Marchenko, V.A.: Sturm–Liouville Operators and Their Applications, Naukova Dumka, Kiev, 1977; English transl. Birkh¨ auser, Basel (1986) [15] McLaughlin, J.R.: Inverse spectral theory using nodal points as data—a uniqueness result. J. Differ. Equ. 73, 354–362 (1988) [16] P¨ oschel, J., Trubowitz, E.: Inverse Spectral Theory. Academic Press, Orlando (1987) [17] Shen, C.L.: On the nodal sets of the eigenfunctions of the string equations. SIAM J. Math. Anal. 19, 1419–1424 (1988) [18] Shen, C.L., Shieh, C.T.: An inverse nodal problem for vectorial Sturm–Liouville equation. Inverse Probl. 16, 349–356 (2000) [19] Shieh, C.T., Yurko, V.A.: Inverse nodal and inverse spectral problems for discontinuous boundary value problems. J. Math. Anal. Appl. 347, 266–272 (2008) [20] Watson, B.A.: Inverse spectral problems for weighted Dirac systems. Inverse Probl. 15, 793–805 (1999) [21] Yang, X.F.: A solution of the inverse nodal problem. Inverse Probl. 13, 203– 213 (1997) [22] Yurko, V.A.: Inverse Spectral Problems for Differential Operators and Their Applications. Gordon and Breach, Amsterdam (2000) [23] Yurko, V.A.: Integral transforms connected with discontinuous boundary value problems. Integral Transforms Spec. Funct. 10(2), 141–164 (2000) [24] Yurko, V.A.: Inverse nodal problems for Sturm–Liouville operators on star-type graphs. J. Inv. Ill-Posed Probl. 16, 715–722 (2008) Chuan-Fu Yang Department of Applied Mathematics Nanjing University of Science and Technology Nanjing 210094, Jiangsu People’s Republic of China e-mail: [email protected]

Vol. 66 (2010)

Reconstruction of the Dirac Operator

Zhen-You Huang Department of Applied Mathematics Nanjing University of Science and Technology Nanjing 210094, Jiangsu People’s Republic of China e-mail: [email protected] Received: May 6, 2009. Revised: September 11, 2009.

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Integr. Equ. Oper. Theory 66 (2010), 553–564 DOI 10.1007/s00020-010-1764-0 Published online March 17, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Eigenvalues of Weighted Composition Operators on the Bloch Space Takuya Hosokawa and Quang Dieu Nguyen Abstract. This note discusses eigenvalues of weighted composition operators uCϕ on the Bloch space. The main result provides a class of uCϕ for which computation of eigenvalues is possible. We also construct an example of a non-compact operator Cϕ whose eigenvalues can be determined precisely. Mathematics Subject Classification (2000). 47B33, 47B38. Keywords. Eigenvalues, weighted composition operator, Bloch space.

1. Introduction Let D be the unit disk in C and ϕ : D → D be a holomorphic self map. Then for a given holomorphic function u on D we may define the weighted composition operator uCϕ on the space of holomorphic functions on D as (uCϕ )(f )(z) = u(z)f (ϕ(z)). This operator may be regarded as a generalization of a multiplication operator and a composition operator. The goal of this work is to compute the set E(uCϕ ) of eigenvalues of uCϕ on the Bloch space B. Recall that a holomorphic function f on D is said to be in B if sup (1 − |z|2 )|f (z)| < ∞.

z∈D

Under the norm f B := |f (0)| + sup (1 − |z|2 )|f (z)|, z∈D

B becomes a Banach spaces. For f ∈ B we have the following growth estimate |f (z)| ≤

2 f B log . log 2 1 − |z|2

The authors are partially supported by the Korean Research Foundation Grant KRF-2005070-C00007. The first author is also partially supported by KRF-2008-314-C00012.

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We refer the reader to the monograph [6] for more details on the Bloch space. It follows easily from the closed graph theorem that the operator uCϕ is bounded on B if and only if it maps B into B. For technical reasons, we only treat the case where ϕ has a fixed point inside D and ϕ is not an automorphism. Assume from now on that ϕ(0) = 0 and |ϕ (0)| < 1, we concern with the existence (in B) of holomorphic solutions f of the “weighted” Schr¨ oder equation u(z)f (ϕ(z)) = λf (z) ∀z ∈ D,

(1.1)

where λ = 0 is a complex constant. It is easy to see that all possible values λ must be of the form u(0)ϕ (0)n where n is a non negative integer. The point is to determine which candidates are the real eigenvalues of uCϕ . Note that in the case where uCϕ is compact, according to a general result of Hammond in [1] that pertains to any Banach space of holomorphic functions on D containing the polynomials, all values u(0)ϕ (0)n belong to E(uCϕ ). The outline of our note is as follows. After giving some preliminaries in Sect. 2, we first prove in Proposition 3.3 the existence of holomorphic solution of Eq. (1.1). Next, we prove in Theorem 3.5 that under somewhat strong restrictions on the growths of u and ϕ near ∂D, all the values u(0)ϕ (0)n , n ≥ 0 are eigenvalues of uCϕ . The key element in our proof is the classical Hadamard three circles theorem. As an easy consequence of this result, we show in Corollary 3.6 that conditions given in Theorem 3.5 are verified if ϕ (D) is relatively compact on D and if u satisfies the growth condition supz∈D (1 − |z|2 )α |u (z)| < ∞ for some constant 0 ≤ α < 1. The final result of the article is an example for which we can describe E(Cϕ ) precisely, where ϕ is univalent on D and Cϕ is not compact.

2. Preliminaries We first fix the notation that will be used later on. For r > 0, by Dr we mean the disk {z : |z| < r}. If g is a holomorphic function on D, then we denote Mr (g) := sup |g(z)| |z|=r

and g∞ := sup Mr (g). r<1

Given a self map ϕ of D, we write ϕn for the nth iterate of ϕ (the composition of ϕ with itself n times). For the ease of exposition, we always mean by u a holomorphic function on D and ϕ a holomorphic self-map of D. We recall the following criteria from [3] for boundedness of uCϕ . Theorem 2.1 [3]. uCϕ is bounded on B if and only if the following are satisfied: 2 (a) supz∈D ((1 − |z|2 )|u (z)| log( 1−|ϕ(z)| 2 )) < ∞. (b)

2

1−|z| supz∈D ( 1−|ϕ(z)| 2 |u(z)ϕ (z)|) < ∞.

Remark. (i)

The condition (a) of Theorem 2.1 implies that u ∈ B and lim (1 − |z|2 )u (z) = 0.

|ϕ(z)|→1

(ii)

In the unweighted case i.e., u ≡ 1, Theorem 2.1 follows easily from Schwarz–Pick’s lemma (see [2]).

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Note. From now on, we make the standing assumption that ϕ(0) = 0 and |ϕ (0)| < 1. The famous Koenigs’ theorem about solutions to Schr¨ oder’s equation (cf. [4], [5]) is fundamental for our work. Theorem 2.2 (Koenigs). Assume that ϕ(0) = 0, and 0 < |ϕ (0)| < 1. Then the following assertions hold. (a) The sequence of holomorphic functions ϕk (z) ϕ (0)k converges uniformly on compact sets of D to a non-constant holomorphic function σ. Moreover, the function σ satisfies the equation σk (z) :=

(f ◦ ϕ)(z) = ϕ (0)f (z), (b)

∀z ∈ D.

Given 0 < r ≤ 1. Assume that f is a holomorphic function on Dr and λ is a non zero complex number satisfying (f ◦ ϕ)(z) = λf (z), ∀z ∈ Dr . Then there exists a positive integer n such that λ = ϕ (0)n and f is a constant multiple of σ n . In particular, f can be extended to a holomorphic function on D.

Remark. The proof in [5, p. 32–33] gives the assertion (b) of Theorem 2.2 The following classical Hadamard three circles theorem is quite useful for estimating growth of holomorphic functions. Theorem 2.3 (Hadamard). Let g be a holomorphic function on D. Then log Mr (g) is a convex function of log r i.e., if 0 < r0 < r1 < r2 < 1 and if log r1 = α log r0 + (1 − α) log r2 , then log Mr1 (g) ≤ α log Mr0 (g) + (1 − α) log Mr2 (g).

3. Results We record the following simple observation. Lemma 3.1. Let λ = 0 be a complex number. Assume that u(0) = 0 and that u, λ satisfy Eq. (1.1). Then f ≡ 0. Proof. Since λ = 0, by evaluating both sides of (1.1) at z = 0, we obtain f (0) = 0. Assume for the sake of seeking a contradiction that f ≡ 0. Then we may write f (z) = z k f ∗ (z),

u(z) = z l u∗ (z),

ϕ(z) = z m ϕ∗ (z)

where k, l, m are positive integers and f ∗ (0) = 0, u∗ (0) = 0, ϕ∗ (0) = 0. Inserting into (1.1) one gets z l+km u∗ (z)ϕ∗ (z)k ϕ∗ (z)k f ∗ (ϕ(z)) = λz k f ∗ (z),

∀z ∈ D.

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Since the vanishing order of the right hand side at z = 0 is at least l+ km > k, we arrive at a contradiction. The proof is complete. Convention. In the rest of the paper, we only consider the case where u(0) = 0. Lemma 3.2. There exists a unique holomorphic function v on D satisfying the following conditions. (a) v(0) = 1. (b) u(z)v(ϕ(z)) = u(0)v(z), ∀z ∈ D. (c) The following estimate holds for every 0 < r < 1 log Mr (v) ≤

Mr (u ) . |u(0)|(r − Mr (ϕ))

Proof. First, suppose that there exist holomorphic functions v1 , v2 on D satisfying the conditions (a) and (b). Set v˜ := v1 − v2 . Then we have u(z)˜ v (ϕ(z)) = u(0)˜ v (z),

∀z ∈ D.

Since v˜(0) = 0, we may apply Lemma 3.1 to f := v˜, λ := u(0) to get that v˜ ≡ 0. The uniqueness of v follows. For the existence of v, consider the sequence of holomorphic functions vk (z) =

u(z)u(ϕ(z)) · · · u(ϕk−1 (z)) . u(0)k

From the elementary inequality log |a| ≤ |1 − a|, we infer 1 log |vk (z)| ≤ |u(ϕn (z)) − u(0)| . |u(0)| n≥0

Fix 0 < r < 1. By Schwarz’s lemma, we have ϕ(Dr ) ⊂ Dr , and, moreover |ϕ(z)| ≤

Mr (ϕ) |z|, r

∀z ∈ Dr .

By iterating this inequality we obtain for each n ≥ 1 n Mr (ϕ) |ϕn (z)| ≤ r , ∀z ∈ Dr . r

(3.1)

It follows from (3.1) that for every z ∈ Dr

Mr (ϕ) n Mr (u ) u(ϕn (z)) − u(0) ≤ rMr (u ) . = r r − Mr (ϕ)

n≥0

n≥0

Thus the series n≥0 |u(ϕn (z)) − u(0)| converges uniformly on compact sets of Dr . Therefore the sequence {vk } tends to a holomorphic function v on D uniformly on compact sets of D. By the above estimate we obtain (c). It is also immediate to check that v satisfies (b). Finally, since vk (0) = 1 for all k we infer that v(0) = 1. The proof is complete.

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Remark. If u is bounded on D and if ϕn ∞ < 1 for some n ≥ 1 then v∞ < ∞. Indeed, we set v˜k := v ◦ ϕk for k ≥ 0. Notice that u(ϕn−1 (z))˜ vn (z) = u(0)˜ vn−1 (z),

∀z ∈ D.

This implies v˜n−1 is bounded on D. Continuing this process, we finally have that v is bounded on D. The above lemma enables us to obtain the following fact about solutions of Eq. (1.1). Proposition 3.3. Let f be a holomorphic function on D and λ = 0 be a complex number. Let v be the function in Lemma 3.2. Then the following assertions are equivalent. (a) f and λ satisfy Eq. (1.1). (b) f = v f˜, λ = u(0)ϕ (0)n , where n ≥ 0 is an integer and f˜ is the solution of the Schr¨ oder equation f˜ ◦ ϕ ≡ ϕ (0)n f˜. Proof. (b) ⇒ (a) follows from the following direct computation. λ ˜ f (z) = λf (z). u(z)f (ϕ(z)) = u(z)v(ϕ(z))f˜(ϕ(z)) = u(0)v(z) u(0) (a) ⇒ (b). Let S := {z ∈ D : v(z) = 0}. Then on D\S we may write f = v f˜, where f˜ is a holomorphic function. Inserting into (1.1) and using the properties of v, we obtain u(0)f˜(ϕ(z)) = λf˜(z)

∀z ∈ D\(S ∪ ϕ−1 (S)).

(3.2)

Since v(0) = 1 and ϕ(0) = 0, we infer that 0 ∈ D\(S ∪ ϕ−1 (S)). Thus there exists r0 > 0 such that Dr0 ⊂ D\(S ∪ ϕ−1 (S)). On the other hand, by Schwarz’s lemma, ϕ is a holomorphic self map of Dr0 . Therefore, the equation (3.2) holds in Dr0 . Now we apply Theorem 2.2 (b) to see that f˜|Dr0 can be extended to a holomorphic solution on D (still denoted by f˜) of (1.1). So the uniqueness property for holomorphic functions implies that f and v f˜ must agree on D. Finally the assertion on λ also follows immediately from Theorem 2.1 (b). Corollary 3.4. Assume that u∞ < ∞ and ϕk ∞ < 1 for some integer k ≥ 1. Then for every integer n ≥ 1, the holomorphic solution of the equation u(z)f (ϕ(z)) = u(0)ϕ (0)n f (z) is bounded on D. Proof. By Proposition 3.3, f = v f˜, where f˜ is the holomorphic solution of the Schr¨ oder equation f˜ ◦ ϕ ≡ ϕ (0)n f˜. Since ϕk ∞ < 1, we have f˜ is bounded on D (see [5, p. 23]). On the other hand, by the remark following Lemma 3.2, the function v is also bounded. This completes the proof. We now deal with the basic question on determining when a solution f of the weighted Schr¨ oder equation (1.1) belongs to B.

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Theorem 3.5. Assume that uCϕ is bounded on B. For 0 < r < 1, we set ar := sup {|u (z)ϕ(z)| + |u(z)ϕ (z)|}. |z|=r

Assume the following conditions hold. (a) limr→1 log(1 − r) log Mr (ϕ) = ∞. (b) log |ar | < ε log(1 − r) log Mr (ϕ), where ε > 0 is a constant satisfying ε log ϕ∞ > −1. Then E(uCϕ ) = u(0), u(0)ϕ (0), . . . , u(0)ϕ (0)k , . . . , . Several remarks regarding our assumptions (a) and (b) are in order. Remark. (i) The condition (a) is automatically satisfied if ||ϕ||∞ < 1. It excludes, however, cases when the growth of Mr (ϕ) is like rk , k ≥ 1. (ii) The condition (b) implies log |ar | < −ε log(1 − |r|) for some constant 0 < ε < 1. (iii) By Theorem 2.1, there exists a constant C > 0 such that |u (z)ϕ(z)| < C − log(1 − |z|),

|u(z)ϕ (z)| < C − log(1 − |z|),

∀z ∈ D.

Consequently, or some constant C > 0 log |ar | < C − log(1 − r),

∀0 < r < 1.

This estimate is, however, much weaker than our assumption (b). Proof. Given λ ∈ {u(0), u(0)ϕ (0), . . . , u(0)ϕ (0)k , . . . , }. Then by Proposition 3.3, there exists a holomorphic function f on D satisfying (1.1). It remains to prove that f ∈ B. By differentiating both sides of Eq. (1.1) we get u (z)f (ϕ(z)) + ϕ (z)u(z)f (ϕ(z)) = λf (z),

∀z ∈ D.

(3.3)

Since ϕ(0) = 0, for |z| = r we have |f (ϕ(z))| ≤ |ϕ(z)|MMr (ϕ) (f ) + |f (0)|.

(3.4)

Note that, by the maximum principle |f (ϕ(z))| ≤ MMr (ϕ) (f ),

∀|z| = r.

(3.5)

By putting (3.3), (3.4) and (3.5) together, we obtain |λ|Mr (f ) ≤ ar MMr (ϕ) (f ) + |f (0||u (z)|.

(3.6)

Since uCϕ is bounded on B, by the remark following Theorem 2.1 we have u ∈ B. Thus we can find a constant A > 0 such that A , ∀|z| = r. (3.7) |u (z)| < 1−r We infer from (3.6) and (3.7) that there exists a constant A > 0 such that (1 − r)|λ|Mr (f ) ≤ (1 − r)ar MMr (ϕ) (f ) + A ,

∀0 < r < 1.

(3.8)

We now assume for the sake of seeking a contradiction that u ∈ B. Fix 0 < r0 < ||ϕ||∞ . Then, there exists a sequence {rk }k≥1 ↑ 1 such that r0 < Mrk (ϕ) < rk for all k ≥ 1 and that lim (1 − rk )Mrk (f ) = ∞.

k→1

(3.9)

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It follows from (3.8) and (3.9) that lim (1 − rk )ark MMrk (ϕ) (f ) = ∞.

k→∞

Hence we can choose a constant λ such that 0 < |λ | < |λ| and that |λ |Mrk (f ) < ark MMrk (ϕ) (f ) ∀k ≥ 1. In other words, log |λ | + log Mrk (f ) ≤ log ark + log MMrk (ϕ) (f ).

(3.10)

By applying Hadamard’s three circles theorem (Theorem 2.3) to r0 < Mrk (ϕ) < rk , we arrive at log MMrk (ϕ) (f ) ≤

log rk − log Mrk (ϕ) log Mr0 (f ) log rk − log r0 log Mrk (ϕ) − log r0 + log Mrk (f ). log rk − log r0

(3.11)

Combining (3.10), (3.11) and rearranging we obtain the following crucial estimate log rk − log r0 (log ark − log |λ |) , ∀k ≥ 1. log Mrk (f ) ≤ log Mr0 (f ) + log rk − log Mrk (ϕ) Now, we claim that for some appropriately chosen r0 ∈ (0, ||ϕ||∞ ), the following assertions holds log r − log r0 (log ar − log |λ |) < ∞. sup log(1 − r) + log r − log Mr (ϕ) r<1 For this, we note that by L’Hospital’s rule r(log r)2 = 0. (3.12) r→1 r→1 r − 1 Pick ε > ε such that ε log ||ϕ||∞ > −1. It follows from (3.12) and the assumptions (a), (b) that for r close to 1, the following inequality holds lim log(1 − r) log r = lim

log |ar | − log |λ | < −ε log(1 − r) (log r − log Mr (ϕ)). By the choice of ε, we can find r0 such that 0 < r0 < ||ϕ||∞ and that ε log r0 > −1. Hence for r ∈ (0, 1) close enough to 1 we have log r − log r0 (log ar − log |λ |) log r − log Mr (ϕ) < log(1 − r) (1 − ε (log r − log r0 )) < 0.

log(1 − r) +

The desired claim follows immediately. We get a contradiction. The proof is complete. Remark. We actually show in the above proof that if f is a holomorphic solution of (1.1) then lim|z|→1 (1 − |z|2 )f (z) = 0 i.e., f belongs to the little Bloch space B0 .

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As an application of the above result, we have Corollary 3.6. Assume that ϕ ∞ < 1 and that sup (1 − |z|2 )α |u (z)| < ∞,

(3.13)

z∈D

for some constant 0 ≤ α < 1. Then uCϕ is bounded on B. Moreover, E(uCϕ ) = {u(0), u(0)ϕ (0), . . . , u(0)ϕ (0)n , . . .}. Proof. Since ϕ(0) = 0 and ϕ ∞ < 1, we have ϕ∞ < 1. For boundedness of uCϕ , it suffices to apply Theorem 2.1. More precisely, we have 2 2 2 (1 − |z| )|u (z)| log ≤ log uB < ∞. 1 − |ϕ(z)|2 1 − ϕ2∞ Next, since u ∈ B we have |u(z)| ≤

2 uB log , log 2 1 − |z|2

∀z ∈ D.

It follows that there exists a constant A > 0 such that 1 − |z|2 uB ϕ ∞ (1 − |z|2 ) log |u(z)ϕ (z)| ≤ 1 − |ϕ(z)|2 log 2(1 − ϕ2∞ ) < A, ∀z ∈ D.

2 1 − |z|2

Putting together, we see that the two conditions in Theorem 2.1 are satisfied. Consequently uCϕ is bounded on B. We will apply Theorem 3.5 to reach the conclusion on E(uCϕ ). Since ϕ∞ < 1, the condition (a) of Theorem 3.5 is satisfied. Now we observe that (3.13) and the assumption ϕ ∞ < 1 imply the following fact: There exists a constant B > 0 such that for |z| close enough to 1 we have max{|u (z)ϕ(z)|, |u(z)ϕ (z)|} 0 is a constant. Thus, we may find ε > 0 satisfying ε log ϕ∞ > −1 such that for r close enough to 1 log |ar | < ε log(1 − r) log Mr (ϕ).

The proof is complete. We move to the case ϕ is univalent and u ≡ 1. Proposition 3.7. Assume that ϕ is univalent on D. Set 1 − |z|2 |ϕ (z)| ≤ 1. 2 |ϕ(z)|→1 1 − |ϕ(z)|

:= lim sup

Then for every k ≥ 1 satisfying 4 < |ϕ (0)|k+1 we have ϕ (0)k ∈ E(Cϕ ).

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Remark. (i) If = 0 then by Proposition 3.7, E(Cϕ ) = {ϕ (0)k : k ≥ 0}. However, in this case, Cϕ is compact by Theorem 2 in [2]. As mentioned at the beginning of the article, the description of E(Cϕ ) was then obtained earlier in [1]. (ii) We will construct at the end of the article, an example of ϕ such that > 0 whereas E(Cϕ ) = {ϕ (0)k : k ≥ 0}. Proof. We will use ideas from Section 5 in [5]. Since ϕ is univalent, by Hurwitz’s theorem and Theorem 2.1 (a), there exists a univalent holomorphic function σ on D such that σ(0) = 0 and that σ ◦ ϕ ≡ λσ,

(3.14)

where λ = ϕ (0). For k ≥ 1, we set fk := σ k . Then fk are holomorphic functions on D and by direct computation (or Theorem 2.1 (b)) fk satisfies the Schr¨ oder equations fk ◦ ϕ ≡ λk fk . It suffices to show that fk ∈ B for all k ≥ 1. We may assume σ is unbounded on D, otherwise the conclusion is trivial. Denote G := σ(D). Then G is an unbounded, simply connected domain in C with 0 ∈ G, λG ⊂ G, G = C. The rest of the proof is split into three steps. Step 1. We show that 4 dG (w) ≤ , d (λw) |λ| G |w|→∞,w∈G lim sup

where dG (z) denotes the distance from z ∈ G to ∂G. Since σ is conformal, we may apply Koebe’s distortion theorem to obtain 1 (1 − |z|2 )|σ (z)| ≤ dG (σ(z)) ≤ (1 − |z|2 )|σ (z)|, ∀z ∈ D. (3.15) 4 Furthermore, by differentiating (3.14) we have ϕ (z)σ (ϕ(z)) = λσ (z),

∀z ∈ D.

Now we may invoke the following computation which is essentially Lemma 5.1 in [5]: 1 − |z|2 (1 − |z|2 )|σ (z)| |ϕ (z)| = |λ| 1 − |ϕ(z)|2 (1 − |ϕ(z)|2 )|σ (ϕ(z))| |λ| dG (σ(z)) |λ| dG (σ(z)) = , ≥ 4 dG (σ(ϕ(z)) 4 dG (λσ(z))

∀z ∈ D.

Here we apply Koebe’s distortion theorem (cf. (3.15)) for the next to last inequality, whereas the last equality follows from (3.14). Putting together, we arrive at dG (σ(z)) 4 lim sup ≤ . (3.16) d (λσ(z)) |λ| G |ϕ(z)|→1 Finally, notice that by (3.14), if {zk } ⊂ D is a sequence satisfying |σ(zk )| → ∞ then |ϕ(zk )| → 1. Thus the desired conclusion follows from (3.16) and conformality of σ.

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Step 2. Given an integer k ≥ 1 such that 4 < |λ|k+1 . We claim sup |w|k dG (w) < ∞.

(3.17)

w∈G

It follows from the result obtained in the previous step that there is a constant A > 0 such that dG (w) < λk dG (λw),

∀|w| > A.

Iterating this inequality we obtain dG (w) < λkj dG (λj w),

∀|w| >

A . |λ|j−1

Given w ∈ G, |w| > A, we can choose j0 ≥ 1 such that A A < |w| ≤ . j −1 0 |λ| |λ|j0 Thus |w|k dG (w) <

Ak |λ|kj0 dG (λj0 w) ≤ Ak sup dG (ξ). |λ|j0 k ξ∈G,|ξ|
This proves the claim. Step 3. Completion of the proof. Fix k ≥ 1 such that 4 < |λ|k+1 . Using Koebe’s distortion theorem again we get (1 − |z|2 )|fk (z)| = (1 − |z|2 )σ (z)|σ(z)|k−1 < 4dG (σ(z))|σ(z)|k−1 . It follows immediately from (3.17) that fk ∈ B. The proof is complete.

We close this article by providing the example promised in a remark following Proposition 3.7. Proposition 3.8. Let λ ∈ (0, 1). Then there exists an univalent function ϕ ∈ B such that (a) (b) (c)

ϕ(z)∞ = 1. 1−|z|2 lim sup|ϕ(z)|→1 1−|ϕ(z)| 2 |ϕ (z)| ≥ E(Cϕ ) = {ϕ (0)k : k ≥ 0}.

Proof. Set

λ2 4 .

G := w = x + iy : |x| < λ−1 , |y| < 1

w = x + iy : λ−j+1 ≤ |x| < λ−j , |y| < j −j . j≥2

Then G is an unbounded simply connected domain in C and satisfies λG ⊂ G. Next, we prove the following assertions: (i) (ii)

supw∈G |w|k dG (w) < ∞ for every k ≥ 1. (w) ≥ λ. lim sup|w|→∞ ddGG(λw)

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For (i), notice that for w = x + iy ∈ G with |x| ≥ λ−1 , we can find j ≥ 1 such that λ−j+1 ≤ |x| < λ−j ,

|y| < j −j .

It follows that dG (w) ≤ 2j −j for such points w. Since λ−jk j −j → 0 as j → ∞ for every k ≥ 1, we obtain (i). For (ii), consider the sequence wj = λ−j , j ≥ 1. Then dG (wj ) = λ−j . This proves (ii). Now we let σ : D → G be a conformal map such that σ(0) = 0. Set ϕ := σ −1 (λσ). Then ϕ (0) = λ and σ ◦ ϕ = λσ. Set fk := σ k . Then we have fk ◦ ϕ = λk fk for every k ≥ 1. In view of the assertions (i) and (ii), by following the proof of Proposition 3.7 we can check that 1 − |z|2 λ2 |ϕ (z)| ≥ 2 4 |ϕ(z)|→1 1 − |ϕ(z)| lim sup

and that fk ∈ B. The latter fact implies that λk ∈ E(Cϕ ). The proof is complete. Acknowledgements This note is written while the authors are postdoctoral fellows at Korea University (Hosokawa) and Chonnam National University (Nguyen) in the academic year 2008-2009. It is our pleasure to thank these institutions for hospitality and to the BK21 program for financial aid. Finally, we are grateful to the referee for bringing to our attention the work of Hammond [1].

References [1] Hammond, C.: On the norm of a composition operator. PhD dissertation presented at the Graduate Faculty of the University of Virginia (2003) (http://oak. conncoll.edu/cnham) [2] Madigan, K., Matheson, A.: Compact composition operators on Bloch spaces. Trans. Am. Math. Soc. 347, 2679–2687 (1995) [3] Ohno, S., Zhao, R.: Weighted composition operators on the Bloch space. Bull. Austral. Math. Soc. 63(2), 177–185 (2001) [4] Shapiro, J.: Composition Operators and Classical Function Theory, Universitext: Tracts in Mathematics. Springer, New York (1993) [5] Shapiro, J., Smith, W., Stegenga, D.: Geometric models and compactness of composition operators. J. Funct. Anal. 127(1), 21–62 (1995) [6] Zhu, K.: Operator Theory in Function Spaces. Marcel-Deker, New York (1990)

Takuya Hosokawa Faculty of Engineering, Ibaraki University Hitachi, Ibaraki 316-8511, Japan e-mail: [email protected]

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Quang Dieu Nguyen Department of Mathematics Hanoi University of Education (Dai Hoc Su Pham Hanoi) Hanoi, Vietnam e-mail: dieu [email protected] and Department of Mathematics, Chonnam National University Gwangju 700-757, Korea Received: May 8, 2009. Revised: June 7, 2009.

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Integr. Equ. Oper. Theory 66 (2010), 565–592 DOI 10.1007/s00020-010-1765-z Published online March 19, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

On a Hardy Type General Weighted Inequality in Spaces Lp(·) Farman I. Mamedov and Aziz Harman Abstract. A Hardy type two-weighted inequality is investigated for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces Lp(·) . Equivalent necessary and sufficient conditions are found for the Lp(·) −→ Lq(·) boundedness of the Hardy operator when exponents q(0) < p(0), q(∞) < p(∞). It is proved that the condition for such an inequality to hold coincides with the condition for the validity of two-weighted Hardy inequalities with constant exponents if we require of the exponents to be regular near zero and at infinity. Mathematics Subject Classification (2000). 26D10, 45P05. Keywords. Hardy operator, Hardy inequality, variable exponent, weighted inequality.

1. Introduction In the present paper, we investigate a Hardy type two-weighted inequality 1/q(x) Hf (x) ≤ C ω 1/p(x) f (x) (1.1) v q(·)

p(·)

p(·)

n

in the norms of generalized Lebesgue spaces L (R ) for the multidimen sional Hardy operator Hf (x) = {t∈Rn :|t|<|x|} f (t)dt. It is proved that the condition for the inequality (1.1) to hold coincides with the validity of two ordinary weighted Hardy inequalities near zero and at inﬁnity, respectively. We require of the exponents q(x), p(x) to be regular near zero and at inﬁnity. Equivalent necessary and sufﬁcient conditions for the inequality (1.1) to hold are found when q(0) < p(0), q(∞) < p(∞). Some statement was proved in [24] for the case q(x) ≥ p(x). For constant exponents the Hardy inequality is a classical topic. It can be said that by this time quite a complete weight theory has been established for the Hardy operator. An attractive chronology of the well-known results can be found, for example, in [21,22,32,39]. In [4,26,27] it is shown that the The first author was supported by INTAS grant of South-Caucasian Republics, No. 8792.

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case 1 < p ≤ q < ∞ reduces to a one-parameter problem on maximum, while in [26,37] the case 0 < q < p, 1 ≤ p < ∞ reduces to the convergence of an improper integral. It is appropriate to note that different criteria are available for characterizing of the same inequality (see [21,22,30–32,38]). This fact is also reﬂected in our work. With a recent introduction of novel ideas and approaches of Lebesgue spaces to the theory of integral operators [8–11,19,23,28,29], and to the theory of non-Newtonian systems [1,3,16,15,34,40] it has become possible to consider Hardy type inequalities in the norms of generalized Lebesgue spaces. Under such approaches the classical results can be formulated eﬀectively in terms of the norms of Lp(·) (Rn ) spaces (see the survey, for example, in [7,12,19]). Hardy type inequalities were also considered in the works of a number of authors [13,14,17,19,24,25,35,36]. However, the relatively difﬁcult case q(x) ≤ p(x) has not been considered before. The aim pursued by this paper is to obtain a solution for this problem. To characterize the behavior of exponents we consider the logarithmic conditions near zero and at inﬁnity. (There is an example proving the necessity of such a condition for the maximal operator [33]); in the regularity condition |x| is replaced by some functions. The method of investigation is based on the weight theory of the Hardy operator. Systematic use is also made on the recent results on weighted Hardy inequalities in Lebesgue norms.

2. Auxiliary Statements, Definitions and Notation Denote p+ = supx∈Rn p(x), p− = inf x∈Rn p(x) for measurable functions p : En → R. The space Lp(·) (Ω), where Ω ⊂ Rn and 1 ≤ p− , p+ < ∞, is introduced as the class of all measurable functions on Ω which have the ﬁnite modular Ip,Ω (f ) := Ω |f (x)|p(x) dx (Ip (f ) := Ip,Rn (f )); the norm in Lp(·) (Ω) is deﬁned as f f Lp(·) (Ω) = inf λ > 0 : Ip ≤1 . λ Set f p(·);Ω := f Lp(·) (Ω) and f p(·) := f p(·);Rn . We refer to [20] for the basic properties of the variable exponent Lebesgue spaces Lp(·) . Throughout the paper it is assumed that the exponent functions p(x), q(x) are measurable and take their values in the interval (0, ∞). The weight functions v(x), ω(x) are assumed measurable and taking positive and ﬁnite values almost everywhere in Rn . For the weight functions the following natural conditions 1

σ = ω − p(x)−1 ∈ L1 (B(0, a)),

v(x) ∈ L1 (Rn \B(0, a))

are assumed for any a > 0. We use the notation v(y)dy, W (x) = V (x) = |y|>|x|

σ(y)dy.

|y|<|x|

For these functions we assume that ˜ (∞) = ∞. V˜ (0) = ∞, W

(2.1)

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It is clear that the functions V (x), W (x) are radial, i.e., depend only ˜ (|x|) = W (x), where the on |x|. They will be written as V˜ (|x|) = V (x), W functions with the upper symbol ∼ are functions of one variable. We denote by P the class of measurable functions p : Rn → (0, ∞) with 0 < p− ≤ p+ < ∞. ˜ (m) < 1, V˜ (m) > 1 and Let 0 < m < 1 and M > 1 be such that W ˜ (M ) > 1. V˜ (M ) < 1, W We say that the function f satisfy to the condition Λ0 if the condition holds: ∃f (0), C1 ∈ R sup x∈B(0,m)

|f (x) − f (0)| ln

1 ≤ C1 , ˜ W (|x|)

(2.2)

and by Z∞ the condition: ∃f (∞), C2 ∈ R sup x∈Rn \B(0,M )

|f (x) − f (∞)| ln

1 ≤ C2 . ˜ V (|x|)

(2.3)

We denote by C, C3 , C4 different positive constants which may depend ˜ (m), only on C1 , C2 , p(0), q(0), p(∞), q(∞), p+ , p− , q + , q − , V˜ (m), V˜ (M ), W ˜ W (M ). We write u ∼ v to state that there exist two positive constants C3 and C4 such that C3 u ≤ v ≤ C4 u. We use the following technical lemmas. Lemma 2.1. Let s ∈ P and s ∈ Λ0 . Then e−C1 W (x)s(0) ≤ W (x)s(x) ≤ eC1 W (x)s(0) ,

|x| ≤ m.

(2.4)

|x| ≥ M.

(2.5)

Let s ∈ P and s ∈ Z∞ . Then e−C2 V (x)s(∞) ≤ V (x)s(x) ≤ eC2 V (x)s(∞) ,

Proof. Since the proof of Lemma 2.1 is elementary, we omit the proof of its ˜ (m) < 1 second part. Let |x| ≤ m and s(x) ≥ s(0), then by the condition W we have W (x)s(x)−s(0) ≤ 1. ˜ (m) < 1 we have If |x| ≤ m and s(x) < s(0). Then by s ∈ Λ0 and W W (x)s(x)−s(0) = (1/W (x))s(0)−s(x) ≤ (1/W (x))C1 / ln(1/W (x)) = eC1 . Therefore for |x| ≤ m we have the right estimate (2.4) W (x)s(x)−s(0) ≤ eC1 . The same inequality also holds for the function W (x)s(0)−s(x) . Using the representation W (x)s(x) = W (x)s(0) · W (x)s(x)−s(0) we obtain estimates (2.4).

When proving our results, essential use will be made of the following Hardy inequality results [39, Sect. 6] (see also [22], Theorems 1.1 , 1.2, and [21, Sect. 5]).

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Theorem 2.1. Let 0 ≤ a 1, r11 = The measurable functions v1 , ω1 : Rn → [0, ∞) so that 1 1 −1

−p

σ 1 = ω1

∈ L1 (B(0, t)\B(0, a);

1 q1

−

1 p1 .

v1 ∈ L1 (B(0, b)\B(0, t))

for any a < t < b. Then the following conditions are equivalent: (1)

There is a positive constant C such that the inequality ⎛ ⎜ ⎝

⎛

⎞q1

⎜ ⎝

⎟ ⎟ f1 (y)dy ⎠ v1 (x)dx⎠

{y∈Rn :a<|y|<|x|}

a<|x|
⎛

⎜ ≤C⎝

⎞1/q1

⎞1/p1

⎟ (f1 (x))p1 ω1 (x)dx⎠

(2.6)

a<|x|
(2)

holds; The condition ⎛ ⎜ BP S (p1 , q1 ) = ⎜ ⎝

⎛

⎜ ⎝

a<|x|
⎛

⎜ v1 (y) ⎝

a<|y|<|x|

⎞q1

⎞r1 /q1

⎟ ⎟ σ1 dt⎠ dy ⎠

a<|t|<|y|

⎞1/r1

⎞−r1 /q1

⎜ ×⎝

⎛

⎟ σ1 (x)dx⎟ ⎠

⎟ σ1 dt⎠

<∞

(2.7)

a<|t|<|y|

(3)

holds; The condition ⎛ ⎜ AM R (p1 , q1 ) = ⎜ ⎝

⎛

⎜ ⎝

a<|x|
⎛ ⎜ ×⎝

⎟ v1 (y)dy ⎠

b>|y|>|x|

a<|t|<|x|

holds;

⎞r1 /q1

⎞r1 /q1 ⎟ σ1 dt⎠

⎞1/r1 ⎟ σ(x)dx⎟ ⎠

<∞

(2.8)

Vol. 66 (2010) (4)

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The condition

⎛

⎜ DM K (p1 , q1 ) = ⎜ ⎝

⎛

a<|x|
⎛ ⎜ ×⎝

⎞r1 /p1

⎜ ⎝

569

⎟ v1 (y)dy ⎠

b>|y|>|x|

⎞r1 /p1 ⎟ σ1 dt⎠

⎞1/r1 ⎟ v1 (x)dx⎟ ⎠

<∞

(2.9)

a<|t|<|x|

holds. Moreover, the best constant C > 0 in (2.6) is estimated as C ∼ BP S (p1 , q1 ) ∼ AM R (p1 , q1 ) ∼ DM K (p1 , q1 ).

(2.10)

3. Main Result Main result of the paper is the following statement. (m) < 1 Theorem 3.1. Let 0 < m < 1 and M > 1 be such that V (m) > 1, W (M ) > 1. Let p, q ∈ P ∩ Λ0 ∩ Z∞ , and let v, σ = and V (M ) < 1, W 1 − p(·)−1 be nonnegative locally integrable functions on Rn \{0}. Set V (x) = ω v(y)dy; W (x) = |y|<|x| σ(y)dy. Suppose that |y|>|x| p− > 1, 0 < q(0) < p(0) < ∞, 0 < q(∞) < p(∞) < ∞. (1)

Then the following statements are equivalent. There is a positive constant C which depends only ˜ (m), W ˜ (M ), V˜ (m), C1 , C2 , p− , p+ , q − , q + , p(0), q(0), p(∞), q(∞), W V˜ (M ) such that the weighted norm inequality v 1/q(·) Hf q(·) ≤ Cw1/p(·) f p(·)

(2)

holds for any measurable function f (x) ≥ 0. There exist constants C3 and C4 such that both weighted inequalities ⎛ ⎛ ⎞q(0) ⎞1/q(0) ⎟ ⎜ ⎜ ⎟ ⎜ v(x) f (y)dy ⎠ dx⎟ ⎝ ⎠ ⎝ B(0,m)

B(0,|x|)

⎛ ⎜ ≤ C3 ⎝

⎞1/p(0) ⎟ σ(x)1−p(0) f (x)p(0) dx⎠

B(0,m)

and

⎛ ⎜ ⎜ ⎝

|x|>M

⎛ ⎜ v(x) ⎝

B(0,|x|)\B(0,M )

⎞q(∞) ⎟ f (y)dy ⎠

⎞1/q(∞) ⎟ dx⎟ ⎠

on

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F. I. Mamedov and A. Harman ⎛

⎞1/p(∞)

⎜ ≤ C4 ⎝

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⎟ σ(x)1−p(∞) f (x)p(∞) dx⎠

|x|>M

(3)

hold for any measurable function f (x) ≥ 0. Both conditions m

1 1 ˜ (t) q(0) V˜ (t) q(0) W

r(0)

˜ (t) < ∞ dW

(3.1)

0

and ∞

1 1 ˜ (t) q(∞) V˜ (t) q(∞) W

r(∞)

˜ (t) < ∞ dW

(3.2)

1 1 1 1 1 1 hold. Here r(0) = q(0) − p(0) , r(∞) = q(∞) − p(∞) . The condition r(x) 1 1 V (x) q(x) W (t) q (x) σ(x)dx < ∞

(3.2 )

M

(4)

Rn

(5)

1 = holds, where r(x) Both conditions

−

m

1 q(x)

−

1 p(x) .

1

1 ˜ (t) p (0) V˜ (t) p(0) W

r(0)

dV (t) < ∞

0

and ∞ r(∞) 1 1 ˜ (t) p (∞) V˜ (t) p(∞) W − dV (t) < ∞ M

(6)

hold. The condition

1

1

V (x) p(x) W (t) p (x)

r(x)

v(x)dx < ∞

Rn

holds. Proof of (3) =⇒ (1). Let f (x) ≥ 0 be an arbitrary function satisfying the condition f ω 1/p p(·) ≤ 1. Then Ip(·) (f ω 1/p ) ≤ 1. To prove (3) =⇒ (1) we need to obtain the estimate Iq(·) (v 1/q(·) Hf ) ≤ C. The proof of the latter inequality takes three steps.

(3.3)

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Step 1. Estimation Near Zero Let us show that the condition (3.1) yields the inequality W (x) ≤ p (0)

CV (x)− q(0) . This inequality could be derived directly from the condition (3.1) (see below the demonstration of (4)=⇒(3.1), and (3.45)). But here we choose another way. By Theorem 2.1, the condition (3.1) implies the condition (2.8) for the inequality ⎛ ⎛ ⎞q(0) ⎞1/q(0) ⎟ ⎜ ⎜ ⎟ ⎜ v(t) ⎝ f (y)dy ⎠ dt⎟ ⎠ ⎝ B(0,m)

B(0,|x|)

⎛

⎞1/p(0)

⎜ ≤C⎝

⎟ σ 1−p(0) f (y)p(0) dy ⎠

(3.4)

B(0,m) 1

to hold. Fix any x ∈ B(0, m). Put f = σ(y)W − p(0) (x)χB(0,|x|) (y) into the inequality (3.4). We obtain ⎛ ⎞1/q(0) 1 ⎜ ⎟ v(y)dy ⎠ ·W p (0) (x) ≤ C x ∈ B(0, m). (3.4 ) ⎝ |x|<|y|<m

|x| ≤

Let us explain this inequality in terms of the functions V (x), W (x). For m 2 we have V˜ (m) V (x). (3.5 ) v(y)dy ≥ 1 − V˜ (m/2) |x|<|y|<m

To prove this inequality we use the elementary inequality V˜ (m) V˜ (m) ≤ V˜ (|x|), |x| ≤ m/2, v(y)dy m/2<|x|<m in the equality

V˜ (m) +

v(y)dy = V˜ (|x|).

|x|<|y|<m

Then for |x| ≤

m 2

we have

V˜ (m) 1+ ˜ V (m/2) − V˜ (m)

v(y)dy ≥ V˜ (|x|),

|x|<|y|<m

from which the inequality (3.5 ) readily follows. Using the latter inequality in (3.4 ), we see that p (0)

W (x) ≤ CV (x)− q(0) for B(0, m/2).

(3.5)

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To estimate the Hardy operator, we write it as a sum of two summands. The ﬁrst operator arises as a result of the application of the Hardy operator to the “largest” part of the test function, while the second one—to the “smallest” part. We further derive the corresponding estimate for each of the summands by using the known results in the weight theory of the Hardy operator. Let us choose a sufﬁciently small number δ > 0 as the smallest number among the numbers m/2 in the conditions (2.2) and (3.1). Let x ∈ B(0, δ) be any ﬁxed point. Denote p− x = inf t∈B(0,|x|) p(t). It is obvious that

f (t)dt = B(0,|x|)

f (t)χf (t)/σ(t)≥1 dt + B(0,|x|)

f (t)χf (t)/σ(t)<1 dt.

(3.6)

B(0,|x|)

Estimation of the largest part. By virtue of (3.3) and H¨ older’s inequality we have f (t)χ(f /σ)≥1 dt = (f /σ)χ(f /σ)≥1 σ(t)dt B(0,|x|)

B(0,|x|)

−

(f /σ)p(t)/px σdt

≤ B(0,|x|)

⎛ ⎜ ≤⎝

⎛ ⎞1/p− x ⎜ ⎝

⎟ (f /σ)p(t) σdt⎠

B(0,|x|)

⎛ ⎜ ≤⎝

⎞1/(p− x)

⎟ σdt⎠

B(0,|x|) ⎞1/(p− x)

⎟ σdt⎠

.

(3.7)

B(0,|x|)

Applying the second inequality in (3.7) we obtain ⎛ ⎜ ⎝

⎞q(x) − ⎟ (f /σ)p(t)/px χ(f /σ)≥1 σdt⎠

B(0,|x|)

⎛

− − ⎜ = W (x)q(x)/(px ) ⎝W (x)−1/(px )

⎞q(x)

− ⎟ (f /σ)p(t)/px χ(f /σ)≥1 σdt⎠

B(0,|x|) − q(x)−qx − px

≤ W (x) ( )

⎛ ⎜ ⎝

B(0,|x|)

⎞qx− − ⎟ (f /σ)p(t)/px σdt⎠

.

(3.8)

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Thus ⎛ ⎜ ⎝

⎞q(x)

⎛

− q(x)−qx − px

⎜ ≤ W (x) ( ) ⎝

⎟ f χ(f /σ)≥1 (t)dt⎠

B(0,|x|)

⎞qx−

− ⎟ (f /σ)p(t)/px σdt⎠

.

B(0,|x|)

By virtue of Holder’s inequality we have ⎛

⎞q(x)

⎜ ⎝

⎟ f χ(f /σ)≥1 (t)dt⎠

B(0,|x|)

− qx

≤ W (x)

− q(x)−qx − px

W (x) ( )

⎛ ⎜ ⎝

1 W (x)

⎞ p−−qx−

p/p−

(f /σ)

⎟ σdt⎠

px

. (3.9)

B(0,|x|)

Hence it follows that ⎛

⎜ v(x) ⎝

B(0,δ)

⎞q(x)

⎟ f χ(f /σ)≥1 (t)dt⎠

dx

B(0,|x|)

⎛

− q(x)−qx − px

W (x) ( ) W (x)

≤

− p− x −p − px

− qx

B(0,δ)

⎜ ⎝

⎞ p−−qx− −

(f /σ)p/p

⎟ σdt⎠

px

v(x)dx.

B(0,|x|)

− q(x)−qx

x ≤ 1, x ∈ B(0, δ). By the condiUsing W (m) < 1 we have W (x) p− tion (2.2) and Lemma 2.1, for the exponents p(x), q(x) we have

− p− −p− qx x − px

(

W (x)

)

≤ CW (x)

q(0)(p(0)−p− ) p(0)

,

x ∈ B(0, δ).

(3.9 )

To prove (3.9 ) it sufﬁces to verify the condition (2.2) for the exponent − − s(x) = qz− (p− x − p )/px . Condition (2.2) for s(x) easily follows from the equality − qx− (p− q(0)(p(0) − p− ) x −p ) − − p(0) px − 1 − − − px − p(0) + p− p(0) q(0) − qx− = − p(0)px qx − q(0) + q(0)p p(0)px

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F. I. Mamedov and A. Harman

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and the fact that the same condition is fulﬁlled for p(x) and q(x). Thus ⎛

⎜ v(x) ⎝

B(0,δ)

⎞q(x)

⎟ f χ(f /σ)≥1 (t)dt⎠

dx

B(0,|x|)

⎛

q(0) p(o)−p−

(

≤C

p(0)

W (x)

)⎜ ⎝

B(0,δ)

⎞ p−−qx−

p/p−

(f /σ)

⎟ σdt⎠

px

.

(3.10)

B(0,|x|)

Using H¨older inequality and (3.3), we obtain ⎛

− ⎜ (f /σ)p/p σdt ≤ ⎝

B(0,|x|)

⎞1/p−

⎟ f p ωdy ⎠

1

1

W (x) (p− ) ≤ W (x) (p− ) . (3.11)

B(0,|x|)

For a further estimation of the right-hand side of (3.10) we need to give more details. Denoting − − 1 g(x) = W (x) (p− ) (f /σ)p/p σdt, (3.11 ) B(0,|x|)

by (3.11) we have 0 ≤ g(x) ≤ 1. Then by virtue of (3.9) we obtain ⎛

⎜ v(x) ⎝

B(0,δ)

≤C

⎞q(x) ⎟ f χ(f /σ)≥1 (t)dt⎠

dx

B(0,|x|) q(0) p(0)−p−

(

W (x)

)

p(0)

g

− p− qx − px

− p− qx − px

W (x)

1 . (p− )

v(x)dx. (3.12)

B(0,δ)

Let λ > 0 be some number. It is not difﬁcult to verify that the condition (2.2) holds for the exponent s(x) = qx− /p− x . Then, by Lemma 2.1, − − W (x)qx /px ≤ CW (x)q(0)/p(0) , the right-hand side of (3.12) does not exceed

q(0)

W (x) p (0) g

C

− p− qx − px

v(x)dx

{x:g(x)≥W λ (x)}∩B(0,δ)

q(0)

W (x) p (0) g

+C

− p− qx − px

v(x)dx := C(I1 + I2 ).

{x:g(x)<W λ (x)}∩B(0,δ)

By virtue of (3.5) and W (x) < 1, for the integral I2 we obtain the estimates

Vol. 66 (2010)

On a Hardy Type General Weighted Inequality

q(0)

I2 ≤

W (x) p (0)

+λ

B(0,δ)

(0) − pq(0)

≤C

− p− qx − px

V (x)

575

v(x)dx

− q− q(0) +λ p p+ p (0)

v(x)dx

B(0,δ)

δ = −C

− V˜ (t)

−

(0) q 1+λ pq(0) . p+

−

(0) q −λ pq(0) · p+

dV˜ (t) = C V˜ (δ)

0

= C V˜ (δ)−λ1 ; λ1 =

p (0) q − · . q(0) p+

(3.13)

For the integral I1 , by partitioning the domain of integration, we have

− p− qx − px

q(0)

W (x) p (0) g

I1 =

v(x)dx

{x:g(x)≥W λ (x)}∩B(0,δ)

=

W (x)

q(0) p (0)

g

− qx − px

q(0) − p(0) p−

q(0)

−

g p(0) p v(x)dx

− q(0) x:g(x)≥W λ (x), qx − ≥ p(0) ∩B(0,δ) px

W (x)

+

q(0) p (0)

g

− qx − px

q(0) − p(0) p−

q(0)

−

g p(0) p v(x)dx

− q(0) x:g(x)≥W λ (x), qx − < p(0) ∩B(0,δ) px

:= I11 + I12 . Due to 0 ≤ g(x) ≤ 1 , for I11 we have the estimates I11 ≤

q(0)

W (x) p (0) g

p− q(0) p(0)

v(x)dx

B(0,δ)

⎛

≤

W (x) B(0,δ)

q(0)(p(0)−p− ) p(0)

⎜ ⎝

q(0) ⎞ p−p(0) − ⎟ (f /σ)p/p σdt⎠

v(x)dx.

(3.14)

B(0,|x|)

Let us now apply Theorem 2.1 to obtain the inequality (2.6) using the condition (2.7) where it is assumed that a = 0, b = δ, q1 = q(0)p− /p(0), p1 = p− , 1/r1 = 1/q1 − 1/p1 , p/p− q(0)(p(0)−p− ) − f p(0) f1 = σ, v1 = W (x) v(x), ω1 = σ(x)1−p . σ

576

F. I. Mamedov and A. Harman

IEOT

Then for the right-hand side of (3.14) we obtain the estimate f p ωdx ≤ C3 BP S (p1 , q1 ) = C4 . ≤ CBP S (p1 , q1 )

(3.15)

B(0,δ)

Let us now make sure that the value BP S (p1 , q1 ) is ﬁnite, i.e., verify the fulﬁllment of the condition (2.7): BP S (p1 , q1 ) ⎡⎛ ⎢⎜ ⎢⎝ = ⎣

B(0,δ)

|y|<|x|

⎡⎛ ⎢⎜ ⎢⎝ ⎣

= B(0,δ)

⎤r1

⎞1/q1

1 ⎥ W1 (x)− q1 ⎥ ⎦ σ1 (x)dx

⎟ v1 (y)W1 (y)q1 dy ⎠

⎞p(0)/q(0)p−

W (y)

− q(0)(p(0)−p− ) + q(0)p p(0) p(0)

⎟ v(y)dy ⎠

|y|<|x|

⎤r(0)p− /p(0) p(0) − q(0)p −

× W (x) ⎡⎛ ⎢⎜ ⎢⎝ ⎣

= B(0,δ)

⎥ ⎦

σ(x)dx ⎤r(0)

⎞1/q(0)

1 ⎥ W (x)− q(0) ⎥ ⎦

⎟ v(y)W (y)q(0) dy ⎠

σ(x)dx

|y|<|x|

≤C

1

1

V (x) q(0) W (x) q (0)

r(0)

σ(x)dx ≤ C0 ,

(3.16)

B(0,δ)

where the last inequality follows from the conditions (3.1) and (2.10). Since the condition (2.7) is fulﬁlled, the above-given passage from (3.14) to (3.15) is legitimate. It can be easily veriﬁed that the power s(x) = qx− /p− x satisﬁes the condition Λ0 . Then by Lemma 2.1 we have the estimate g(x)

− qx − px

q(0) − p(0)

≤ CW (x)

λC1 1 W (x)

log

≤ C6 ,

x ∈ B(0, δ),

for the corresponding term in the integral I12 . Again, repeating the previous procedure of passing from (3.14) to (3.15), by (3.11 ) and Theorem 2.1 we have q(0) p− q(0) W (x) p (0) g p(0) v(x)dx ≤ C7 f p ωdx = C7 . (3.17) I12 ≤ B(0,δ)

B(0,δ)

Therefore I1 ≤ C4 + C7 .

(3.18)

Vol. 66 (2010)

On a Hardy Type General Weighted Inequality

Applying estimates (3.13), (3.18), we get ⎛ ⎞q(x) ⎜ ⎟ f χf /σ≥1 dt⎠ v(x)dx ≤ C8 . ⎝ B(0,δ)

577

(3.19)

B(0,|x|)

Estimation of the smallest part. Applying Theorem 2.1 with the condition (2.8) and assuming that q1 = q(0), f1 = σ,

p1 = p(0), a = 0,

v1 = v(x),

ω1 = σ(x)1−p(0) ,

b = δ,

we come to ⎛

⎜ ⎝

B(0,δ)

⎞q(x)

⎟ f χ(f /σ)<1 dt⎠

B(0,|x|)

⎛

⎜ ⎝

≤ B(0,δ)

v(x)dx

⎞q(x) ⎟ σdt⎠

v(x)dx

B(0,|x|)

W (x)q(x) v(x)dx (by (2.2))

= B(0,δ)

W (x)q(0) v(x)dx (by Theorem 2.1)

≤C B(0,δ)

≤ C9

σdx = C10 .

(3.20)

B(0,δ)

To verify the condition (2.8) of Theorem 2.1, it is sufﬁces to apply the previous procedure to (3.16) and use the condition (3.1). Estimates (3.19) and (3.20) complete the estimation near zero: ⎛ ⎞q(x) ⎜ ⎟ f dt⎠ v(x)dx ≤ C. (3.21) ⎝ B(0,δ)

B(0,|x|)

Step 2. Estimation at Infinity It is not difﬁcult to show that by Theorem 2.1 and the condition (2.8), the condition (3.2) implies the inequality ⎞1/q(∞) ⎛ ⎛ ⎞q(∞) ⎟ ⎜ ⎜ ⎟ ⎜ f (y)dy ⎠ vdx⎟ ⎝ ⎠ ⎝ Rn \B(0,M )

M <|y|<|x|

578

F. I. Mamedov and A. Harman ⎛

⎞1/p(∞)

⎜ ≤C⎝

IEOT

⎟ σ 1−p(∞) f p(∞) dx⎠

.

Rn \B(0,M ) 1

Let |x| > M be ﬁxed. Then substituting f = σ(y)W (x)− p(∞) ·χB(0,|x|)\B(0,M ) (y), as in (3.4 ) above, into the latter inequality for |x| > M , we have the estimate (∞) 1 pq(∞) ) σ(y)dy ≤ C( . V (x) M <|y|<|x|

Now let us show that the inequality ˜ (M ) W W (x) σ(y)dy ≥ 1 − ˜ (2M ) W

(3.21 )

M <|y|<|x|

holds for |x| ≥ 2M . To prove this inequality it sufﬁces to apply the obvious inequality ˜ (M ) W ˜ (M ) ≤ σ(y)dy, |x| > 2M, W ˜ (2M ) − W ˜ (M ) W M <|y|<|x|

in the equality

˜ (|x|) = W ˜ (M ) + W

σdy.

M <|y|<|x|

Thus, for |x| ≥ 2M we have the following auxiliary estimate which enables us to proceed with our approach discussed above: (∞) pq(∞) 1 W (x) ≤ C. (3.22) V (x) Let N > 1 be the largest number of the numbers 2M in the conditions (3.2) and (2.3). Again writing the Hardy operator as a sum of two summands, we obtain the estimates ⎛ ⎞q(x) ⎜ ⎟ f (t)dt⎠ v(x)dx ⎝ Rn \B(0,N )

≤ 2q

+

B(0,|x|)\B(0,N )

⎛

−1

⎜ ⎝

Rn \B(0,N )

+ 2q

+

−1

Rn \B(0,N )

:= 2q

+

−1

(i1 + i2 ).

⎞q(x)

⎟ f χf /σ≥1 dt⎠

B(0,|x|)\B(0,N )

⎛ ⎜ ⎝

B(0,|x|)\B(0,N )

v(x)dx ⎞q(x)

⎟ f χf /σ<1 dt⎠

v(x)dx

Vol. 66 (2010)

On a Hardy Type General Weighted Inequality

579

Estimation of the largest part. Taking assumption (3.3) into account, we ﬁnd that ⎛ ⎞q(x) ⎜ ⎟ i1 = (f /σ)χf /σ≥1 σdt⎠ v(x)dx ⎝ Rn \B(0,N )

B(0,|x|)\B(0,N )

⎛

⎜ ⎝

≤ Rn \B(0,N )

⎟ (f /σ)p(t) χf /σ≥1 σdt⎠

v(x)dx

B(0,|x|)\B(0,N )

≤

⎞q(x)

v(x)dx = C11 .

(3.23)

Rn \B(0,N )

Estimation of the smallest part. Let us now estimate ⎛

⎜ ⎝

i2 = Rn \B(0,N )

⎞q(x)

⎟ (f /σ)χf /σ<1 dt⎠

v(x)dx

B(0,|x|)\B(0,N )

assuming that F (x) =

f (t)χf /σ<1 dt. B(0,|x|)\B(0,N )

F (x) + Then F (x) ≤ B(0,|x|) σdt = W (x). Let G(x) = W (x) , x ∈ R , where 0 ≤ G(x) ≤ 1. We have the following elementary estimates derived by partitioning the domain of integration q(x) q(x) F (x)q(x) v(x)dx = G W v(x)dx i2 = Rn \B(0,N )

Rn \B(0,N )

q(x)

=

G

W

q(x)

v(x)dx

{x:G(x)<1/W (x),|x|≥N }

q(x)

+

G

{x:G(x)≥1/W (x),|x|≥N }

≤

W

q(x)

Gq(∞) Gq(x)−q(∞) W

v(x)dx + Rn \B(0,N )

v(x)dx q(x)

v(x)dx

{x:G(x)≥1/W (x)}

Gq(∞) Gq(x)−q(∞) W

≤ V˜ (N ) +

q(x)

v(x)dx

{x:G(x)≥1/W (x),q(x)>q(∞)}

+ {x:G(x)≥1/W (x),q(x)≤q(∞)}

Gq(∞) Gq(x)−q(∞) W

q(x)

v(x)dx.

580

F. I. Mamedov and A. Harman

IEOT

According to Lemma 2.1, for s ∈ Z∞ , by (3.22), we have W s(x) ≤ C.W . Indeed, s(∞)

C2

W (x)s(x) ≤ W (x)s(∞) · W (x)s(x)−s(∞) ≤ W (x)s(∞) · W (x) ln(1/V (x)) C ·q(∞)

2 − p (∞)·ln(1/V (x))

≤ W (x)s(∞) · V (x)

=e

C2 q(∞) p (∞)

· W s(∞) . (3.23 )

Using (3.23 ) with s := q for the integral term over the set {x : G(x) ≥ ≤ q(∞)} we have q(x) ˜ Gq(∞) Gq(x)−q(∞) W v(x)dx i2 ≤ V (N ) +

1 W (x) , q(x)

{x:G(x)>1/W (x), q(x)>q(∞)}

Gq(∞) W

+

q(x)

v(x)dx

{x:G(x)>1/W (x), q(x)>q(∞)}

(by the fact 0 ≤ G(x) ≤ 1)

Gq(∞) W

≤ V˜ (N ) +

q(x)

v(x)dx

{x:G(x)>1/W (x), q(x)>q(∞)}

F q(∞) W q(x)−q(∞) v(x)dx

≤ V˜ (N ) + {x:G(x)>1/W (x), q(x)>q(∞)}

(by (3.22), (2.3) and Lemma 2.1) ≤ V˜ (N ) + C

F q(∞) v(x)dx.

(3.24)

Rn \B(0,N )

Let us now apply Theorem 2.1 with the condition (2.8) and assume that f1 = f χf /σ<1 ,

ω1 = ω (p(∞)−1)/(p(x)−1)

v1 = v,

q1 = q(∞)

p1 = p(∞),

where the condition (2.8) follows from the condition (3.2). Then we have ⎛

⎜ F q(∞) vdx ≤ C⎝

Rn \B(0,N )

⎞q(∞)/p(∞) ⎟ f p(∞) ω (p(∞)−1)/(p(x)−1) χf /σ<1 dx⎠

,

Rn \B(0,N )

i.e., ⎛ ⎜ i2 ≤ C ⎝

⎞q(∞)/p(∞) ⎟ f p(∞) ω (p(∞)−1)/(p(x)−1) χf /σ<1 dx⎠

+ V˜ (N ).

Rn \B(0,N )

(3.25) By virtue of (3.23 ) with s := p and by partitioning the domain of integration, we have the estimates

Vol. 66 (2010) f

On a Hardy Type General Weighted Inequality

p(∞)

Rn \B(0,N )

ω

p(∞)−1 p−1

581

(f /σ)p(∞) σχf /σ<1 dx

χf /σ<1 dx = Rn \B(0,N )

(f /σ)p(∞) σχf /σ<1 dx

= {x:f /σ<W −2/p(∞) (x)}

(f /σ)p(∞) σχf /σ<1 dx

+ {x:1≥f /σ≥W −2/p(∞) (x)}

W −2 σdx

≤ Rn \B(0,N )

(f /σ)p(x) (f /σ)p(∞)−p(x) σχf /σ<1 dx

+ {x: 1≥f /σ>W −2/p(∞) (x),p(x)

(f /σ)p(x) (f /σ)p(∞)−p(x) σχf /σ<1 dx

+ {x: 1≥f /σ≥W −2/p(∞) (x),p(x)≥p(∞)}

1 + W 2(p−p(∞)) (f /σ)p(x) σdx ≤ ˜ (N ) W {x:f /σ≤1,p(x)>p(∞)} +C σ 1−p(x) f p(x) dx Rn \B(0,N )

(by (3.23 ) with s := p) ≤

1 +C ˜ (N ) W

σ 1−p(x) f p(x) dx

Rn \B(0,N )

= C + C12

f p(x) ωdx ≤ C13 .

(3.26)

Rn \B(0,N )

From estimates (3.23), (3.24), (3.25), (3.26) we obtain ⎛ ⎞q(x) ⎜ ⎟ f (t)dt⎠ v(x)dx ≤ C. ⎝ Rn \B(0,N )

(3.27)

B(0,|x|)\B(0,N )

Step 3. Estimation of the middle part Since ˜ (N ) ≤ C14 , ω −1/(p−1) dt := W Ip (·);B(0,N ) (ω −1/p ) = B(0,N )

by virtue of (3.3) and H¨ older’s inequality, for the p(·)-norms we obtain f (t)dt ≤ Cf ω 1/p p(·);B(0,N ) ω 1/p p (·);B(0,N ) ≤ C15 . (3.28) B(0,N )

582

F. I. Mamedov and A. Harman

We have

⎛

Rn \B(0,N )

⎛

⎞q(x)

⎜ ⎝

⎟ f (t)dt⎠

B(0,N )

⎛

⎜ ⎜ ≤ ⎝1 + ⎝

IEOT

v(x)dx

⎞q+ ⎞

⎟ ⎠

⎟ f (t)dt⎠

v(x)dx.

(3.29)

Rn \B(0,N )

B(0,N )

By (3.28) and (3.29), ⎛ ⎞q(x) ⎜ ⎟ q+ ˜ f (t)dt⎠ v(x)dx ≤ (1 + C15 )V (N ) = C16 . (3.30) ⎝ Rn \B(0,N )

B(0,N )

Further, using the elementary inequality (a + b)p ≤ 2p−1 (ap + bp ), where a ≥ 0, b ≥ 0, p ≥ 1, by virtue of estimates (3.21), (3.27), (3.30) we obtain the following elementary estimates: ⎛ ⎛ ⎞q(x) ⎞q(x) ⎜ ⎜ ⎟ ⎟ f (t)dt⎠ v(x)dx ≤ f (t)dt⎠ v(x)dx ⎝ ⎝ Rn

B(0,|x|)

⎛

⎜ ⎝

+ B(0,N )\B(0,δ)

+ 2q

+

−1

+

−1

⎞q(x)

⎟ f (t)dt⎠

B(0,|x|)

Rn \B(0,N )

+ 2q

B(0,δ)

Rn \B(0,N )

⎛ ⎜ ⎝

v(x)dx ⎞q(x)

⎟ f (t)dt⎠

B(0,N )

⎛ ⎜ ⎝

B(0,|x|)

v(x)dx ⎞q(x)

⎟ f (t)dt⎠

v(x)dx

B(0,|x|)\B(0,N )

≤ C. The part (3) =⇒ (1) of Theorem 3.1 is proved.

(3.31)

Proof of (1) =⇒ (3). Proof of (1) =⇒ (3.1). Now, we will construct an auxiliary function. Using it in the inequality (1.1), a necessity condition will be derived. Let us choose a sufﬁciently small δ > 0 as the smallest one among the numbers m 2 in the conditions (2.2) and (3.1). Assume that W (δ) < 1, V (δ) > 1 and t ∈ B(0, δ) is ﬁxed. So, let the inequality (1.1) be fulﬁlled for any function f ≥ 0. Denote by q + , q − , p+ , p− respective values taken with respect to the ball B(0, δ). Using q(0) < p(0) we can assume q + < p− if δ > 0 is a sufﬁciently small number. Set

Vol. 66 (2010)

On a Hardy Type General Weighted Inequality

583

1 1 A = B(0,m) [V (x) q(0) W (x) q (0) ]r(0) σ(x)dx. We can assume that A > 1, otherwise (1)=⇒(3.1) is proved. To prove the necessity of the condition (3.1) we need to have the estimation V (x)s(x) ∼ V (x)s(0) for s(x) ∈ Λ0 . For this, below we estimate V (x) via W (x) for |x| < δ. Let a function s(x) ∈ Λ0 . According to Lemma 2.1, we have the estimate W (x)s(x) ∼ W (x)s(0) near zero. Let 0 < t < δ be ﬁxed. Testing (1.1) with the function ⎛

1 ⎞− p(x)

⎜ f (x) = ⎝

⎟ σ(y)dy ⎠

σ(x)χB(0,t) (x),

B(0,t)

we have

Ip(·) ω 1/p ft =

σ(x) dx = 1. (t) W

B(0,δ)

Hence

1 Iq(·) v q Hft ≤ 1.

(t) < 1, for t < |x| < δ we have Using W ⎛

⎜ ⎝

Hft (x) ≥ B(0,t)

1 ⎞− p(y)

⎟ σ(z)dz ⎠

1

(t)1− p+ . σ(y)dy ≥ W

(3.32)

B(0,t)

Using (3.32), we obtain 1 C ≥ Iq(·) v q Hft ≥ ≥

v(x)(Hft )q(x) dx

|x|>t

(t) v(x)W

q+ (p+ )

q+

(t) (p+ ) . dx = V (t)W

(3.33)

|x|>t

Hence q+

(t)− (p+ ) , V (t) ≤ C W

0 < t < δ.

(3.34)

Now, using (3.34), let us show V (x)p(x) ≤ CV (x)p(0) . By using (3.34) and V (δ) > 1, for |x| < δ we have C1

1 ln V (x)p(x) ≤ V (x)p(0) V (x)p(x)−p(0) ≤ V (x)p(0) V (x) ( W (x) )

p(0)

≤ V (x)

W (x)

C1 q + 1 (p+ ) ln W (x)

(

C1 q +

) = V (x)p(0) e (p+ ) .

(3.35)

584

F. I. Mamedov and A. Harman

IEOT

We also need the following inverse inequality with exponent q(x), i.e. q(x)

using (3.34) to prove that V (x)

−

≥e

C1 q + (p+ )

V (x)q(0) : C1

q(0)

V (x)

q(x)

= V (x)

q(0)−q(x)

q(x)

≤ V (x)

V (x)

C q+ − + 1 1 (p ) ln W (x)

(

≤ V (x)q(x) W (x)

1 V (x) ( W (x) ) ln

C1 q +

) = V (x)q(x) e− (p+ ) .

(3.36)

Let us proceed proving the necessity of the condition 3.1. In the inequality (1.1) put the test function r(0)

r(0)

f (x) = V (x) p(0)q(0) W (x) p(0)q (0) σ(x)χB(0,δ) (x)A−1/p(x) ,

x ∈ B(0, δ).

According to Lemma 2.1, p ∈ Λ0 and (3.35), we have 1 r(0)(p(x) r(0)p(x) p V (x) p(0)q(0) W p(0)q (0) (x)σ(x)A−1 dx Ip(·) ω f = B(0,δ)

r(0)

r(0)

V (x) q(0) W q (0) (x)σ(x)A−1 dx ≤ C.

≤ C

(3.37)

B(0,δ)

For x ∈ B(0, δ) we have r(0) f (y)dy ≥ V (x) p(0)q(0) · B(0,|x|)

r(0)

W p(0)q (0) (y)

B(0,|x|)

= V (x)

r(0) q(0)p(0)

·

1 A1/p−

−1 +1

σ(y)dy A1/p−

|x| r(0) (t) p (0)q(0) dW (t) W 0

r(0) r(0) r(0) 1 (|x|) p (0)q(0) q(0)p(0) W V (x) − p (0)q(0) A1/p r(0) r(0) C ≥ 1/p− V (x) q(0)p(0) W (x) p (0)q(0) . A

=

Hence, using (1.1), (3.37) and q ∈ Λ0 , we have ⎛

⎜ ⎝

C ≥ B(0,δ)

⎞q(x)

⎟ f (y)dy ⎠

v(x)dx

B(0,|x|) r(0)

r(0)

v(x)V (x) q(0)p(0) q(x) W (x) p (0)q(0)

≥ B(0,δ)

⎛ ⎜ ≥ C17 ⎝

Aq+ /p−

+ ⎞1− q−

B(0,δ)

1

q(x)

q

v(x)V (x)

r(0) p(0)

W (x)

r(0) p (0)

⎟ dx⎠

dx

Vol. 66 (2010)

On a Hardy Type General Weighted Inequality

585

(integration by parts or using Theorem 2.1 and (2.10) gives) ⎛ ⎜ ≥ C18 ⎝

+ ⎞1− q−

q

σ(x)V (x)

r(0) q(0)

W (x)

r(0) q (0)

⎟ dx⎠

+

1− qq−

= C18 A

.

B(0,δ)

Since the exponents q + < p− the latter inequality implies A ≤ C. Proof of (1)=⇒(3.2). Let this time q + , q − , p+ , p− denote the corresponding values taken with respect to the set Rn \B(0, N ). Again, using (2.3) and q(∞) < p(∞) we can assume that q + < p− if the number N > 1 is sufﬁciently 1 1 large. Set B = |x|>N [[V (x) q(∞) W (x) q (∞) ]r(0) σ(x)dx]. We can assume that B > 1, otherwise (1)=⇒(3.2) is proved. To prove the necessity of the condition (3.2) we need to estimate W (x)s(x) ∼ W (x)s(∞) , |x| > N for s(x) ∈ Z∞ . In this way, below we estimate W (x) via V (x) for |x| > N. Let t > N . Testing (1.1) again by the function 1 ⎛ ⎞− p(x) ⎜ ⎟ σ(y)dy ⎠ σ(x)χB(0,t) (x), t > N, ft (x) = ⎝ B(0,t)

we have

Ip(·) ω 1/p ft = B(0,t)

Hence

σ(x) dx = 1. (t) W

1 Iq(·) v q Hft ≤ C.

˜ (t) > 1 , for |x| > N we have Using W 1 ⎛ ⎞− p(y) 1 ⎜ ⎟ (t)1− p− Hft (x) ≥ σ(y) ⎝ σ(z)dz ⎠ dy ≥ W . B(0,t)

B(0,t)

˜ (t) > 1, we obtain Using (3.38), and W 1 C ≥ Iq(·) v q Hft ≥ v(x)(Hft )q(x) dx ≥ |x|>t

(t) = V (t)W

q− (p− )

(3.38)

q−

(t) (p− ) dx v(x)W

|x|>t

.

(3.39)

Hence q−

(t)− (p− ) , V (t) ≤ C W

t > N.

(3.40)

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Now, let us show W (x)s(x) ∼ W (x)s(∞) for s(x) ∈ Z∞ and |x| > N. ˜ (x) > 1, for |x| > N we have Using (3.40) and W C2

1 ln W (x)s(x) ≤ W (x)s(∞) W (x)s(x)−s(∞) ≤ W (x)s(∞) W (x) ( V (x) )

s(∞)

≤ W (x)

−

V (x)

C2 q − (p− ) ln( 1 ) V (x)

=e

C2 q − (p− )

W (x)s(∞) .

(3.41)

We need also the inverse inequality with exponent q(x), i.e. using (3.40) let us prove that −

W (x)q(x) ≥ e

C2 q − (p− )

W (x)q(∞)

for |x| > N.

We have C2

1 ln W (x)q(x) = W (x)q(x) W (x)q(∞)−q(x) ≤ W (x)q(x) W (x) ( W (x) )

−

≤ W (x)q(x) V (x)

C2 q (p− ) ln

−

C2 q −

1 ( V (x) ) = e (p− ) W (x)q(x) .

(3.42)

Let us proceed to proving the necessity of the condition (3.2). Testing the inequality (1.1) with the function r(∞)

r(∞)

f (x) = V (x) p(∞)q(∞) W (x) p(∞)q (∞) σ(x)χRn \B(0,N ) (x)B −1/p(x) . According to Lemma 2.1, p ∈ Z∞ and (3.41), we have r(∞)p(x) r(∞)p(x) V p(∞)q(∞) W (x) p(∞)q (∞) σ(x)B −1 dx Ip(·) ω 1/p ft = |x|>N

r(∞)

r(∞)

V (x) q(∞) W (x) q (∞) σ(x)B −1 dx ≤ C16 .

≤C |x|≥N

For |x| > N , using (3.42) and p ∈ Z∞ we have

Hf (x) =

f (y)dy B(0,|x|)\B(0,N )

r(∞)

r(∞)

V (y) q(∞)p(∞) W (y) p(∞)q (∞) σ(y)B −1/p(y) dy

= B(0,|x|)\B(0,N ) r(∞)

≥ V (x) q(∞)p(∞) .

r(∞)

= V (x) q(∞)p(∞) .

1 B 1/p− 1

r(∞)

W (x) p (∞)q(∞) σ(y)dy B(0,|x|)\B(0,N )

|x| r(∞) (t) p (∞)q(∞) dW (t) W

B 1/p− N −1 r(∞) 1 r(∞) q(∞)p(∞) +1 = − V (x) 1/p p (∞)q(∞) B r(∞) r(∞) (|x|) p (∞)q(∞) − W (N ) p (∞)q(∞) × W (by using (3.21 ) and (2.1)) ≥

C B 1/p−

r(∞)

r(∞)

V (x) q(∞)p(∞) W (x) p (∞)q(∞) .

Vol. 66 (2010)

On a Hardy Type General Weighted Inequality

587

Therefore

C ≥ Iq(·) v 1/q Hf r(∞)q(x) r(∞) v(x)V (x) q(∞)p(∞) q(x) W (x) p (∞)q(∞) ≥ Rn \B(0,N )

⎛ ⎜ ≥ C17 ⎝

1 dx B q+ /p−

+ ⎞1− q−

p

v(x)V (x)

r(∞) p(∞)

W (x)

r(∞) p (∞)

⎟ dx⎠

= C17 B 1−q

+

/p−

.

Rn \B(0,N )

Integrating by parts or using Theorem 2.1 and (2.10) gives ⎛

+ ⎞1− q−

⎜ ≥ C18 ⎝

p

V (x)

r(∞) q(∞)

W (x)

r(∞) q (∞)

⎟ σ(x)dx⎠

.

Rn \B(0,N )

Hence B ≤ C, i.e. (1) =⇒ (3). Thus (1) ⇔ (3) is proved.

The implications (3) ⇔ (2) and (2) ⇔ (5) follow by a simple application of Theorem 2.1 under the conditions (2.8) and (2.9), respectively. Proof of (4) ⇔ (3). To prove this implication it sufﬁces to show the following estimates: r(x) 1 1 V (x) q(x) W (t) q (x) σ(x)dx B(0,m)

∼

1

1

V (x) q(0) W (t) q (0)

r(0) σ(x)dx

(3.43)

B(0,m)

and

|x|>M

1

1

V (x) q(x) W (t) q (x)

r(x) σ(x)dx

r(∞) 1 1 V (x) q(∞) W (t) q (∞) σ(x)dx.

∼

(3.44)

|x|>M

Let us demonstrate (4)=⇒(3.1). Let q(0) < p(0) be fulﬁlled, and p+ , p− , q , q denote the respective values of p, q taken on the set B(0, m). We can suppose q + < p− if m is sufﬁciently small. Let also m be such that V (m) > 1 (m) < 1. We need to estimate V (x)s(x) ∼ V (x)s(0) for |x| < m and and W s(x) = p(q−1) p−q . For this, we will prove the upper estimate V (t) via W (t) for any ﬁxed t ∈ (0, m). Let k ∈ N be such that +

−

(t) ≤ 2−k . 2−k−1 < W

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(m) < 1 and (4) we have Using V (m) > 1 and W C ≥

V (x)

p p−q

W (x)

p(q−1) p−q

σ(x)dx ≥

B(0,m)

W (x)

q + −1 q+ p−

1−

σ(x)dx

B(0,m)

(m) W

q + −1

1

+ q− 1− (s) 1− qp− dW (s) dx V (s) p+ W (s) W

≥ 0 2−k

≥

V (s)

1 q− 1− + p

(s) W

q + −1 q+ p−

1−

2−k−1 +

−

(p −1) − q p− −q +

≥2

V (x)

1 q− 1− + p

(t) ln 2 · W

+ (p− −1) (s) p+ dW − (k+1)q p− −q + dx ≥ 2 ln 2 · V (t) p+ −q− (s) W

q + (p− −1) p− −q +

p+

V (t) p+ −q− .

Hence (t)−β , V (t) ≤ C W

where β =

q + (p− − 1) p+ − q − · , p− − q + p+

t ∈ (0, m).

(3.45)

Using the previous inequality, according to Lemma 2.1 (p, q ∈ Λ0 −→ s = p p−q ∈ Λ0 ) in a manner similar to in(3.35) we have the estimate p(0)

p

V (x) p−q ∼ V (x) p(0)−q(0) , Using p(0) < q(0) and p, q ∈ Λ0 , we have W (x)

p(q−1) p−q

∼ W (x)

x ∈ B(0, m).

p(q−1) p−q

p(0)(q(0)−1) p(0)−q(0)

,

(3.46)

∈ Λ0 . Hence, by Lemma 2.1,

(x) ∈ B(0, m).

(3.47)

Estimates (3.46) and (3.47) prove (3.43). Hence (4)=⇒ (3.1) is proved. Let us prove (4)=⇒(3.2). Let p− , p+ and q − , q + denote the respective values of the exponents p and q taken on the set Rn \B(0, M ). Let q(∞) < p(∞) be fulﬁlled. We can suppose q + < p− if M is a sufﬁciently larger number. Let also M be such (M ) > 1. We need to estimate W (x)s(x) ∼ W (x)s(∞) that V (M ) < 1 and W for |x| > M and s(x) = p(q−1) p−q . For this, we will prove the upper estimate W (t) via V (t) for any ﬁxed t > M. Let k ∈ N be such that (t) ≤ 2k . 2k−1 < W (M ) > 1 and (4) we have Using V (M ) < 1 and W p(q−1) p C ≥ V (x) p−q W (x) p−q σ(x)dx |x|>M

≥

V (x) |x|>M

1 q+ 1− − p

W (x)

q − −1 q− 1− + p

σ(x)dx

Vol. 66 (2010)

On a Hardy Type General Weighted Inequality ∞

p−

q − (p+ −1) p+ −q −

p−

q − (p+ −1) p+ −q −

(s) V (s) p− −q+ W

≥ (M ) W −k

2

(s) V (s) p− −q+ W

≥ 2−k−1

≥2

(k−1)q − (p+ −1) p+ −q − −

+

(p −1) − q p− −q +

≥2

589

(s) dW dx (s) W (s) dW dx W (s)

p−

(t) p− −q+ ln 2 · W

(t) ln 2 · W

q − (p+ −1) p+ −q −

p−

V (t) p− −q+ .

Hence (t) ≤ C V (t)−α , W

where α =

p+ − q − p− · p− − q + q − (p+ − 1)

(3.48)

Using this estimate and p, q ∈ Z∞ as in (3.41), we have W (x)

p(q−1) p−q

∼ W (x)

p(∞)(q(∞)−1) p(∞)−q(∞)

,

|x| > M.

(3.49)

Also, according to Lemma 2.1 and p, q ∈ Z∞ , we have p(∞)

p

V (x) p−q ∼ V (x) p(∞)−q(∞) ,

|x| > M.

(3.50)

Using (3.49) and (3.50) we have (3.44). (4) =⇒ (3.2) is proved. Hence (4) =⇒ (3) is also proved. The proof of (3) =⇒ (4) is in fact the same as that of (4)=⇒ (3). Proof of (6) =⇒ (5). Let p− , p+ , q − , q + denote the respective values of p, q taken on the set B(0, m). As q(0) < p(0), we can suppose q + < p− if m is (m) < 1. Let (6) sufﬁciently small. Let also m be such that V (m) > 1 and W be fulﬁlled. Let t ∈ (0, m) be any ﬁxed number. Let also 2k−1 < V (t) ≤ 2k (m) < 1, we have for some k ∈ N. Using V (m) > 1 and W r q(p−1) 1 q 1 V p W p vdx = V p−q W p−q vdx C ≥ B(0,m)

B(0,m)

≥

V

1 p+ −1 − q

W

p+ −1 p− −1 q+

∞ vdx ≥ − 2−m0

B(0,m)

∞ ≥ −

p+

(s) V (s) p+ −q− W

q + (p+ −1) p− −q +

2−m0 +

− p+p−q−

≥2

p+

(s) V (s) p+ −q− W

(t) . ln 2 · W

q + (p+ −1) p− −q +

q + (p+ −1) p− −q +

dV (s) V (s)

(k−1)p+ q + (p+ −1) dV (s) (t) p− −q+ ≥ 2 p+ −q− ln 2 · W V (s) p+

V (t) p+ −q− .

Hence (t)−θ , V (t) ≤ C W

where θ =

q + (p+ − 1) p+ − q − · , p− − q + p+

t ∈ (0, m).

(3.51)

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F. I. Mamedov and A. Harman

Using V (m) > 1, by Lemma 2.1 we have s(x) =

q p−q

IEOT

∈ Λ0 . Hence C2

s(x)

V (x)

s(0)

= V (x)

s(x)−s(0)

V (x)

−

≤ V (x)s(0) W (x)

C2 θ 1 ln W (x)

(

s(0)

≤ V (x)

1 V (x) ( W (x) ) ln

) = eC2 V (x)s(0) ,

x ∈ B(0, m).

In the same manner as above, V (x)s(0) ≤ eC2 V (x)s(x) . Hence V (x)s(x) ∼ V (x)s(0) ;

x ∈ B(0, m).

(3.52)

Also using p, q ∈ Λ0 , by Lemma 2.1, W

q(p−1) p−q

∼W

q(0)(p(0)−1) p(0)−q(0)

The equivalence (3.52) and (3.53) gives q(p−1) q V p−q W p−q vdx ∼ V B(0,m)

q(0) p(0)−q(0)

W

q(0)(p(0)−1) p(0)−q(0)

(3.53)

vdx.

B(0,m)

The same approach easily proves that q(p−1) q p−q p−q V W vdx ∼ V |x|>M

x ∈ B(0, m).

,

q(∞) p(∞)−q(∞)

W

q(∞)(p(∞)−1) p(∞)−q(∞)

vdx.

|x|>M

Hence the implication (6) =⇒ (5) is shown.

The proof of (5) =⇒ (6) is similar to (6) =⇒ (5). The implication (3) =⇒ (4) is similar to (4) =⇒ (3). The implication (5) =⇒ (2) follows by using Theorem 2.1 under the conditions (2.9) and (2.8). Acknowledgement The authors express their thanks to the referee for reading carefully the paper manuscript and making some useful suggestions.

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[7] Cruz-Uribe, D., Fiorenza, A., Martell, J.M., P´erez, C.: The boundedness of classical operators on variable LP spaces. Ann. Acad. Sci. Fenn. Math. 31(1), 239–264 (2006) [8] Cruz-Uribe, D., Fiorenza, A., Neigebauer, C.J.: The maximal function on variable Lp spaces. Ann. Acad. Sci. Fenn. Math. 28(1), 223–238 (2003) [9] Diening, L.: Maximal function on generalized Lebesgue spaces Lp(·) . Math. Inequal. Appl. 7(2), 245–253 (2004) [10] Diening, L.: Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces Lp(·) and W k,p(·) . Math. Nachr. 268, 31–43 (2004) [11] Diening, L., R˚ uˇziˇcka, M.: Calderon–Zigmund operators on generalized Lebesgue spaces Lp(·) and problems related to fluid dynamics. J. Reine Angew. Math. 563, 197–220 (2003) [12] Diening, L., H¨ ast¨ o, P., Nekvinda, A.: Open problems in variable exponent Lebesgue and Sobolev spaces. In: FSDONA 04 Proceedings (P. Drabek and J. Rakosnik eds.); Milovy, Czech Republic, 38–58. Academy of Sciences of the Czech Republic, Prague, 2005 [13] Diening, L., Samko, S.: Hardy inequality in variable exponent lebesgue spaces. Fract. Calc. Appl. Anal. 10(1), 1–18 (2007) [14] Edmunds, D.E., Kokilashvili, V., Meskhi, A.: On the boundedness and compactness of weighted hardy operators in spaces Lp(x) . Georgian Math. J. 12(1), 27–44 (2005) [15] Fan, X.-L., Zhao, D.: A class of De Giorgi type and holder continuity. Nonlinear Anal. 36 (3):295–318 (1999) (Ser. A: Theory, Methods & Applications) [16] Fan, X.-L., Zhao, D.: The Quasi-Minimizer of Integral Functionals with m(x) Growth Conditions. Nonlinear Anal. 39(7):807–816 (2000) (Ser. A: Theory, Methods & Applications) [17] Harjulehto, P., H¨ ast¨ o, P., Koskenoja, M.: Hardy’s inequality in a variable exponent Sobolev space. Georgian Math. J. 12(3), 431–442 (2005) [18] Kokilashvili, V., Samko, S.G.: Singular integrals in weighted Lebesgue spaces with variable exponent. Georgian Math. J. 10(1), 145–156 (2003) [19] Kokilashvili, V., Samko, S.G.: Maximal and fractional operators in weighted Lp(x) spaces. Rev. Math. Iberoamericana 20(2), 493–515 (2004) [20] Kovaˇcik, O., Rakosnik, J.: On spaces Lp(x) and W k,p(x) . Czechoslov. Math. J. 41(116)(4), 592–618 (1991) [21] Kufner, A., Maligranda, L., Persson, L.-E.: The Hardy inequality. About its history and some related results. Vydavatelsk y ´ Servis, Plzeˇ n (2007) [22] Kufner, A., Persson, L.-E.: Weighted inequalities of Hardy type. World Scientific Publishing , River Edge (2003) [23] Lerner, A.K.: On modular inequalities in variable Lp spaces. Arch. Math. (Basel) 85(6), 538–543 (2005) [24] Mamedov, F.I., Harman, A.: On a weighted inequality of Hardy type in spaces Lp(·) . J. Math. Anal. Appl. 353(2), 521–530 (2009) [25] Mashiyev, R.A., C ¸ eki¸c, B., Mamedov, F.I., Ogras, S.: Hardy’s inequality in power-type weighted Lp(·) (0, ∞) spaces. J. Math. Anal. Appl. 334(1), 289–298 (2007) [26] Maz’ya, V.G.: Sobolev Spaces. (Translated from Russian) Springer series in Soviet Mathematics. Springer, Berlin (1985)

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[27] Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972) [28] Musielak, J.: Orlicz spaces and modular spaces. Lecture Notes in Mathematics, 1034. Springer, Berlin (1983) [29] Nekvinda, A.: Hardy–Littlewood maximal operator on Lp(x) (R). Math. Inequal. Appl. 7(2), 255–265 (2004) [30] Okpoti, C.A., Persson, L.-E., Sinnamon, G.: An equivalence theorem for some integral conditions with general measures related to Hardy’s inequality. J. Math. Anal. Appl. 326(1), 398–413 (2007) [31] Okpoti, C.A., Persson, L.-E., Sinnamon, G.: An equivalence theorem for some integral conditions with general measures related to Hardy’s inequality. II. J. Math. Anal. Appl. 337(1), 219–230 (2008) [32] Opic, B., Kufner, A.: Hardy-type inequalities. Pitman Research Notes in Mathematics Series, vol. 219. Longman Scientific & Technical, Harlow, 1990 [33] Pick, L., R˚ uˇziˇcka, M.: An example of a space Lp(x) on which the Hardy–Littlewood maximal operator is not bounded. Expo. Math. 19(4), 369–371 (2001) [34] R˚ uˇziˇcka, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Mathematics, vol. 1748. Springer, Berlin (2000) [35] Samko, S.G.: Hardy–Littlewood–Stein–Weiss inequality in the Lebesgue spaces with variable exponent. Fract. Calc. Appl. Anal. 6(4), 421–440 (2003) [36] Samko, S.G.: Hardy inequality in the generalized Lebesgue spaces. Fract. Calc. Appl. Anal. 6(4), 355–362 (2003) [37] Sinnamon, G., Stepanov, V.D.: The weighted hardy inequality: new proofs and the case p = 1. J. Lond. Math. Soc. (2) 54(1), 89–101 (1996) [38] Stepanov, V.D.: Weighted norm inequalities of Hardy type for a class of integral operators. J. Lond. Math. Soc. (2) 50(1), 105–120 (1994) [39] Wedestig, A.: Weighted inequalities of Hardy type and their limiting inequalities. Ph.D. Thesis, Department of Mathematics, Lule˚ a University of Technology, Sweden, 2003 [40] Zhikov, V.V.: On some variational problems. Russ. J. Math. Phys. 5(1), 105– 116 (1998) Farman I. Mamedov and Aziz Harman Dicle University 21280 Diyarbakir, Turkey e-mail: [email protected]; [email protected] Received: May 25, 2009. Revised: January 27, 2010.

Integr. Equ. Oper. Theory 66 (2010), 593–611 DOI 10.1007/s00020-010-1768-9 Published online March 13, 2010 c Birkh¨ auser / Springer Basel AG 2010

Integral Equations and Operator Theory

Toeplitz Operators on the Fock Space Josh Isralowitz and Kehe Zhu Abstract. We study Toeplitz operators on the Fock space with positive measures as symbols. Main results include characterizations of Fock– Carleson measures, bounded Toeplitz operators, compact Toeplitz operators, and Toeplitz operators in the Schatten p-classes. Mathematics Subject Classification (2000). Primary 47B35; Secondary 47B10. Keywords. Fock space, Toeplitz operator, Schatten class.

1. Introduction Throughout the paper we ﬁx a positive parameter α and consider the Gaussian measure 2 α dλα (z) = e−α|z| dA(z) π on the complex plane C, where dA(z) = dxdy is ordinary area measure. For any p > 0, the Fock space Fαp consists of all entire functions f with 2 the property that the function f (z)e−α|z| /2 is in Lp (C, dA). For f ∈ Fαp , we write ⎡ ⎤1/p p 2 pα f p = ⎣ f (z)e−α|z| /2 dA(z)⎦ . 2π C

In particular, Fα2 is just the subspace (with inherited norm and inner product) of all entire functions in L2 (C, dλα ). The inner product in L2 (C, dλα ) (and hence in Fα2 as well) will be denoted by f, g = f (z)g(z) dλα (z). C

It is well known that the orthogonal projection P : L2 (C, dλα ) → Fα2

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IEOT

is an integral operator, namely, P f (z) = K(z, w)f (w) dλα (w), C

where K(z, w) = eαzw is the reproducing kernel of the Hilbert space Fα2 . See [1–3,6]. Given ϕ ∈ L∞ (C), we can deﬁne a linear operator Tϕ : Fα2 → Fα2 by Tϕ (f ) = P (ϕf ),

f ∈ Fα2 .

We call Tϕ the Toeplitz operator on Fα2 with symbol ϕ. It is clear that Tϕ is bounded with Tϕ ≤ ϕ∞ . It is also clear that for any complex numbers a and b and for any bounded functions ϕ and ψ we have Tϕ = Tϕ∗ , Taϕ+bψ = aTϕ + bTψ , and Tϕ ≥ 0 whenever ϕ ≥ 0. By the integral representation for the projection operator P , we can write Tϕ (f )(z) = K(z, w)f (w)ϕ(w) dλα (w). C

This motivates us to deﬁne Toeplitz operators on Fα2 with much more general symbols. More specifically, if µ is a complex Borel measure on C, we deﬁne the Toeplitz operator Tµ as follows: 2 Tµ (f )(z) = K(z, w)f (w)e−α|w| dµ(w), z ∈ C. C

The careful reader will notice a slight deviation of our definition of Tµ from the traditional definition of Toeplitz operators on weighted Bergman spaces that was begun in [7]. Namely, we have included the extra weight factor 2 e−α|w| here. This minor modiﬁcation makes several arguments significantly easier than their counter-parts in the Bergman space theory. Note that Tµ is very loosely deﬁned here, because it is not clear when the integrals above will converge, even if the measure µ is ﬁnite, as the kernel function K(z, w) is unbounded for any ﬁxed z = 0. Suppose that µ is a Borel measure that satisﬁes the condition 2 |K(z, w)|e−α|w| d|µ|(w) < ∞ (1.1) C

for all z ∈ C. Then because of the exponential form of the kernel function, it is clear that condition (1.1) is equivalent to 2 |K(z, w)|2 e−α|w| d|µ|(w) < ∞ (1.2) C

for all z ∈ C.

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If µ satisﬁes condition (1.1), then the Toeplitz operator Tµ is well-deﬁned on a dense subset of Fα2 . In fact, if f (w) =

N

ck K(w, zk )

k=1

is any ﬁnite linear combination of kernel functions in Fα2 , then it follows from (1.2) and the Cauchy–Schwarz inequality that Tµ (f ) is well deﬁned. It is easy to check that the set of all ﬁnite linear combinations of kernel functions is dense in Fα2 . All measures used in the paper will be assumed to satisfy condition (1.1), so that all Toeplitz operators are well-deﬁned. It also follows from condition (1.1) that we can deﬁne a function µ on C as follows: 2 µ (z) = |kz (w)|2 e−α|w| dµ(w), z ∈ C, (1.3) C

where

2 α kz (w) = K(w, z)/ K(z, z) = eαw¯z− 2 |z|

are called normalized reproducing kernels in Fα2 . Alternatively, we can write 2 |K(z, w)|2 dµ(w) = e−α|z−w| dµ(w). (1.4) µ (z) = K(z, z)K(w, w) C

C

We call µ the Berezin transform of µ. If the Toeplitz operator Tµ happens to be a bounded operator on Fα2 , then µ (z) = Tµ kz , kz ,

z ∈ C.

(1.5)

Again, the careful reader will notice that the definition of µ in (1.3) contains 2 an extra factor e−α|w| when compared to the traditional definition of the Berezin transform in the Bergman space setting. In the special case when α dµ(z) = ϕ(z) dA(z), π for µ . In this case we call ϕ the Berezin we have Tµ = Tϕ and we will write ϕ transform of ϕ and observe that 2 α ϕ(z) = ϕ(w)e−α|z−w| dA(w). π C

Therefore, if the measure µ is absolutely continuous as above, all our results can be formulated in terms of the density function ϕ. For any z ∈ C and r > 0 we use B(z, r) = {w ∈ C : |w − z| < r} to denote the Euclidean disc centered at z with radius r. If µ is a locally ﬁnite Borel measure, the function z → µ(B(z, r)) is the constant πr2 times the average of µ over B(z, r). Thus, we will call µ(B(z, r)) an averaging function of µ.

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The purpose of this paper is to provide characterizations for Toeplitz operators Tµ on Fα2 , µ ≥ 0, that are bounded, compact, or in the Schatten classes Sp . These problems have been widely studied in the contexts of Hardy and Bergman spaces on various domains, and a large number of techniques and methods have been developed over the past twenty years or so; see [7,11, 13,14] and references there.. In particular, our characterizations of boundedness and compactness for positive Tµ would have been anticipated by anyone who is familiar with the Bergman space theory. But the characterization of Schatten class Toeplitz operators in terms of the Berezin transform is different in the Fock space setting. More specifically, the cut-oﬀ phenomenon that is often seen in the Bergman space theory (see [13,14]) disappears in the Fock space setting. The main results of the paper can easily be extended to the Fock space in Cn . We chose to develop the theory in dimension one mainly for the sake of clarity. The extension to higher dimensions requires no new ideas. There is only some additional combinatorial complexity in handling the lattices in Cn .

2. Fock–Carleson Measures In this section we characterize Carleson-type measures for Fock spaces. The following pointwise estimate for functions in Fαp will be needed on several occasions later. Lemma 2.1. For any r > 0 and p > 0, there exists a positive constant C such that 2 2 |f (w)e−α|w| /2 |p dA(w) |f (z)e−α|z| /2 |p ≤ C B(z,r)

for all z ∈ C. Proof. We apply a change of variables to the integral p 2 I(z) = f (w)e−α|w| /2 dA(w) B(z,r)

to obtain

|f (z + u)|p e−

I(z) = |u|
= e−

pα 2 2 |z|

pα 2 2 |z+u|

dA(u)

2 f (z + u)e−α¯zu p e− pα 2 |u| dA(u).

|u|
By polar coordinates and the subharmonicity of the function p h(u) = f (z + u)e−α¯zu ,

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we have 2 − pα 2 |z|

I(z) ≥ e

p

|f (z)|

e−

pα 2 2 |u|

597

dA(u).

|u|

This gives the desired result.

The following elementary estimate will also be used many times later on. Lemma 2.2. For any r > 0 there exists a positive constant C = Cr such that µ(B(z, r)) ≤ C µ (z) for all z ∈ C. Proof. Given z ∈ C we have 2 µ(B(z, r)) = dµ(w) ≤ eαr B(z,r)

= eαr

2

≤ eαr

2

2

e−α|z−w| dµ(w)

B(z,r) 2

|kz (w)|2 e−α|w| dµ(w)

B(z,r)

2

|kz (w)|2 e−α|w| dµ(w)

C αr 2

= e

µ (z). 2

Thus, the result holds with C = eαr .

Given any two points z and w in C that are not on a same ray emanating from the origin, the set of points nz + mw, where n and m are arbitrary integers, is called the lattice generated by z and w. For convenience we will write every such lattice as a sequence. The main result of this section is the following. Theorem 2.3. Suppose µ ≥ 0, p > 0, r > 0, and {an } is the lattice in C generated by r and ri. Then the following conditions are equivalent. (a)

There exists a positive constant C such that 2 p 2 p α α f (w)e− 2 |w| dµ(w) ≤ C f (w)e− 2 |w| dA(w) C

(b) (c)

C

for all entire functions f . There exists a constant C > 0 such that µ(B(z, r)) ≤ C for all z ∈ C. There exists a constant C > 0 such that µ(B(an , r)) ≤ C for all n.

Proof. For any entire function I(f ) = C

f we set 2 p α f (w)e− 2 |w| dµ(w).

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Then I(f ) ≤

∞

IEOT

2 p α f (w)e− 2 |w| dµ(w).

n=1 B(an ,r)

By Lemma 2.1 and the triangle inequality, there exists a constant C1 > 0 such that 2 p 2 p α α f (u)e− 2 |u| dA(u) f (w)e− 2 |w| ≤ C1 B(an ,2r)

for all w ∈ B(an , r). If condition (c) holds, then we can ﬁnd a positive constant C2 (independent of f ) such that ∞ 2 p α I(f ) ≤ C2 f (u)e− 2 |u| dA(u) n=1 B(an ,2r)

for all entire functions f . It is clear that there exists a positive integer m such that every point in the complex plane belongs to at most m of the disks B(an , 2r). Therefore, 2 p α I(f ) ≤ C2 m f (u)e− 2 |u| dA(u). C

This shows that condition (c) implies (a). To show that condition (a) implies (b), we consider the normalized reproducing kernels α

2

kz (w) = eαwz− 2 |z| , Since

w ∈ C.

2 p α 2π kz (w)e− 2 |w| dA(w) = pα C

for every z ∈ C, we see that condition (a) implies that there exists a positive constant C such that pα 2 e− 2 |z−w| dµ(w) ≤ C C

for all z ∈ C. In particular,

e−

pα 2 2 |z−w|

dµ(w) ≤ C

B(z,r)

for all z ∈ C. This clearly implies that µ(B(z, r)) ≤ Ce

pα 2 2 r

for all z ∈ C, so that condition (a) implies (b). It is trivial that condition (b) implies (c).

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Note that Lemma 2.1 and the equivalence of (a) and (b) in Theorem 2.3 were proven in both [8,9], where the authors considered Fock–Carleson measures with respect to more general weighted Fock spaces. We have included proofs of these two results, because our arguments are more elementary and less complicated than the ones found in [8,9]. It is interesting to note that conditions (b) and (c) are independent of p and α. It follows that if condition (a) holds for some p > 0 and some α, then it holds for every p and every α (with the constant C dependent on p and α). It is also interesting to note that condition (a) is independent of r. Therefore, if condition (b) or condition (c) holds for some r > 0, then it holds for every r > 0 (with the constant C dependent on r). In view of the remarks above, we will call any positive Borel measure µ that satisﬁes condition (a), (b), or (c) a Fock–Carleson measure. Similarly, we say that a positive Borel measure µ on C is a vanishing Fock–Carleson measure if 2 p α lim fn (z)e− 2 |z| dµ(z) = 0 n→∞

C

whenever {fn } is a bounded sequence in Fαp that converges to 0 uniformly on compact subsets of the complex plane. Theorem 2.4. Suppose µ ≥ 0, p > 0, r > 0, and {an } is the lattice in C generated by r and ri. Then the following conditions are equivalent. (i) µ is a vanishing Fock–Carleson measure. (ii) µ(B(z, r)) → 0 as z → ∞. (iii) µ(B(an , r)) → 0 as n → ∞. The proof can easily be adapted from that of Theorem 2.3, following essentially the same ideas used in [13] for the Bergman space theory. The routine details are omitted here.

3. Boundedness and Compactness The next result characterizes bounded Toeplitz operators with positive symbols and gives several other equivalent conditions for a positive Borel measure on C to be Fock–Carleson. Theorem (a) The (b) The (c) The

3.1. The following conditions are equivalent for µ ≥ 0. Toeplitz operator Tµ is bounded on Fα2 . Berezin transform µ is bounded on C. measure µ is Fock–Carleson.

Proof. If Tµ deﬁnes a bounded linear operator on Fα2 , then it is easy to check that 2 (3.1) Tµ f, g = f (w)g(w)e−α|w| dµ(w) C

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J. Isralowitz and K. Zhu

for all f and g ∈ Fα2 . In particular, 2 2 2 α Tµ f, f = |f (w)|2 e−α|w| dµ(w) = f (w)e− 2 |w| dµ(w) C

IEOT

(3.2)

C

Fα2 .

If we set f = kz in (3.2), where kz are the normalized reprofor all f ∈ ducing kernels in Fα2 , then we obtain µ (z) = Tµ kz , kz ,

z ∈ C.

It follows from the Cauchy–Schwarz inequality that the boundedness of Tµ on Fα2 implies that 0 ≤ µ (z) ≤ Tµ kz kz ≤ Tµ for all z ∈ C, which proves that condition (a) implies condition (b). That condition (b) implies (c) follows from Lemma 2.2 and Theorem 2.3. Finally, if condition (c) holds, then by the definition of µ ˜, we have that µ ˜ is bounded. By an argument that is identical to the proof of Lemma 14 in [2], we have that Tµ : Fα2 → L2 (C, dλα ) is bounded, with operator norm comparable to sup{˜ µ(z) : z ∈ C}. However, it is easy to check, by Fubini’s theorem, that P Tµ = Tµ , which means that Tµ : Fα2 → Fα2 is bounded. As a consequence of the characterization of vanishing Fock–Carleson measures in Theorem 2.4, we obtain several characterizations for compact Toeplitz operators on the Fock spaces, which in turn gives additional characterizations for vanishing Fock–Carleson measures. Theorem 3.2. The following conditions are equivalent for µ ≥ 0. (a) Tµ is compact on Fα2 . (b) µ (z) → 0 as z → ∞. (c) µ is a vanishing Fock–Carleson measure. We omit the proof of the above theorem, again because it can be easily adapted from the proof of Theorem 3.1 following the same ideas from the Bergman space theory. Carefully examining the proof of Theorem 3.1, we see that the following quantities are equivalent. (a) µ1 = Tµ , the norm of Tµ on Fα2 . µ(z) : z ∈ C}. (b) µ2 = sup{ (c) µ3 = sup{µ(B(z, r)) : z ∈ C}, where r is any ﬁxed positive radius. The above quantities are also equivalent to ⎧ ⎫ p ⎨ ⎬ 2 µ4 = sup f (z)e−α|z| /2 dµ(z) : f p = 1 . ⎩ ⎭ C

From now on, whenever unspeciﬁed, the notation µ will denote (any) one of the above quantities. Theorem 3.3. The following conditions are equivalent for µ ≥ 0. (a) µ is a vanishing Fock–Carleson measure.

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µ − µR → 0 as R → ∞, where µR is the truncation of µ on the disk B(0, R). There exists a sequence of finite Borel measures µn , each with compact support, such that µ − µn → 0 as n → ∞.

Proof. If µ is a vanishing Fock–Carleson measure, then µ(B(z, r)) → 0 as z → ∞. It follows that µ − µR 3 → 0 as R → ∞. This shows that condition (a) implies (b). Since each µR has compact support, it is obvious that condition (b) implies (c). If µn is a ﬁnite Borel measure with compact support, then it is clear that Tµn is compact. Condition (c) implies that Tµn − Tµ → 0 as n → ∞, which in turn implies that Tµ is compact. Thus condition (c) implies (a).

4. Toeplitz Operators in Sp with p ≥ 1 We are going to determine when a Toeplitz operator Tµ on Fα2 belongs to the Schatten class Sp . This section is devoted to the case p ≥ 1, while the next section concerns the case 0 < p ≤ 1. Background information about the Schatten classes Sp can be found in [13] for example. For any bounded linear operator T on Fα2 , we can deﬁne the Berezin transform T by. T(z) = T kz , kz ,

z ∈ C,

(4.1)

where kz are the normalized reproducing kernels in Fα2 . If T is positive on Fα2 , then mimicking the proof of Theorem 6.4 in [13] we can show that α T(z) dA(z). (4.2) tr (T ) = π C

In particular, T is in the trace class S1 if and only if the integral above converges. As a consequence of (4.2), we obtain the following trace formula for Toeplitz operators on Fock spaces. Proposition 4.1. Suppose µ ≥ 0. Then Tµ is in the trace class S1 if and only if µ is finite on C. Moreover, tr (Tµ ) = µ(C). Proof. Since all integrands below are nonnegative, we use Fubini’s theorem to obtain α tr (Tµ ) = µ (z) dA(z) π C 2 2 α|z|2 = e dλα (z) |eα¯zw |2 e−α(|z| +|w| ) dµ(w) C

C

602

J. Isralowitz and K. Zhu =

−α|w|2

e

dµ(w)

C

|eα¯zw |2 dλα (z)

C

=

IEOT

dµ(w) = µ(C). C

This also shows that tr (Tµ ) < ∞ if and only if µ(C) < ∞.

Lemma 4.2. If p ≥ 1 and ϕ ∈ Lp (C, dA), then Tϕ ∈ Sp . Proof. This is proved in exactly the same way that Proposition 7.11 in [13] was proved. Lemma 4.3. Suppose r > 0, µ is a positive Borel measure on C, and µ r (z) = µ(B(z, r))/(πr2 ),

z ∈ C.

If µ r is in Lp (C, dA), then Tµ and Tµr are bounded on Fα2 . Moreover, there exists a positive constant C (independent of µ) such that Tµ ≤ CTµr . Proof. Since µ r is in Lp (C, dA), an easy application of Theorem 3.1 gives us that Tµ is bounded. Moreover, another application of Theorem 3.1 tells us that Tµr is bounded. Thus, given f ∈ Fα2 , we use Fubini’s theorem to obtain 2 2 πr Tµr f, f = πr |f (z)|2 µ r (z) dλα (z) =

C

|f (z)|2 µ(B(z, r)) dλα (z)

C

2

|f (z)| dλα (z)

= C

|f (z)|2 χB(w,r) (z) dλα (z)

dµ(w)

= C

α = π

χB(z,r) (w) dµ(w) C

C

dµ(w) C

|f (z)e−α|z|

2

/2 2

| dA(z).

B(w,r)

Combining the above identity with Lemma 2.1, we obtain a positive constant C such that 2 CTµr f, f ≥ |f (w)|2 e−α|w| dµ(w) = Tµ f, f . C

This proves the desired result.

We are now ready to prove the main result of the section. Theorem 4.4. Suppose µ ≥ 0, p ≥ 1, r > 0, and {an } is the lattice in C generated by r and ri. Then the following conditions are equivalent. (a) The operator Tµ is in the Schatten class Sp . (b) The function µ (z) is in Lp (C, dA).

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The function µ(B(z, r)) is in Lp (C, dA). The sequence {µ(B(an , r))} is in lp .

Proof. That (a) implies (b) follows from the ﬁrst part of Theorem 6.6 in [13]. Lemma 2.2 shows that condition (b) implies (c). If the averaging function µ r (z) is in Lp (C, dA), then it follows from Lemma 4.2 that Tµr is in Sp . Combining this with Lemma 4.3, we conclude that Tµ is in Sp . This proves that (c) implies (a). Hence conditions (a), (b), and (c) are equivalent. To prove that condition (d) is equivalent to the other conditions, we ﬁrst assume that condition (b) holds, which implies that the function µ(B(z, 2r)) is in Lp (C, dA). Choose a positive integer m such that each point in the complex plane belongs to at most m of the discs B(an , r). Then ∞ µ(B(z, 2r))p dA(z). m µ(B(z, 2r))p dA(z) ≥ n=1 B(an ,r)

C

For each z ∈ B(an , r) we deduce from the triangle inequality that µ(B(z, 2r)) ≥ µ(B(an , r)). Therefore, m

µ(B(z, 2r))p dA(z) ≥ πr2

∞

µ(B(an , r))p .

n=1

C

This shows that condition (b) implies (d). To ﬁnish the proof, we assume that condition (d) holds, that is, ∞

µ(B(an , r))p < ∞.

n=1

It is easy to see that we also have ∞

µ(B(zn , r))p < ∞

n=1

where {zn } is the lattice generated by r/2 and ir/2. In fact, for each point zk that is not in the lattice {an }, the disc B(zk , r) is covered by six adjacent discs B(ak , r). Therefore, ∞ µ(B(z, r/2))p dA(z) ≤ µ(B(z, r/2))p dA(z) n=1 B(zn ,r/2)

C

≤

∞

µ(B(zn , r))p dA(z)

n=1 B(zn ,r/2)

=

∞ πr2 µ(B(zn , r))p < ∞. 4 n=1

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This shows that condition (d) implies (c), as the equivalence of (c) to (b) implies that if condition (c) holds for one positive radius, then it will hold for any other positive radius. This completes the proof of the theorem.

5. Toeplitz Operators in Sp with 0 0, and {an } is the lattice in C generated by r and ri. Then the following conditions are equivalent. (a) The function µ (z) is in Lp (C, dA). (b) The function µ(B(z, r)) is in Lp (C, dA). (c) The sequence {µ(B(an , r))} is in lp . Proof. We begin with the inequality ∞ 2 µ (z) = e−α|z−w| dµ(w) ≤

2

e−α|z−w| dµ(w).

n=1 B(an ,r)

C

For w ∈ B(an , r) we have |z − w|2 ≥ (|z − an | − |an − w|)2 ≥ |z − an |2 − 2r|z − an |. It follows that µ (z) ≤

∞

e−α|z−an |

2

+2αr|z−an |

µ(B(an , r)).

n=1

Since 0 < p ≤ 1, H¨ older’s inequality gives µ (z)p ≤

∞

e−pα|z−an |

2

+2prα|z−an |

µ(B(an , r))p .

n=1

It follows from this and Fubini’s theorem that ∞ 2 µ (z)p dA(z) ≤ µ(B(an , r))p e−pα|z−an | +2prα|z−an | dA(z). n=1

C

C

By an obvious change of variables, the integral above equals 2 e−pα|z| +2prα|z| dA(z), C

which is easily seen to be convergent. This shows that {µ(B(an , r))} ∈ lp implies that µ ∈ Lp (C, dA). On the other hand, there exists a positive integer m such that every point in the complex plane belongs to at most m of the discs B(an , r). Thus, ∞ m µ (z)p dA(z) ≥ µ (z)p dA(z). C

n=1 B(an ,r)

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For any z ∈ B(an , r), we have 2 µ (z) = e−α|z−w| dµ(w) ≥

C

B(an ,r)

605

2

e−α|z−w| dµ(w)

2

≥ e−4αr µ(B(an , r)). It follows that

m

µ (z)p dA(z) ≥ πr2 e−4pαr

2

∞

µ(B(an , r))p .

n=1

C

Thus µ ∈ Lp (C, dA) implies that {µ(B(an , r))} ∈ lp , which proves the equivalence of conditions (a) and (c). That condition (a) implies (b) follows from Lemma 2.2. To prove that condition (b) implies (c), we assume that the function µ(B(z, r)) is in Lp (C, dA). Consider the lattice generated by r/2 and ri/2 and arrange it into a sequence {zn }. There exists a positive integer m such that every point in the complex plane belongs to at most m of the discs B(zn , r/2). Therefore, ∞ µ(B(z, r))p dA(z). m µ(B(z, r))p dA(z) ≥ n=1 B(zn ,r/2)

C

For each z ∈ B(zn , r/2), the triangle inequality gives us that µ(B(z, r)) ≥ µ(B(zn , r/2)). Thus

m

µ(B(z, r))p dA(z) ≥

C

∞ πr2 µ(B(zn , r/2))p . 4 n=1

By the equivalence of conditions (a) and (c), the function µ belongs to Lp (C, dA), and applying the equivalence of (a) and (c) once more, we con clude that {µ(B(an , r))} ∈ lp . This completes the proof of the lemma. Lemma 5.2. Suppose µ ≥ 0, 0 < p ≤ 1, and the function µ belongs to Lp (C, dA). Then the operator Tµ belongs to Sp . Proof. Since µ belongs to Lp (C, dA), it is not difﬁcult to see that Tµ is µ = µ ˜. Thus the lemma can be proven in exactly the bounded, so that T same way that part (b) of Theorem 6.6 in [13] was proved. We will need the following Lemma, whose proof can be found in [13]. Lemma 5.3. If 0 < p ≤ 1, then for any orthonormal basis {en } of a separable Hilbert space H and any compact operator T on H, we have that T pSp ≤

∞ ∞ n=1 k=1

|T en , ek |p .

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We are now ready to characterize Toeplitz operators Tµ in Sp when 0 < p ≤ 1. The well-informed reader will notice that several key ideas in the proof of the following theorem are borrowed from the Bergman space theory, but several new ideas and techniques are needed here. For example, the wellknown Forelli–Rudin integral estimate that is so often used in the Bergman space theory is lacking in the Fock space setting. Theorem 5.4. Suppose µ ≥ 0, 0 0, and {an } is the lattice in C generated by r and ri. Then the following conditions are equivalent. (a) (b) (c) (d)

Tµ belongs to the Schatten class Sp . µ belongs to Lp (C, dA). µ r belongs to Lp (C, dA). { µr (an )} belongs to lp .

Proof. The equivalence of (b), (c), and (d) was proved in Lemma 5.1. That condition (b) implies condition (a) was proved in Lemma 5.2. Therefore, to ﬁnish the proof, we will show that condition (a) implies (d). In what follows, C1 , C2 , . . . will denote positive constants that only depend on p, α, and r. For convenience, we will use the norm |z|∞ = max{|x|, |y|}, where z = x+iy, and for the rest of the proof, we will let B(z, r) denote the closed ball centered at z with radius r in this norm. If we can prove that condition (a) implies ∞

µ(B(2an , r))p < ∞,

n=1

then condition (d) will easily follow. To this end, ﬁx some large R > 0 and partition {2an } into N subsequences such that the Euclidean distance between any two points in each subsequence is at least R. Let {ζn } be such a subsequence and let ν=

∞

µχn ,

n=1

where χn is the characteristic function of B(ζn , r). Since Tµ ∈ Sp and µ ≥ ν, we have Tν ≤ Tµ , and so Tν ∈ Sp with Tν Sp ≤ Tµ Sp . Let {en } be an orthonormal basis for Fα2 and deﬁne a bounded linear operator A on Fα2 by Aen = kζn , n ≥ 1. Let T = A∗ Tν A so that T Sp ≤ Tµ Sp . We split the operator T as T = D + E where D is the diagonal operator deﬁned on Fα2 by Df =

∞

T en , en f, en en ,

n=1

and E = T − D. By the triangle inequality, we have T pSp ≥ DpSp − EpSp .

(5.1)

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Since D is a positive diagonal operator, we have DpSp = =

∞

T en , en p =

n=1 ∞ n=1

⎛ ⎝ ⎛

⎞p

2

e−α|z−ζn | dν(z)⎠

C

⎞p 2 ⎟ e−α|z−ζn | dν(z)⎠

B(ζn ,r)

∞

≥ C1

Tν kζn , kζn p

n=1

∞ ⎜ ≥ ⎝ n=1

∞

ν(B(ζn , r))p .

(5.2)

n=1

On the other hand, by Lemma 5.3, we have EpSp ≤

∞ ∞

|Een , ek |p =

n=1 k=1

|Tν kζn , kζk |p

n=k

p 2 −α|z| = dν(z) . kζn (z)kζk (z)e n=k

(5.3)

C

A straightforward calculation shows that 2

kζn (z)kζk (z)e−α|z| = e−

α|z−ζn |2 2

e−

α|z−ζk |2 2

eαi Im (zζn +¯zζk ) ,

so that (5.3) gives us EpSp

≤

n=k

⎛ ⎞p α|z−ζk |2 α|z−ζn |2 ⎝ e− 2 e− 2 dν(z)⎠ .

(5.4)

C

If n = k, then |ζn −ζk | ≥ R. Thus for |z −ζn | ≤ R2 the triangle inequality gives us |z − ζk | ≥ R2 . Therefore, for each z ∈ C, we have e−

α|z−ζn |2 2

e−

α|z−ζk |2 2

≤ e−

αR2 16

e−

α|z−ζn |2 4

e−

α|z−ζk |2 4

.

Plugging this into (5.4), we obtain ⎛ ⎞p α|z−ζk |2 α|z−ζn |2 pαR2 p ⎝ e− 4 ESp ≤ e− 16 e− 4 dν(z)⎠ . n=k

C

For each m in {0, 1, 2, . . .} and n ∈ N, let Em,n = {z : r(2m − 1) ≤ |z − ζn |∞ < r(2m + 1)}.

(5.5)

608

J. Isralowitz and K. Zhu

IEOT

Since 0 < p ≤ 1, we have that ⎛ ⎞p α|z−ζk |2 α|z−ζn |2 ⎝ e− 4 e− 4 dν(z)⎠ n=k

C

⎛

≤

∞

⎜ ⎝

n=k m=0 ∞

≤ C2

⎞p e−

α|z−ζn |2 4

Em,n

e−

pαr 2 (2m−1)2 4

m=0

⎟ dν(z)⎠

α|z−ζk |2 − 4

e

⎛ ⎜ ⎝ n=k

⎞p α|z−ζk |2 − 4

e

⎟ dν(z)⎠

(5.6)

Em,n

for some constant C2 . For any ﬁxed m and n we write N = Ω1m,n ∪ Ω2m,n , where Ω1m,n = {k ∈ N : |ζn − ζk |∞ ≤ 2rm}, and Ω2m,n = {k ∈ N : |ζn − ζk |∞ > 2rm}. Therefore, we have that ∞

e−

pαr 2 (2m−1)2 4

m=0

≤

⎛ ⎜ ⎝ n=k

∞

2 2 − pαr (2m−1) 4

e

m=0

+

∞

⎞p α|z−ζk |2 − 4

e

Em,n

⎛ ∞ ⎜ ⎝ n=1 k∈Ω1m,n

e−

pαr 2 (2m−1)2 4

m=0

⎟ dν(z)⎠

∞ n=1

⎞p α|z−ζk |2 − 4

e

⎟ dν(z)⎠

Em,n

⎛

⎜ ⎝ k∈Ω2m,n

⎞p

α|z−ζk |2 − 4

e

⎟ dν(z)⎠

Em,n

= S1 + S2 .

(5.7)

Since card Ω1m,n ≤ C3 (m + 1)2 for some C3 > 0, we have that S1 ≤ C4 ≤ C4

∞

(m + 1)2 e−

m=0 ∞

(m + 1)2 e−

pαr 2 (2m−1)2 4

pαr 2 (2m−1)2 4

m=0

≤ C5

∞

n=1 ∞

ν(Em,n )p

n=1 {k : |2ak −ζn |=2rm} 4 −

(m + 1) e

m=0

∞

pαr 2 (2m−1)2 4

∞ k=1

ν(B(2ak , r))p

ν(B(2ak , r))p

Vol. 66 (2010) ≤ C6 = C6

Toeplitz Operators on Fock Spaces

∞

609

ν(B(2ak , r))p

k=1 ∞

ν(B(ζk , r))p .

(5.8)

k=1

Now we will estimate the sum S2 . Observe that if k ∈ Ω2m,n , then z ∈ Em,n implies that |z − ζk |∞ ≥ |ζk − ζn |∞ − |z − ζn |∞ > |ζk − ζn |∞ − r(2m + 1) > 0. So we have that ∞ ∞ 2 2 pα(|ζk −ζn |∞ −2rm)2 − pαr (2m−1) 4 4 S2 ≤ e ν(Em,n )p e− m=0

≤ C7 ≤ C8

n=1

∞

e−

∞

pαr 2 (2m−1)2 4

m=0 ∞

k∈Ω2m,n

ν(Em,n )p

n=1 2 2 − pαr (2m−1) 4

(m + 1)2 e

m=0

≤ C9

∞

≤ C10 = C10

k=1 ∞

e−

pα(2rk)2 4

(m + k + 1)2

k=1

∞

ν(B(2ak , r))p

n=1 {k : |2ak −ζn |=2rm}

(m + 1)4 e−

m=0 ∞

∞

pαr 2 (2m−1)2 4

∞

ν(B(2ak , r))p

k=1

ν(B(2ak , r))p ν(B(ζk , r))p .

(5.9)

k=1

Combining (5.8) and (5.9) with (5.5), (5.6), and (5.7), we obtain EpSp ≤ C11 e

−pαR2 16

∞

ν(B(ζk , r))p .

k=1

Going back to (5.1) and (5.2), we deduce that Tµ pSp ≥ DSp − ESp ≥ (C1 − C11 e

−pαR2 16

)

∞

ν(B(ζk , r))p .

k=1

Since C1 and C11 do not depend on R, setting R > 0 large enough gives us ∞ k=1

ν(B(ζk , r))p ≤ C12 Tµ pSp .

Since this holds for each of the N subsequences of {2an }, we obtain ∞ n=1

µ(B(2an , r))p ≤ C12 N Tµ pSp

(5.10)

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J. Isralowitz and K. Zhu

IEOT

for all positive Borel measures µ such that ∞ µ(B(2an , r))p < ∞. n=1

However, an easy approximation argument shows that (5.10) holds for all positive Borel measures µ with Tµ ∈ Sp . This completes the proof that condition (a) implies (d), and thus completes the proof of Theorem 5.4.

6. Some Examples In this section we will present examples to illustrate that several results proved earlier in the paper are sharp. First, we show that the conclusion in Lemma 4.2 is false for every 0 < p < 1. To see this, consider the set K ⊂ R2 = C given by ∞ 1 1 2k, 2k + k − 2p × 2k, 2k + k − 2p , K= k=1

and consider the characteristic function ϕ = χK of K. Clearly, ϕ is in Lp (C, dA) for 0 < p < 1. However, ∞

ϕ r (an )p = +∞,

n=1

which means that Tϕ is not in Sp . Similar examples can easily be constructed in higher dimensions. By an argument that is identical to the proof of Proposition 7.15 in [13], we have the following result. Proposition 6.1. Suppose ϕ ≥ 0, 0 1. If fact, if p > 1 and if we deﬁne 2

ϕ(z) = χ[0,1] (|z|)|z|− p ,

(6.1)

p

operator then ϕ ∈ / L (C, dA). On the other hand, as ϕ is radial, the Tϕ is k α k of diagonal with respect to the standard orthonormal basis n! z k≥0

Fα2 . One can then easily check that Tϕ ∈ Sp for each p > 1. Finally in this section we show that if we drop the assumption that f be non-negative, then Theorems 3.2, 4.4, and 5.4 can fail dramatically. To see this, we use the following example from [3], 1

2

2

f (z) = e( 5 +i 5 )|z| . It can be checked that the diagonal operator Tf (with respect to the standard monomial basis) is unitary, and that 3 4 (− 15 +i 25 )|z|2 +i f(z) = . e 5 5

Vol. 66 (2010)

Toeplitz Operators on Fock Spaces

611

Clearly f vanishes at inﬁnity and f is in Lp (C, dA) for each 0 < p < ∞. This tells us that the Berezin transform of f vanishing at inﬁnity is in general not sufﬁcient for Tf to be compact on Fα2 , and the Berezin transform being in Lp (C, dA) is in general not sufﬁcient for Tf to be in the Schatten class Sp .

References [1] Berger, C., Coburn, L.: Toeplitz operators and quantum mechanics. J. Funct. Anal. 68, 273–299 (1986) [2] Berger, C., Coburn, L.: Toeplitz operators on the Segal-Bargmann space. Trans. Am. Math. Soc. 301, 813–829 (1987) [3] Berger, C., Coburn, L.: Heat flow and Berezin-Toeplitz estimates. Am. J. Math. 116, 563–590 (1994) [4] Berger, C., Coburn, L., Zhu, K.: Toeplitz operators and function theory in n-dimensions. Springer Lect. Notes Math. 1256, 28–35 (1987) [5] Coburn, L., Isralowitz, J., Li, B.: Toeplitz operators with BMO symbols on the Segal-Bargmann space. Preprint (2008) [6] Janson, S., Peetre, J., Rochberg, R.: Hankel forms and the Fock space. Rev. Math. Iber. 3, 61–138 (1987) [7] Luecking, D.: Trace ideal criteria for Toeplitz operators. J. Funct. Anal. 73, 345–368 (1987) [8] Oleinik, V.: Embedding theorems for weighted classes of harmonic and analytic functions. J. Soviet Math. 9, 228–243 (1978) [9] Ortega-Cerda, J.: Sampling measures. Publ. Mat. 42, 559–566 (1998) [10] Tung, Y.: Fock spaces. Ph.D. dissertation, University of Michigan (2005) [11] Zhu, K.: Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains. J. Oper. Theory 20, 329–357 (1988) [12] Zhu, K.: On certain unitary operators and composition operators. Proc. Symp. Pure Math. 51(part 2), 371–385 (1990) [13] Zhu, K.: Operator Theory in Function Spaces, 2nd edn. Mathematical Surveys and Monographs, vol. 138. American Mathematical Society (2007) [14] Zhu, K.: Schatten class Toeplitz operators on weighted Bergman spaces of the unit ball. N. Y. J. Math. 13, 299–316 (2007) Josh Isralowitz Department of Mathematics State University of New York at Buffalo Buffalo, NY 14260, USA e-mail: [email protected] Kehe Zhu Department of Mathematics State University of New York at Albany Albany, NY 12222, USA e-mail: [email protected] Received: June 10, 2009. Revised: October 28, 2009.