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Analysis in Theory and Applications
A BMO ESTIMATE FOR THE MULTILINEAR SINGULAR INTEGRAL OPERATOR Qihui Zhang (University of Information Engineering, China) Received Dec. 14, 2005;
Revised June 15, 2006
Abstract The behavior on the space L∞ (Rn ) for the multilinear singular integral operator defined by Z ´ Ω(x − y) ` A(x) − A(y) − ∇A(y)(x − y) f (y) dy TA f (x) = n+1 |x − y| Rn is considered, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishing moment of order one, A has derivatives of order one in BMO(Rn ). It is proved that if Ω satisfies some minimum size condition and an L1 -Dini type regularity condition, then for f ∈ L∞ (Rn ), TA f is either infinite almost everywhere or finite almost everywhere, and in the latter case, TA f ∈ BMO(Rn ).
Key words
multilinear singular integral operator, L1 -Dini type regularity condition
AMS(2000) subject classification
42B20
1
Introduction
We will work on Rn , n ≥ 2. Let Ω be homogeneous of degree zero, integrable on the unit sphere S n−1 and satisfy the vanishing condition Ω(θ)θ dθ = 0.
(1)
S n−1
Let A be a function whose derivatives of order one belong to the space BMO(Rn ). For x, y ∈ Rn , set R(A; x, y) = A(x) − A(y) − ∇A(y)(x − y).
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Define the multilinear singular integral operator TA by Ω(x − y) R(A; x, y)f (y) dy. TA f (x) = p. v. |x − y|n+1 n R
(2)
A well known result of Cohen[1] states that if Ω ∈ Lip1 (S n−1 ), then TA is a bounded operator on Lp (Rn ) with bound C∇ABMO(Rn ) for 1 < p < ∞, and TA f exists almost everywhere for q n−1 f∈ Lp (Rn ). Hofmann[2] improved the result of Cohen, and proved that Ω ∈ L (S ) 1
q>1
is a sufficient condition such that TA is bounded on Lp (Rn ) for 1 < p < ∞. Hu[3] observed that if Ω satisfies the size condition sup |Ω(θ)| logβ (|θ · ξ|−1 ) dθ < ∞ for some β > 3, ξ∈S n−1
S n−1
then TA is bounded on L2 (Rn ). There are many papers concerning the operator TA , among other things, we refer the references [4,5]. The purpose of this paper is to consider the existence and the behavior on L∞ (Rn ) for the multilinear singular integral operator TA . It is easy to see that if f ∈ L∞ (Rn ), the integral Ω(x − y) R(A; x, y)f (y) dy n+1 |x−y|> |x − y| may be divergent and hence the above definition no longer makes sense. We now define TA f by Ω(x − y) R(A; x, y)f (y) dy, (3) TA f (x) = lim n+1 →0 <|x−y|
where the limit is taken in pointwise sense and such a definition coincides with the earlier one for f ∈ Lp (Rn ). We will show that for f ∈ L∞ (Rn ), TA f (x) is either infinite almost every1
where or finite almost everywhere. We remark that in this paper, we are very much motivated by the work of Sun and Zhang[6] . Some ideas are from Hu’s paper [7]. Before stating our result, we first give some definitions. The operators defined by Ω(x − y) R(A; x, y)f (y) dy TA∞ f (x) = lim N →∞ 1<|x−y|
→0
<|x−y|<1
Ω(x − y) R(A; x, y)f (y) dy |x − y|n+1
will be useful in the proof of our theorem. Note that TA0 f (x) exists almost everywhere in Rn , thus if TA f (x) exists almost everywhere, then TA∞ f (x) also exists almost everywhere, and vice versa. For Ω ∈ L1 (S n−1 ), define the L1 modulus of continuity of Ω by |Ω(ρx) − Ω(x)| dx, ω(δ) = sup |ρ|≤δ
S n−1
Q. H. Zhang : A BMO Estimate for the Multilinear Singular Integral Operator
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where |ρ| denotes the distance of ρ from the identity rotation, and the supremum is taken over all rotations on the unit sphere and |ρ| ≤ δ. Our main result can be stated as follows. Theorem 1. Let TA be the multilinear singular integral operator defined by (3), Ω be homogeneous of degree zero, integrable on the unit sphere and satisfy the vanishing condition (1), A have derivatives of order one in BMO(Rn ). Suppose for some q > 2, Ω ∈ L(logL)q (S n−1 ),namely, |Ω(x)| logq (2 + |Ω(x)|) dx < ∞, S n−1
1
and the L modulus of continuity of Ω satisfies 1 1 dδ < ∞. ω(δ) log 2 + δ δ 0 Then for any f ∈ L∞ (Rn ), TA f (x) is either infinite almost everywhere or finite almost everywhere. Furthermore, if f ∈ L∞ (Rn ) such that TA∞ f (x0 ) exists for some x0 ∈ Rn , then TA f (x) is finite almost everywhere and TA f BMO(Rn ) ≤ C∇ABMO(Rn ) f L∞(Rn ) . Remark 1. We point out the relation of the conditions needed for Ω in the references mentioned above and in Theorem 1 as follows. (i) It is easy to see that Lip1 (S n−1 ) ⊂
Lq (S n−1 ) ⊂
q>1
(ii) If Ω ∈
q>1
L(log L)β (S n−1 ).
β>1
Lq (S n−1 ), then for any β > 3, the following inequality holds sup ξ∈S n−1
S n−1
|Ω(θ)| logβ (|θ · ξ|−1 ) dθ < ∞.
(iii) If Ω ∈ Lip1 (S n−1 ) and the kernel function K(x, y) = Ω(x − y)/(|x − y|n ), then the conditions needed for K(x, y) in [5] hold directly. Throughout this paper, C denotes the constants that are independent of the main parameters involved but whose values may differ from line to line. For a measurable set E, χE denotes the characteristic function of E. For a ball B = B(x, r), kB denotes the ball B(x, kr). For a local integrable function f and a ball B, mB (f ) denotes the mean value of f over B, namely, 1 mB (f ) = f (y) dy. For a suitable function f , a bounded measurable set E and γ ≥ 1, |B| B define
|f (y)| 1 |f (y)| logγ 2 + f L(log L)γ , E = inf λ > 0 : dy ≤ 1 , |E| E λ λ
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and
f exp L1/γ , E
1 = inf λ > 0 : |E|
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1/γ |f (y)| exp dy ≤ 2 . λ E
Recall that Φ(t) = t logγ (2 + t) is a Young function on [0, ∞) and its complementary Young function is Ψ(t) ≈ exp(t1/γ ), so the generalization of the H¨ older inequality 1 |f (x)h(x)| dx ≤ Cf L(log L)γ , E hexp L1/γ , E |E| E
(4)
holds for any bounded measurable set E, suitable functions f and h, see [3, p. 278] for details. Also, note that for any t1 , t2 with 0 < t1 , t2 < ∞,
t1 t2 ≤ C t1 log(2 + t1 ) + exp t2 . Consequently, for any a > 0 and suitable functions f and h, −1 |f (x)h(x)| dx ≤ C |f (x)| log(2 + a|f (x)|) dx + Ca exp(|h(x)|) dx. E
E
(5)
E
2
Proof of Theorem 1
We begin with some preliminary lemmas. Lemma 1[3] . Let A be a function on Rn with derivatives of order one in Lp (Rn ) for some p > n. Then
1 1/p |A(x) − A(y)| ≤ C|x − y| |∇A(z)|p dz , y |Ix | Ixy
√ where Ixy is the cube centered at x with sides parallel to the axes and side length is 2 n|x − y|. Lemma 2.
Let Ω be homogeneous of degree zero, integrable on the unit sphere. If there
exists a constant α > 0 such that |y| < αR, then |Ω(x − y) − Ω(x)| dδ dx ≤ C ω(δ) . n |x| δ R<|x|<2R |y|/(2R)<δ<|y|/R For the proof of this lemma, see Lemma 5 in [9]. Lemma 3. Under the hypotheses of Theorem 1, set K(x, z) = then for any x, x0 ∈ Rn ,
Ω(x − z) R(A; x, z), |x − z|n+1
lim
N →∞
|x−z|>N
|K(x, z) − K(x0 , z)| dz = 0.
Q. H. Zhang : A BMO Estimate for the Multilinear Singular Integral Operator
Proof.
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The case x = x0 is obvious. Let us consider the case x = x0 . Set r = |x − x0 |.
Without loss of generality, we may assume that ∇ABMO(Rn ) = 1 and N ≥ 4r. Note that for each fixed x, x0 ∈ Rn and z ∈ Rn \B(x, 4r), |x0 − z|/2 < |x − z| < 2|x0 − z|. A straightforward computation shows that |K(x, z) − K(x0 , z)| ≤
|Ω(x − z) − Ω(x0 − z)| |R(A; x, z)| |x − z|n+1 |Ω(x0 − z)| + |x − x0 ||R(A; x, z)| |x − z|n+2 |Ω(x0 − z)| + |A(x) − A(x0 ) − ∇A(z)(x − x0 )|. |x − z|n+1
For each fixed integer k, set Ak (z) = A(z) − mB(x, 2k N ) (∇A)z. We know from Lemma 1 that for each z ∈ B(x, 2k N )\B(x, 2k−1 N ), |Ak (x) − Ak (z)| ≤ ≤ ≤
1 1/p p C|x − z| |∇A(˜ z ) − m (∇A)| d˜ z k B(x, 2 N ) |Ixz | Ixz 1/p 1 p C|x − z| |∇A(˜ z ) − m z k N ) (∇A)| d˜ B(x, 2 |B(x, 2k N )| B(x, 2k N ) C|x − z|.
On the other hand, another application of Lemma 1 tells us that 1 1/p p |∇A(˜ z ) − m k (∇A)| d˜ z |Ak (x) − Ak (x0 )| ≤ C|x − x0 | B(x, 2 N ) |Ixx0 | Ixx0 1 1/p p x0 (∇A)| d˜ ≤ Cr |∇A(˜ z ) − m z Ix |Ixx0 | Ixx0 +Cr mB(x, 2k N ) (∇A) − mIxx0 (∇A) N . ≤ Cr k + log r With the aid of the formula R(A; x, z) = R(Ak ; x, z), x, z ∈ Rn and Ak (x) − Ak (x0 ) − ∇Ak (z)(x − x0 ) = A(x) − A(x0 ) − ∇A(z)(x − x0 ), we have that for each fixed x, x0 ∈ Rn and x = x0 , |K(x, z) − K(x0 , z)| dz
|x−z|>N ∞
≤C
k=1
B(x, 2k N )\B(x, 2k−1 N )
|Ω(x − z) − Ω(x0 − z)| 1 + |∇A(z) − mB(x, 2k N ) (∇A)| dz |x − z|n
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+Cr
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∞
N |Ω(x0 − z)| k + log 1 + |∇A(z) − mB(x, 2k N ) (∇A)| dz n+1 r B(x, 2k N ) |x − z|
k=1
= EN + FN . We first consider the term EN . Invoking the inequality b ≤ t + t1−θ bθ ,
b > 0,
(see [7, p. 444]), we can write (choose t = k + log EN
t > 0,
θ>1
(6)
N in (6)) r
∞ N |Ω(x − z) − Ω(x0 − z)| dz k + log r |x − z|n k k−1 B(x, 2 N )\B(x, 2 N) k=1
1−q ∞ q k + log Nr |Ω(x − z)|∇A(z) − mB(x, 2k N ) (∇A) dz +C k |B(x, 2 N )| B(x, 2k N ) k=1
1−q ∞ q k + log Nr +C |Ω(x0 − z)|∇A(z) − mB(x, 2k N ) (∇A) dz |B(x, 2k N )| B(x, 2k N )
≤
C
=
E1N + E2N + E3N .
k=1
The term E1N is easy to deal with. In fact, it follows from Lemma 2 that E1N
≤
C
≤
C
∞ k k−1 N ) k=1 Cr/(2 N )<δ
0
ω(δ) log(2 + δ
)
δ
ω(δ) log(2 + δ −1 )
dδ δ
.
The generalization of the H¨older inequality (4) gives that for some q > 2, E2N
∞ N 1−q k + log Ω(x − ·)L(log L)q , B(x, 2k N ) r k=1
q × ∇A − mB(x, 2k N ) (∇A) 1/q k
≤ C
∞ N 1−q k + log ≤ C , r
exp L
, B(x, 2 N )
k=1
where in the last inequality, we have invoked the well-known John-Nirenberg inequality, which states that
∇A − mB(x, 2k N ) (∇A) q
exp L1/q , B(x, 2k N )
Similarly, E3N ≤ C
q = ∇A − mB(x, 2k N ) (∇A)exp L, B(x, 2k N ) ≤ C.
∞ k=1
k + log
N 1−q . r
Q. H. Zhang : A BMO Estimate for the Multilinear Singular Integral Operator
· 277 ·
Combining the estimates for the terms E1N , E2N , E3N , we thus obtain that lim EN = 0. N →∞
We now turn our attention to the term FN . The inequality (5) together with the JohnNirenberg inequality shows that for x, x0 ∈ Rn and x = x0 , ∞ (k + log Nr )r FN ≤ C |Ω(x0 − z)| log(2 + |Ω(x0 − z)|) dz |B(x, 2k N )|1+1/n B(x, 2k N ) k=1
∞ |∇A(z) − mB(x, 2k N ) (∇A)| (k + log Nr )r +C exp dz C2 |B(x, 2k N )|1+1/n B(x, 2k N ) k=1
≤
C
∞ k + log Nr k=1
2k Nr
.
It is obvious that lim FN = 0. Thus, the desired result follows from the estimates for EN and N →∞
FN directly. Proof of Theorem 1. Without loss of generality, we assume that ∇ABMO(Rn ) = 1 and f L∞ (Rn ) = 1. At first, we claim that if f ∈ L∞ (Rn ) such that TA∞ f (x0 ) exists for some x0 ∈ Rn , then TA f (x) exists almost everywhere in Rn . To prove this, let B be any ball containing x0 . Decompose f as f (x) = f (x)χ4B (x) + f (x)χRn \4B (x) = f1 (x) + f2 (x). The fact that f1 (x) ∈ L2 (Rn ) together with the L2 (Rn ) boundedness of TA (see Theorem 1 in [7]) implies that TA f1 (x) exists almost everywhere in Rn . If we can show that TA f2 (x) exists almost everywhere in B, our claim then follows directly. We now show the existence of TA f2 (x) for x ∈ B. Denote by r and xB the radius and the center of B, respectively. Set TAr, ∞ f (x0 )
= lim
N →∞
7r<|x0 −z|
Ω(x0 − z) R(A; x0 , z)f (z) dz. |x0 − z|n+1
It is obvious that TAr, ∞ f (x0 ) is finite for any r with 0 < r < ∞. Set SN = {z ∈ Rn : |x − z| < N, |xB − z| > 4r}. For each < r and N > 10r, write K(x, z)f2 (z) dz = <|x−z|
SN
=
SN
=
K(x, z)f (z) dz
K(x, z) − K(x0 , z) f (z) dz +
IN (x) + IIN (x).
SN
K(x0 , z)f (z) dz
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Note that Lemma 3 implies that limN →∞ IN (x) exists. It suffices to consider the term IIN (x). Recall that |x − z| ∼ |x0 − z| ∼ |xB − z| for x, x0 , xB ∈ B and z ∈ Rn \4B. It follows SN = D1
D2
D3
D4 ,
where D1 = {z ∈ Rn : 7r < |x0 − z| < N − 2r}, D2 = {z ∈ Rn : N − 2r ≤ |x0 − z|, |xB − z| < N − r}, D3 = {z ∈ Rn : N − r ≤ |xB − z|, |x − z| < N }, D4 = {z ∈ Rn : |xB − z| > 4r, |x0 − z| ≤ 7r}. Note that Di ∩ Dj = ∅ for any 1 ≤ i, j ≤ 4 and i = j. Therefore, K(x0 , z)f (z) dz + K(x0 , z)f (z) dz IIN (x) = D1 D2 K(x0 , z)f (z) dz + K(x0 , z)f (z) dz. + D3
Obviously,
lim
N →∞
D1
D4
K(x0 , z)f (z) dz = TAr, ∞ f (x0 ).
Let A∗N (z) = A(z) − mB(x0 , N ) (∇A)z. A familiar argument involving Lemma 1 states that for N −r ≤ |x0 −z| < N , |A∗N (x0 )−A∗N (z)| ≤ C|x0 − z|. Then by the generalization of the H¨older inequality (4), we get |Ω(x0 − z)| |R(A; x0 , z)| dz n+1 N −r≤|x0 −z|
|Ω(x0 − z)| 1 + |∇A(z) − mB(x0 , N ) (∇A)| dz ≤C n N −r≤|x0 −z|
N n−1 r N . + C log n (N − r) N −r
Thus it is easy to see that lim
N →∞
N −r≤|x0 −z|
|K(x0 , z)| dz = 0,
Q. H. Zhang : A BMO Estimate for the Multilinear Singular Integral Operator
which in turn implies lim N →∞
D2
K(x0 , z)f (z) dz ≤ lim N →∞
N −2r≤|x0 −z|
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|K(x0 , z)| dz = 0,
and
lim K(x0 , z)f (z) dz ≤ lim |K(x0 , z)| dz = 0. N →∞ N →∞ N −2r≤|x −z|
when N → ∞. Then we obtain the existence of TA f2 (x) for x ∈ B. We can conclude the proof of Theorem 1. Our claim tells us that if f ∈ L∞ (Rn ), then TA f (x) is either finite almost everywhere in Rn , or infinite almost everywhere in Rn . Thus, it remains to show that if f ∈ L∞ (Rn ) such that TA f (x) exists almost everywhere, then for any ball B, 1 |TA f (x) − mB (TA f )| dx ≤ C. (7) |B| B To prove (7), just as before, we denote by r the radius of B, and set f (x) = f (x)χ4B (x) + f (x)χRn \4B (x) = f1 (x) + f2 (x). The L2 (Rn ) boundedness of TA via the H¨older inequality tells us 1 1/2 1 1/2 1 |TA f1 (x)| dx ≤ |TA f1 (x)|2 dx ≤C |f (x)|2 dx ≤ C. |B| B |B| B |B| 4B On the other hand, since TA f (x) and TA f1 (x) exist almost everywhere, we can find a point x∗ ∈ B such that TA f2 (x∗ ) exists. Then it is clear that TAr,∞ f (x∗ ) exists. Now our goal is to show that for any x ∈ B, |TA f2 (x) − TAr,∞ f (x∗ )| ≤ C.
(8)
If we can do this, a trivial computation gives that 1 2 |TA f (x) − mB (TA f )| dx ≤ |TA f (x) − TAr,∞ f (x∗ )| dx |B| B |B| B 2 |TA f1 (x)| dx + |TA f2 (x) − TAr,∞ f (x∗ )| dx ≤ |B| B B ≤ C. We now prove the inequality (8). By the same method used above, it is easy to check that for any x ∈ B,
= ≤
|TA f2 (x) − TAr,∞ f (x∗ )|
K(x, z) − K(x∗ , z) f (z) dz + lim K(x∗ , z)f (z) dz − TAr,∞ f (x∗ ) lim N →∞ S N →∞ S N N
∗ lim |K(x∗ , z)| dz. (9) K(x, z) − K(x , z) f (z) dz +
N →∞
SN
23 B\22 B
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k (z) = A(z) − m2k B (∇A)z. Again, Lemma 1 states that for x ∈ B For each fixed integer k, set A and z ∈ 2k B\2k−1 B, k (z)| ≤ C|x − z|, |A k (x) − A k (x∗ )| ≤ C|x − x∗ | k + log k (x) − A |A
2r . |x − x∗ |
An argument similar to that used in the proof of Lemma 3 shows that for x ∈ B,
≤ ≤
K(x, z) − K(x∗ , z) f (z) dz S N |K(x, z) − K(x∗ , z)| dz Rn \4B ∞
|Ω(x − z) − Ω(x∗ − z)| 1 + |∇A(z) − m2k B (∇A)| dz n |x − z| k k−1 B k=3 2 B\2 ∞
2r |Ω(x∗ − z)| k + log +C|x − x∗ | 1 + |∇A(z) − m2k B (∇A)| dz ∗ n+1 |x − x | 2k B\2k−1 B |x − z| C
k=3
=
G + H.
Applying the inequality (6) here we choose t = k + log
2r , Lemma 2 and the general|x − x∗ | ization of the H¨older inequality (4), we obtain that for x ∈ B and some q > 2,
G ≤
≤
2r |Ω(x − z) − Ω(x∗ − z)| dz ∗ |x − x | 2k B\2k−1 B |x − z|n k=3 1−q 2r ∞ k + log |x−x ∗| q +C |Ω(x − z)|∇A(z) − m2k B (∇A) dz |2k−1 B| k 2 B k=3 1−q 2r ∞ k + log |x−x ∗| q +C |Ω(x∗ − z)|∇A(z) − m2k B (∇A) dz k−1 |2 B| 2k B k=3 ∞ dδ ω(δ) log(2 + δ −1 ) C δ ∗ k ∗ k−1 C|x−x |/(2 r)<δ
C
∞ k + log
k=3 ∞
+C +C ≤
C.
k=3 ∞ k=3
q k 1−q Ω(x − ·)L(log L)q , 2k B ∇A − m2k B (∇A) exp L1/q , 2k B
q k 1−q Ω(x∗ − ·)L(log L)q , 2k B ∇A − m2k B (∇A) exp L1/q , 2k B
To estimate the term H, an application of the inequality (5) and the John-Nirenberg inequality
Q. H. Zhang : A BMO Estimate for the Multilinear Singular Integral Operator
· 281 ·
shows that for x ∈ B, H
≤ C ≤ C
2r ∞ k + log |x−x∗ | k=3 ∞
2k r k 2k |2k B|
k=3 ∞
+C ≤ C.
k=3
Consequently,
2k B
k k 2 |2k B|
|x − x∗ |
2k B
|Ω(x∗ − z)| 1 + |∇A(z) − m2k B (∇A)| dz n |x − z|
|Ω(x∗ − z)| log(2 + |Ω(x∗ − z)|) dz
|∇A(z) − m2k B (∇A)| dz exp C2 2k B
K(x, z) − K(x∗ , z) f (z) dz ≤ C.
SN
(10)
On the other hand, a familiar argument involving Lemma 1 tells us that for z ∈ 2k B\2k−1 B, k (x∗ ) − A k (z)| ≤ C|x∗ − z|. |A This inequality together with the generalization of the H¨ older inequality (4) gives us
|Ω(x∗ − z)| 1 + |∇A(z) − m23 B (∇A)| dz |K(x∗ , z)| dz ≤ C ∗ n 23 B\22 B 23 B\22 B |x − z| ≤ CΩ(x∗ − ·)L log L, 23 B ∇A − m23 B (∇A) 3 exp L, 2 B
≤
C.
(11)
Combining the estimates (9), (10) and (11) yields the inequality (8), and then completes the proof of Theorem 1. Acknowledgement The author would like to thank the referee for some valuable suggestions and corrections.
References
[1] Cohen, J., A Sharp Estimate for a Multilinear Singular Integral in Rn , Indiana Univ. Math. J., 30(1981), 693-702. [2] Hofmann, S., On Certain Non-standard Calder´ on-Zygmund Operators, Studia Math., 109(1994), 105-131. [3] Hu, G., L2 (Rn ) Boundedness for a Class of Multilinear Singular Integral Operators, Acta Math. Sinica, English Series, 30(2003), 693-702.
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[4] Hu, G. and Yang, D., Sharp Function Estimates and Weighted Norm Inequalities for Multilinear Singular Integral Operators, Bull. London. Math. Soc., 35(2003), 759-769. [5] Hu, G., Lp and Endpoint Estimates for Multi-linear Singular Integral Operators, Proc. Royal Soc. Edin.(A), 134(2004), 501-514. [6] Sun, Y. and Zhang, Z., A Note on the Existence and Boundedness of Singular Integrals, J. Math. Anal. Appl., 273(2002), 370-377. [7] Hu, G., Lp Boundedness for the Multilinear Singular Integral Operator, Integr. Equ. Oper. Theory, 52(2005), 437-449. [8] Admas, R. A., Sobolev Spaces, Academic Press, New York, 1975. [9] Kurtz, D. S. and Wheeden, R. L., Results on Weighted Norm Inequalities for Multipliers, Trans. Amer. Math. Soc., 255(1979), 343-362.
Department of Applied Mathematics University of Information Engineering P. O. Box 1001-7410, Zhengzhou, 450002 P. R. China E-mail: z
[email protected]
