Function Spaces, 3 intern. conf. to the memory of W.Orlicz, Poznan, 1992

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Collectanea Mathematica University of Barcelona Gran Via 585 08071 Barcelona, Spain Fax: 34-93 4021601 [email protected]

Volume 44 (1993) PREFACE

The Third International Conference "Function Spaces" held in Poznan (August 30 - September 4, 1992) was organized by the Institute of Mathematics of Adam Mickiewicz University of Poznan. The Conference was devoted to the memory of Professor Wladyslaw Orlicz (1903-1990) - one of the founders of functional analysis and the organizer of modern mathematical school in Poznan. The main subject of the Conference was the theory of function spaces and related topics in functional analysis. During the Conference there were held 15 plenary lectures and 57 short communications. This volume contains 29 articles devoted to various topics in the theory of function spaces and Orlicz spaces, geometry of Banach spaces, the approximation theory, the interpolation theory and the general theory of linear topological spaces. Special thanks are due to Doc. M. Jaroszewska, Doc. S. Stoinski, Dr. M. Mikosz, Dr. A. Borucka-Cieslewicz, Dr. L. Skrzypczak, Dr. T. Kubiak and others for assistance in the organization of the Conference. We are pleased to express our thanks and gratitude to Prof. J. Cerdà and the Editors of Collectanea Mathematica for agreement to publishing the Proceedings and kind assistance during their preparation. The Organizing Committee Julian Musielak (Chairman) Henryk Hudzik Marian Nowak

A.G. Aksoy and J.B. Baillon, Measures of non-compactness in Orlicz modular spaces, 1-11. .

Józef Banas and Krzysztof Fraczek, Locally nearly uniformly smooth Banach spaces, 13-22. .

Richard Becker, Opérateurs sommants sur un cone normal d'un espace de Banach, 23-40. .

M. Berkolaiko and I. Novikov, On infinitely smooth almost-wavelets with compact support, 41-46. .

Zbigniew Binderman, Applications of sequential shifts to an interpolation problem, 47-57. .

Jurie Conradie, Generalized precompactness and mixed topologies, 59-70. .

Shutao Chen and Huiying Sun, On weak topology of Orlicz spaces, 71-79. .

Susanne Dierolf, On the three-space-problem and the lifting of bounded sets, 81-89. . Janusz Dronka, Remarks on the Istratescu measure of noncompactness, 91-103.

.

Leonhard Frerick, On complete, precompact and compact sets, 105-114. .

Alejandro García del Amo, On reverse Hardy's inequality, 115-123. .

H.P. Heinig, A Fourier inequality with $A_p$ and weak-$L^1$ weight, 125127. .

H. Hudzik, Every nonreflexive Banach lattice has the packing constant equal to 1/2, 129-134. .

Henryk Hudzik and Marek Wisla, On extreme points of Orlicz spaces with Orlicz norm, 135-146. .

Alois Kufner, Higher order Hardy inequalities, 147-154. .

W. Kurc, A dual property to uniform monotonicity in Banach lattices, 155-165. .

Grzegorz Lewicki, Minimal projections onto subspaces of $l^{(n)}_\infty$ of codimension two, 167-179. .

L. Maligranda and L.E. Persson, Inequalities and interpolation, 181-199. .

Mario Milman, A commutator theorem with applications, 201-210. .

S.Ya. Novikov, Boundary spaces for inclusion map between rearrangement invariant spaces, 211-215. .

Marian Nowak, Order continuous seminorms and weak compactness in Orlicz spaces, 217-236. .

Paulina Pych-Taberska, Properties of some bivariate approximants, 237-246. .

Yves Raynaud, On complemented subspaces of rearrangement invariant function spaces, 247-260. . E.M. Semenov, Random rearrangements in functional spaces, 261-268. .

Leszek Skrzypczak, Besov spaces and function series on Lie groups II, 269277. .

Tingfu Wang, Zhongrui Shi and Quandi Wang, On $W^*UR$ point and $UR$ point of Orlicz spaces with Orlicz norm, 279-299. .

Tingfu Wang and Quandi Wang, On the weak star uniformly rotund points of Orlicz spaces, 301-306. .

Ye Yining and Huang Yafeng, $P$-convexity property in Musielak-Orlicz sequence spaces, 307-325. .

Zhao Linsheng, Bounded variation functions of order $k$ on sequence spaces, 327-337. .

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 1–11 c 1994 Universitat de Barcelona

Measures of non-compactness in Orlicz modular spaces

A.G. Aksoy Department of Mathematics, Claremont McKenna College Claremont, California U.S.A.

J.B. Baillon Universit´e Lyon I-IMI, 69622 Villeurbonne Cedex, France

Abstract In this paper we show that the ball measure of non-compactness of a norm bounded subset of an Orlicz modular space Lψ is equal to the limit of its n-widths. We also obtain several inequalities between the measures of noncompactness and the limit of the n-widths for modular bounded subsets of Lψ which do not have ∆2 -condition. Minimum conditions on ψ to have such results are specified and an example of such a function ψ is provided.

Introduction and Preliminaries We start by recalling the usual deﬁnitions of Orlicz and modular spaces. Definition A. Let X be a vector space. A functional ρ: X → [0, ∞] is called a modular; if for f, g ∈ X the following is true: (i) ρ(f ) = 0 if and only if f = 0 (ii) ρ(af ) = ρ(f ) if |a| = 1 AMS(1980) Subject Classification (1985 revision) 46E30, 46A50, 46B99 Keywords: Modular space, Orlicz space, measure of non-compactness, n-widths, Kolmogorov diameters, ∆2 -condition

1

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Aksoy and Baillon

(iii) ρ(af + bg) ≤ ρ(f ) + ρ(g) for a + b = 1 and a, b ≥ 0. If (iii) is replaced by

(iii)

ρ(af + bg) ≤ as ρ(f ) + bs ρ(g) for f, g ∈ X, 0 ≤ a, b, as + bs = 1

with 0 < s ≤ 1 ﬁxed, ρ is called s-convex modular (convex if s = 1). For example, everymonotone F -norm ρ is a modular, every norm is a convex modular, and ρ(x) = |x| for x ∈ R is a 12 -convex modular. To a modular we associate a modular space. Let X be a real vector space and ρ be a modular on X. We deﬁne the modular space Xρ by Xρ = f ∈ X: lim ρ(αf ) = 0 . α→0

Obviously Xρ is a vector subspace of X. Definition B. An Orlicz function ψ: R → R+ is a continuous nondecreasing function with ψ(0) = 0, ψ(t) → ∞ as t → ∞ and ψ(−x) = ψ(x), i.e. ψ behaves similarly to power function ψ(t) = tp . Let ψ be an Orlicz function and let (X, M, µ) be a σ-ﬁnite measure space. Then for every measurable real valued function f on X, we deﬁne the Orlicz modular by ψ(|f (x)|)dµ. ρ(f ) = X

ρ is convex if ψ is convex. The Orlicz space is the space of all (equivalence classes of) measurable real valued functions f on X so that lim ρ(λf ) = 0. Obviously, an Orlicz space Lψ λ→0

is a generalization of the classical Lp -spaces. Although ψ behaves similarly to the power function ψ(t) = tp , the convexity of the Orlicz function can be omitted; two examples of such functions are: ψ(t) = et − 1 and ψ(t) = ln(1 + t). The vector space Lψ can be equipped with an F -norm deﬁned by f ≤λ .

f ρ = inf λ > 0: ρ λ If ρ is convex, then

f ρ = inf λ > 0: ρ

f λ

≤1

will deﬁne a norm on Lψ , in either case the norm is called a Luxemburg norm.

Measures of non-compactness in Orlicz modular spaces

3

With this norm (Lψ , ρ ) is a Banach space in case ρ is convex [9]. One has two structures on Lψ ; one is that of Banach space induced by the norm ρ , and the other is the structure of a modular space induced by the Orlicz modular ρ. Although the study of structure of Lψ spaces is interesting in itself, many applications to diﬀerential and integral equations with kernels of nonpower types are the basic reason for the development of Orlicz spaces. Also, it should be noted that the most commonly used rearrangement invariant functions spaces, beside Lp space are the Orlicz function spaces. (See e.g. J. Lindenstrauss and L. Tzafriri [11].) Let ρ be an Orlicz modular on Lψ , a sequence (fk ) of elements Lψ is called modular convergent (or ρ-convergent) to f ∈ Lψ if ρ(fk − f ) → 0 as h → ∞. Norm-convergence in Lψ implies ρ-convergence, but ρ-convergence does not imply norm-convergence. In case the measure space is σ-ﬁnite, the following theorem gives the equivalence. We say ψ satisﬁes ∆2 -condition if (i) lim sup ψ(2u)/ψ(u) < ∞ and lim sup ψ(2u)/ψ(u) < ∞ in case the measure µ is u→∞

u→0

atomless and inﬁnite. (ii) lim sup ψ(2u)/ψ(u) < ∞ in case the measure µ is atomless and ﬁnite. u→∞

(iii) lim sup ψ(2u)/ψ(u) < ∞ in case the measure µ in case the measure µ is purely u→0

atomic. All of them imply that there exists K, c > 0 such that for all f ∈ Lψ we have ρ(2f ) ≤ Kρ(f ) + c. Theorem ([9]) Norm convergence and ρ-convergence are equivalent in Lψ if and only if ψ satisﬁes the ∆2 -condition. It should be remarked that Orlicz spaces Lψ with the ∆2 -condition are not far from Lp -spaces in the sense that there are analogous theorems about separability. However, in the spaces which lack a ∆2 -condition, the fact that ρ-convergence is not reducible to norm convergence makes modular convergence interesting. For further theory of Orlicz modular spaces, we refer to [8], [9], [12] and [14]. As for the measures of non-compactness [2], they are of interest in many spaces. They are used in ﬁxed point theory (see Darbo [3], Sadovskii [17], Reich [15], [16]), and also in the study of the essential spectrum (see Nussbaum [13], Lebow-Schechter [10], Aksoy [1]). Measures of non-compactness of embeddings in the context of

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Aksoy and Baillon

Sobolev spaces are given by D.E. Edmunds and W.D. Evans [6]. The (ball) measure of non-compactness α(T ) of T is deﬁned to be: α(T ) = inf ε > 0: T (Bx ) can be covered by ﬁnitely many balls of radius ε . The estimates of α for embedding maps can be found in [4]. Two types of measures of non-compactness, namely entropy and approximation numbers of embeddings in Orlicz spaces, are also studied in [5]. In [7], one can ﬁnd ﬁxed point theorems in Orlicz modular spaces. The purpose of this paper is to study measures of non-compactness in the context of Orlicz spaces, where the Orlicz space under consideration is either equipped with the norm or just an Orlicz modular. We will investigate equality of certain measures of non-compactness in Lψ even if ψ does not satisfy the ∆2 -condition. From this point on, ψ is assumed to be convex. Definition 1. Let ξ > 0 be a ﬁxed real number and let f ∈ Lψ . We deﬁne f , the norm of f , as:

f =

ξ where s(f ) = sup{s: ρ(sf ) ≤ ξ} > 0 . s(f )

Proposition 1

f =

ξ s(f )

satisﬁes the properties of a norm.

Proof. Suppose f = 0, then using the fact that ψ(0) = 0, we obtain ρ(sf ) = 0 ≤ ξ which implies that f = 0. On the other hand, if f = 0, from the deﬁnition there is sn → ∞ such that ρ(sn f ) ≤ ξ or equivalently ξ ≥ ψ(|sn f (x)|)dµ. Since ψ is lower semi-continuous, we have ξ ≥ lim ψ(|sn f (x)|)dµ ≥ limψ(|sn f (x)|)dµ ≥ ψ(lim|sn f (x)|)dµ = + {x:f (x)=0} {x:f (x)=0} = ψ(0)dµ + ψ(∞)dµ {x:f (x)=0}

{x:f (x)=0}

= µ({x: f (x) = 0}) · ψ(0) + µ({x: f (x) = 0}) · ψ(∞) Again using the facts that ψ(0) = 0 and ψ(∞) = ∞, we obtain: ≥ 0 + ∞ · µ({x: f (x) = 0}) which implies µ({x: f (x) = 0}) = 0 or f = 0 a.e. µ.

Measures of non-compactness in Orlicz modular spaces To show λf = |λ| f , consider

s(λf ) = sup{s: ρ(sλf ) ≤ ξ} = sup =

s s :ρ · λf ≤ ξ |λ| λ

5

1 1 sup{s: ρ(sf ) ≤ ξ} = s(f ) |λ| |λ|

so

|λ|ξ ξ = = |λ| f . s(λf ) s(f ) To show the triangle inequality f + g ≤ f + g , let sf and sg denote the s(f ) and s(g), respectively. Then sf · sg sg sf (f + g) = (sf · f ) + (sg · g) ≤ ξ . sf + sg sf + sg sf + sg Since ψ is convex sf · sg sg sf ρ (f + g) = ρ(sf · f ) + ρ(sg · g) ≤ ξ . sf + sg sf + sg sf + sg

λf =

Thus s(f + g) ≥

sf ·sg sf +sg

f + g =

and hence s(f + g) ≥

s(f )·s(g) s(f )+s(g) .

Now

s(f ) + s(g) ξ ≤ξ· = f + g . s(f + g) s(f ) · s(g)

Remark 1. Proof of Proposition 1 can be shortened if one makes the following observations: Let v > 0 and consider the new modular ρv = vρ, then f 1

f ρv = inf t > 0: ρ ≤ . t v ξ 1 Let i(f ) = inf{t: ρ( ft ) ≤ ξ}, then i(f ) = f ξ1 ρ and since s(f ) = i(f1 ) , s(f ) = ξ f ξ ρ holds and clearly deﬁnes a norm. For any modular, it is known that ρ(f ) ≤ 1 if and only if f ρ ≤ 1. Therefore, using the above Remark 1, we can conclude that: 1

f ξ1 ρ ≤ 1 iﬀ ρ(f ) ≤ 1 iﬀ ρ(f ) ≤ ξ. ξ Therefore, ξ f ξ1 ρ ≤ ξ iﬀ ρ(f ) ≤ ξ and hence in our notation:

f ≤ ξ iﬀ ρ(f ) ≤ ξ. Notations B (r) = {f : f ≤ r}, Bρ (r) = {f : ρ(f ) ≤ r} denotes the norm-ball and ρ-ball centered at 0 and radius r, respectively, where · is the norm deﬁned in Proposition 1 and ρ is the Orlicz modular on Lψ . Furthermore, we will use r± for any number of the form r ± ε for any ε > 0 small enough, if there is no ambiguity.

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Proposition 2 f ∈ B· (r) if and only if ρ( ξf r ) ≤ ξ. ξ Proof. If ρ( ξf r ) ≤ ξ, then s(f ) ≥ r , thus f =

ξ s(f )

≤ r. Conversely let f ∈ B· (r),

then ≤ s(f ). First suppose = s(f ), then since ψ is increasing there is sf = s(f )− such that ρ(s− f f ) ≤ ξ. Now using the Fatou property and the fact that ψ is lower semi-continuous, we have − limρ(s− f ) ≥ limψ(s f ) ≥ ψ(lims− f f f f ) = ρ(s(f ) · f ) . ξ r

ξ r

There, ρ( rξ f ) ≤ ξ. Secondly, suppose rξ < s(f ), then there is sf = s(f )− such that ρ(s− f f ) ≤ ξ, ξ − but r < sf ≤ s(f ). Again using the fact that ψ is increasing we obtain ξ ρ f ≤ ρ(s− f f ) ≤ ξ. r The following result uses Proposition 2 to illustrate the relationship between ρ-balls and norm-balls of Lψ . Proposition 3 (i) When r ≤ ξ, we have B· (r) ⊆ Bρ (r). (ii) When ξ ≤ r, we have Bρ (r) ⊆ B · (r). Proof. (i) If f ∈ B· (r), then by Proposition 2, ρ( ξf r ) ≤ ξ. Since ψ(0) = 0 and ψ is convex, we have r ξf ξf r r ρ(f ) = ρ · ≤ ρ + 1− ρ(0) ξ r ξ r ξ r ≤ ·ξ =r ξ which shows that f ∈ Bρ (r). (ii) If f ∈ Bρ (r), then ρ(f ) ≤ r. Again using Proposition 2 together with the fact ψ(0) = 0 and ψ convex will yield: ξ ξ ξ f ≤ ρ(f ) + 1 − ρ(0) ≤ ξ ρ r r r thus f ∈ B· (r). Remark 2. Notice that, although in the above proof of Proposition 3 we are using the facts ψ(0) = 0 and ψ is convex, in fact what we need is ψ satisfying ψ(ax) ≤ aψ(x) for 0 ≤ a ≤ 1 . Corollary Let D be a subset of Lψ . Then D is ψ-bounded implies D is · -bounded.

Measures of non-compactness in Orlicz modular spaces

7

Proof. Since ψ is increasing if r1 ≤ r2 , then D ⊂ Bρ (r1 ) implies D ⊂ Bρ (r2 ). Now if D ⊆ Bρ (r) by (ii) of the previous proposition, we have

D ⊂ Bρ max(r, ξ) ⊆ B· max(r, ξ) . Definition 2. Let D be a norm-bounded subset of Lψ . The norm n-th width of D in the sense of Kolmogorov is denoted by dn · and deﬁned as dn· (D) = inf{r > 0: D ⊆ B· (r) + An where An is a vector space with dim of An ≤ n} and norm-ball measure of non-compactness α α· (D) = inf

r > 0: D ⊆

· (D) k

is deﬁned as

B· (xi ; r) .

i=1

Here k is arbitrary but ﬁnite; notice that k

b· (xi ; r) = B· (r) +

i=1

k

{xi } .

i=1

Theorem 1 Let D be a · -bounded subset of Lψ . Then α· (D) = lim dn· (D). n

Proof. We obviously have α· (D) ≥ lim dn· (D). To show the reverse inequality, n

suppose we choose an admissible r and An such that D ⊂ B· (r) + An , then we can write D ⊆ D1 + D2 where D1 ⊂ B· (r) and D2 ⊂ An . Observe that D2 is

· -bounded, because for every f ∈ D one has f = f1 + f2 where f2 ∈ D2 and

f2 = f − f1 ≤ f + f1 . Now if we use the seminorm property in An which is ﬁnite, we obtain: for every ε > 0, there exists a ﬁnite covering for D2 by balls of radius ε, i.e. D2 ⊆ B· (xi ; ε) . ﬁnite

So B· (r) + B· (xi ; ε) ⊆ B· (xi ; r + ε) which implies α· (D) ≤ r + ε.

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Aksoy and Baillon

Remark 3. In the above proof to show B· (r) + B· (xi ; ε) ⊆ B· (xi ; r + ε) we used the triangle property of our norm. But all we need is

f + g ≤ f + C g for ﬁxed C > 0. This inequality holds if the Orlicz function ψ satisﬁes the condition: ψ(ax + by) ≤ aψ(x) + bCψ(y) with a + bC = 1, a, b ≥ 0 C > 0 ﬁxed. Notice that by replacing norm-balls by ρ-balls in Deﬁnition 2, one can similarly deﬁne modular n-width of D, dnρ (D) and modular-ball measure of non-compactness αρ (D) for a ρ-bounded subset D as follows: dnρ (D) = inf{r > 0: D ⊆ Bρ (r) + An where An is a vector space with dim of An ≤ n} k αρ (D) = inf r > 0: D ⊂ Bρ (xi ; r) . i=1

Obviously we have lim dnρ (D) ≤ αρ (D). Therefore, one can ask whether Then orem 1 type of equality holds with respect to modular, too. Following Theorem 2 gives an aﬃrmative answer to this question in case ψ satisﬁes ∆2 -condition. Later by Theorem 3 we give partial answers to the same questions in case ψ does not satisfy ∆2 -condition. Theorem 2 Suppose that ψ satisﬁes the ∆2 -condition, then for a ρ-bounded subset D of Lψ we have: lim dnρ (D) = αρ (D).

n→∞

Measures of non-compactness in Orlicz modular spaces

9

Proof. Since D is ρ-bounded, there exists M with ρ(d) ≤ M for all d ∈ D. On the other hand, since ψ satisﬁes ∆2 -condition, there are K, C > 0 such that ρ(2f ) ≤ Kρ (f ) + C. Now choose an admissible r and An such that D ⊂ Bρ (r) + An . Next we claim that there is K1 such that ρ(x) ≤ K1 for each x ∈ An ∩ D, because if x ∈ An ∩ D, then x = x1 − x2 with x1 ∈ D and x2 ∈ D ∩ Bρ (r) and hence 2x1 − 2x2 1 1 ≤ ρ(2x1 ) + ρ(2x2 ) ρ(x) = ρ 2 2 2 1 1 1 1 Kρ(x1 ) + Kρ(x2 ) + 2C ≤ KM + KM + 2C. 2 2 2 2 Now choose K1 so that

k1 ξ

≥ 1, then using convexity we have ρ( Kξ1 x) ≤

≤ ξ which implies that x ≤ K1 . Since An ∩ D is norm-bounded and ﬁnite dimensional, for any ε > 0, there exists {yi }ni=1 such that B (yi ; ε), 0 < ε < 1. An ∩ D ⊂ ξ K1 ρ(x)

ﬁnite

Using Proposition 3 (i), (take ξ = 1) we obtain D ⊂ Bρ (r) + Bρ (yi ; ε) ⊆ B(yi ; r + ε) ﬁnite

ﬁnite

which implies αρ (D) ≤ lim dnρ (D). n→∞

Lemma 1 Suppose D is a ρ-bounded subset of Lψ , then we have one of the following: (i) αρ (D) ≥ α· (D) ≥ ξ (ii) αρ (D) ≤ α· (D) < ξ (iii) α· (D) = ξ . Proof. Case 1. Suppose αρ (D) ≥ ξ. Then using Proposition 3(ii), we have r+ ≥ α − ρ(D) ≥ ξ such that D⊂ Bρ (xi ; r+ ) ⊆ B· (xi ; r+ ) ﬁnite

ﬁnite

which implies α· (D) ≤ αρ (D). Case 2. Suppose α· (D) < ξ. Then there is r+ such that α· (D) ≤ r+ < ξ. But by Proposition 3 (i) we have D⊂ B· (xi , r+ ) ⊆ Bρ (xi , r+ ) ﬁnite

and therefore αρ (D) ≤ α· (D) < ξ.

ﬁnite

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Aksoy and Baillon

Lemma 2 Suppose D is a ρ-bounded subset of Lψ , then we have only one of the following: (i) δρ (D) ≥ δ· (D) ≥ ξ (ii) δρ (D) ≤ δ· (D) < ξ (iii) δ· (D) = ξ where δ· (D) = lim dn· (D) and δρ (D) = lim dnρ (D). n

n

Proof. Case 1. Suppose δρ (D) ≥ ξ, then there is r+ ≥ δρ (D) ≥ ξ and An such that D ⊂ Bρ (r+ ) + An ⊆ B· (r+ ) + An . In the last inclusion we used Proposition 3(ii) again. Thus we have δ· (D) ≤ δρ (D) δρ (D) ≥ ξ Case 2 of this Lemma is similar to Case 2 of Lemma 1. Combining the results in Lemma 1 and Lemma 2 and Theorem 1 we obtain: Theorem 3 Let D be a ρ-bounded subset of Lψ , then we have one of the following: (i) αρ (D) ≥ δρ (D) ≥ α· (D) = δ· (D) ≥ ξ (ii) δρ (D) ≤ αρ (D) ≤ α· (D) = δ· (D) < ξ (iii) δ· (D) = α· (D) = ξ . Remark 4. Combining Remark 2 after Proposition 3 and Remark 3 after Theorem 1 we deduce that the conditions we need to put on ψ in order for the above theorem to hold are: 1. ψ(0) = 0 2. ψ(ax + by) ≤ aψ(x) + bψ(y) with a + bC = 1 a, b ≥ 0, C > 0 ﬁxed. These two conditions together imply that ψ(ax) ≤ aψ(x), which implies that ψ is increasing. Also condition 2 above implies ψ is lower semicontinuous. It is clear that condition 2 is satisﬁed by convex functions but does not imply convexity for ψ. Therefore Theorem 1 and 3 are valid for larger classes of functions than convex functions. For example, the function 1 3 1 x, x − ψ(x) = min x, max 2 2 2 satisﬁes the conditions given in Remark 4 for C = 2.

Measures of non-compactness in Orlicz modular spaces

11

Acknowledgments We would like to thank Professors M.A. Khamsi and A. Kaminska for their remarks. The ﬁrst-named author would like to thank the Pomona College Department of Mathematics and Professor Sandy Grabiner for their hospitality and encouragement during the summer of 1991.

References 1. A.G. Aksoy, The radius of the essential spectrum, J. Math. Anal. and Appl. Vol. 128, No. 1 (1987), 101–107. 2. J. Banas and K. Goebel, Measures of noncompactness in Banach spaces, Notes in Pure and Applied Math., Marcel Dekker, Vol. 60 (1980), New York. 3. G. Darbo, Punti uniti in transformazioni a condominio non campatto, Rend. Sem. Mat. Univ. Padua 24 (1955), 84–92. 4. D.E. Edmunds, Embeddings of Sobolev spaces, Proceedings of spring school in non-linear analysis, function spaces, and applications, Teubner-Texte Mathematischen 19, Teubner, Leibzig, (1979). 5. D.E. Edmunds and R.M. Edmunds, Entropy and approximation numbers of embeddings in Orlicz spaces, J. London Math. Soc. (2), 32, No. 3, (1985), 528–538. 6. D.E. Edmunds and W.D. Evans, Spectral Theory and Differential Operators, Oxford Science Publications, Clarendon Press, Oxford, (1987). 7. M.A. Khamsi, W.M. Kozlowski and S. Reich, Fixed point theory in modular function spaces, Journal of non-linear analysis, 14 (1990), 935–953. 8. W.M. Kozlowski, Modular function spaces, Marcel dekker, New York and Basel, 1988. 9. M.A. Krasnoselskii and Ya.B. Rutickii, Convex functions and Orlicz spaces, (translation), P. Noordhoff Ltd., Groningen, 1961. 10. A. Lebow and M. Schechter, Semigroup of operators and measures of non-compactness, J. Func. Anal. 7 (1971), 1–26. 11. J. Lindenstrauss and J. Tzafriri, Classical Banach spaces II, Springer-Verlag, Berlin-Heidelberg-New York, 1979. 12. J. Musielak. Orlicz spaces and modular spaces, Lecture Notes in Mathematics, Vol. 1034, Springer-Verlag, Berlin, 1983. 13. R.D. Nussbaum, The radius of the essential spectrum, Duke Math. J. 37 (1970), 473–479. 14. M.M. Rao and Z.D. Ren, Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991. 15. S. Reich (1973), Fixed points of condensing functions, J. Math. Anal. and Appl. 41 (1973), 460–467. 16. S. Reich, Fixed points in locally convex spaces, Math. Z. 125 (1972), 17–31. 17. B.N. Sadovskii, Limit-compact and condensing operators, Russian Math. Surveys 27 (1972), 85–155.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 13–22 c 1994 Universitat de Barcelona

Locally nearly uniformly smooth Banach spaces*

´ zef Bana´ Jo s and Krzysztof Fra ¸ czek Department of Mathematics, Technical University of Rzesz´ow W. Pola 2, 35–959 Rzesz´ow, Poland

Abstract The aim of this paper is to study the relationships between the concepts of local near uniform smoothness and the properties H and H ∗ .

1. Introduction The aim of this paper is to study relationships between the concepts of local near uniform smoothness and the properties H and H ∗ . These notions play very signiﬁcant role in some recent trends of the geometric theory of Banach spaces. These trends depend upon the study of classical notions of the geometry of Banach spaces from the view point of compactness conditions (cf. [1,2,7,10,11,16,17,18,19], for example). Let us mention that such an approach in the geometric theory of Banach spaces was initiated by the papers of Huﬀ [11], Partington [17] and Goebel and S¸ekowski [10]. In these papers the authors introduced an interesting generalization of the classical Clarkson’s notion of uniform convexity in Banach spaces [5]. The generalization of such a type was realized with help of the notion of a measure of noncompactness. After the papers [10,11,17] there have appeared a lot of papers (cf. the papers cited before) devoted to the study of other notions and properties of Banach spaces which can be formulated with help of compactness conditions. The fairly recent state of this theory is presented in the papers [2,3], for instance. * Research supported by Grant Nr 2 1001 91 01 from KBN Keywords and phrases: Drop property, locally nearly uniformly convex Banach space, locally nearly uniformly smooth Banach space, Radon - Riesz property, H ∗ property.

13

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Bana´ s and Fra ¸ czek

The investigations of this paper are continuation of the study from the paper [1], where the concept of local near uniform smoothness was introduced and some properties of this concept were derived.

2. Notation, definitions and auxiliary facts Let E be a given real Banach space with the norm denoted by · E or · and the zero element θ. The dual space of E will be denoted by E ∗ and the second dual by E ∗∗ . Let B(x, r) denote the closed ball in E centered at x and with radius r. Moreover, we write B = BE = B(θ, 1), B ∗ = BE ∗ , B ∗∗ = BE ∗∗ . The symbol S = SE stands for the unit sphere in E while S ∗ = SE ∗ , S ∗∗ = SE ∗∗ . The canonical embedding from E into E ∗∗ is denoted by κ. For a bounded and nonempty subset X of E by the symbol α(X) we will denote the Kuratowski measure of noncompactness of X: α(X) = inf ε > 0:X can be covered by a ﬁnite family of sets having diameters smaller than ε . For the properties of the function α we refer to [4]. In what follows assume that f ∈ S ∗ is arbitrary ﬁxed. For ε ∈ [0, 1] denote by F (f, ε) the slice deﬁned in the following way F (f, ε) = {x ∈ B: f (x) ≥ 1 − ε} . Similarly, for x ∈ S we deﬁne the slice F ∗ (x, ε) in the space E ∗ as F ∗ (x, ε) = {f ∈ B ∗ : f (x) ≥ 1 − ε}. Now we recall two deﬁnitions which are important for our purposes (cf. [1,16]). Definition 1. We say that a Banach space E is referred to as locally nearly uniformly convex (LNUC) if lim α(F (f, ε)) = 0 for every f ∈ S ∗ . ε→0

In other words, E is LNUC if for any f ∈ S ∗ and ε > 0 there exists δ > 0 such that

α F (f, δ) ≤ ε.

It is worthwhile to mention that E is LNUC if any only if the norm · E has the so-called drop property [16, 18].

Locally nearly uniformly smooth Banach spaces

15

Definition 2. A Banach space E is called locally nearly uniformly smooth (LNUS) if for any x ∈ S and ε > 0 there exists δ > 0 such that α F ∗ (x, δ) ≤ ε. Thus, E is LNUS if and only if lim α(F ∗ (x, ε)) = 0 for every x ∈ S. ε→0

The basic relationship between the concepts of LNUC and LNUS spaces was established in [1]. It is contained in the theorem given below. Theorem 1 A Banach space is LNUC if and only if E ∗ is LNUS. Observe that in the light of Corollary 2 from [3], the proof of this theorem given in [1] is entirely correct. Thus Remark formulated in [2] is not true.

3. Main results The following two deﬁnitions will be essential for our further considerations. Definition 3 [6,8]. We say that the norm · in a Banach space E has property H (this property is also known as the Kadec-Klee property or the Radon-Riesz property) whenever for any sequence (xn ) in E converging weakly to some x ∈ E with lim xn = x we have that (xn ) converges to x in norm. n→∞

The dual property to H is formulated in the next deﬁnition which is taken from [1]. Definition 4. We say that the norm · in a Banach space E has property H ∗ whenever for any sequence (fn ) ⊂ E ∗ converging weakly star to f ∈ E ∗ with fn E ∗ → f E ∗ we have that (fn ) converges to f in the norm of E ∗ (i.e. fn − f E ∗ → 0). The fundamental result obtained in [1] asserts that under the assumption of reﬂexivity of a space E we have that E is LNUS if and only if the norm · in E has the property H ∗ . Analyzing the proof of this result given in [1] it has been observed that the implication “if the norm in E has property H ∗ then E is LNUS” is true without the assumption on reﬂexivity of E. Unfortunately, this observation was suggested by the false theorem given in the books [13, 15]. That “theorem” says that the unit

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ball B ∗ in the space E ∗ is weakly star sequentially compact (cf. [13], p.3 and [15], p. 8). Obviously, such an assertion is generally not true [9]. The ﬁrst author would like to thank professor L. Vesely for indicating this error [20]. Now let us notice that the proof given in [1] is correct under an additional assumption. More precisely, we have the following result being improved version of Corollary 1 from [1]. Theorem 2 Let E be a Banach space with the norm having H ∗ property and such that the ball B ∗ in E ∗ is weakly star sequentially compact. Then E is LNUS. Let us point out some classes of spaces having the dual ball weakly star sequentially compact. Recall [9] that the Banach space E is said to be weakly compactly generated (WCG) whenever there exists a weakly compact set K ⊂ E such that the linear span of K is dense in E. For example, all separable or reﬂexive Banach spaces are WCG [9]. It can be shown [9] that if E is WCG Banach space then B ∗ is weakly star sequentially compact. Thus the spaces c0 , c and l1 have the dual ball weakly star sequentially compact. On the other hand the space l∞ has no longer this property [9]. In order to illustrate our considerations let us pay attention to the below given examples. Example 1: It was shown in [2] that the space c0 is NUS so it is obviously LNUS. On the other hand, using the same argumentation as in [2] we can easily seen that c is not LNUS. This means, in the light of Theorem 2, that c does not have property H ∗ . Hence we can infer that c0 does not have property H ∗ . Indeed, suppose the contrary. Then, for any sequence (fn ) ⊂ S(c0 )∗ = Sl1 and for f ∈ Sl1 such that (fn ) is weakly star convergent to f i.e. fn (x) → f (x)

as n → ∞

(1)

for any x ∈ c0 , we have that fn → f in the norm of l1 . Now, let us assume that fn (x) → f (x) as n → ∞

(2)

for every x ∈ c. Since (2) implies (1) this yields that fn → f in the norm of l1 . This means that c has the property H ∗ . Thus we get a contradiction.

Locally nearly uniformly smooth Banach spaces

17

Example 2: It is well known that the space l1 has property H. But on the other hand this space does not have property H ∗ . Indeed, let us consider the sequence (fn ) ⊂ S(l1 )∗ = Sl∞ such that fn = (1, 1, . . . , 1, 0, 1, 1, . . .) (0 on n-th place). Further, let f = (1, 1, . . .) ∈ Sl∞ . Taking an arbitrary x = (x1 , x2 , . . .) ∈ l1 we have ∞ xk , fn (x) = −xn + k=1

f (x) =

∞

xk .

k=1

Thus fn (x) − f (x) = −xn → 0 as n → ∞. Hence we infer that (fn ) is weakly star convergent to f . On the other hand we have fn − f = 1(n = 1, 2, . . .) in the norm of l∞ . It allows us to deduce that l1 has not property H ∗ . In what follows let us observe that the assertion being partially converse to that from Theorem 2 is no longer true. More precisely, if we assume that E is LNUS Banach space with B ∗ being weakly star sequentially compact then the norm · E is not in general H ∗ . In fact, let us take the space c0 . Then in virtue of Example 1 we see that the norm in c0 has not property H ∗ although this space is LNUS and B(c0 )∗ = Bl1 is weakly star sequentially compact. Thus the assumption on reﬂexivity in the result quoted immediately after Definition 4, is essential. Now let us recall [1] that if we assume that the norm in E ∗ has property H ∗ then the norm in E has property H. The converse assertion is true under the additional assumption on reﬂexivity, for example [1]. In the sequel we are going to study some further connections between the properties H and H ∗ . We start with the following theorem. Theorem 3 If the norm in E has property H ∗ then the norm in E ∗ has property H. Proof. Take a sequence (fn ) ⊂ S ∗ and f ∈ S ∗ such that fn → f weakly. Then fn → f weakly star what, in view of our assumption, allows us to conclude that fn → f in the norm of E ∗ . Observe that the converse theorem is not true. In fact, putting E = c0 we see (cf. Example 1) that the norm in E ∗ has property H but the norm in E has not property H ∗ . Nevertheless, we can prove that at least partially converse assertions to that from Theorem 3 are valid.

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Theorem 4 Let E be a Grothendieck space such that the norm in E ∗ has property H. Then the norm in E has property H ∗ . Let us recall [9] that a Banach space E is called a Grothendieck space whenever the weak star and weak convergence of sequences in E ∗ are the same. Proof. Let us take an arbitrary sequence (fn ) ⊂ S ∗ and f ∈ S ∗ such that fn → f weakly star. Then, by the assumption, the sequence (fn ) converges weakly to f . Since the norm in E ∗ has property H we can infer that (fn ) converges to f in norm and the proof is complete. Because every reﬂexive Banach space is Grothendieck, thus as an immediate consequence of the above result we derive the following theorem. Theorem 5 Assume that E is a reﬂexive Banach space and that the norm in E ∗ has property H. Then the norm in E has property H ∗ . Our next results give the criterion for the existence of a predual space. Theorem 6 Assume that E is LNUS space. Then E has a predual space if and only if E is reﬂexive. Proof. Assume that E has a predual space F, F ∗ = E. Then, by Theorem 1 we have that F is LNUC. Hence, by a result from [16] we deduce that F is a reﬂexive space. The converse implication is obvious. As an immediate consequence of the above theorem we obtain the following corollary. Corollary 1 The spaces l1 and l∞ are not LNUS. Observe now that from Theorem 1 and 2 we can obtain the following criterion for the space E to be LNUC. Theorem 7 E is LNUC if and only if the norm in E ∗ has property H ∗ and the ball B ∗∗ is weakly star sequentially compact in E ∗∗ .

Locally nearly uniformly smooth Banach spaces

19

Further on we shall use the result due to Klee [12]. We express this result in the terminology accepted before. Lemma 1 Let E be a Banach space with a norm · such that E ∗ is separable. Then there exists a norm · 1 equivalent to the norm · which has the property H ∗ . We have mentioned before that every separable Banach space has the property that B ∗ is weakly star sequentially compact. Keeping in mind this result and Theorem 2 we can derive the following theorem. Theorem 8 Let E be a Banach space with a norm · . Assume that E ∗ is separable. Then there is a norm · 1 equivalent to the norm · such that the space (E, · 1 ) is LNUS. 4. Remarks concerning product spaces In this section we are going to discuss brieﬂy the properties introduced before in the so-called lp product of a sequence of Banach spaces. Assume that (Ei , i ) (i = 1, 2, . . .) is a sequence of Banach spaces. Fix a number p ∈ (1, ∞) and consider the set lp (Ei ) = lp (E1 , E2 , . . .) consisting of all sequences x = (x1 ), xi ∈ Ei for i = 1, 2, . . ., such that ∞

xi pi < ∞.

i=1

It is well-known that lp (Ei ) forms a Banach space with respect to the norm xp = (xi )p =

∞

p1 xi pi

.

i=1

In the paper [14] it is shown that if the space E has property H then the product space lp (E, E, . . .) has also this property. It is not diﬃcult to see that the same argumentation may be used to show that the space lp (Ei ) = lp (E1 , E2 , . . .) has property H whenever every space Ei has this property. Further, let us notice that with the help of a similar reasoning we can prove the following theorem.

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Theorem 9 Assume that the norm i in the space Ei has the property H ∗ for all i = 1, 2, . . . Then the space lp (Ei ) has also the property H ∗ . Moreover, we can infer also the following assertion. Theorem 10 Let (Ei ) be a sequence of Banach spaces such that the ball Bi∗ in Ei∗ is weakly star sequentially compact for any i = 1, 2, . . . Then the ball Bp∗ in the space (lp (Ei ))∗ is weakly star sequentially compact. Proof. Let us take an arbitrary sequence (xn ) ⊂ Bp∗ . Since (lp (Ei ))∗ = lq (Ei∗ ), where p1 + 1q = 1 (cf. [14]), we obtain that Bp∗ = Blq (Ei∗ ) i.e. Bp∗ is the unit ball in the space lq (Ei∗ ). Next, let us represent any term xn of our sequence in the form xn = (xn1 , xn2 , xn3 , . . .) = (xnk ) (k = 1, 2, . . .) . Observe that for any ﬁxed k the sequence (xnk ) = (x1k , x2k , x3k , . . .) is contained in the ball Bk∗ . By the assumption we infer that there exists a subsequence of the sequence (xnk ) which is weakly star convergent to an element xk ∈ Ek∗ . Hence, applying the standard diagonal procedure we can select a subsequence (xjn ) of the sequence (xn ) having the following property: If we denote xxjn = (xj1n , xj2n , xj3n , . . .) = (xjkn )

(k = 1, 2, . . .)

and if we ﬁx arbitrarily k ∈ N then the sequence (xjkn ) (n = 1, 2, . . .) is a subsequence of (xnk ) which is weakly star convergent to the element xk ∈ Bk∗ . On the other hand we have that (xjn ) is contained in the ball Bp∗ = Blq (Ei∗ ) . Hence, applying a result from [14] we can deduce that the subsequence (xjn ) is weakly star convergent to the element x = (x1 , x2 , x3 , . . .). In view of the weak star lowersemicontinuity of the norm [8] we infer that x ∈ Bp∗ . Thus the proof is complete. In what follows let us observe that taking into account Theorems 2, 9 and 10 we can derive the following corollary.

Locally nearly uniformly smooth Banach spaces

21

Corollary 2 Assume that (Ei , i ) is a sequence of Banach spaces such that i has property H ∗ and the ball BEi∗ is weakly star sequentially compact (for all i = 1, 2, . . .). Then the product space lp (Ei ) (1 < p < ∞) is LNUS. Let us recall that the conclusion of the above corollary was obtained in [3] under the assumption that Ei is reﬂexive and LNUS for i = 1, 2, . . .. But then we infer (in view of the result from [1] quoted before) that the norm i has property H ∗ (i = 1, 2, . . .). Moreover, reﬂexivity of Ei implies that the ball BEi∗ is weakly star sequentially compact. Thus we conclude that the result from [3] is a particular case of Corollary 2.

References 1. J. Bana´s, On drop property and nearly uniformly smooth Banach spaces, Nonlinear Analysis 14 (1990), 927–933. 2. J. Bana´s, Compactness conditions in the geometric theory of Banach spaces, Nonlinear Analysis 16 (1991), 669–682. 3. J. Bana´s and K. Fr¸aczek, Conditions involving compactness in the geometry of Banach spaces, (preprint). 4. J. Bana´s and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, Vol. 60, Marcel Dekker, New York (1980). 5. J.A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414. 6. D.F. Cudia, Rotundity, Proc. Symp. Pure Math. 7, Amer. Math. Soc., Providence, R.I., (1963), 73–97. 7. J. Danes, A geometric theorem useful in nonlinear functional analysis, Boll. Un. Math. Ital. 6 (1972), 369-372. 8. M.M. Day, Normed Linear Spaces, Springer, Berlin (1973). 9. J. Diestel, Sequences and Series in Banach Spaces, Springer, New York (1984). 10. K. Goebel and T. Sekowski, The modulus of noncompact convexity, Ann. Univ. Mariae CurieSklodowska, Sect. A, 38 (1984), 41–48. 11. R. Huff, Banach spaces which are nearly uniformly convex, Rocky Mount. J. Math. 10 (1980), 743–749. 12. V.L. Klee, Mappings into normed linear spaces, Fund. Math. 49 (1960), 25-34. 13. V. Lakshmikantham and S. Leela, Nonlinear Differential Equations in Abstract Spaces, Pergamon Press (1981). 14. I.E. Leonard, Banach sequence spaces, J. Math. Anal. Appl. 54 (1976), 245–265. 15. R.H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, J. Wiley and Sons, New York (1976). 16. V. Montesinos, Drop property equals reflexivity, Studia Math. 87 (1987), 93–100.

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17. J.R. Partington, On nearly uniformly convex Banach spaces, Math. Proc. Camb. Phil. Soc. 93 (1983), 127–129. 18. S. Rolewicz, On drop property, Studia Math. 85 (1987), 27–35. 19. T. Sekowski and A. Stachura, Noncompact smoothness and noncompact convexity, Atti. Sem. Mat. Fis. Univ. Modena 36 (1988), 329–338. 20. L. Vesely, Personal communication.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 23–40 c 1994 Universitat de Barcelona

Operateurs ´ sommants sur un cone normal d’un espace de Banach

Richard Becker Equipe d’Analyse, Universit´e Pierre et Marie Curie-Paris 6 75252 Paris, France

Abstract The aim of this paper is to develop a theory of p-summing operators (between Banach spaces) in presence of an order structure given by a convex normal cone.

On ´etudie la structure des cones normaux, contenus dans un espace de Banach ou son dual, (th´eor`emes 14, 17, 18, 19, 20, 22). Pour cela, on tente d’´etendre, au cas des cˆones normaux ou faiblement complets (classe S), contenus dans un espace de Banach ou son dual, la th´eorie de la factorisation des op´erateurs d´eﬁnis sur un espace de Banach, ` a valeurs dans un espace Lp ([6]). Ici nous prendrons p = 1. Nous utiliserons la remarque suivante: Soit E un e.l.c.s et 1 < q < ∞; si x1 , . . . , xn , sont dans E l’application x → q1 n q |xi (x)| , o` u x ∈ E, est homog`ene de degr´e 1 sur E. Elle est donc int´egrable 1

pour toute mesure conique ≥ 0 sur E. Le caract´ere, normal ou faiblement complet, des cˆones consid´er´es jouera un role important dans cette th´eorie. De plus il y a un lien avec la th´eorie de la repr´esentation int´egrale, puisque les propri´et´es que nous allons consid´erer ne d´ependront que des mesures coniques maximales, au sens de G. Choquet. ([4], §30.11). Pour chaque cˆone X consid´er´e nous allons d´eﬁnir un indice IX , comme pour les espaces de Banach ([6] page 99 et [7] expos´e XXI); mais, dans notre cas, cet indice pourra ˆetre > 2; cela tient au fait qu’il n’y a pas de th´eor´eme de Dvoretzky pour les cˆones. Cette th´eorie pr´esente un double aspect: 23

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Becker

D’une part on peut ´etudier un cˆ one particulier, plong´e dans un espace de Banach; les r´esultats obtenus sont ind´ependants du plongement. D’autre part, un espace de Banach ´etant donn´e, on peut ´etudier la famille des cˆones qu’il contient. 1. Rappelons quelques d´eﬁnitions ([4], 30.38.40): Si E est un e.l.c.s de dual E on note h(E, E ) l’espace vectoriel r´eticul´e de fonctions sur E engendr´e par E et M + (E, E ) le cˆone des mesures coniques ≥ 0 sur E, i.e. le cˆone des formes ≥ 0 sur h(E.E ). On ordonne M (E, E ) = M + (E, E ) − M + (E, E ) par le cˆ one M + (E, E ). Si µ ∈ M + (E, E ) on nˆ ote r(µ) l’´el´ement de (E )∗ tel que l(r(µ)) = µ(l) pour toute l∈E. Soit Kµ = {r(λ): 0 ≤ λ ≤ µ}. Si Γ est un cˆ one convexe ferm´e de E, on dit que µ ∈ M + (E, E ) est port´ee par Γ si µ(f ) ne d´epend que de f |Γ , pour toute f ∈ h(E, E ). On notera M + (Γ, E ) le cˆone des tels µ. ˆ = inf(f (x): f ∈ E et f ≥ x sur Γ); Si h ∈ h(E, E ) on pose, pour tout x ∈ Γ: h(x) ˆ on a, lorsque X ∈ S: h(x) = sup(λ(h): λ ∈ M + (Γ, E ) et r(λ) = x) . ([4] Problem 30.1) 2. Rappelons la formule suivante, dˆ ue `a G. Choquet ([4] 40.1): pour toute µ ∈ M + (E, E ) on a , pour toute l ∈ E : µ(l+ ) = sup{l(x): x ∈ Kµ }. 3. Notons ´egalement que, si E est faiblement complet, pour toute µ ∈ M + (E, E ) n et toute f ∈ h(E, E ) il existe une d´ecomposition de µ en µ = µi telle que: 1

µ(f ) =

n

f (r(µi )).

([4] 30.1, 2).

1

4. On sait que toute µ ∈ M + (E, E ) est une int´egrale de Daniell sur (E )∗ ([4] 38.13) et, de plus, qu’il existe une mesure m ≥ 0 sur (E )∗ , muni de sa tribu cylindrique, telle que, pour toute h ∈ h(E, E ) on ait: µ(h) = m(h). ([5] page 12 et [1] Th´eor`eme 21).

Op´erateurs sommants sur un cone normal d’un espace de Banach

25

Notons, en particulier, que µ est le supr´emun des mesures cylindriques sur E major´ees par µ sur h(E, E ). On note C(E, E ) l’ensemble des mesures cylindriques sur E. 5. Faisons maintenant le lien entre mesures coniques sur un e.l.c.s. et applications lin´eaires, du dual de cet e.l.c.s. dans un espace L1 . Soit E un e.l.c.s; si u est une application lin´eaire (pas forc´ement continue) de E dans un espace L1 (Ω, m), il existe une unique λu ∈ M + (E, E ) telle que λu

1≤i≤n

xi

=

Ω

u(xi )

dm, pour x1 , . . . , xn ∈ E .

1≤i≤n

Si B est un espace de Banach et si u est continue on a Kλu ⊂ B , d’apr´es la formule du 2. R´eciproquement, si λ ∈ M + (B, E ) v´eriﬁe Kλ ⊂ B , on peut consid´erer l’injection canonique i de E dans le compl´et´e s´epar´e de h(B, E ), pour la semi-norme f → ˆ λ cet espace; c’est un L-espace, donc un espace de la forme L1 (Ω, m). λ(|f |); soit h Comme Kλ ⊂ B on voit, d’apr´es la formule du 2, que l’application i est continue de E dans L1 (Ω, m). On a λi = λ. 6. Th´ eor` eme Soit E un e.l.c.s. et u une application lin´eaire (pas forc´ement continue) de E dans un espace L1 (Ω, m). Si X est un cˆ one convexe ferm´e contenu dans E, les deux propri´et´es suivantes sont ´equivalentes: 1o λu est port´ee par X. 2o u est ≥ 0 sur le polaire X 0 de X dans E . Preuve: On note λ au lieu de λu . 1o ⇒ 2o . Soit x ∈ X 0 ; comme λ est port´ee par X on a λ(x− ) (u(x ))− dm = 0, d’ou 2o . o 2 ⇒ 1o . On proc´ede en deux ´etapes:

=

0, donc

xi = a) On montre d’abord que toute 0 ≤ µ ≤ λ est de la forme µ 1≤i≤n

u(xi ) gµ dm, pour une certaine gµ ∈ L∞ (Ω, m) telle que 0 ≤ gµ ≤ 1: Ω

1≤i≤n

26

Becker Il existe en eﬀet un homomorphisme d’espace vectoriel r´eticul´e, soit p, de

h(E, E ) dans L1 (Ω, m), tel que p xi = u(xi ) . 1≤i≤n

1≤i≤n

On a λ(h) = m(p(h)), pour toute h ∈ h(E, E ). Comme on a 0 ≤ µ ≤ λ on peut d´eﬁnir une forme lin´eaire µp ≥ 0 sur l’image de p par µp (p(h)) = µ(h), pour toute h ∈ h(E, E ). On a µp ≤ m sur l’image de p. D’apr´es le th´eor`eme de Hahn-Banach ([3] 3.1.1) on peut alors prolonger µp en une forme lin´eaire ≥ 0 sur L1 (Ω, m), major´ee par m. D’ou l’existence de gµ , si on a suppos´e que (Ω, m) est une somme d’espaces mesur´es de masse ﬁnie, ce qui est toujours possible. Nous n’avons pas utilis´e dans ce a) l’hypoth´ese du 2o . b) Soit x ∈ X 0 ; pour toute µ telle que 0 ≤ µ ≤ λ on a, d’apr´es a): x (r(µ)) = µ(x ) = u(x )gµ dm qui est ≥ 0, d’apr´es l’hypoth´ese du 2o . Ω

On a donc r(µ) ⊂ (fermeture de X dans (E )∗ ). D’apr´es la d´ecomposition rappel´ee en 3, on a bien λ port´ee par X. 7. Factorisation d’op´ erateurs. ([6] Chapitre I) Rappelons que, si u est une application lin´eaire continue d’un espace de Banach u 1 < q ≤ ∞, B dans un espace L1 (Ω, m), on dit que u se factorise par Lq (Ω, m), o` s’il existe une fonction g ≥ 0 de Lq (Ω, m), avec 1q + q1 = 1, telle que:

u(x) : x ∈ B1 g

soit une partie born´ee de Lq (Ω, m).

On note C1,q (u) l’inf des bornes possibles quand g ≥ 0 varie avec gq = 1. 8. D´ eﬁnition de N et de N0 ([6] page 32) Si E est un espace de Banach et q v´eriﬁe 1 ≤ q < ∞, on d´esigne par N (E, lq ) l’ensemble des applications T de E dans lq de la forme T (x) = (xn (x))n ou les xn 1 q sont dans E et v´eriﬁent xn < ∞ on pose Nq (T ) = ( xn q ) q . On d´esigne par N0 (E, lq ) le sous-espace de N (E, lq ) pour lequel les xn = 0 sont en nombre ﬁni. On peut alors ´enoncer le th´eor`eme individuel suivant, qui est une adaptation du th´eor`eme 19 de [6] .

Op´erateurs sommants sur un cone normal d’un espace de Banach

27

9. Th´ eor` eme individuel Soit B un espace de Banach et λ ∈ M + (B, E ) telle que Kλ ⊂ B ; soit u une application lin´eaire continue de B dans un espace L1 (Ω, m), telle que λu = λ. Pour tout q > 1, les quatre propri´et´es suivantes sont ´equivalentes: n 1 n q 1 q q xi pour toutes x1 , . . . , xn dans B . a) On a λ ( |xi | ) q ≤ C 1

1

b) L’op´erateur u se factorise par Lq (Ω, m) avec C1,q (u) ≤ C. c) Pour tout espace de Banach F et tout op´erateur q-sommant v de B dans F , on a: v(λ)( F ) ≤ CΠq (v). De plus v(λ) est de Radon sur F . d) Pour tout op´erateur v ∈ N0 (E, lq ) on a: v(λ)( F ) ≤ CNq (v). Preuve: Il semble int´eressant de prouver b) ⇒ c) dans un cas ´el´ementaire: n n Prenons λ = εxi et Ω = 1, . . . , n, muni de m = εi , avec x1 , . . . , xn ∈ B. 1

1

On aura u(x ) = (x (xi ))1≤i≤n . u L’hypoth´ese b) se traduit par l’existence de (ϕ(i))1≤i≤n de norme 1 dans lq , o` x (xi ) 1 1 q q + q = 1, telle que ( ϕ(i) )1≤i≤n soit de norme ≤ Cx B dans l . Dans le cadre n de c) on a: v(λ) = εv(xi ) d’o` u 1

v(λ)( F ) =

n

v(xi )F

1 n

x i ϕ(i)v par H¨ older ϕ(i) F 1 n x q 1/q i (v est q-sommant) ≤ v ϕ(i) F 1 n 1/q xi q ≤ C, par b) ≤ sup x ϕ(i) x ∈B1 1 =

dans le cas g´en´eral le th´eor`eme r´esulte de ([6] 18,19,22) lorsque m est une probabilit´e, sinon: a) ⇔ b) c’est une application de ([6] 5). b) ⇒ c) On se ram´ene ` a une probabilit´e, comme dans la preuve de ([6] 23). c) ⇒ d) c’est ´evident puisque Πq (v) ≤ Nq (v). d) ⇒ a) c’est identique au c) ⇒ a) de ([6] 19).

28

Becker

Nous allons maintenant ´enoncer un th´eor`eme (th´eor`eme 14) valable pour toutes les mesures coniques port´ees par un cˆone contenu dans un espace de Banach. 10. Rappels sur les cones, contenus dans un espace de Banach Un cone convexe X, contenu dans un espace de Banach B, est dit normal si l’ensemble (x: x ∈ X et ∃y ∈ B1 tel que (y − x) ∈ X) est born´e. ([8] V§3) Si X ∈ S pour σ(B, E ) il est normal ([2] preuve de 5). Si X est normal, son adh´erence dans B , pour σ(B , E ), est dans S. ([2] Proposition 5) Si X ⊂ E est dans S pour σ(B , B), il est normal ([2] preuve de 5). Si X ⊂ B est normal son adh´erence pour σ(E , B) est dans S. ([8] V.3.5) 11. Applications sommantes sur les cˆ ones Soit X un cˆ one convexe contenu dans un espace de Banach B et F un autre espace de Banach. On dit que T ∈ L(B, F ) est p-sommant sur X, o` u p ≥ 1, si: Pour toute suite x1 , . . . , xn de X on a: n

1/p T (xi )p

≤ C sup

x ∈B1

1

n

1/p |x (xi )|p

.

1

On note Πp (X, T ) la plus petite constante C possible. Rappelons qu’une suite (xn ) u p ≥ 1, si: dans un espace de Banach B est dite scalairement lp , o` Mp ((xn )) = sup

x ∈B1

|x (xn )|p

1/p

< ∞.

12. Th´ eor` eme Soit X ⊂ B un cˆ one normal; il existe alors une constante MX telle que: 10 Pour toute suite x1 , . . . , xn de X on a: n n xi ≤ M1 ((xi )) ≤ MX xi . 1

1

20 Pour tout espace L1 (Ω, m), toute u ∈ L(B , L1 ), qui est ≥ 0 sur X 0 , on a: r(λu ) ≤ u ≤ MX r(λu ). 30 On a MX = 1, lorsque B1 =conv ((B1 ∩ X 0 ) ∪ (−(B1 ∪ X 0 ))).

Op´erateurs sommants sur un cone normal d’un espace de Banach

29

Preuve: 10 L’in´egalit´e de gauche est triviale; pour celle de droite, il suﬃt de savoir que toute x ∈ B peut s’´ecrire x = y − z avec y , z ≥ 0 sur X et y + z ≤ cte. x ; cela provient du fait que X est un cˆone normal de B ([8] V.3.5) 0 2 On sait que u = sup (λu (|x |)), d’o` u r(λu ) = sup (x (r(λu ))) ≤ u; de x ∈B1

x ∈B1

plus, X ´etant un cˆ one normal de B, on a u = sup (λu (|x |)) ≤ Cter(λu ), x ∈B1

0

grˆ ace au raisonnement du 1 . D’o` u le r´esultat. 3 est ´evident. 0

13. Lemme ˆ l’adh´erence de Soit B un espace de Banach et X ⊂ B un cˆone convexe; soit X X dans B pour σ(B , E ). ˆ il existe un ﬁltre de suites xα , . . . , xα de X tel que Pour toute suite x1 , . . . , xn de X, n 1 n n α α xi → xi pour i = 1, . . . , n, au sens σ(B , E ), et xi ≤ xi . 1

1

Preuve: On suppose d’abord X ferm´e dans B. Soit An ⊂ B n l’ensemble des suites n z1 , . . . , zn de X telles que zi ≤ 1. 1

An est le polaire dans B n de l’ensemble Un ⊂ (B )n d´eﬁni par Un = (X 0 ) ∪ (diagonale de (B1 )n ). Le bipolaire de Un , pour σ(E , B), est ((X 0 )n + (diagonale de(B1 )n ), puisque la diagonale est compacte pour σ(B , B). Pla¸cons nous maintenant dans (B )n , muni de σ(B , B ); le polaire de An est donc ((X 0 )n + (diagonale de(B1 )n )); le bipolaire de An est donc l’ensemble des suites n ˆ telles que zi ≤ 1. z1 , . . . , zn de X 1

Cela suﬃt `a conclure lorsque X est ferm´e dans B; sinon on raisonne sur la fermeture de X dans B. 14. Th´ eor` eme ˆ son Soit B un espace de Banach et X ⊂ B un cone convexe normal. Soit X adh´erence dans σ(B , E ). Pour tout q > 1 les propri´et´es suivantes sont ´equivalentes: a) Pour toute suite (xn ) de X, scalairement l1 , il existe une suite (yn ) de X, scalairement lq , et une suite (αn ) de lq , de norme 1, telles que: xn = αn yn pour tout n et Mq ((yn )) ≤ Ca M1 ((xn )).

30

Becker

b) Pour tout espace de Banach F , toute v ∈ L(B, F ), qui est q-sommant sur X, est 1-sommant sur X et on a: Π1 (X, v) ≤ Cb Πq (X, v). c) Pour toute λ ∈ M + (X, E ) et toute suite x1 , . . . , xn de E on a: λ

n

|xi |q

1/q

n

≤ Cc r(λ)

1

xi q

1/q

ˆ , o` u r(λ) ∈ X.

1

d) Pour tout espace L1 (Ω, m), toute u ∈ L(E , L1 ), qui est ≥ 0 sur X 0 , est factorisable par Lq (Ω, m) et on a C1,q (u) ≤ Cd u. e) Pour toute suite x1 , . . . , xn de B on a, si h =

n

|xi |q

1

1/q

ˆ , en se pla¸cant dans X:

n

ˆ h(x) ≤ Ce x

xi q

1/q

ˆ , pour tout x ∈ X.

1

f) Mˆeme condition que e), mais en se limitant `a des x ∈ X. Preuve: Elle suit de pr´es celle du th´eor`eme 23 de [6], sauf pour b)⇒c) et c)⇒e). a)⇒b). (Avec Cb ≤ Ca ). Soit x1 , . . . , xn une suite de X et F un espace de Banach et T ∈ L(B, F ); on a: n

T (xi ) =

1

n

αi T (yi )

1

≤

n

T (yi )q

1

(par H¨ older) 1/q

≤

≤ (X, T )Ca M1 (xi )

(X, T )Mq (yi ) q

avec les hypoth´eses de a).

q

D’o` u Π1 (X, T ) ≤ Ca Πq (X, T ). b)⇒c). (Avec Cc ≤ Cb si la condition de 12,30 est v´eriﬁ´ee.)

Op´erateurs sommants sur un cone normal d’un espace de Banach

31

n 1 10 ) On sait que h = ( |xi |q ) q est limite ﬁltrante croissante sur B des fonctions 1

yi , o` ∈ E , qui minorent h. On a donc, u y1 , . . . , ym de h(B , E ) de la forme 1≤i≤m

pour tout ε > 0, une hε = yi telle que 1≤i≤m

0 ≤ hε ≤ h et λ(hε ) ≥ λ(h) − ε. Il existe alors, d’apr´es 3, une d´ecomposition λ =

m

λi telle que λ(hε ) =

1

m

hε (r(λi )).

1

ˆ mais pas forc´ement dans X; grace au lemme 13, il 20 ) Les r(λi ) sont dans X, existe x1 , . . . , xm dans X tels que m m xi ≤ r(λi ) = r(λ) 1

m m hε r(λi ) − hε (xi ) ≤ ε . 1

(

n 1

et

1

1

30 ) Soit v ∈ L(B, lq ) tel que v(x) = (x1 (x), . . . , xn (x), 0, 0, ); on a

q (X, v)

≤

1 q

xi q ) .

Appliquons b) a` la suite x1 , . . . , xm et `a v; il vient: m

v(xi )q ≤ Cb

1

n

xi q

1/q

M1 (xi ) .

1

Or on a: λ(hε ) =

m 1

m m m hε r(λi ) ≤ hε (xi ) + ε ≤ h(xi ) + ε = v(xi )q + ε , 1

1

1

d’o` u le r´esultat, compte tenu de la norme choisie. c) et d) sont ´equivalents compte tenu de 5 et de 6 et du th´eoreme 9, a),b). ˆ c) et e) sont ´equivalents, en raison de la formule g´en´erale suivante: h(x) = sup(λ(h): r(λ) = x et λ port´ee par X), valable dans un cadre tr´es large. (§1) d) ⇒ a). (Avec Ca ≤ Cd ). Si (xn ) est une suite de X scalairement l1 , on lui associe une u ∈ L(E , l1 ), qui est ≥ 0 sur X 0 , par x → (x (x1 ), . . . , x (xn ), . . .) et on applique d).

32

Becker

e) et f) sont ´equivalents: il suﬃt de prouver que f) ⇒e). ˆ on utilise la formule rappel´ee dans l’´equivalence de c) et de e), la formule Soit x ∈ X; du 3 et le lemme 13. 15. Nous allons montrer maintenant que les conditions du th´eor`eme pr´ec´edent ne d´ependent de X que de mani´ere intrins´eque, c’est `a dire abstraction faite de B: Examinons en eﬀet la condition a) du th´eor`eme pr´ec´edent: Pour toute suite x1 , . . . , xn de X on a, d’apr´es le th´eor`eme 12: n mM1 (x1 ) ≤ xi ≤ M1 (xi ) et 1

n mMq (xi ) ≤ sup ai xi : ai q ≤ 1

et ai ≥ 0 ≤ Mq (xi ) , avec m > 0.

1

est un autre cˆone normal, contenu dans un espace de Banach Par cons´equent, si X tel que X et X soient aﬃnement isomorphes de sorte que X1 = X ∩ B1 et B, X1 = X ∩ B1 s’absorbent mutuellement par cet isomorphisme, la condition a) du th´eor`eme pr´ec´edent restera la mˆeme. 16. Cas des cˆ ones ` a bases, faiblement compactes Nous allons voir que les conditions du th´eor`eme 14 sont remplies pour tout q > 1, d´es que X ⊂ B est un cˆ one ´a base σ(B, E ) compacte. C’est une diﬀ´erence fondamentale avec le cas des espaces de Banach de dimension inﬁnie, pour lesquels une telle propri´et´e n’est jamais v´eriﬁ´ee pour q > 2, en raison du th´eor`eme de Dvoretzky ([6]) Remarque 89). Preuve: Soit u comme dans le d) du th´eor`eme 14; on note λ au lieu de λu . Soit l ∈ B1 telle que Xl = (l−1 (1)) ∩ X soit une base σ(B, E ) compacte de X. On a u = sup (λ(|x |)); donc λ est localisable sur Xl , par une mesure de Radon x ∈B1

mλ ≥ 0 de masse mλ (1) = λ(1) ≤ u. Si x1 , . . . , xn sont dans B on aura, pour tout x ∈ Xl : n n 1/q 1/q |xi (x)|q ≤ xi q sup (x) d’o` u 1

n

λ

|xi |q

1/q

x∈Xl

1

≤ u sup (x)

1

x∈Xl

n

xi q

1/q .

1

Donc les conditions du th´eor`eme 14 sont v´eriﬁ´ees, avec Ca = sup (x). x∈Xl

On note IX l’ensemble des q > 1 tels que q v´eriﬁe les conditions du th´eor`eme 14. Le th´eor`eme suivant est fondamental.

Op´erateurs sommants sur un cone normal d’un espace de Banach

33

17. Th´ eor` eme Soit B un espace de Banach et X ⊂ B un cone normal. Il existe une partie convexe born´ee, h´er´editaire de X, soit K, qui absorbe X1 , admettant une jauge j, telle que, pour tout q > 1, avec q ∈ IX , on ait: 1) Pour tout ε > 0 et tout entier n il existe des ´el´ements z1 , . . . , zn de X tels que n n j(zi ) ≥ 1 − ε et j( ai zi ) ≤ 1, pour tous ai ≥ 0 v´eriﬁant (ai )q ≤ 1. 1

1

2) De plus, il existe x1 , . . . , xn dans X 0 , major´ees par 1 sur K, telles que xi (zi ) ≥ 1 − ε et xi (zj ) ≤ ε pour tout i et tous i = j. Preuve: Elle suit celle du lemme de Rosenthal, des lemmes 1 et 2, du th´eor`eme de ([7] expos´e XXI), avec certaines diﬀ´erences que nous allons indiquer. On proc´ede par ´etapes: 10 ) Comme X est un cone normal on peut munir B d’une norme ´equivalente telle que tout ´el´ement de la boule unit´e duale, soit U , v´eriﬁe U = conv ((U ∩ X 0 ) ∪ (−(U ∩ X 0 ))). ([8] V.3.5). On prendra pour K la trace de X sur cette boule unit´e. Dans toute la d´emonstration on supposera que B est muni d’une telle norme. 20 ) On raisonne par l’absurde; par cons´equent les conditions du th´eor`eme 14 ne sont pas v´eriﬁ´ees: Pour tout entier p > 0 il existe donc une suite x1 , . . . , xn , de X telle que, en posant: Ω = {1, . . . , n, . . .} ∞ εi (mesure sur Ω) m= 1

et, en introduisant l’op´erateur u de B dans L1 (Ω, m), d´eﬁni par (u(f ))(i) = f (xi ), on ait C1,q (u) > pu. ∞ Soit λ = εxn la mesure conique sur X, associ´ee `a u. 1

Il existe donc x1 , . . . , xm ∈ E telles que: m 1

d’apr´es ([6] th´eor`eme 8).

|xi |q

1/q dλ > pu

m 1

xi q

1/q ,

34

Becker

Il existe donc un entier n tel que: m n m 1/q 1/q q |xi | d εxi > pu xi q ; 1

1

1

par cons´equent, pour toute entier p > 0 il existe une suite ﬁnie x1 , . . . , xr de X telle que, en posant: Ω = {1, . . . , r} r m= εi (mesure sur Ω) 1

et, en introduisant l’op´erateur u, de B dans L1 (Ω, m), d´eﬁni par u(f )(i) = f (xi ), on ait C1,q (u) > pu (c.f. th´eoreme 12). Notons que l’int´eret d’avoir une suite x1 , . . . , xr ﬁnie est que C1,q (u) est ﬁni. ([7] expos´e XXI, lemme de Rosenthal). 30 ) Nous allons suivre maintenant la d´emonstration du lemme de Rosenthal de ([7] expos´e XXI), en vue de d´emontrer les analogues des lemmes 1 et 2 de ([7] expos´e XXI), mais. Il n’y aura pas vraiment d’analogue du th´eor`eme de Rosenthal. Les notations sont cependant celles de ce lemme, avec p = 1. Toutefois le ε et le α de ([7] expos´e XXI) ne seront pas utilis´es, donc (1) et (2) de cet expos´e n’interviennent pas. Le rˆ ole de Y sera jou´e par E , qui n’est pas de dimension ﬁnie, mais cette diﬀ´erence ne sera pas un obstacle; soit x1 , . . . , xk une suite ﬁnie de X telle que, en posant Ω = {1, . . . , k} k µ= εi (mesure sur Ω) 1

et en notant u l’op´erateur de Y (= E ) dans L1 (Ω, µ), d´eﬁni par u(f )(i) = f (xi ), on ait: C1,q (u) ≥ N u. On suppose u ≤ 1. Ici Y n’est pas de dimension ﬁnie, mais C1,q (u) est tout de mˆeme ﬁnie, puisque Ω est ﬁni. On introduit la fonction ϕ, ν et u1 comme dans ([7] expos´e XXI); on a (5) et (6) de ([7] expos´e XXI): c’est `a dire: ϕq dµ = 1 et ν = ϕq µ et u1 d´eﬁni de Y dans L1 (Ω, ν) par u1 (y) = u(y) ; on a: (ϕq ) u1 = u ≤ 1 et C1,q (u1 ) = C1,q (u) = N ((5) de [7] expos´e XXI) u(y) q q dµ ≤ N q yq ((6) de [7] expos´e XXI). |u1 (y)| dν = ϕ

Op´erateurs sommants sur un cone normal d’un espace de Banach

35

Lemme 1 n Soit A une partie mesurable de Ω, telle que ν(A) ≤ K . Il existe une partie mesurable E de Ω et un vecteur y de norme 1, dans Y , qui est ≥ 0 sur X, tel que: 1 |u1 (y)|q dν ≥ δ q N q . E ∩ A = ∅, ν(E) ≤ , K E Preuve: La d´emonstration est celle de ([7] expos´e XXI), mais l’´el´ement y ´evoqu´e dans la preuve du lemme 1 sera ≥ 0 sur X, ce qui est possible, en raison de la norme choisie sur Y (= B ). Lemme 2 Il existe n sous-ensembles (Ai ) de Ω, deux-´a-deux disjoints et n ´el´ements (yi ) de norme 1 dans Y , qui sont ≥ 0 sur X, tels que q u1 (yi ) dν ≥ δ q N q . Ai

Preuve: Elle est triviale, comme dans ([7] expos´e XXI). 40 On ach´eve maintenant la preuve du th´eor`eme 17: Pour chaque i tel que 1 ≤ i ≤ n on choisit une fonction ϕi ≥ 0, nulle hors de Ai , et telle que, en posant fi = u1 (yi ), on ait: (ϕi )q = 1 et δN ≤ fi ϕi dν ≤ N . On consid´ere l’op´erateur v de Lq (Ω, ν) dans lnq , d´eﬁni par: (g, ϕi ) ; on a v ≤ N1 . v(g) = N Soit uq l’op´erateur u1 , consid´er´e comme op´erateur de Y dans Lq (Ω, ν). L’op´erateur vuq est de norme ≤ 1 et on a: (vuq )∗ (ei ) = r´esultante de (( N1 )( ϕϕqi )dν) = r´esultante de (( ϕNi )dµ). C’est donc un ´el´ement de X, puisque x1 , . . . , xk est une suite ﬁnie de X. On pose zi = (vuq )∗ (ei ). n Comme vuq ≤ 1, on a ai zi ≤ 1 pour toute suite a1 , . . . , an ≥ 0, de norme

1

≤ 1 dans lnq . On a de plus yi (zi ) ≥ δ, donc zi ≥ δ. 1 D’apr´es le lemme 2 on a (fj , ϕi ) ≤ (1 − δ q ) q N , pour i = j. Le th´eor`eme en r´esulte, en prenant δ assez voisin de 1. Voici une r´eciproque partielle du th´eor`eme pr´ec´edent.

36

Becker

18. Th´ eor` eme Soit q > 1 v´eriﬁant la 1ere condition du th´eor`eme pr´ec´edent; alors aucun q1 > q n’est ´el´ement de IX . Preuve: On raisonne par l’absurde, en utilisant le th´eor`eme 14: Avec les notations du th´eor`eme pr´ec´edent on pourrait ´ecrire: z1 = α1 z1 , . . . , zn = αn zn avec (αi )q1 = 1 u et Mq1 ((zi )) ≤ Cb M1 ((zi )), d’o` n 1

zi ≤

n

zi q1

1/q1

≤ Mq1 (zi ) n1/q1

1

≤ Cb M1 (zi ) n1/q1 . 1

Par hypoth´ese M1 ((zi )) ≤ Cte.n q ; ( 1 +1) d’ou n ≤ 2.Cb .Cte.n q1 q , ce qui est impossible, puisque 1 > n → ∞.

1 q1

+

1 q ,

quand

19. Corollaire L’ensemble IX est un segment de ]1, +∞[, admettant 1 pour extr´emit´e, ou bien vide. Preuve: Si IX = ∅ soit q ∈ IX ; soit qo > 1 tel que qo < q; si qo ∈ IX le th´eor´eme pr´ec´edent prouve que q ∈ IX , contradiction. Comme dans ([7] expos´e XXI) le cas ou IX est vide est remarquable. 20. Th´ eor` eme (Avec les notations du th´eor`eme 17) Les deux propri´et´es suivantes sont ´equivalentes: a) IX = ∅. b) Pour tout ε > 0 et tout entier n il existe z1 , . . . , zn ∈ X, de jauge 1, tels que: n zi ≤ 1 + ε. 1

Preuve: a)⇒b). Cela d´ecoule du th´eor`eme 17: on a (

n

1

1

1q ) q = n q , qui peut ˆetre rendu arbitraire-

1

ment voisin de 1, pour q > 1 suﬃsammant voisin de 1, l’entier n ´etant ﬁx´e.

Op´erateurs sommants sur un cone normal d’un espace de Banach

37

b)⇒a). Soit ε > 0 et n donn´es; on a donc des z1 , . . . , zn comme dans b). Pour tout q > 1, n la condition aqi ≤ 1 entraine sup(ai ) ≤ 1; donc la conclusion 10 du th´eor`eme 17 1

est r´ealis´ee, d’apr`es l’hypoth´ese b), puisque K = j −1 (1) est une partie h´er´editaire de X. On conclut par le th´eor`eme 18. p et Lp+ 21. Cas des cˆ ones l+ p a) Si X = l+ , pour 1 < p < ∞, on a IX =]1, p ].

Preuve: On prend B = lp et B = lp . On montre d’abord que p ∈ IX : On sait, d’apres le th´eor`eme 14 e) qu’il suﬃt d’envisager le cas des mesures coniques maximales sur X. ([4] 30.11) Soit x = (xi ) ∈ X et λx la mesure conique maximale qui repr´esente x. Soient x1 , . . . , xn ∈ E ; il faut montrer que λx

n

|xi |p

1/p

n

xi p

≤ Cte.x.

1

1/p .

1 ∞

On notera xi = (xi,m ). On sait que λx =

xi εi o` u εi est la mesure de Dirac sur X

1

plac´ee en ei (i-i´eme vecteur de la base canonique de lp ). on a : λx

n

|xi |p

1/p

=

1

xj

i=n

j

|xi,j |p

1/p

i=1

 1/p i=n 1/p  ≤ (xj )p |xi,j |p  j

j

n

= x

1/p

xi p

i=1

,

1

ce qu’il fallait prouver. Montrons que tout q > p est hors de IX :

Soit e∗i ∈ lp la i-i`eme coordonn´ee et λn =

n 1

n 1 εi ; on a λn (( |e∗i |q ) q ) = n et 1

n n 1 1 1 1 1 ( e∗i q ) q = n q ; de plus ei = n p , donc n ≤ Cte.n( q + p ) est impossible pour 1

n assez grand, si q > p .

1

38

Becker

On notera que, contrairement au cas des espaces de Banach, on peut avoir IX > 2. b)

Si X = Lp+ , pour 1 < p < ∞, on a IX =]1, p ].

Preuve: On prend B = Lp et E = Lp , sur un espace (Ω, A, m). On montre d’abord que p ∈ IX : Soit f ∈ Lp+ et λf la mesure maximale sur X qui repr´esente f : Si g1 , . . . , gn ∈ Lp on a pour tout q ≥ 1, en notant g1 , . . . , gn les formes lin´eaires correspondantes sur Lp : n n 1/q 1/q λf |gi |q f (ω) |gi (ω)|q m(dω), d’o` u: = Ω

1

λf

n

|gi |p

1/p

1

≤ f p Ω

1

= f p

n

n

1/p

|gi (ω)|p m(dω)

1

gi pp

1/p .

1

Donc p ∈ IX . Pour montrer que tout q > p est hors de IX on distingue deux cas: Cas de X = Lp+ ([0, 1]): p , on abouti a` une contradiction, en prenant des g1 , . . . , gn Comme dans le cas de l+ (i−1) i telles que gi = 1([ n , n ]) et f = 1. Cas de X = Lp+ (R): On prendra des g1 , . . . , gn telles que gi = 1([i − 1, i]) et f = 1([0, n]). 22. Cas des cˆones X contenus dans le dual d’un espace de Banach B, appartenant `a S pour σ(B , B). Les r´esultats pr´ec´edents s’´etendent `a cette classe de cˆones, notamment les th´eor`emes 9,12,14,17,18,19,20. Le cone X est alors normal dans E ([2] preuve de 5), mais on suppose toujours que X ∈ S, car on n’a pas l’analogue du lemme 13. Dans ce cadre, on a X ⊂ E et u est d´eﬁnie sur B, `a valeurs dans un espace L1 et ≥ 0 sur X 0 ; on a λu ∈ M + (E , B). L’exemple 16 des cˆones `a bases, faiblement compactes se retrouve ici dans un cadre plus g´en´eral: one de S tel qu’il existe m > 0 et L ∈ B v´eriﬁant mx ≤ L(x) ≤ Soit X ⊂ B un cˆ x sur X; on a alors IX =]1, ∞[:

Op´erateurs sommants sur un cone normal d’un espace de Banach

39

La d´emonstration est analogue ` a celle de 16, puisque toute mesure conique ≥ 0 sur X, ayant sa r´esultante dans X1 , est localisable en une mesure de Radon ≥ 0 sur X1 , de masse major´ee par une constante absolue. Nous allons maintenant donner une caract´erisation des op´erateurs p-sommants d´eﬁnis sur un cˆ one normal, situ´e dans un espace de Banach; (c.f.§11 pour les d´eﬁnitions). 23. Th´ eor` eme Soit X un cone normal contenu dans un espace de Banach B et T un op´erateur lin´eaire d´eﬁni sur X, a` valeurs dans un espace de Banach F ; les deux propri´et´es suivantes sont ´equivalentes pour tout p ≥ 1: a) T est p-sommant de X dans F . b) Il existe une mesure de Radon m ≥ 0 sur X 0 ∩ B1 telle que l’on ait, pour tout x ∈ X: p p T (x) ≤ x(x ) m(dx ).

Preuve: On note d’abord que tout ´el´ement x ∈ B s’´ecrit x = y − z o´ u y , z ∈ X 0 et y + z ≤ Cte.x . ([8] V.3.5). On proc´ede ensuite comme dans ([9] expos´e II th´eor`eme 4.5). 24. Remarque. Il peut bien arriver que l’application identique de X dans B soit p-sommante. Dans le cas ou p = 1, le cˆone est alors `a base faiblement compacte (introduire le barycentre de m), s’il est dans S. 25. Th´ eor` eme Soit X un cone normal suit´e dans le dual d’un espace de Banach B, et T un op´erateur lin´eaire d´eﬁni sur X, a` valeurs dans un autre espace de Banach F ; les deux propri´et´es suivantes sont ´equivalentes pour tout p ≥ 1: a) T est p-sommante de X dans F . b) Il existe une mesure de Radon m ≥ 0 sur la trace de X 0 , (pris dans B ), sur B1 telle que l on ait, pour toute x ∈ X: p x (x ) m(dx ). T (x )p ≤ Preuve: Elle est analogue ` a celle du th´eor`eme 23; on se place dans B pour des raisons de compacit´e.

40

Becker

26. Remarque. Si l’application identique de X dans B est 1-sommante, alors on retrouve la classe des cˆones consid´er´es au §22, en introduisant le barycentre de m. 1 Ces cˆones ne sont pas n´ecessairement `a base compacte (par exemple X = l+ avec B = c0 ).

Bibliographie 1. R. Becker, Sur l’int´egrale de Daniell, Revue Roumaine 26 (2) (1981), 189–206. 2. R. Becker, Mesures coniques sur un espace de Banach ou son dual, J. Austral. Math. Soc. (s´erie A) 39 (1985), 39–50. 3. N. Bourbaki, E.V.T chapitre I et II, 2 i´eme e´ d. Paris Hermann. 4. G. Choquet, Lectures on Analysis, Vol. 1-3, Math. Lectures Notes Series, New York, Amsterdam, Benjamin 1969. 5. I. Kluvanek, Vector measures and control systems, Mathematics Studies, No 20, North-Holland. 6. B. Maurey, Th´eor`emes de factorisation pour les op´erateurs lin´eaires a` valeurs dans les espaces Lp , Ast´erisque No 11 (1974). 7. B. Maurey, S´eminaire Maurey-Schwartz (72–73), Ecole Polytechnique. 8. H.H. Schaeffer, Topological Vector Spaces, Springer Verlag, Berlin, Heidelberg, New York 1971. 9. L. Schwartz, S´eminaire Maurey-Schwartz (72–73), Ecole Polytechnique.

Bibliographie complementaire ´ 10. O. Blasco, Positive p-summing operators on Lp spaces, Proc. Amer. Math. Soc. 100 (1987), 275–280. 11. H.H. Schaeffer, Banach lattices and positive operators, Springer Verlag, Berlin, Heidelberg, New York 1974.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 41–46 c 1994 Universitat de Barcelona

On infinitely smooth almost-wavelets with compact support

M. Berkolaiko Department of Mathematics, Voronezh Ing. Constr. Institute 394006 Voronezh, Russia

I. Novikov Department of Mathematics, Voronezh State University 394693 Voronezh, Russia

Abstract In 1985 Y. Meyer has constructed the infinitely smooth function ψ(t), t ∈ R, j

with compact spectrum such that the system of functions 2 2 ψ(2j t−k), j, k ∈ Z, forms an orthonormal basis for L2 (R) [1]. Now such systems are called wavelets. There are known wavelets with exponential decay on infinity [2,3,4] and wavelets with compact support [5]. But these functions have finite smoothness. It is known that there does not exist infinitely differentiable compactly supported wavelets

In the article we present the system of inﬁnitely smooth functions ψ = {ϕ0k , ψjk , j = 0, 1, 2, . . . ; k ∈ Z} with the following properties: 1) ψ forms an orthonormal basis for L2 (R); 2) ϕ0k (t) = ϕ00 (t − k); ψjk (t) = ψj0 (t − k2−j ); t ∈ R, j = 0, 1, 2, . . . ; k ∈ Z; 3) supp ϕ00 = (−3, 0); supp ψj0 = [−(j + 3)2−j , j2−j ]. In contrast to wavelets, the system ψ is not generated by dilations and translations of a single function. However, the measure of ψjk -supports converges to zero if j tends to inﬁnity and under the ﬁxed j the functions ψjk , k = 0, are obtained by translations of ψj0 . These properties allow to call the system ψ as almost-wavelets. 41

42

Berkolaiko and Novikov

Construction of ψ. Let N = 1, 2, 3, . . .. Deﬁne the function 1 + eiξ N QN (eiξ ), ξ ∈ R, mN (ξ) := 2 where the polynomial N −1

QN (ξ) =

qN (l)ξ l ,

qN (0) = 0,

l=0

satisﬁes the identity |QN (eiξ )|2 =

N − 1 + l 2 ξ l sin ( ) , l 2

n−1 l=0

and is determined as in [5]. The functions mN (ξ) meet the identity |mN (ξ)|2 + |mN (ξ + π)|2 = 1,

ξ ∈ R.

(1)

As usual F ϕ denotes the Fourier transform of ϕ and F −1 ϕ its inverse, respectively. One can prove that inﬁnite products ∞ Gj (ξ) := mN (2−N ξ), j = 0, 1, 2, . . . N =j+1

converge absolutely for any ξ ∈ R, and the convergence is uniform on any bounded set. And what is more the following inequality is true 1, |ξ| ≤ 2j π; |Gj (ξ)| ≤ (2) |ξ|−α ln |ξ| , |ξ| > 2j π; where α is a positive constant depending only on j. Deﬁne ϕj (t) := (2π)−1/2 2−j/2 (F −1 Gj )(t), j ∈ {0, 1, 2, . . .}. By virtue of (2), the deﬁnition is correct. Let ϕjk (t) := ϕj (t − k2−j ), k ∈ Z. The functions ψj (t), j = 0, 1, 2, . . . are determined by their Fourier transforms F ψj (ξ) = 21/2 e−i(2

−j−1

ξ+π)

mj+1 (2−j−1 ξ + π)F ϕj+1 (ξ), ξ ∈ R.

By virtue of (2-3), the deﬁnition is correct. Consider the functions ψjk (t) = ψj (t − k2−j ), j = 0, 1, . . . ; k ∈ Z; and denote ψ = {ϕ0k , ψjk , j = 0, 1, 2, . . . ; k ∈ Z}. Properties of ψ. From (1-2) it follows:

On infinitely smooth almost-wavelets with compact support

43

Lemma 1 Functions ϕjk and ψjk , j = 0, 1, . . . ; k ∈ Z are inﬁnitely smooth. Lemma 2 For any j = 0, 1, 2, . . . ; k, k ∈ Z (ϕjk , ϕjk ) = δkk ; (ψjk , ψjk ) = δkk ; (ϕjk , ψjk ) = 0 ,

(3)

where δkk is the Kronecker symbol, (f, g) denotes inner product in L2 (R). Let [xl , l ∈ L] designate the closure of the linear span of {xl , l ∈ L} in L2 -norm, L-some index set. Introduce the vector spaces Vj := [ϕjk , k ∈ Z], Wj := [ψjk , k ∈ Z], j = 0, 1, 2, . . . By virtue of Lemma 2, systems {ϕjk , k ∈ Z} and {ψjk , k ∈ Z} are orthonormal bases of Vj and Wj , respectively, j = 0, 1, 2, . . . . Lemma 3 For any j = 0, 1, 2, . . . Vj ⊂ Vj+1 , Vj ⊕ Wj = Vj+1 ; ∞

(4)

Vj = L2 (R).

j=0

Lemma 4 For any f ∈ L2 (R)

|(f, ϕ0k )| + 2

∞

|(f, ψjk )|2 = f 2L2 .

j=0 k∈Z

k∈Z

Let j = 0, 1, . . . ; r = 1, 2, . . .. Denote (j)

ηr,0 (t) := 2j/2 χ(−2−j−1 ,2−j−1 ) (t), where χe is an indicator function of the set e.

44

Berkolaiko and Novikov (j)

Deﬁne functions ηr,ν , ν = 1, . . . , r; by recursion (j) ηr,ν (t) :=

√

(2j+2r+1−2ν)

2

hj+r+1−ν (l)ηr,ν−1 (2t + 2−j l), (j)

l=0

where numbers hN (l), l = 0, 1, . . . , 2N − 1, are determined by the identity −1/2

mN (ξ) ≡ 2

2N −1

hN (l)eilξ .

l=0

Lemma 5 (j)

For any j = 0, 1, 2, . . . the functions ηr,r converge to ϕj pointwise and in L2 (R) as r → ∞. Lemma 5 implies Lemma 6 The function ϕj and ψj , j = 0, 1, . . . have compact supports: supp ϕj = [−(2j + 3)2−j , 0], supp ψj = [−(j + 3)2−j , j2−j ]. Finally we have Theorem The system Ψ, which consists of inﬁnitely smooth functions with compact supports, forms an orthonormal basis for L2 (R). Remark 1. Let N = 1, 2, . . .. In [4] Ingrid Daubechies has considered the functions

∞ (N ) mN (2−l ξ) (t), Φ1 (t) = (2π)−1/2 F −1 (N ) Φ2 (t)

1/2

= (2π)

l=1 (N ) gN (ν)Φ1 (2t

+ ν)

ν∈Z

where gN (ν) := (−1)ν hN (−ν + 1), ν ∈ Z.

On infinitely smooth almost-wavelets with compact support

45

She has proved that the system

(N )

2j/2 Φ2 (2j t − k), j ∈ Z, k ∈ Z

(N )

is an orthonormal basis for L2 (R) and supp Φ2 = [−N + 1, N ]. Besides, the (N ) (N ) smoothness of Φ1 , Φ2 is ﬁnite and grows with increase of N . 2. Using the formula ∞

cos(2−N ξ) =

N =1

sin ξ ξ

it is easy to show that for any j = 0, 1, . . . F ϕj (ξ) = (2π)−1/2 2−j/2 ei2

−j−1

(j+2)ξ

Rj (ξ)

∞

QN (2−N ξ),

N =j+1

where

Rj (ξ) =

sin 2−j−1 ξ 2−j−1 ξ

j+1 ∞ N =j+2

sin 2−N ξ . 2−N ξ

Remark that functions upj (t) = (2π)−1/2 (F −1 Rj )(t), j = 0, 1, 2, . . . were introduced and studied in [6]. Comments 1,2 show that our approach develops the ideas of [5-6]. 3. The construction of functions ψjk , j = 0, 1, . . . ; k ∈ Z, satisfying (3-4) uses the concepts of multiresolution analysis [7]. 4. It is interesting to investigate the basis properties of the almost-wavelets in Besov-Lizorkin-Triebel spaces. The authors intend to devote a separate paper to this.

46

Berkolaiko and Novikov References

1. Y. Meyer, Principe d’incertitude, bases hilbertiennes et alg`ebres d’operatours, Sem. Bourbaki, No 662 (1985-1986), 1–15. 2. J.O. Stromberg, A modified Franklin system and higher order spline systems on Rn as unconditional basis of Hardy spaces, Repts. Der. Math. Univ. Stockholm, v. 21, 1981. 3. G.Battle, A block spin construction of ondelettes, Part 1: Lemarie functions, Comm. Math. Phys. v. 110 (1987), 601–615. 4. P.G. Lemarie, Ondelettes a` localisation exponentielle, J. Math. Pures Appl. v. 67 (1988), 227–236. 5. I. Daubechies, Orthonormal bases of wavelets with compact support, Comm. Pure Appl. Math. v. 41 (1987), 909–996. 6. V.L. Rvachev, V.A. Rvachev, Nonclassical methods in approximate theory for boundary value problems, Kiev: Naukova dumka, 1979, (in Russian). 7. S.G. Mallat, Multiresolution approximation and wavelet orthonormal bases of L2 (R), Trans. Amer. Math. Soc. v. 315 (1989), 69–87.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 47–57 c 1994 Universitat de Barcelona

Applications of sequential shifts to an interpolation problem

Zbigniew Binderman Academy of Agriculture, Nowoursynowska 166, 02-766 Warszawa, Poland

Abstract In the present, paper initial operators for a right invertible operator, which are induced by sequential shifts and have the property c(R) (cf. [23]) are constructed. An application to the Lagrange type interpolation problem is given. Moreover, an example with the Pommiez operator is studied.

§ 0. Let X be a linear space over the ﬁeld C of the complex numbers. Denote by L(X) the set of all linear operators with domains and ranges in X and by L0 (X) the set of those operators from L(X) which are deﬁned on the whole space X. We denote by R(X) the set of all right invertible operators belonging to L(X), by RD – the set of all right inverses of a D ∈ R(X) and by FD – the set of all initial operators for D, i.e. RD := {R ∈ L0 (X): DR = I}, FD := {F ∈ L0 (X): F 2 = F, F X = ker D and ∃ R ∈ RD : F R = 0}. In the sequel, we shall assume that dim ker D > 0, i.e. D is right invertible but not invertible. The theory of right invertible operators and its applications can be found in the book of D. Przeworska-Rolewicz [17]. Here and in the sequel we admit that 00 := 1. We also write: N for the set of all positive integers and N0 := {0} ∪ N. 1991 Mathematics Subject Classification: 47B99, 47E05, 47G99. Keywords and phrases: right invertible operators, sequential shifts, Lagrange interpolation problem, Pommiez operator.

47

48

Binderman For a given operator D ∈ R(X) we shall write (cf. [17]): S :=

∞

ker Di .

(0.1)

i=1

If R ∈ RD then the set S is equal to the linear span P (R) of all D-monomials, i.e. S = P (R) :=lin {Rk z: z ∈ ker D, k ∈ N0 }.

(0.2)

Evidently, the set P (R) is independent of the choice of the right inverse R. § 1. Suppose that Y = (s) is the set of all sequences a = {an }, where an ∈ C (n ∈ N0 ). In the sequel, a non-empty set Ω ⊆ C containing a number diﬀerent from zero and a sequence a = {an } ∈ Y are arbitrarily ﬁxed. Definition 1.1. Suppose that D ∈ R(X) and dim ker D > 0. We say that Ta,Ω = {Ta,h }h∈Ω ⊂ L0 (X) is a family of sequential shifts for the operator D induced by the sequence a if the following conditions are satisﬁes: Ta,h =

∞

an hn Dn

on the set S,

n=0

for all h ∈ Ω; k ∈ N0 , where S is deﬁned by Formula (0.2). We should point out that by deﬁnition of the set S, the last sum has only a ﬁnite number of members diﬀerent from zero. The listed properties and other information about shifts for right invertible operators can be found in the author’s papers [1]-[11] (cf. also works of D. PrzeworskaRolewicz [16]-[20], [22]). Theorem 1.1 (cf. [5]) Suppose that D ∈ R(X) and dim ker D > 0, F is an initial operator for D corresponding to an R ∈ RD and a family TΩ = {Th }h∈Ω ⊂ L0 (X). Then the following two conditions are equivalent: a) TΩ is a family of sequential shifts for the operator D induced by the sequence a = {an }, k b) Th Rk F = aj hj Rk−j F for all h ∈ Ω; k ∈ N0 . j=0

Applications of sequential shifts to an interpolation problem

49

Proposition 1.1 (cf. [5]) Suppose that D ∈ R(X), dim ker D > 0 and Ta,Ω = {Ta,h }h∈Ω is a family of sequential shifts for the operator D induced by the sequence a = {an }. Let F be an initial operator for D corresponding to an R ∈ RD . Then (i) For all h ∈ Ω : z ∈ ker D; k ∈ N0 k

Ta,h R z =

k

aj hj Rk−j z .

(1.1)

j=0

(ii) The operators Ta,h (h ∈ Ω) are uniquely determined on the set S. (iii) If X is a complete linear metric space, S = X and Ta,h are continuous for h ∈ Ω then Ta,h are uniquely determined on the whole space. (iv) For all h ∈ Ω the operator Ta,h commute on the set S with the operator D. Proposition 1.2 Suppose that all assumptions of Proposition 1.1 are satisﬁed and am = 0 for a number m ∈ N. For an arbitrary ﬁxed h ∈ Ω \ {0} we deﬁne the operator Fm,h := α(h)F Ta,h Rm ,

(1.2)

α(h) := h−m a−1 m .

(1.3)

where Let an operator A ∈ L0 (X) be arbitrary ﬁxed. Then (i) The operator Fm,h is an initial operator for D corresponding to the right inverse Rm,h := R − Fm,h R .

(1.4)

Dm,h := D + AFm,h

(1.5)

(ii) The operator is right invertible and Rm,h ∈ RDm,h . Proof. (i) Theorem 1.1. and the equality F R = 0 together imply 2 Fm,h = [αm (h)F Ta,h Rm ][αm (h)F Ta,h Rm ] 2 2 (h)F Ta,h Rm F [Ta,h Rm ] = αm (h)F = αm

m

am−j hm−j Rj F [Ta,h Rm ]

j=0

=

2 αm (h)am hm F 2 Ta,h Rm

= αm (h)F Ta,h Rm = Fm,h .

50

Binderman

Moreover, the operator Fm,h is a projection onto ker D. Indeed, for all z ∈ ker D by Formula (1.1) we have m

Fm,h z = αm (h)F Ta,h R z = αm (h)F

m

am−j hm−j Rj z

j=0 m

= αm (h)am h F z = F z = z . The operator Fm,h is an initial operator for D corresponding to the right inverse determined by Formula (1.4) (cf. [17]). (ii) Consider the operator Dm,h := D + AFm,h . Point (i) and the deﬁnition together implies Fm,h Rm,h = 0, DFm,h = 0 . This yields that on X Dm,h Rm,h = [D + AFm,h ]Rm,h = DRm,h + AFm,h Rm,h = DRm,h = I, i.e. Rm,h ∈ RD ∩ RDm,h . Following [23], an initial operator F0 for D has the property c(R) for an R ∈ RD if there exist scalars ck such that F0 R k z =

ck z for all z ∈ ker D; k ∈ N k!

(1.6)

0 ⊆ FD has and ck = 0 for all k ∈ N if F0 = F . We shall write: F0 ∈ c(R). A set FD 0 the property (c) if for every F0 ∈ FD there exists an R ∈ RD such that F0 ∈ c(R). The set FD of all initial operators has the property (c) if and only if dim ker D = 1 (cf. [23]).

Proposition 1.3 Suppose that all assumptions of Proposition 1.1 are satisﬁed. Let the operator Fm,h be deﬁned by Formula (1.2), where 0 = h ∈ Ω is arbitrarily ﬁxed. Then Fm,h ∈ c(R) and the coeﬃcients ck have the form ck = βk hk were βk = k!am+k a−1 m .

(k ∈ N),

(1.7)

Applications of sequential shifts to an interpolation problem

51

Proof. Let z ∈ ker D; k ∈ N be arbitrary ﬁxed. Then by Formula (1.1) we have Fm,h Rk z = [αm (h)F Ta,h Rm ]Rk z = αm (h)F Ta,h Rm+k z = αm (h)F

m+k

am+k−j hm+k−j Rj z = αm (h)am+k hm+k F z

j=0

am+k k = h z. am By Proposition 1.2, Fm,h ∈ FD . Proposition 1.3 implies Proposition 1.4 Suppose that D ∈ R(X) and dim ker D > 0, F is an initial operator for D corresponding to an R ∈ RD and let 0 = h ∈ C be arbitrarily ﬁxed. Then there exists Fh ∈ L0 (X), which is an initial operator for D corresponding to a right inverse Rh := R − Fh R such that Fh Rn z = hn z

for all z ∈ ker D (n ∈ N).

(1.8)

The operator Fh is deﬁned by the formula Fh := F Th ,

(1.9)

where Th is an extension of the operator Th ∈ L0 (S): Th :=

∞

hn D n .

(1.10)

n=0

Proof. Consider the operator Fh determined by Formula (1.9), where Th is an extension of the operator Th deﬁned by Formula (1.10). By Proposition 1.2, Fh is an initial operator for D corresponding Rh determined by Formula (1.4). Proposition 1.3 implies that Fh ∈ c(R) and Formula (1.8) holds. We have also (cf. Proposition 2.3.-[23], Theorem 5.25.-[17]):

52

Binderman

Proposition 1.5 Suppose that all assumptions of Proposition 1.4 are satisﬁed. Then there exists Fh ∈ L0 (X), which is an initial operator for D corresponding to Rh = R − Fh R, such that hn z for all z ∈ ker D (n ∈ N). Fh R n z = n! The operator Fh is deﬁned by the formula Fh := F Th , where Th is an extension of the operator Th ∈ L0 (S): ∞ hn n Th := D . n! n+0

§ 2. Let D ∈ R(X) and dim ker > 0. We consider the following Lagrange type interpolation problem (cf. Przeworska-Rolewicz [23], [17], also Nguyen Van Mau [14], Tasche [24]): N −1 Find a D-polynomial of degree N − 1 (N > 1), i.e. an element u = R k zk , k=0

where R ∈ RD ; z0 , z1 , . . . , zN −1 ∈ ker D which admits, for given N diﬀerent initial operators F0 , F1 , . . . , FN −1 ∈ FD , the given values j = 0, 1, . . . , N − 1,

Fj u = uj ,

(2.1)

where uj ∈ ker D. Theorem 2.1 (cf. [23], Theorem 3.1) Suppose that D ∈ R(X), R ∈ RD and F0 , F1 , . . . , FN −1 ∈ c(R) such that Fj R k z =

djk z for j = 0, 1, . . . , N − 1, k ∈ N. k!

If V = det(djk )j,k=0,1,...,N −1 = 0 then the considered interpolation problem has a unique solution for every u0 , u1 , . . . , uN −1 ∈ ker D of the form u=

N −1 N −1 1 (−1)k+j Vjk Rk uj , V j=0

(2.2)

k=0

where Vjk is the minor determinant obtained by canceling in V the k-th column and the j-th row; j, k = 0, 1, . . . , N − 1.

Applications of sequential shifts to an interpolation problem

53

Proposition 1.4 implies that there exist initial operators F0 , F1 , . . . , FN −1 ∈ FD ∩ c(R) such that Fk Rn z = hnk z for all z ∈ ker D, n ∈ N, (k = 0, 1, . . . , N − 1), Fk := Fhk = F Thk , where hk ∈ C are arbitrarily ﬁxed (0 ≤ k ≤ N −1), Th is an extension of the operator Th ∈ L0 (S) deﬁned by Formula (1.10). Evidently, for diﬀerent hk , k = 0, 1, . . . , N −1 the determinant V = det k!hjk j,k=0,1,...,N −1 = 0. In particular we take hk = εk , where εk = exp(2πik/N ), k = 0, 1, . . . , N − 1. Then (2.3) Fk = Fεk and Fk Rn z = εnk z = εnk z, where ε := ε1 = exp(2πi/N ). We deﬁne the vectors: R := I, R, R2 , . . . , RN −1 , uT := u0 , u1 , . . . , uN −1 , (where as usually AT denotes the matrix transposed to A). Theorem 2.2 Suppose that D ∈ R(X) and dim ker D > 0, F is an initial operator for D corresponding to an R ∈ RD . Then the interpolation problem with Fk (k = 0, 1, . . . , N − 1) deﬁned by Formula (2.3) has a unique solution of the form u = RBu, where



1 1  1  B= 1 N . 1

1

ε−1 ε−2 .

ε−(N −1)

1

ε−2 ε−4 .

ε−2(N −1)

(2.4)  ... 1 . . . ε−(N −1)   . . . ε−2(N −1)  .  . . 2 . . . ε−(N −1)

54

Binderman

Proof. We are looking for a solution of the interpolation problem of the form u = Rz, satisfying the conditions (2.1), where the vector zT := [z0 , z1 , . . . , zN −1 ], z0 , z1 , . . . , zN −1 ∈ ker D is to be determined, we obtain the equation B−1 z = u, where



B−1

1 1  = 1  . 1

1 ε1 ε2 .

ε(N −1) (N −1)(3N −2)

1 ε2 ε4 .

ε2(N −1)

 ... 1 . . . ε(N −1)   . . . ε2(N −1)  .  . . 2 . . . ε(N −1)

2 The determinant |B−1 | = i N 2 = 0 and BB−1 = B−1 B = I. This implies that the problem has a unique solution determined by Formula (2.4). N

Example 2.1: Let X = H(K) be the class of all functions analytic in the disk K = {h ∈ C: |h| < r, r > 0}. We deﬁne operators D, R as follows: [Dx](t) := where

x(t) − x(0) ; [Rx](t) := tx(t); x ∈ X, t ∈ K, t x(t) − x(0) := x (0). t=0 t

The operators D, R are uniquely determined on the whole space X, i.e. D, R ∈ L0 (X), dim ker D = 1, codim RX = 1 (cf. [13]). The operator D is called a Pommiez operator (cf. Pommiez [15]). We can prove (cf. [3]) that R ∈ RD , [F x](t) = [(I − RD)x] = x(0),

(2.5)

P (R) =lin {Rk 1: k = 0, 1, 2, . . .}.

(2.6)

Evidently, FD ⊂ c(R) and S = P (R) = X. In order to construct the operators Fh deﬁned in Proposition 1.4., we observe that Rk 1 = tk , Rk F x = (F x)Rk 1 = x(0)tk , x ∈ X, t ∈ K, k ∈ N0 . We take Tf,h x :=

∞ n=0

hn Dn x for x ∈ S; h ∈ K.

(2.7)

Applications of sequential shifts to an interpolation problem

55

Clearly, T = {Th }h∈K is a family of sequential shifts for the operator D induced by the sequence a = {1, 1, . . . , 1, . . .}. Proposition 1.1 and Formula (2.5) together imply: Th Rk F x =

k

hj Rk−j F x = x(0)

j=0

=

k

hj tk−j

j=0

k+1 −tk+1 x(0) h h−t

k

x(0)(k + 1)h

for t = h

(2.8)

for t = h .

Evidently, Th R F x ∈ X. Equality (2.6) implies that every element x ∈ P (R) can be written in the form m bk Rk 1 (m ∈ N0 ), x(t) = k

k+0

where bk (k = 0, 1, . . . , m) are scalars, in one and only one manner. Let Th be deﬁned by Formula (2.7) for arbitrarily ﬁxed h ∈ K and let x ∈ P (R). Then k m m m k k 1 1 T h x = Th bk R bk T h R = bk hj tk−j = k=0

k=0

k=0

j=0

  tx(t)−hx(h) for t = h t−h =  d [tx(t)] = x(h) + hx (h) for t = h dt t=h This follows from Formula (2.8). We take Th ∈ L0 (X): tx(t)−hx(h) for t = h t−h Th x (t) := x(h) + hx (h) for t = h for all x ∈ X, h ∈ K. Hence for h ∈ K, x ∈ X Fh x (t) = F Th x (t) = x(h).

In this case the conditions (2.1) with Fj = Fhj , where hj = hm for j = m = 0, 1, . . . , N − 1, have the form Fj u (t) = Fhj u (t) = u(hj ) = uj (j = 0, 1, . . . , N − 1), where u ∈ X, uj are scalars. The interpolation problem has a unique solution for every scalars u0 , u1 , . . . , uN −1 of the form (2.2), where V = det(k!hkj ) = 0. In particular, the interpolation problem with the knots hj = εj , εj = exp(2πij/N ) (0 ≤ j ≤ N − 1) on the unit circle has a unique solution for every scalars u0 , u1 , . . . , uN −1 of the form (2.4).

56

Binderman References

1. Z. Binderman, Complex R-shifts for right invertible operators, Demonstratio Math. 23:4 (1990), 1043–1053. 2. Z. Binderman, On some properties of convex R-shifts, Demonstratio Math. 25:1-2 (1992), 207–217. 3. Z. Binderman, Some properties of operators of complex differentation and shifts, Zeszyty Naukowe Politechniki L´odzkiej, Matematyka 24 (1993), 5–18. 4. Z. Binderman, Cauchy integral formula induced by right invertible operators, Demonstratio Math. 25:3 (1992), 671–690. 5. Z. Binderman, Functional shifts induced by right invertible operators, Math. Nachr. 157 (1992), 211–214. 6. Z. Binderman, On periodic solutions of equations with right invertible operators induced by functional shifts, Demonstratio Math. 26:3-4 (1993), 535–543. 7. Z. Binderman, A unified approach to shifts induced by right invertible operators, Math. Nachr. 161 (1993), 239–252. 8. Z. Binderman, On summation formulas induced by functional shifts of right invertible operators, Demonstratio Math. 27 (1994), (to appear). 9. Z. Binderman, Periodic solutions of equations of higher order with a right invertible operator induced by functional shifts, Ann. Univ. M. Curie-Sklodowska 46 (1992), 9–22. 10. Z. Binderman, On some functional shifts induced by operators of complex differentation, Opuscula Math., (to appear). 11. Z. Binderman, Some remarks on sequential shifts induced by right invertible operators , Functiones et Approximatio, (to appear). 12. Z. Binderman, On singular boundary value problems for generalized analytic functions, Zeszyty Naukowe Politechniki L´odzkiej, Matematyka, 23 (1993), 11–16. 13. M.K. Fage and N.I. Nagnibida, An equivalence problem of ordinary linear differential operators, (Russian), Nauka, Novosybirsk, 1987. 14. Nguyen van Mau, Boundary value problems and controllability of linear systems with right invertible operators, Dissertations Mathematicae 316, Warszawa, 1992. 15. M. Pommiez, Sur la suite differences divis´ees successives relatives a une finction analytique, C.R. Acad. Sci. 260:20 (1965), 5161–5164. 16. D. Przeworska-Rolewicz, Shift and periodicity for right invertible operators, Research Notes in Mathematics 43, Pitman Advanced Publish. Program, Boston-London-Melbourne, 1980. 17. D. Przeworska-Rolewicz, Algebraic Analysis, PWN – Polish Scientific Publishers and D. Reidel Publish. Comp. Warszawa-Dordrecht, 1988. 18. D. Przeworska-Rolewicz, Spaces of D -paraanalytic elements, Dissertationes Mathematicae 302, Warszawa, 1990. 19. D. Przeworska-Rolewicz, True Shifts, J. Math. Analysis Appl. 170 (1992), 27–48. 20. D. Przeworska-Rolewicz, Generalized Bernoulli operator and Euler-Maclaurin formula, Lecture Notes in Economics and Math. Systems 382, Springer-Verlag, Berlin-Heidelberg-New York, (1992), 355–368.

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21. D. Przeworska-Rolewicz, Advantages of one-dimensional kernels, Math. Nachr. 149 (1990), 133–147. 22. D. Przeworska-Rolewicz, The operator exp(hD) and its inverse formula, Demonstratio Math. 26 (1993), 545–552. 23. D. Przeworska-Rolewicz, Property (c) and interpolation formulae induced by right invertible operators, Demonstratio Math. 21 (1988), 1023–1044. 24. M. Tasche, A unified approach to interpolation methods, Journ. of Integral Eqs. 4 (1982), 55–75. 25. H. von Trotha, Structure properties of D-R spaces, Dissertationes Math. 180, 1981.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 59–70 c 1994 Universitat de Barcelona

Generalized precompactness and mixed topologies

Jurie Conradie Department of Mathematics, University of Cape Town Rondebosch, 7700 South Africa

Abstract The equicontinuous sets of locally convex generalized inductive limit (or mixed) topologies are characterized as generalized precompact sets. Uniformly preLebesgue and Lebesgue topologies in normed Riesz spaces are investigated and it is shown that order precompactness and mixed topologies can be used to great advantage in the study of these topologies.

Notions of smallness play an important role in analysis, and one of the best known and most useful of these is precompactness. Many smallness properties appearing in the literature can in fact be thought of as “precompactness-like” conditions. A generalized form of precompactness was introduced in [4] in order to prove an extension of Grothendieck’s precompactness lemma. The aim of this paper is to show that in the context of locally convex spaces there is an intimate relationship between generalized precompactness and generalized inductive limit (or mixed) topologies. Generalized precompactness is deﬁned in the ﬁrst section, and examples are given. Mixed topologies are introduced in the second section and it is shown that the equicontinuous sets of locally convex mixed topologies can often be characterized as generalized precompact sets. In the third section the special case of order precompact sets in normed Riesz spaces and the related mixed topologies are explored in more detail. These topologies turn out to be closely related to the uniformly Lebesgue topologies introduced by Nowak in [11].

59

60

Conradie 1. Generalized precompactness

Let E be a vector space (real or complex). A bornology B on E is a collection of subsets of E which covers E, is closed under ﬁnite unions and has the property that if B ∈ B and A ⊂ B, then A ∈ B. Now let E be a topological vector space. Following [4], we call a subset A of E B-precompact if for every neighborhood U of 0 in E, there is a B ∈ B such that A ⊂ B + U . The reader is referred to [4] for the elementary properties of B-precompact sets. Some examples can also be found there; we give some more. Example 1.1: Let E be a Banach space and B the collection of relatively weakly compact subset of E. It follows from a result of Grothendieck ([5], Chapter 13, Lemma 2) that the B-precompact sets are exactly the relatively weakly compact sets in E. Example 1.2: Let E be a normed space and B be the collection of bounded weakly metrizable subsets of E. It follows from [14], Lemma 1.2 that the B-precompact sets are exactly the bounded weakly metrizable sets. Example 1.3: Let E be a locally convex space. If U is a closed absolutely convex neighborhood of 0 in E, we shall denote by EU the locally convex space obtained by equipping E with the gauge of U as seminorm. Let B be the collection of bounded subsets of E. Then E is quasinormable if for every closed absolutely convex neighborhood U of 0 in E, there is another such neighborhood V such that V is B-precompact in EU (cf. [9], 10.5.2). Example 1.4: Let A be an ideal of operators between Banach spaces in the sense of Pietsch ([13]). For a Banach space F let B be the collection of all subsets of B of F such that there is a Banach space G and an S ∈ A(G, F ) such that B ⊂ S(BG ), where BG denotes the closed unit ball of G. A bounded linear operator T : E → F belongs to the surjective hull of the closure of A if and only if T maps the unit ball of the Banach space E into a B-precompact set. (cf. [10]). A bornology B on a vector space E is a vector bornology if it is closed under sums, scalar multiples and balanced hulls. A subset B0 of a bornology B is a basis for B if for every B ⊂ B, there is a B0 ∈ B0 such that B ⊂ B0 . A vector bornology will be called convex if it has a basis consisting of absolutely convex sets. It is easy to check that if B is a bornology on a topological vector space E, the collection Bp of all B-precompact sets is again a bornology on E. If B is a vector bornology, so is Bp ; if B is convex, Bp is closed under the formation of convex hulls.

Generalized precompactness and mixed topologies

61

2. Mixed Topologies The mixed topologies we consider will be a special case of the generalized inductive limit topologies introduced by Garling [7], and a slight generalization of the mixed topologies of Persson [12]. If E is a vector space, B a convex bornology on E and τ a vector topology on E such that every B ∈ B is τ -bounded, we shall call the triple (E, B, τ ) a mixed space. If in addition B has a basis consisting of τ -closed sets, the mixed space is called normal. The mixed topology γτ (B) is the ﬁnest locally convex topology coinciding with τ on the sets in B. If the bornology B is clear from the context, we shall abbreviate γτ (B) to γτ . We write β for the ﬁnest locally convex topology for which every B ∈ B is bounded; when equipped with this topology E is a bornological space. The space of all linear functionals on E which are bounded on the sets in B will be denoted by E b . This is also the dual (E, β) of E equipped with the topology β. The space E b will always have the topology τb of uniform convergence on the sets of B. It is easy to check that if τ is locally convex, we have τ ≤ γτ ≤ β and hence (E, τ ) ⊂ (E, γτ ) ⊂ E b . Furthermore, (E, γτ ) is a complete subspace of E b ([12], Corollary 1.1, Theorem 2.1). In the case where (E, B, τ ) is normal, it follows from Grothendieck’s completeness theorem ([15], Chapter VI, Theorem 2) that (E, γτ ) is the closure of (E, τ ) in E b . It follows from [7], Proposition 1, that if (E, B, τ ) is a mixed space, B0 a basis for B and τ locally convex, then a basis for the γτ -neighborhoods of 0 is given by the collection of absolutely convex hulls of the sets ∪{(B ∩ UB ): B ∈ B0 }, where (UB )B∈B0 ranges over families of absolutely convex τ -neighborhoods of 0 in E. This description enables us to generalize a result of Cooper ([Co], Proposition 1.22) characterizing the γτ -equicontinuous sets. Theorem 2.1 Let (E, B, τ ) be a normal mixed space, with τ locally convex and B0 a basis for B consisting of τ -closed sets. A subset A of (E, γτ ) is γτ -equicontinuous if and only if for every ε > 0 and every B ∈ B0 there is a τ -equicontinuous set A(ε, B) such that A ⊂ A(ε, B) + εB 0 , where the polar B 0 is taken in E b .

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Proof. We ﬁrst note the every B ∈ B0 is σ(E, (E, τ ) )-closed, and hence also σ(E, (E, γτ ) )-closed, by [15], Chapter VI, Theorem 2, Corollary 3. If U is an absolutely convex closed τ -neighborhood of 0 in E, it can then be shown as in the proof of [15], Chapter VI, Theorem 2 that (U ∩ B)0 ⊂ U 0 + B 0 ⊂ 2(U ∩ B)0 . If A is a γτ -equicontinuous set and ε > 0, it follows from the characterization of the γτ -neighborhoods of 0 that we can ﬁnd a family (UB )B∈B0 of absolutely convex closed τ -neighborhoods of 0 such that A ⊂ ε[ac ∪ {(B ∩ UB ): B ∈ B0 }]0 , (where “ac” denotes the absolutely convex hull) and hence for every B ∈ B0 , A ⊂ ε(B ∩ UB )0 ⊂ εUB0 + εB 0 . The result then follows from the fact that εUB0 is τ -equicontinuous. Conversely, suppose A ⊂ (E, γτ ) satisﬁes the given condition. It follows easily from the deﬁnition of γτ that A will be γτ -equicontinuous if (and only if) the restrictions of the functionals in A to B is τ -equicontinuous for every B ∈ B0 . If ε > 0 and B ∈ B0 , then it follows from the assumption that we can ﬁnd a closed absolutely convex τ -neighborhood UB of 0 such that A ⊂ 12 ε[UB0 + B 0 ] ⊂ ε[UB ∩ B]0 . Hence |f (x)| ≤ ε for every f ∈ A, x ∈ UB ∩ B, as required. As was pointed out in [4], the fact that for a bornology B on E, ∪B = E, is needed to show that every precompact set is B-precompact. This still holds even if we only assume that ∪B is dense in E. This slightly generalized version of Bprecompactness allows us to restate the above theorem. Corollary 2.2 Let (E, B, τ ) be a normal mixed space, E the collection of τ -equicontinuous subsets of (E, τ ) and let (E, γτ ) have the topology τb . Then a subset of (E, γτ ) is γτ -equicontinuous if and only if it is E-precompact. It follows from the corollary that B-precompact sets in a duality-setting may well signify the presence of a normal mixed space. We illustrate this using Example 1.1. Let E denote the dual of the Banach space E, B the bornology of norm bounded sets in E and τ the Mackey topology τ (E , E). Then it is easy to see that (E , B , τ ) is a mixed space, and it follows from the fact that the closed unit ball in E is weak∗ compact that it is in fact a normal mixed space. It now follows from Example 1.1 and Theorem 2.1 that the associated mixed topology is in fact the Mackey topology τ (E , E). A further example will be explored in more depth in the next section.

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63

3. Order Precompact Sets and Uniformly Lebesgue Topologies In this section we look at an example of the duality discussed in the previous section in the setting of Riesz spaces. We brieﬂy summarize some of the notions to be used; more information may be found in [1]. All Riesz spaces will be assumed to be Archimedean. If E is a Riesz space, its set of positive elements will be denoted by E + , the space of all order bounded linear functionals on it by E ∼ , and the space of order continuous linear functionals by E × . A subset A of E is solid if x ∈ E, y ∈ A and |x| ≤ |y| implies x ∈ A. An ideal in E is a solid linear subspace; the ideal of E generated by a subset of A of E is denoted by IA . If x, y ∈ E, we write [x, y] = {z ∈ E: x ≤ z ≤ y} and call such a set an order interval. A set is order bounded if it is contained in an order interval. A linear space topology τ on a Riesz space E is locally solid if it has a basis for the neighborhoods of 0 consisting of solid sets; if it is in addition locally convex, it will be called locally convex-solid. The space of all τ -continuous linear functionals is denoted by (E, τ ) (or E for short); if τ is locally solid, (E, τ ) is an ideal in E ∼ . If E is a Banach lattice with dual E , E = E ∼ . A locally solid topology τ is preLebesgue if every disjoint order-bounded sequence is τ -convergent to 0, Lebesgue if every decreasing net with inﬁmum 0 is τ -convergent to 0, and Fatou if it has a basis for the neighborhoods of 0 consisting of solid order-closed sets. A pre-Lebesgue topology is Lebesgue if and only if it is Fatou. If τ is a Hausdorﬀ Fatou topology, every solid order-closed set is τ -closed. If E is a normed lattice, we shall write Ea (respectively Ea× ) for the largest ideal of E (respectively E ∩ E × ) on which the topology induced by the norm of E is Lebesgue. If E is a locally solid Riesz space and B the bornology of order bounded subsets of E, the B-precompact sets were called Riesz precompact in [2]. These sets are closely related to the order precompact and quasi-order precompact sets of [6]. We recall that a subset A of E is order precompact if for every solid neighborhood U of 0 in E, there is a positive x in the ideal of E generated by A such that A ⊂ [−x, x]+U . We refer to [2] for more detailed information on these notions. Uniformly Lebesgue topologies were introduced by Nowak in [11] in the setting of normed function spaces. In order to generalize this notion to a large class of normed Riesz spaces, we need to generalize the topology of convergence in measure on sets of ﬁnite measure. This is done in [3]; we give the facts pertinent to this paper here. Let E be a Riesz space which contains an order-dense Riesz subspace F which admits a Hausdorﬀ Lebesgue topology τ . The topology τ has a set P of deﬁning

64

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Riesz pseudonorms. If p ∈ P and 0 ≤ u ∈ F , we can deﬁne a Riesz pseudonorm pu by pu (x) = p(|x| ∧ u) (x ∈ E). The topology τm deﬁned by the pseudonorms pu (p ∈ P, 0 ≤ u ∈ F ) is independent of F and τ . It is a Hausdorﬀ Lebesgue topology, and is in fact the coarsest such topology on E. If (xn ) is a disjoint sequence in E + , (xn ) is τm -convergent to 0. In the case where E is a Riesz space of measurable functions on a semi-ﬁnite measure space, τm is the topology of convergence in measure on sets of ﬁnite measure. In the rest of this section whenever the existence of the topology τm on a Riesz space E is assumed, it will be assumed that E has an order-dense Riesz subspace which admits a Hausdorﬀ Lebesgue topology. It is known that all Hausdorﬀ Lebesgue topologies induce the same topology on the order bounded subsets of a Riesz space ([1], Theorem 12.9). Using this result it is easy to show that a Hausdorﬀ locally solid topology τ on E is Lebesgue if and only if every order bounded net (xα ) which is τm -convergent to 0 is also τ -convergent to 0. This motivates the following deﬁnition (see also [11], Deﬁnition 1.1). Let E be a normed Riesz space with unit ball BE . A locally solid topology τ is uniformly Lebesgue if every net in BE which is τm -convergent to 0 is also τ convergent to 0; and τ is uniformly pre-Lebesgue if every disjoint sequence in BE is τ -convergent to 0. It follows at once that every uniformly (pre-) Lebesgue topology is (pre-) Lebesgue. The converse holds in L∞ -spaces. Clearly τm is a uniformly Lebesgue topology. Since disjoint sequences are τm -convergent to 0, every uniformly Lebesgue topology is uniformly pre-Lebesgue. The following duality result will play a crucial role in the rest of this section. Theorem 3.1 Let E be a normed Riesz space with closed unit ball B and A a solid σ(E ∼ , E)bounded subset of E ∼ . Deﬁne the seminorm pA on E by pA (x) = sup{|f (x)|: f ∈ A}. Consider the following statements: (1) B is order pA -precompact and A is order |σ|(E ∼ , E)-precompact. (2) A is order precompact for the norm on E and B is order |σ|(E, IA )-precompact. (3) fn → 0 for every disjoint sequence (fn ) in A. (4) pA (xn ) → 0 for every disjoint sequence (xn ) in B. Then (1) ⇒ (4), and if E is a Banach lattice, or if the topology τm can be deﬁned in E, (4) ⇒ (1). If A is a subset of the norm dual E of E, (1) ⇐⇒ (2) ⇐⇒ (3).

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65

Proof. This is a special case of [2], Theorem 3.3. If E is a Banach lattice, (4) ⇒ (1) follows from the fact that the norm topology on E is a complete topology for which B is bounded. In the case where the topology τm can be deﬁned on E, it is a Hausdorﬀ Lebesgue topology on E. To see that B is τm -bounded, it suﬃces to observe that it can be shown (cf. [3], Theorem 5.6) that τm is coarser than the norm topology on E. Corollary 3.2 Let E be a normed Riesz space and F an ideal in E . Then the topology induced by the norm of E on F is Lebesgue if and only if the closed unit ball BE of E is order |σ|(E, F )-precompact. Proof. Let BE be order |σ|(E, F )-precompact and f ∈ F + . Then (1) ⇒ (3) of 3.1, with A = [−f, f ], shows that every disjoint sequence in A is norm convergent to 0, and it follows that the norm of E induces a pre-Lebesgue topology on F . The norm topology is Fatou on E , and since F is an ideal, also on F . It follows that the norm topology on F is Lebesgue. Conversely, if the norm topology on F is Lebesgue, hence pre-Lebesgue, it follows from (3) ⇒ (1) of 3.1 that BE is |σ| (E, F )-precompact. Corollary 3.3 Let τ be a locally convex pre-Lebesgue topology on a normed Riesz space E. If BE is order τ -precompact, τ is uniformly pre-Lebesgue. Conversely, if τ is uniformly Lebesgue, or E is a Banach lattice and τ is uniformly pre-Lebesgue, BE is order τ -precompact. Proof. Since τ is pre-Lebesgue, every τ -equicontinuous subset A of F = (E, τ ) is order |σ|(F, E)-precompact ([2], Theorem 2.7); also F is an ideal in E ∼ . If BE is order τ -precompact, it follows from (1) ⇒ (4) of 3.1 that τ is uniformly pre-Lebesgue. Conversely, under the stated conditions it follows as before that (4) ⇒ (1) of 3.1 holds, and the result follows. Corollary 3.4 If E is a Banach lattice and τ a uniformly pre-Lebesgue locally convex topology on E, then F = (E, τ ) ⊂ Ea .

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Proof. This follows from 3.3, 3.2 and the fact that τ is ﬁner than |σ|(E, F ). The converse of 3.4 does not hold. As an example, let E = L2 [0, 1] and τ be the usual norm topology of E. Then (E, τ ) = L2 [0, 1] = Ea . If τ were uniformly pre-Lebesgue, BE would be order τ -precompact (by 3.3), and it would then follow from [8], Lemma 4.4 that L2 [0, 1] is ﬁnite-dimensional. Theorem 3.5 Let τ be a locally convex-solid topology on a normed Riesz space E. If every τ equicontinuous set is an order precompact subset of Ea , τ is uniformly pre-Lebesgue, and the converse holds if E is a Banach lattice. Proof. If every τ -equicontinuous set A is order precompact in Ea , then in particular F = (E, τ ) ⊂ Ea . It follows from 3.2 that BE is order |σ|(E, F )-precompact, and the result then follows from (2) ⇒ (4) of 3.1. Conversely, if E is a Banach lattice, we have F = (E, τ ) ⊂ E ∼ = E . Also, (4) ⇒ (2) of 3.1 shows that every τ -equicontinuous set A in E is order precompact in E and that BE is order |σ|(E, F )-precompact. It follows from 3.2 that F ⊂ Ea . We immediately obtain a partial converse to 3.4: Corollary 3.6 Let E be a normed Riesz space and F an ideal in Ea . Then |σ|(E, F ) is a uniformly pre-Lebesgue topology. The following result, reminiscent of the fact that Hausdorﬀ Lebesgue topologies coincide on order bounded sets, will be needed to analyse uniformly Lebesgue topologies: Theorem 3.7 Let τ be a Lebesgue topology on the Riesz space E. Then τm induces a ﬁner topology than τ on the order τ -precompact sets of E. If τ is Hausdorﬀ, the two topologies coincide on the order τ -precompact sets. Proof. Let A be τ -precompact and suppose (xα ) is a net in A which is τm -convergent to x ∈ A. It follows from [1], Theorem 12.8 that the topology induced by τm on order intervals of E is ﬁner than that induced by τ . Let p be a τ -continuous pseudonorm + and ε > 0. Choose u ∈ IA such that p(|x| − |x| ∧ u) < ε for every x ∈ A. Since (xα ) is τm -convergent to x, (|x − xa | ∧ 2u) is τm -convergent to 0, and so it follows a

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67

priori that we can ﬁnd an α0 such that p(|x − xa | ∧ 2u) < ε for α ≥ α0 . Therefore, for α ≥ α0 p(x − x0 ) ≤ p(|x − xα | − |x − xα | ∧ 2u) + p(|x − xα | ∧ 2u) ≤ p(|x| − |x| ∧ u) + p(|xα | ∧ u) + p(|x − xα | ∧ 2u) < 3ε and so (xα ) is τ -convergent to x. If τ is Hausdorﬀ, τm ≤ τ on E, and the result follows. Proposition 3.8 Let τ be a locally solid topology on a normed Riesz space E. If τ is Lebesgue and BE is order τ -precompact, τ is uniformly Lebesgue. The converse holds if τ is locally convex. Proof. The ﬁrst part is immediate from Theorem 3.7, and the second follows from Corollary 3.3. Corollary 3.9 Let τ be a locally convex uniformly pre-Lebesgue topology on a normed Riesz space E. Then the following are equivalent: (a) τ is uniformly Lebesgue (b) τ is Lebesgue (c) τ is Fatou. Proof. The implications (a) ⇒ (b) ⇐⇒ (c) are all easy, and (b) ⇒ (a) follows from 3.8 and (4) ⇒ (1) of 3.1, recalling that by convention we are assuming that the topology τm can be deﬁned on E. Corollary 3.10 If E is a normed Riesz space, |σ|(E, Ea× ) is a uniformly Lebesgue topology. Proof. Immediate from 3.6 and 3.9. It is now possible to give a non-trivial example of a uniformly pre-Lebesgue topology which is not uniformly Lebesgue. Let E be a Banach lattice such that E × ⊂ E = Ea , (E × = E ). Then τ = |σ|(E, E ) = |σ|(E, Ea ) is uniformly preLebesgue (by 3.6), but since (E, τ ) = Ea ⊂ E × , (Ea = E × ), τ is not Lebesgue and

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therefore not uniformly Lebesgue. An example of such a Banach lattice is the direct sum L∞ [0, 1] ⊕ L2 [0, 1], equipped with the coordinate-wise ordering and the norm f ⊕ g = max{f ∞ , g2 }. The dual space is

L∗∞ [0, 1]

⊕ L2 [0, 1], with norm ϕ ⊕ ψ = ϕ∗ + ψ2 ,

where · ∗ denotes the norm of L∗∞ [0, 1]. Proposition 3.11 Let τ be a uniformly Lebesgue topology on a normed Riesz space E. Then (E, τ ) ⊂ Ea× . Proof. Let f ∈ (E, τ ) and suppose the sequence (xn ) converges in norm to 0. Then there is no loss in generality in assuming that xn ∈ BE for all n ∈ N, and the same argument as in the proof of 3.1 shows that (xn ) is τm -convergent to 0. It then follows from 3.3 and 3.7 that (xn ) is τ -convergent to 0, and so f (xn ) → 0. Hence (E, τ ) ⊂ E and the result now follows in the same way as in 3.4. Theorem 3.12 A locally convex-solid topology τ on a normed Riesz space E is uniformly Lebesgue if and only if every τ -equicontinuous set is an order-precompact subset of Ea× . Proof. If ever τ -equicontinuous subset is an order precompact subset of Ea× ⊂ Ea , it follows from 3.5 that τ is uniformly pre-Lebesgue; since (E, τ ) ⊂ Ea× ⊂ E × , τ is also Lebesgue and hence by 3.9 uniformly Lebesgue. Conversely, if τ is uniformly Lebesgue, (E, τ ) ⊂ Ea ∩ E × = Ea× (by 3.11) and the result then follows from (4) ⇒ (1) of 3.1. It is clear from the above result that if the topology τm can be deﬁned on a normed Riesz space E, then there is a ﬁnest uniformly Lebesgue locally convex topology on E, namely the topology of uniform convergence on the order precompact subsets of Ea× . Likewise it follows from 3.5 that there is a ﬁnest uniformly pre-Lebesgue topology on a Banach lattice E, namely, the topology of uniform convergence on the order precompact subsets of Ea . We now show that these topologies are often mixed topologies. It is clear from 2.2 that the appropriate mixed spaces to consider are (E, B, |σ|(E, Ea× )) and (E, B, |σ|(E, Ea )) respectively, where B is the bornology of norm-bounded subsets of E. Lemma 3.13 Let E be a Fatou-normed Riesz space and B its bornology of norm-bounded sets. If τ is a Hausdorﬀ Fatou topology on E such that every B ∈ B is τ -bounded, then (E, B, τ ) is a normal mixed space.

Generalized precompactness and mixed topologies

69

Proof. Clearly B0 = {nBE : n ∈ N} is a basis for B consisting of solid order-closed sets, and since τ is a Hausdorﬀ Fatou topology, each such set is τ -closed ([1], Theorem 12.7). Theorem 3.14 Let E be a Fatou-normed Riesz space, B the bornology of norm-bounded subsets of E and F an ideal in Ea× which separates the points of E. Then (E, B, |σ|(E, F )) is a normal mixed space and γ|σ|(E,F ) is the topology of uniform convergence on the order precompact sets of the norm closure F of F in E . In particular, if Ea× separates the points of E, the ﬁnest uniformly Lebesgue topology is a mixed topology. Proof. It is enough to note that |σ|(E, F ) is a Hausdorﬀ Lebesgue, hence Fatou, topology with dual F . If the topology τm can be deﬁned on a Fatou-normed Riesz space E, 3.13 shows that (E, B, τm ) is also a normal mixed space. Since τm is in general not a locally convex topology, Theorem 2.1 cannot be used to identify the mixed topology γτm . However, Theorem 3.7 comes to the rescue in many cases. Lemma 3.15 Let E be a normed Riesz space such that Ea× separates the points of E. Then τm and |σ|(E, Ea× ) coincide on the closed unit ball BE of E. Proof. The topology |σ|(E, Ea× ) is Hausdorﬀ and hence τm can be deﬁned on E. The result now follows from 3.2 and 3.7. To formulate the next result, we introduce the notation γτ for the ﬁnest vector topology which coincides with a vector topology τ on the norm-bounded subsets of a normed Riesz space E. It follows from [7], Proposition 5 that if τ is locally convex γτ = γτ . Theorem 3.16 Let E be a normed Riesz space such that Ea× separates the points of E. Then . γτm = γ|σ|(E,Ea× ) = γ|σ|(E,E × = γτ m ) a

Proof. The result follows at once from 3.15 and the above remark. Corollary 3.17 (cf. [11], Theorem 3.3) Let E be a normed Riesz space. The following are equivalent: (a) γτ m is locally convex (b) Ea× separates the points of E (c) τm and |σ|(E, Ea× ) coincide on the closed unit ball BE of E.

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Proof. (a) ⇒ (b): Since γτ m is clearly a uniformly Lebesgue topology, (E, γτ m ) ⊂ Ea× by 3.11. If γτ m is locally convex, it follows that Ea× separates the points of E. (b) ⇒ (c) : This is 3.15. (c) ⇒ (a) : This follows as in 3.16. Proposition 3.18 Let E be a normed Riesz space. Then γτ m is the ﬁnest uniformly Lebesgue topology on E. Proof. Clearly γτ m is uniformly Lebesgue. If τ is a uniformly Lebesgue topology on E, τm is ﬁner than τ on BE , and so τ ≤ γτ ≤ γτ m . Applications of the material in this section to the characterization of compact sets and compact operators in Banach lattices will be given in a forthcoming paper.

References 1. C.D. Aliprantis and O. Burkinshaw, Locally solid Riesz spaces, Academic Press, New York, 1978. 2. J.J. Conradie, Duality results for order precompact sets in locally solid Riesz spaces. Indag. Math. N.S. 2 (1991), 19–28. 3. J.J. Conradie, The coarsest Hausdorff Lebesgue topology, preprint. 4. J.J. Conradie and J. Swart, A general duality result for precompact sets, Indag. Math. N.S. 1 (1990), 409–416. 5. J. Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics 92, SpringerVerlag, New York-Berlin-Heidelberg-Tokyo, (1984). 6. M. Duhoux, Order precompactness in topological Riesz spaces, J. London Math. Soc. (2) 23 (1981), 509–522. 7. D.J.H. Garling, A generalized form of inductive limit topology for vector spaces, Proc. Lond. Math. Soc. 14 (1964), 1–28. 8. J.J. Grobler, Indices for Banach function spaces, Math. Z. 145 (1975), 99–109. 9. H. Jarchow, Locally convex spaces, Teubner, Stuttgart, 1981. 10. H. Jarchow and U. Matter, Interpolative constructions for operator ideals, Note di Matematica 8 (1988), 45–56. 11. M. Nowak, Mixed topology on normed function spaces, I, Bull. Pol. Ac. Math. 36 (1988), 251–262. 12. A. Persson, A generalization of two-norm spaces, Ark. Mat. 5 (1963), 27–36. 13. A. Pietsch, Operator Ideals, North Holland, Amsterdam, 1980. 14. N. Robertson, Asplund operators and holomorphic maps, Manuscripta Math. 75 (1992), 25–34. 15. A.P. Robertson and W. Robertson, Topological Vector Spaces, Cambridge University Press, 1966.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 71–79 c 1994 Universitat de Barcelona

On weak topology of Orlicz spaces∗

Shutao Chen Harbin Normal University, Harbin, China

Huiying Sun Harbin Institute of Technology, Harbin, China

Abstract This paper presents some properties of singular functionals on Orlicz spaces, from which, criteria for weak convergence and weak compactness in such spaces are obtained.

In [1], T. Ando shows that every linear bounded functional can be decomposed into a function part and a singular part, and the last part is represented by some class of ﬁnite additive set functions. Since very few properties of such set functions are known, most problems concerning weak topology in Orlicz spaces are left open; for instance, even T. Ando himself in [2], leaving the singular functionals aside, discusses only the LN -weak convergence and LN -weak compactness. In this paper, we ﬁrst give criteria for a singular functional on an Orlicz space to be norm attainable and to be an extreme point of the unit ball of the dual space, then, applying Rainwater’s Theorem, we obtain criteria for weak convergence and weak compactness in the space. Throughout this paper, we denote by M : R → R+ an Orlicz function, i.e., it is even, continuous, convex and satisﬁes M (u) = 0 iﬀ u = 0, and Mu(u) → 0 as u → 0, Mu(u) → ∞ as u → ∞. If M is an Orlicz function, then its complemented ∗

To the memory of Professor W. Orlicz. The authors are supported by the National Science Fund of China and Science Fund of Heilongjiang.

71

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Orlicz function N is deﬁned by N (v) = sup{uv − M (u)}. u∈R

Let (G, Σ, µ) be a non-atomic, σ-ﬁnite and complete measurable space. For each µ-measurable function x(t) on G, we deﬁne ρ(x) = ρM (x) =

M (x(t))dµ G

LM = {x: ρM (λx) < ∞ for some

λ > 0}

EM = {x: ρM (λx) < ∞ for all λ > 0} x ≤1 , x = inf λ > 0: ρM λ then the Orlicz space LM and its subspace EM are Banach spaces with the Luxemburg norm · . The following four lemmas can be found in [1]. Lemma 1 ∗ = LN (the Orlicz space generated For any f ∈ L∗M , there exist unique v ∈ EM by the complemented Orlicz function N ) and ϕ ∈ S = {f ∈ L∗M : f (EM ) = {0}} such that f = v + ϕ. Moreover, f = v + ϕ, where v and ϕ are norms of v and ϕ as functionals on LM respectively. + Let L+ M = {|x| = (|x(t)|): x = (x(t)) ∈ LM }. If ϕ ∈ S is nonnegative on LM , then we say ϕ is positive. For any ϕ ∈ S, x ∈ L+ M , let

ϕ+ (x) = sup{ϕ(y): 0 ≤ y(t) ≤ x(t), t ∈ G} ϕ− (x) = − inf{ϕ(y): 0 ≤ y(t) ≤ x(t), t ∈ G} . When x ∈ LM is arbitrary, we denote x+ = (|x|+x) and x− = x − x+ , and deﬁne 2 ϕ± (x) = ϕ± (x+ ) − ϕ± (−x− ), then both ϕ± are positive and ϕ = ϕ+ − ϕ− . Lemma 2 For any ϕ ∈ S, we have ϕ = ϕ+ + ϕ− . Lemma 3 ϕ + ψ = ϕ + ψ for all positive ϕ, ψ ∈ S.

On weak topology of Orlicz spaces

73

Lemma 4 If ϕ ∈ S is positive, then there exists x ∈ L+ M with x = 1 such that ϕ(x) = ϕ, i.e., every positive ϕ ∈ S is norm attainable. Theorem 5 Let f ∈ S, then for any disjoint subsets N , N of G, we have f |N ∪N = f |N + f |N , where for any subset A of G, x ∈ LM , f |A (x) = f (x|A ), and where x|A (t) = x(t), when t ∈ A and = 0, when t ∈ G \ A. Proof. For any ε > 0, since f is singular, we can ﬁnd x, y in LM with their supports in N and N respectively such that ρ(x) ≤ 12 , ρ(y) ≤ 12 and such that f (x) = f |N (x) ≥ f |N − ε, f (y) = f |N (y) ≥ f |N − ε. Let u = x + y, then ρ(u) = ρ(x) + ρ(y) ≤ 1 and hence, f |N + f |N ≥ f |N + f |N = f |N ∪N ≥ f (u) = f |N (x) + f |N (y) ≥ f |N + f |N − 2ε . Theorem 6 For any f ∈ S, if there exists x ∈ LM , x = 1, such that f (x) = f , then for any subset A of G, we have f (x|A ) = f |A . Proof. Let B = G \ A, then by Theorem 5, f = f |A + f |B ≥ f |A (x) + f |B (x) = f . Hence, we must have f (x|A ) = f |A (x) = f |A and f (x|B ) = f |B . Theorem 7 f ∈ S is norm attainable iﬀ there exists a subset A of G such that f + = f |A and f − = −f |B , where B = G \ A.

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Proof. Suﬃciency. According to Lemma 4, there exist x, y ∈ LM such that ρ(x) ≤ 1 1 + − − 2 , ρ(y) ≤ 2 and f |A (x) = f (x) = x, −f |B (y) = f (y) = f . Obviously, we may assume x = x|A and y = y|B , hence, if we deﬁne u = x − y, then ρ(u) ≤ 1 and thus, f = f + + f − = f |A (x) − f |B (y) = f (u). Necessity. Choose x ∈ LM with ρ(x) ≤ 1 such that f (x) = f and let A = {t ∈ G: x(t) ≥ 0}, B = G \ A. It follows from the deﬁnition of f + and Theorem 6 that f + |A ≥ f + |A (x) ≥ f |A (x) = f |A . Hence, by Lemma 3 f + = f + |A + f + |B ≥ f |A + f + |B . Similarly, we have f − ≥ f |B + f − |A . Therefore f = f + + f − ≥ f |A + f |B + f + |B + f − |A . It follows from Theorem 5 that f + |B = f − |A = 0. Thus, for any u ∈ LM , f + (u) = f + (y|A ) − f − (u|A ) = f (u|A ) = f |A (u). In the same way, we have f − (u) = −f |B (u). Theorem 8 The set of all norm attainable singular functionals is dense in S. Proof. Given any ϕ ∈ S and ε > 0, by the Bishop-Phelps Theorem, we can ﬁnd a norm attainable functional f ∈ L∗M such that f − ϕ < ε. By Lemma 1, f = v + ψ for some v ∈ LN and ψ ∈ S. Choose x ∈ LM with x = 1 such that v + ψ = f = v, x + ψ, x then v, x = v and ψ, x = ψ, hence, ψ is norm attainable and ϕ − ψ = ϕ − f − v < ε. Lemma 9 Suppose that f ∈ S, x, y ∈ LM with x ≤ 1, y ≤ 1 and A is a subset of G, then x(t)y(t) ≥ 0, t ∈ A implies f (y|A ) ≥ f (x) − f ; x(t)y(t) ≤ 0, t ∈ A implies f (y|A ) ≤ f − f (x).

On weak topology of Orlicz spaces

75

Proof. If x(t)y(t) ≥ 0, t ∈ A, then ρ(x − y|A ) ≤ ρ(x) + ρ(y) < ∞. Since f is singular, we have f (x − y|A ) ≤ f . By the same way, if x(t)y(t) ≤ 0 on t ∈ A, we have f (y|A ) − f (x) ≤ f . Theorem 10 L∗M

Let f ∈ S and f = 1. Then f is an extreme point of the unit ball B(L∗M ) of iﬀ for any subset A of G, f |A · f |G\A = 0.

Proof. The “only if ” part. If there exists a subset A of G such that f |A > 0 and f |B > 0, where B = G \ A, then ϕ := f |A /f |A , ψ := f |B /f |B ∈ B(L∗M ), and f = f |A ϕ + f |B ψ, which contradicts the condition that f is an extreme point since f |A + f |B = f = 1. The “if ” part. We ﬁrst point out that f + · f − = 0. To show this, by Theorem 8, we may assume that f is norm attainable. It follows from Theorem 7 that f + · f − = 0. Thus, without loss of generality, we may assume that f is positive (otherwise, we consider the positive functional −f = f − ). It follows from Lemma 4 that there exists x ∈ L+ M such that f (x) = f = x = 1. ∗ Suppose f1 , f2 ∈ B(LM ) satisfying f1 + f2 = 2f , we have to show that f1 = f2 . First, by Lemma 1, we can easily deduce that f1 , f2 ∈ S. For each y ∈ LM satisfying f (y) = 0, deﬁne A = {t ∈ G: x(t)y(t) ≥ 0} and B = G \ A, then, without loss of generality, we may assume that f |A = 0. Hence, f1 (x|B ) + f2 (x|B ) = 2f (x) = 2 and therefore, fi (x|B ) = fi = 1, i = 1, 2, which indicates f1 |A = f2 |A = 0. It follows from Lemma 9 that fi (y|B ) ≤ fi − fi (x) = 0,

i = 1, 2.

Hence, fi (y) = fi (y|B ) ≤ 0, i = 1, 2. Since y ∈ ker(f ) is arbitrary, we have fi (y) = 0, i.e., ker(fi ) contains ker(f ), which shows that f = αi fi for some αi ∈ R, and so, we must have f = fi , i = 1, 2. For each x ∈ LM , let θ(x) = inf{α > 0: ρ( αx ) < ∞}, then

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Lemma 11 ([1]) θ(x) =dist (x, EM ) and for each f ∈ S, f (x) f = sup : x ∈ LM \ E M . θ(x) Lemma 12 For any x ∈ LM and a partition {Nk }k≤m of G,

max θ(x|Nk ) = θ(x). k

Proof. It is clear that θ(x|Nk ) ≤ θ(x) for all k ≤ m. If α := max θ(x|Nk ) < θ(x), k

then for any β ∈ (α, θ(x))

x

ρ

β

=

x|N k <∞ ρ β

k≤m

which shows θ(x) ≤ β < θ(x), a contradiction. Let {xn } be a sequence in LM and F a subset of L∗M . We say xn → x ∈ LM F -weakly as n → ∞, provided that f (xn − x) → 0 for all f ∈ F . Theorem 13 The necessary and suﬃcient condition for xn → 0 S-weakly is that for any subsequence {yk } of {xn }

(1) lim θ min |yk | = 0 m

k≤m

where min |yk |(t) = min |yk (t)|, t ∈ G. k≤m

k≤m

Proof. If the condition is not suﬃcient, then by Rainwater’s Theorem (see [3], p. 155), there exist ε > 0, an extreme point f ∈ S of B(L∗M ) and a subsequence {yk } of {xn } such that f (yk ) > ε for all k ∈ N. It follows from Theorem 10 that we may assume that f is positive. From condition (1), we can ﬁnd some m ∈ N such that θ(mink≤m |yk |) < ε. Let Nk = t ∈ G: |yk (t)| = min |yk |(t) , k = 1, 2, . . . , m k≤m

then by Theorem 10, there exists k ≤ m such that f |Nk = f . Hence f (|yk | ) ≥ f (yk |N ) = f (yk ) > ε. Nk

k

On the other hand, by Lemma 11

f (|yk |N ) ≤ θ(|yk |N )f ≤ θ min |yk | < ε k

a contradiction.

k

k≤m

On weak topology of Orlicz spaces

77

If the condition is not necessary, then there exist ε > 0 and a subsequence {yk } of {xn } such that

θ min |yk | > ε, m ∈ N. k≤m

Let N1 (1) = {t ∈ G: y1 (t) ≥ 0}, N1 (2) = G \ N1 (1). If Nk (s) has been found for s = 1, 2, . . . , 2k , k = 1, 2, . . . , m, then let

Nm+1 (2s − 1) = t ∈ Nm (s): ym+1 (t) ≥ 0 , Nm (s) , Nm+1 (2s) = Nm+1 (2s − 1) s = 1, 2, . . . , 2m+1 . By induction, for any k ∈ N, we ﬁnd a partition {Nk (s): s ≤ 2k } of G satisfying for any m ≥ k, yk (t) is nonnegative or nonpositive on Nm (s), s = 1, 2, . . . , 2m . By Lemma 12, there is some sm ≤ 2m such that

θ min |yk | |Nm (sm ) = θ min |yk | > ε. k≤m

k≤m

Hence,by Lemma 11 and the Hahn-Banach Theorem, we can ﬁnd fm ∈ S with fm = 1 such that

fm min |yk | |Nm (sm ) = θ min |yk | |Nm (sm ) > ε, m ∈ N . k≤m

k≤m

+ − − − fm , it is clear by Lemma 2 and Lemma 11 that fm = 0, i.e. (Observe fm = fm ∗ ∗ fm is positive.) Since B(LM ) is w compact, the sequence {fm } has a cluster point f ∈ S. It follows that for each k ∈ N, there exists some m > k such that

|f (yk ) − fm (yk )| < In view of

ε . 2

fm mink≤m |yk | |Nm (sm ) fm |Nm (sm ) ≥

θ mink≤m |yk | |Nm (sm ) = 1 = fm

we ﬁnd fm |G\Nm (sm ) = 0 according to Theorem 5. Therefore ε |f (yk )| ≥ |fm (yk )| − |f (yk ) − fm (yk )| ≥ |fm (yk |Nm (sm ) )| − 2 ε

ε ≥ fm min |yk | |Nm (sm ) − > k≤m 2 2 contradicting the hypothesis that xn → 0 S-weakly.

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Lemma 14 ([5]) xn → 0 LN -weakly iﬀ i) E xn (t)dµ → 0 as n → ∞ for each E ∈ Σ and ii) lim sup λ−1 ρM (λxn ) = 0 . λ→0 n

Lemma 15 ([2]) A subset E of LM is LN -weakly compact iﬀ lim sup λ−1 ρ(λx) = 0 .

λ→0 x∈E

By Theorem 13 and Lemma 14, we obtain. Theorem 16 A sequence {xn } in LM converges to 0 weakly iﬀ a) lim xn (t)dµ = 0 for all E ∈ Σ; n

E

b) lim sup λ−1 ρ(λxn ) = 0

and

λ→0 n

c) for any subsequence {yk } of {xn }, we have lim θ(min |yk |) = 0 . m

k≤m

Theorem 17 A subset K of LM is weakly compact if and only if 1) lim sup λ−1 ρ(λx) = 0 and λ→0 x∈E

2) lim θ(min |xn − x|) = 0 for all sequence {xn } in K satisfying m

n≤m

lim n

xn (t) − x(t) dµ = 0 for each E ∈ Σ .

(∗)

E

Proof. Necessity. The ﬁrst condition follows from Theorem 15. Now, we check the second one. Let K be a weakly compact subset of LM , then for sequence {xn } in K satisfying (∗), we can pick a subsequence {xni } of {xn } weakly convergent to some point x in LM . It follows from Theorem 16 that

0 = lim θ min |xni − x | ≥ lim θ min |xn − x | ≥ 0 . m

i≤m

m

n≤m

Since (∗) implies that xn → x EN -weakly by [1], we have x = x.

On weak topology of Orlicz spaces

79

Suﬃciency. For any sequence {xn } in K, by Theorem 15 and [1], it contains a subsequence, again denoted by {xn }, LN -weakly convergent to some x ∈ LM . For any subsequence {yj } of {xn }, by the second condition and [1],

lim θ min |yj − x| = 0 m

j≤m

it follows from Theorem 13 that xn → x weakly.

References 1. 2. 3. 4.

T. Ando, Linear functionals on Orlicz spaces, Nieuw Arch. Wisk. 8 (3) (1960), 1–16. T. Ando, Weakly compact sets in Orlicz spaces, Canad. J. Math. 14 (1962), 170–176. J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag, New York, 1984. C. Wu, T. Wang, S. Chen, Y. Wang, Geometrical Theory of Orlicz Spaces (in Chinese), Harbin, 1986. 5. C. Wu, T. Wang, Orlicz Spaces and Applications (in Chinese), Harbin, 1983.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 81–89 c 1994 Universitat de Barcelona

On the three-space-problem and the lifting of bounded sets

Susanne Dierolf FB IV - Mathematik, Universit¨at Trier D-54286 Trier, Germany

Dedicated to the memory of Professor Wladyslaw Orlicz.

Abstract We exhibit a general method to show that for several classes of Fr´echet spaces the Three-space-problem fails. This method works for instance for the class of distinguished Fr´echet spaces, for Fr´echet spaces with the density condition and also for dual Fr´echet spaces (which gives a negative answer to a question of D. Vogt). An example of a Banach space, which is not a dual Banach space but the strong dual of a DF-space, shows that there are two really different possibilities of defining the notion of a dual Fr´echet space. If in a Three-spaceproblem the corresponding quotient map is assumed to lift bounded sets, we obtain partial positive answers. Finally, we give this property of lifting bounded sets a special treatment.

Introduction By the Three-Space-Problem we mean the following question: Given a short exact sequence of locally convex spaces (resp. Fr´echet spaces) i

q

0 → F →E →E/F → 0 ,

(∗)

where F is a linear subspace (resp. closed linear subspace) of E, i denotes the inclusion and q the quotient map, we assume that both F and E/F have a certain property P , and ask whether this implies that E also has P . We call a property P 81

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Dierolf

three-space-stable (in the class of locally convex spaces or Fr´echet spaces, respectively) if the above implication holds for all short exact sequences (∗). In the class of locally convex spaces one has three-space-stability for the properties metrizable (Graev [9; p.17]), complete, barrelled, nuclear, and Schwartz-space, whereas many other properties are not three-space-stable, such as bornological, quasibarrelled, DF-space, reﬂexive, Montel, quasinormable, distinguished (see Roelcke, Dierolf [15] and also [16, §12] for topological groups). In the class of Fr´echet spaces there are some more positive results (sometimes contrasting counterexamples in the general locally convex setting): Three-space-stability holds for reﬂexive, Montel, quasinormability (Roelcke, Dierolf [15]), for (DN) and (Ω) (Vogt [18] and [19]), and for quojection (Metafune, Moscatelli [13]). In 1990 Bonet, Dierolf, Fern´ andez showed that distinguishedness and Stefan Heinrich’s density condition (abbreviated nski, to (DC))1 are not three-space-stable (see [3]), and recently D´ıaz, Dierolf, Doma´ Fern´andez proved that also the property of being the strong dual of a DF-space is not three-space-stable (see [8]), which answered a question of D. Vogt. The corresponding counterexamples for these last three properties were all modeled with the help of the following construction, which we developed by adapting the method for counterexamples in the locally convex setting used in Roelcke-Dierolf [15] to the Fr´echet situation. We learned later that this method had already been used by Pisier; therefore we will call it “Pisier-method”. 1. Pisier-method Given a Fr´echet space X, a closed linear subspace L in X and a Fr´echet space Z which is continuously included into the quotient X/L, the map p: X × Z → X/L, (x, z) → q(x) − z, (where q: X → X/L denotes the quotient map) is linear, continuous and surjective, hence a quotient map and therefore generates the following short exact sequence p 0 → Y := ker p → X × Z →X/L → 0. The aim of this construction is to choose X, L and Z such that X/L is some separable Banach space, Y becomes a Montel space and that Z has bad properties, which implies that also X × Z has bad properties. If X is Fr´echet Montel, then Y will be Montel if and only if every bounded sequence in Z which converges in X/L already converges in Z. In view of this we obtain the following: 1

A Fr´echet space E is called distinguished (resp. quasinormable, resp. has (DC)), if its strong dual is bornological (resp. each bounded set in Eb gets its topology from a Banach space continuously included into Eb , resp. the bounded subset of Eb are metrizable).

Eb

On the three-space-problem and the lifting of bounded sets

83

Proposition 2 Let P be a property of Fr´echet spaces, such that every Fr´echet Montel space and also 1 have P . Moreover we assume that one of the two following conditions is satisﬁed: (i) P implies distinguishedness; (ii) Every Fr´echet space with property P is topologically complemented in its bidual. Then P is not three-space-stable. Proof. Let X be the K¨ othe-Grothendieck-Fr´echet-Montel space admitting a quotient 1 map q: X → . (i) we recall that X is also continuously included into 1 , hence Z := In 1case 1 X∩ ( ) provided with the intersection topology is a Fr´echet space (of generalized Moscatelli type), which is nondistinguished as the inclusion X → 1 is not open onto its range (see Bonet, Dierolf, Fern´andez [3; proof of Example 4]). Now Z is continuously included into 1 (1 ) ∼ = 1 (N × N); identifying 1 and 1 (N × N) we thus obtain q: X → 1 , i: Z → 1 as entries of the Pisier-method. As X is Montel, the special choice of Z implies that Y is also Montel. Moreover, as Z is nondistinguished, also X × Z is nondistinguished and a fortiori does not have P . In case (ii) we make use of the fact that 1 contains a closed linear subspace Z which is not complemented in its bidual Z (see Lindenstrauß [10]). Taking q: X → 1 and i: Z → 1 as entries of the Pisier-method, we immediately obtain that Y is Montel, but X × Z is not complemented in this bidual, hence does not have P . As (DC) implies distinguishedness, we obtain as a consequence that distinguishedness and (DC) are not three-space-stable. Moreover, since the strong dual of a quasibarrelled DF-space is a Fr´echet space which is complemented in its bidual, we obtain another application of Proposition 2. In this context we would like to remark that there are two reasonable possibilities to deﬁne the notion of a Fr´echet space F to be a “dual Fr´echet space”. (i) F is the strong dual of a DF-space. (ii) F is the strong dual of a quasibarrelled DF-space (which via the completion is equivalent to F being the strong dual of a barrelled DF-space). If E is a DF-space such that Eb is separable, then E is already quasibarrelled. In fact, every bounded set in (E , σ(E , E )) is metrizable and relatively compact, hence every bounded set in E is separable, which implies the quasibarrelledness of the DF-space E. (See [8; Example 1]). In general, however, the two notions above are diﬀerent as the following example shows.

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Example 3. Let I be a set such that card I ≥ card R. Then 1 (I) provided with the initial topology T with respect to the natural projections 1 (I) → (1 (J), · 1 )

(J ⊂ I, J countable)

is a complete DF-space, whose strong dual is the Banach subspace E := (xι )ι∈I ∈ ∞ (I): {ι ∈ I: xι = 0} is countable of (∞ (I), · ∞ ) . In fact, one has continuous identities

1 (I), · 1 → 1 (I), T → 1 (I), σ(1 (I), c0 (I)) .

Therefore, the T -bounded sets and the · 1 -bounded sets in 1 (I) coincide, from which one easily obtains that (1 (I), T ) is a DF-space whose strong dual is (E, · ∞ ). For the completeness of T note that (1 (I), T ) has a 0-basis of σ(1 (I), c0 (I))-closed sets, which implies the quasicompleteness and hence the completeness of (1 (I), T ). In Bonet, Dierolf, Fern´ andez [5; Example 3] that space had been investigated. One of the purposes had been to show that it is not a Mackey-space. We would like to make use of the opportunity to correct an error in the corresponding argumentation in [5; p. 113, line 1/2]: If (1 (I), T ) were a Mackey-space, the union of the p (I)-unit-balls Bp (p ∈ N) would K and the ∞ (I)be relatively compact in (E, σ(E, 1 (I)). As the intersection of ι∈I unit ball B∞ is contained in 2 Bp and the σ(E, 1 (I))-closure of B∞ ∩ K is p∈N

ι∈I

equal to B∞ ∩ E, this would imply that B∞ ∩ E is σ(E, 1 (I))-compact, which is not true, since, clearly, B∞ ∩ E does not have extreme points. Returning to the Banach space (E, · ∞ ), we have established that (E, · ∞ ) is the strong dual of a DF-space and claim that (E, · ∞ ) is not topologically isomorphic to the strong dual of any quasibarrelled DF-space. In fact, if (E, · ∞ ) were the strong dual of a quasibarrelled DF-space F , we would obtain that F is a topological subspace of Eb , hence a normed space. Therefore (E, · ∞ ) would be a dual Banach space. The following proof, which was given by P. Doma´ nski, shows that this is not true. In fact, let us assume that (E, · ∞ ) is a dual Banach space and a fortiori complemented in its bidual, which is injective by Lindenstrauß, Rosenthal [11; p. 335], as (E, · ∞ ) is clearly an L∞ -space. This implies the injectivity of (E, · ∞ ), whence there exists a continuous linear projector P : (∞ (F ), · ∞ ) → (E, · ∞ ).

On the three-space-problem and the lifting of bounded sets

85

As P |c0 (I) is the identity map, we obtain from Rosenthal [17; Prop. 1.2] that there is J ⊂ I, card J =card I such that P |∞ (J) is a topological isomorphism onto its range. This is wrong, as dim ∞ (J) = 2card J whereas dim E =card J. In particular, we have obtained a Banach space which is the strong dual of a DF-space without being a dual Banach space. Coming back to the three-space-problem and Proposition 2, we observe that all spaces involved in this proposition are separable and thus we obtain Corollary 4 There is a short exact sequence 0 → F → E → 1 → 0 such that F is a Fr´echet Montel space and E is not the strong dual of any DF-space. In the context of D. Vogt’s question about the three-space-stability of dual Fr´echet spaces, it would in a natural way also be interesting to know, whether a short exact sequence of dual Fr´echet spaces 0 → F → E → G → 0 is already a dual sequence, i.e. whether there exists a short exact sequence 0 → Y → X → Z → 0 of DF-spaces (resp. barrelled DF-space) such that the former is dual to the latter. This is not true, even for reﬂexive spaces, as the short exact sequence 0 → ker q → q E →2 → 0, arising from a Fr´echet Montel space E and a quotient map q: E → 2 , shows. In fact, if q were the transpose of a monomorphism Y → X with Y, X DF-spaces, we could assume that Y, X are barrelled and would obtain that q lifts bounded sets (see the deﬁnition below), which is obviously not true. It turned out that this property of “lifting bounded sets” plays quite an essential role in the context of the three-space-problem and several other structural questions. For the sake of completeness let us start with the deﬁnition. Definition 5. Let E be a locally convex space and q: E → E/L a quotient map. We say that q lifts bounded sets (resp. lifts bounded sets with closure) if for every bounded set A in E/L there exists a bounded set B in E such that q(B) ⊃ A (resp. q(B) ⊃ A).

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Remarks. The above quoted quotient map q: E → 1 with E Fr´echet Montel shows that a quotient map between Fr´echet spaces need not lift bounded sets with closure. In fact, a quotient map E → E/F with E Montel and E/F not Montel never lifts bounded sets with closure. The property of lifting bounded sets is a very useful tool for the characterization of quasinormability of Fr´echet spaces: By a result of Palamodov, Merzon, and DeWilde (see [7]), a Fr´echet space F is quasinormable if and only if for every Fr´echet space E containing F as a topological subspace, the corresponding quotient map q: E → E/F lifts bounded sets. Moreover, by a result of Cholodovskij [6], a closed linear subspace F of a quasinormable Fr´echet space E is itself quasinormable if the quotient map q: E → E/F lifts bounded sets. From the Banach-Dieudonn´e-Theorem one gets that a quotient map E → E/F between Fr´echet spaces lifts bounded sets, if the quotient E/F is a Fr´echet Montel space. On the other hand, a quotient map q: E → E/L with E locally convex lifts bounded sets with closure if and only if the transpose q t : (E/L)b → Eb is a topological isomorphism onto its range. Our attention to this property was raised by its role in the distinguishedness of biduals of Fr´echet spaces: Theorem 6 Let E be a Fr´echet space. Then the following are equivalent (i) The bidual E is distinguished. (ii) E and E /E are both distinguished and the quotient map q: E → E /E lifts bounded sets with closure. Proof. For the implication: E distinguished ⇒ E /E is distinguished and q lifts bounded sets with closure and for the implication (ii) ⇒ (i) see Bonet, Dierolf, Fern´andez [4]. The fact that the distinguishedness of E implies the distinguishedness of E follows from a recent result of Meise and Vogt [12; 26.12] (see Proposition 9 c) below). With the help of the above theorem it was possible to ﬁnd a distinguished Fr´echet space whose bidual is not distinguished, which gives a negative answer to a problem of Grothendieck (see Bonet, Dierolf, Fern´ andez [4].) For the two concepts of lifting bounded sets given in Deﬁnition 5, one observes that for general locally convex spaces these notions are not equivalent. In fact, for a non En → E, (xn )n∈N → xn , regular LB-space E =indn En , the canonical map n∈N

n∈N

On the three-space-problem and the lifting of bounded sets is a quotient map, lifts bounded sets with closure (as

87

En is a DF-space), but not

n∈N

without closure. On the other hand, even for a quotient map q between Fr´echet spaces it may happen that a bounded set in the quotient can be lifted with closure but not without, as the following example shows: Example 7. Let G, H be Banach spaces such that G is continuously included into H and the inclusion G → H has proper dense range. Then the continuous linear map

q: G × co (H) → H, n∈N

(xn )n∈N , (yn )n∈N

n∈N

→ (xn + yn )n∈N

is surjective, hence a quotient map. Let BG , BH denote closed unit balls in G and H, respectively. Then q {0} ×

n∈N

is a bounded set in

n∈N

BH ∩ c0 (H)

=

BH ∩ c0 (H) = BH

n∈N

H which is not contained in

n∈N

ρn BG + ρ

BH ∩ c0 (H) for

n∈N

any positive reals ρ, ρn (n ∈ N). (See Bonet, Dierolf [2; Example].) Nevertheless, in contrast to the above example the following statement holds: Theorem 8 Let E be a metrizable locally convex space and F ⊂ E a closed linear subspace. Then the quotient map q: E → E/F lifts bounded sets if and only if q lifts bounded sets with closure. The proof is essentially the proof of the Theorem in Bonet-Dierolf [2]. As a consequence of 7 and 8 one obtains an example of a quotient map q: E → G = E/F where E and G are both countable products of Banach spaces, such that q does not lift bounded sets. In contrast to this, one has the following positive result of Mi˜ narro [14]: q If 0 → F → E →G = E/F → 0 is a short exact sequence of metrizable locally convex spaces, such that E is quasinormable and G is normable, then q lifts bounded sets. We will give a proof (due to Bonet) diﬀerent from the one in [14]: q t maps the Banach space Gb continuously into Eb , which has a representation as a retractive LB-space indn Gn . Thus, q t maps Gb continuously into some Gn . By the retractivity

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of indn Gn , there is m ≥ n such that Gm and Eb induce the same topology on the q t -image of the unit ball of Gb . As q t (G ) is closed in (E , σ(E , E)) and therefore in Gm , the map q t : Gb → Gm is a topological isomorphism onto its range. Taking these statements together, we obtain that also q t : Gb → Eb is a topological isomorphism, which implies that q lifts bounded sets. From Mi˜ narro’s result one obtains that in a short exact sequence 0 → F → E → G → 0 of Fr´echet spaces, where G is a Banach space (or a Montel space), the quasinormability of E is equivalent to the quasinormability of F . We will return now to the three-space-problem and shall point out that under the hypothesis that for a short exact sequence of Fr´echet spaces the quotient map lifts bounded sets, partial positive answers can be obtained. Proposition 9 Let i

q

0 → F →E →E/F → 0

(∗)

be a short exact sequence of Fr´echet spaces such that q lifts bounded sets. a) If F and E/F are both distinguished (resp. have (DC)), then also E is distinguished (resp. has (DC)). (See Bonet, Dierolf, Fern´ andez [3].) b) If F is the strong dual of a barrelled DF-space and if E/F is reﬂexive, then also E is the strong dual of a barrelled DF-space and (∗) is a dual sequence. (See Diaz, Dierolf, Doma´ nski, Fern´ andez [8]). qt

it

c) The dual sequence 0 → (E/F )b →Eb →Fb → 0 is also topologically exact. (See Meise, Vogt [12; 26.12].) In particular, if E/F is Montel and F is a dual Fr´echet space in the sense of b), then (∗) is a dual short exact sequence. Thus, the Montel property, if put on the subspace F is even typical for the counterexamples in Proposition 2, but if put on the quotient, guarantees the above mentioned three-space-stability. The following question remains open: Given a short exact sequence 0 → F → E → G → 0 of Banach spaces such that F and G are dual Banach spaces, is it then true that E is a dual Banach space and that the sequence is a dual sequence?

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References 1. K.D. Bierstedt, J. Bonet, Biduality in Fr´echet and (LB)-spaces. Progress in Functional Analysis (K.D. Bierstedt, J. Bonet, J. Horv´ath. M. Maestre, eds.), North-Holland Math. Studies 170, North-Holland, Amsterdam 1992, p. 113–133. 2. J. Bonet, S. Dierolf, On the lifting of bounded sets. Proc. Edin. Math. Soc. 36 (1993), 277–281. 3. J. Bonet, S. Dierolf, C. Fern´andez, On the three-space-problem for distinguished Fr´echet spaces. Bull. Soc. Roy. Sci. Li`ege 59 (1990), 301–306. 4. J. Bonet, S. Dierolf, C. Fern´andez, The bidual of a distinguished Fr´echet space need not be distinguished. Archiv. Math. 57 (1991), 475–478. 5. J. Bonet, S. Dierolf, C. Fern´andez, On two classes of LF-spaces. Portugaliae Mathematica 49 (1992), 109–130. 6. V.E. Cholodovskij, On quasinormability of semimetrizable topological vector spaces, Funkc. Anal. 7 (1976), 157–160. 7. M. De Wilde, Sur le rel`evement des parties born´ees d’un quotient d’espaces vectoriels topologiques. Bull. Soc. Roy. Sci. Li`ege 5-6 (1974), 299–301. 8. J.C. D´ıaz, S. Dierolf, P. Doma´nski, C. Fern´andez, On the three space problem for dual Fr´echet spaces, Bull. Polish Academy Sciences 40 (1992), 221–224. 9. M.I. Graev, Theory of topological groups I. Uspeki Mat. Nauk 5 (1950), 3–56. 10. J. Lindenstrauss, On a certain subspace of 1 . Bull. Acad. Polon. Sci. 12 (1964), 539–542. 11. J. Lindenstrauss, H.P. Rosenthal, The Lp Spaces. Israel J. Math. 7 (1969), 325–349. 12. R. Meise, D. Vogt, Einf¨uhrung in Funktionanalysis, Vieweg, Wiesbaden, 1992. 13. G. Metafune, V.B. Moscatelli, On the three-space-problem for locally convex spaces, Collect. Math. 37 (1986), 287–296. 14. A. Mi˜narro, Two results on quasinormable spaces, Preprint 1991. 15. W. Roelcke, S. Dierolf, On the three-space-problem for topological vector spaces, Collect. Math. 32 (1981), 3–25. 16. W. Roelcke, S. Dierolf, Uniform structures on topological groups and their quotients, MGrawHill Publ. Co., New York 1981. 17. H.P. Rosenthal, On relatively disjoint families of measures, with some applications to Banach space theory. Studia Math. 37 (1970), 13–36. 18. D. Vogt, Subspaces and quotients of (s). In Funktional Analysis: Surveys and Recent Results (K.D. Bierstedt, B. Fuchssteiner, eds.), North-Holland Math. Studies Amsterdam, 1977, p. 167–187. 19. D. Vogt, On two classes of (F)-spaces. Archiv Math. 45 (1985), 255–266.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 91–103 c 1994 Universitat de Barcelona

Remarks on the Istratescu measure of noncompactness

Janusz Dronka Department of Mathematics, Technical University of Rzesz´ow, 35-959 Rzesz´ow, W. Pola 2, Poland

Abstract In this paper we give estimations of Istratescu measure of noncompactness I(X) of a set X ⊂ lp (E1 , . . . , En ) in terms of measures I(Xj ) (j = 1, . . . , n) of projections Xj of X on Ej . Also a converse problem of finding a set X for which the measure I(X) satisfies the estimations under consideration is considered.

1. Introduction Let E be a real Banach space with the norm . Given a nonempty subset A of E we say that it is ε- separated if for every pair x, y in A we have x − y ≥ ε . For a nonempty and bounded subset X of E we shall consider the Istratescu measure of noncompactness of X, deﬁned in the following way: I(X) = sup ε > 0: there exists an inﬁnite ε-separated subset A of X . This classical measure was studied in many papers and has found many applications (cf. [1,2]). AMS Subject Classification 46B20.

91

92

Dronka

Particularly, in the geometry of Banach spaces the following “separation” constant, often called the Kottman constant, is considered: K(E) = I(B(E)) where B(E) denotes the unit ball of a space E. As it was indicated by Papini (see [13]) the Kottman constant is strictly related to other signiﬁcant notions, for example: K(E) =

2P (E) 1 − P (E)

(cf. [10,12]),

where P (E) is the packing constant of E, deﬁned as follows: P (E) = sup r > 0: inﬁnitely many balls of radius ≥ r can be packed in B(E) . (We say that a collection of balls {B(xi , ri )} is packed in B(E) if B(xi , ri ) ⊆ B(E) for every index i, and moreover the interiors of any two of the balls are disjoint – they do not “overlap”). We have also: K(E) · J(E) ≥ 2 (cf. [13]), where J(E) is the Yung constant of E, deﬁned by J(E) = sup 2r(A)/δ(A); A is bounded, nonempty subset of E , where δ(A) is the diameter of A, and r(A) = inf sup x − a. x∈E a∈A

We also know that K(E) is continuous with respect to the Banach-Mazur distance in any isomorphism class (cf. [11]). If we want to get some numerical results we may start from the Elton-Odell (1 + ε)-separation theorem (cf. e.g. [4]), which gives us: 1 < K(E) ≤ 2 (the right inequality is trivial). From the papers of Kottman, Papini, Dom´ınguez Benavides and other authors it follows that: i) K(H) = 21/2 , if H is a Hilbert space (cf. e.g. [5]), ii) K(lp ) = 21/p , for 1 ≤ p < ∞ (cf. [10]), iii) K(co ) = K(l∞ ) = 2 ,

Remarks on the Istratescu measure of noncompactness

93

iv) K(Lp ) = max{21/p , 21/q } , for 1 ≤ p < ∞, 1/p + 1/q = 1 and µ – not purely atomic measure (cf. [3,12]), v) K(lp (E1 , E2 , . . .)) = max{21/p , sup K(Ei )}, where Ei are Banach spaces i∈N

for i = 1, 2, . . . (with a norm i respectively) and lp (E1 , E2 , . . .), with 1 ≤ p < ∞, denotes the space of all sequences {xi }, xi ∈ Ei , with ∞ xi pi < ∞ (lp (E1 , E2 , . . .) is a Banach space with the natural norm) (cf.

i=1

[9,11]). Recently some new interesting results concerning the Kottman constant in Orlicz and Musielak-Orlicz sequence spaces were obtained by H. Hudzik and others [6,7,8]. The result

v)

suggests the following question:

What can be said about Istratescu measure of a bounded subset of lp (E1 , E2 , . . .) if measures of its projections on subspaces Ei are known? The aim of this paper is to answer this question for the ﬁnite product space lp (E1 , . . . , En ) which may be treated as the special case of the space lp (E1 , E2 , . . .).

2. Notation, definitions and some auxiliary facts This section is devoted to establish some auxiliary results which will be needed further on. Let Ej be a Banach space with a norm j for 1 ≤ j ≤ n. Let us recall that the product space lp (E1 , . . . , En ) is deﬁned as the linear space E1 × . . . × En with a norm: 1/p  n xj pj  if 1 ≤ p < ∞ , x =  j=1

and x = max xj j , 1≤j≤n

for p = ∞ ,

where x = (x1 . . . , xn ) ∈ E1 × . . . × En . The basic properties of product spaces may be found in [9], for example. In the sequel we will use the following fact concerning the Istratescu measure of noncompactness in product spaces.

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Lemma 1 If X is a bounded and nonempty subset of the space lp (E1 , . . . , En ), 1 ≤ p ≤ ∞, then I(X) ≥ I(Xj ) for j = 1, . . . , n, where Xj denotes the projection of X on Ej . Proof. Fix an arbitrary j, 1 ≤ j ≤ n, and ε > 0. Let {xkj } be an I(Xj )−ε-separated sequence in Xj . Then there exists a sequence {xk } of elements of X such that xkj is a projection of xk on the space Ej , for k = 1, 2, . . .. It is easy to see that {xk } is I(Xj ) − ε-separated, too. Indeed, for any k, l ∈ N we get xk − xl =

n

1/p xki − xli pi

1/p ≥ xkj − xlj pj

i=1

≥ I(Xj ) − ε if 1 ≤ p < ∞ , and

xk − xl = max xki − xli ≥ xkj − xlj 1≤i≤n

≥ I(Xj ) − ε for p = ∞ . Thus for any ε > 0 we have I(X) ≥ I(Xj ) − ε , which ends the proof. Now, let us formulate a few properties of the Istratescu measure of noncompactness which will be used later on (cf. [1,2]). For any nonempty and bounded subsets X, Y of a Banach space E we have (1)

X ⊂ Y ⇒ I(X) ≤ I(Y ),

(2)

I(RX) = RI(X) for any R > 0,

(3)

I(X + a) = I(X) for any a ∈ E.

The next useful property of the Istratescu measure I is formulated in the following lemma.

Remarks on the Istratescu measure of noncompactness

95

Lemma 2 Let X be a bounded and convex subset of a Banach space E such that 0 ∈ X, and let f : E → R ∪ {∞} be a function deﬁned by f (x) =

inf{t ≥ 0: x ∈ tX} ∞

if the set is nonempty, otherwise.

Then for any sequence {xk } of elements of X and δ > 0 there exist 0 ≤ R ≤ 1 and a subsequence {yk } of {xk } such that: (4)

lim f (yk ) = R ,

k→∞

and for any k, l ∈ N: yk − yl ≤ RI(X) + δ. Proof. Since xk ∈ X we get 0 ≤ f (xk ) ≤ 1 for k = 1, 2, . . .. Hence using Weierstrass theorem we can choose a subsequence {wk } of {xk } and 0 ≤ R ≤ 1 satisfying (4). Put ε = δ/(I(X) + 1) > 0. From (4) it follows that wk ∈ (R + ε)X for k ≥ k0 , k0 ∈ N. Now, using (2) and Ramsey’s theorem (cf. [4], for example) we claim that there exists a subsequence {yk } of {wk } such that yk − yl ≤ (R + ε)I(X) + ε = RI(X) + δ for any k, l ∈ N. If not, there would exist a subsequence of {wk } for which the opposite inequality holds, which in turn contradicts the fact that I((R + ε)X) = (R + ε)I(X). This completes the proof. Finally, let us formulate a numerical lemma which is a simple consequence of H¨older’s inequality. Lemma 3 Let 1 ≤ p < q < ∞ , Rj ≥ 0 and mj ≥ 0 for j = 1, . . . , n, and R1q + · · · + Rnq ≤ 1.

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Dronka

Then n

(q−p)/q  n pq/(q−p)  Rjp mpj ≤  mj .

j=1

j=1

The equality is attained for:

n −1/q pq/(q−p) p/(q−p) ˆj = m mi , where j = 1, . . . , n. R j i=1

Proof. Indeed, putting xj = Rjp , yj = mpj , s = we have s, t > 1 and n j=1

1 s

+

1 t

q q , t= , p (q − p)

= 1. Hence, applying H¨ older’s inequality we obtain

Rjp mpj =

n

1/s  1/t  n n xj yj ≤  xsj   yjt 

j=1

j=1

j=1

p/q  (q−p)/q  n n pq/(q−p)  = Rjq   mj j−1

j=1

(q−p)/q  n pq/(q−p)  ≤ mj . j=1

The remainder of the proof is an easy calculation and is therefore omitted.

3. Main results Let Ej be a Banach space with a norm j for j = 1, . . . , n. For the Istratescu measure of noncompactness in the product space lp (E1 , . . . , En ) we have the following estimations. Theorem 1 If X is a bounded subset of lp (E1 , . . . , En ) then 1/p  n (5) max I(Xj ) ≤ I(X) ≤  [I(Xj )]p  , 1≤j≤n

j=1

if 1 ≤ p < ∞ .

Remarks on the Istratescu measure of noncompactness

97

and I(X) = max I(Xj ), if p = ∞ ,

(6)

1≤j≤n

where Xj denotes a projection of X on Ej for j = 1, . . . , n. The converse is also true, which is stated in the next theorem. Theorem 2 If X1 , . . . , Xn are bounded and convex subsets of E1 , . . . , En respectively, then for any number µ satisfying 1/p  n max I(Xj ) ≤ µ ≤  [I(Xj )]p  , where 1 ≤ p < ∞,

1≤j≤n

j=1

there exists a subset X of lp (E1 , . . . , En ) such that I(X) = µ and projections of X on each of Ej coincide with Xj . Proof of Theorem 2. Let 1 ≤ p < ∞, E1 , . . . , En and X1 , . . . , Xn be as in the statement of the theorem. Keeping in mind (3), without loss of generality we may assume that 0 ∈ Xj for j = 1, . . . , n . Denote mj = I(Xj ) . We deﬁne fj : Ej → R ∪ {∞} (j = 1, . . . , n) similarly as in Lemma 2, namely: fj (x) =

inf{t ≥ 0: x ∈ tXj }

if the set is not empty,

∞

otherwise .

Consider the sets Dq =

(x1 , . . . , xn ) ∈ lp (E1 , . . . , En ):

n

[fj (xj )]q ≤ 1

j=1

for 1 ≤ q < ∞, and

D∞ = X1 × . . . × Xn .

It is easy to see that

(7)

the projection of Dq onto the space Ej coincides with Xj for any 1 ≤ q ≤ ∞ and j = 1, . . . , n .

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Now we will calculate the Istratescu measure of noncompactness of the sets Dq in the space lp (E1 , . . . , En ). We have: I(Dq ) = max I(Xj ) for 1 ≤ q ≤ p,

(8)

1≤j≤n

 (9)

I(Dq ) = 

n

(q−p)/pq [I(Xj )]pq/(q−p) 

for p < q < ∞,

j=1

 I(Dq ) = 

(10)

n

1/p [I(Xj )]p 

for q = ∞.

j=1

To prove (8) ﬁx p and q, where 1 ≤ q ≤ p < ∞ . The inequality I(Dq ) ≥ max I(Xj ) 1≤j≤n

follows immediately from (7) and Lemma 1. To prove the opposite, since Dq ⊆ Dp for 1 ≤ q ≤ p, taking into account (1) it is suﬃcient to show that I(Dp ) ≤ max I(Xj ). 1≤j≤n

Suppose to the contrary that there exists an α+ε-separated sequence {xk } contained in Dp , where α = max1≤j≤n I(Xj ) and ε > 0 . Denote xk = (xk1 , . . . , xkn ) for k = 1, 2, . . .. Put δ = (α + ε)p − αp > 0 . Applying consecutively Lemma 2 to each of Xj and taking a subsequence of {xk } in place of {xk } if necessary, we may assume that: (11)

lim fj (xkj ) = Rj , where 0 ≤ Rj ≤ 1 for j = 1, . . . , n,

k→∞

and (12)

xkj − xlj pj ≤ Rjp mpj +

δ for l, k ∈ N. 2j

Moreover it is easy to see that (13)

n j=1

Rjp ≤ 1

Remarks on the Istratescu measure of noncompactness

99

(if not, due to (11), xk with suﬃciently large k would not belong to Dp ). Now, using (12) and (13) we get xk − xl = p

n

xkj −

xlj pj

<

j=1

n

Rjp mpj + δ

j=1 n

≤ αp (

Rjp ) + δ ≤ αp + δ = (α + ε)p

j=1

which contradicts the fact that {xk } is α + ε-separated. This implies (8). To prove (9) let 1 ≤ p < q < ∞ . Denote β = (

n

pq

[I(Xj )] q−p )

q−p pq

.

j=1

First we will prove that I(Dq ) ≤ β . Suppose to the contrary that there exists a sequence {xk } of elements of Dq which is β + ε-separated, where ε > 0 . Denote δ = (β + ε)p − β p > 0 . Using the same argumentation as above we may assume without loss of generality that for the sequence {xk } the properties (11), (12) and R1q + · · · + Rnq ≤ 1 hold. Now, taking into account (12) and applying Lemma 3 we obtain xk − xl p =

n

xkj − xlj pj <

j=1

n

Rjp mpj + δ

j=1

≤ β p + δ = (β + ε)p which contradicts the fact that the sequence {xk } is β + ε-separated. Thus we have proved that I(Dq ) ≤ β, To prove the opposite inequality let ε > 0 and δ = β p − (β − ε)p . Consider ˆ n such as in Lemma 3. We have: ˆ1, . . . , R R n

n

ˆ q = 1, R j

j=1

ˆ j ≥ 0 for j = 1, . . . , n. ˆ p mp = β p and R R j j

j=1

From (2) it follows that ˆ j Xj ) = R ˆ j mj for j = 1, . . . , n. I(R ˆ j Xj such that Thus we can choose a sequence {xkj }k∈N of elements of R ˆ p mp − δ for j = 1, . . . , n and k = l. xkj − xlj pj ≥ R j j n

100

Dronka

Let Since xkj

xk = (xk1 , . . . , xkn ) for k = 1, 2 . . . . ˆ j Xj we have fj (xkj ) ≤ R ˆ j for j = 1, . . . , n and ∈R n n q ˆ q = 1. [fj (xkj )] ≤ R j j=1

j=1

Hence xk ∈ Dq for k = 1, 2, . . . Moreover we have: n n ˆ p mp − δ xkj − xlj pj ≥ xk − xl p = R j j j=1 p

= β − δ = (β − ε)

j=1 p

for k, l ∈ N, k = l.

Thus for an arbitrary ε > 0 there exists a sequence of elements of Dq which is β − ε-separated. So I(D) ≥ β. This ends the proof of (9). To prove (10) let q = ∞ . Arguing similarly as above it is easy to check that:  1/p n I(D∞ ) ≥ γ, where γ =  mpj  . j=1

Suppose that the opposite inequality does not hold. Then there exists a sequence {xk } of elements of D∞ which is γ + ε-separated, where ε > 0 . Let xk = (xk1 , . . . , xkn ) where xkj ∈ Xj for k = 1, 2, . . . Put δ = (γ + ε)p − γ p . Using Ramsey’s theorem we can choose a subsequence of the sequence {xk } (without loss of generality all {xk }) such that: δ xkj − xlj pj ≤ I(Xj )p + for j = 1, . . . , n, k, l ∈ N. 2n Therefore n n δ xkj − xlj pj ≤ mpj + < γ p + δ = (γ + ε)p . xk − xl p = 2 j=1 j=1 The obtained contradiction proves (10). Finally, observe that  (q−p)/pq n lim  [I(Xj )]pq/(q−p)  = max I(Xj ) q→p

and

j=1

1≤j≤n

 (q−p)/pq  1/p n n =  [I(Xj )]p  , lim  [I(Xj )]pq/(q−p) 

q→∞

j=1

j=1

which in conjunction with (8), (9) and (10) completes the proof of Theorem 2.

Remarks on the Istratescu measure of noncompactness

101

Since the proof of (10) is valid for any 1 ≤ p < ∞ and bounded (not necessarily convex) sets we obtain the following corollary. Corollary 1 If X1 , . . . , Xn are bounded and nonempty subsets of Banach spaces E1 . . . , En respectively, then for the measure I in the space lp (E1 , . . . , En ), where 1 ≤ p < ∞, we have  1/p n I(X1 × . . . × Xn ) =  [I(Xj )]p  . j=1

Proof of Theorem 1. Let X be a bounded subset of lp (E1 , . . . , En ). Observe, that in view of Lemma 1 the inequality I(X) ≥ max I(Xj )

(14)

1≤j≤n

holds for any 1 ≤ p ≤ ∞, where Xj denotes a projection of X on Ej for j = 1, . . . , n. On the other hand, we have X ⊆ X1 × . . . × Xn . Thus, using (1) and Corollary 1 we obtain:  1/p n I(X) ≤ I(X1 × . . . × Xn ) =  [I(Xj )]p  , j=1

for 1 ≤ p < ∞, which in conjunction with (14) gives (5). Now, we will show that for p = ∞: I(X) ≤ max I(Xj ). 1≤j≤n

Given ε > 0 let {xk } be an I(X) − ε-separated sequence of X. Denote xk = (xk1 , . . . , xkn ), for k = 1, 2, . . . We have xk − xl = max xkj − xlj j ≥ I(X) − ε, for k, l ∈ N, k = l. 1≤j≤n

From Ramsey’s theorem it follows that for at least one of the indices j, 1 ≤ j ≤ n, there exists a subsequence {yk } of {xk } such that yk − yl = ykj − ylj j

for all k, l ∈ N.

The sequence {ykj } of elements of Xj is therefore I(X) − ε-separated and we have I(X) − ε ≤ I(Xj ) ≤ max I(Xi ), 1≤i≤n

for any ε > 0, which completes the proof of (6) and Theorem 1. As an immediate consequence of Theorems 1 and 2 we obtain the following corollary concerning the Kottman constant of product spaces.

102

Dronka

Corollary 2 Let Ej be a Banach space for j = 1, . . . , n and let 1 ≤ p ≤ ∞. Then: (15)

K lp (E1 , . . . , En ) = max K(Ej ). 1≤j≤n

Proof. For p = ∞ the statement simply follows from the equality (6) of Theorem 1. Indeed, putting X = B(l∞ (E1 , . . . , En )) in (6) we have:

I B l∞ (E1 , . . . , En ) = max I B(Ej ) 1≤j≤n

which is (15). If 1 ≤ p < ∞ then putting in (8) of the proof of Theorem 2 Xj = B(Ej ) for 1 ≤ j ≤ n, and using the same notation, we have: fj (x) = xj for x ∈ Ej , and Dp = B(lp (E1 , . . . , En )), so (15) is an immediate consequence of (8). This completes the proof. Finally, let us mention that analogous estimations as in Theorems 1 and 2 may be obtained for the space lp (E1 , E2 , . . .), however they are not satisfactory since the series which appears on the right hand side of (5) need not be convergent.

References 1. R.R. Akmerov, M.I. Kamenskii, A.S. Potapov, A.E. Rodkina, B.N. Sadovskii, Measures of noncompactness and condensing operators, Nauka, Novosibirsk 1986 [English translation: Operator Theory, Advances and Applications 55, Birkh¨asuser Verlag, Basel-Boston-Berlin 1992]. 2. J. Bana´s, K. Goebel, Measures of noncompactness in Banach spaces, Lecture Notes in Pure and Applied Mathematics 60, Marcel Dekker, Inc., New York and Basel 1980. 3. C.E. Cleaver, Packing spheres in Orlicz spaces, Pacific J. Math. 65 (1976), 325–335. 4. J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag New York, 1984. 5. T. Dom´ınguez Benavides, Some properties of the set and ball measures of non-compactness and applications, J. London Math. Soc. (2) 34 (1986), 120–128. 6. H. Hudzik, Any nonreflexive Banach function lattice has packing constant equal to 1/2, preprint. 7. H. Hudzik, C. Wu, Y. Ye, Packing constant in Musielak-Orlicz sequence spaces equipped with the Luxemburg norm, to appear.

Remarks on the Istratescu measure of noncompactness

103

8. H. Hudzik, T. Landes, Packing constant in Orlicz spaces equipped with the Luxemburg norm, to appear. 9. G. K¨othe, Topological Vector Spaces I, Springer-Verlag, Berlin-Heidelberg-New York, 1969. 10. C.A. Kottman, Packing and reflexivity in Banach spaces, Trans. Amer. Math. Soc. 150 (1970), 565–576. 11. C.A. Kottman, Subsets of the unit ball that are separated by more than one, Studia Math. 53 (1975), 15–27. 12. P.L. Papini, Some parameters of Banach spaces, Rend. Sem. Mat. Fis. Milano 52 (1983), 131–148. 13. P.L. Papini, Jung’s constant and around, preprint (1989).

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 105–114 c 1994 Universitat de Barcelona

On complete, precompact and compact sets

Leonhard Frerick Universit¨at Trier, FB IV, Mathematik, D-54286 Trier, Germany

Abstract Completeness criterion of W. Robertson is generalized. Applications to vector valued sequences and to spaces of linear mappings are given.

In this paper we prove a result generalizing the very useful completeness criterion of W. Robertson (cf. [7] or [5, p. 210]), which uses the closedness of the elements of a zero basis in a coarser vector space topology to guarantee the completeness of subsets of a topological vector space. At the same time we can apply our result to vector valued sequence spaces and to spaces of linear mappings. Moreover, with a similar argument used in the completeness criterion, we prove suﬃcient and necessary conditions for compactness and precompactness of subsets of topological vector spaces, which we apply to sets of vector valued sequences and vector valued continuous functions. The underlying idea of our proceeding is the following: We regard a Hausdorﬀ topological vector space (in the following called X1 ), which is embedded in a second Hausdorﬀ topological vector space (called Y1 ) such that the topology on X1 depends (in a special sense) on the preimages under uniformly continuous mappings of zero neighborhoods of a third Hausdorﬀ topological vector space (called X2 ). In this situation we get the desired criteria for subsets of X1 using properties of the spaces Y1 and X2 . More precisely let (X1 , t1 ), (X2 , t2 ), (Y1 , s1 ), (Y2 , s2 ) be Hausdorﬀ topological vector spaces, such that Xr is a subspace of Yr with sr ∩ Xr ⊂ tr (r = 1, 2). Let M be a set of continuous (not necessary linear) mappings m from (Y1 , s1 ) into (Y2 , s2 ) with m(0) = 0, such that M |X1 := {m|X1 : m ∈ M } contains only uniformly continuous mappings from (X1 , t1 ) into (X2 , t2 ). 105

106

Frerick m∈M

−−−−−−−−−→

(Y1 , s1 )    

(Y2 , s2 )    

m|X ∈M |X

1 −−−−1→ (X2 , t2 ) (X1 , t1 ) −−−−−

Under these conditions we get criteria for completeness, precompactness and compactness of subsets of X1 : Proposition 1 α)

Let A ⊂ X1 have a complete closure in (Y1 , s1 ). For all m ∈ M and (X2 ,t2 )

be a complete subset of (X2 , t2 ) and let X1 = for all x ∈ X1 let m(A + x) −1 m (X2 ). Moreover, let V be a basis of the zero neighborhood ﬁlter U0 (X1 ) m∈M

which satisﬁes: For all V ∈ V and all z ∈ V there are m ∈ M and U ∈ U0 (X2 ) such (X1 ,t1 )

that V ∩ (y − (m|X1 )−1 (U )) = ∅. Then A is a complete subset of (X1 , t1 ). β) Let B ⊂ X1 . If {{x1 ∈ X1 : m(x1 ) ∈ U }: m ∈ M, U ∈ U0 (X2 )} is a basis of U0 (X1 ), then: i) B is precompact in (X1 , t1 ) if B is precompact in (Y1 , s1 ) and m(B − B) is relatively compact in (X2 , t2 ) for all m ∈ M . ii) B is compact in (X1 , t1 ) if and only if B is compact in (Y1 , s1 ) and m(B − B) is compact in (X2 , t2 ) for all m ∈ M . Proof. α) 1) Let F be a Cauchy ﬁlter in (A, t1 ∩ A). Because s1 ∩ X1 ⊂ t1 there (Y1 ,s1 )

exists y ∈ A such that F → y in (Y1 , s1 ). From the continuity of m and from m(0) = 0 we obtain that m(F) → m(y) and m(F − y) → 0 in (Y2 , s2 ) ∀m ∈ M . 2)

The uniform continuity of m|X1 and s2 ∩ X2 ⊂ t2 guarantee the existence (X2 ,t2 )

of xm ∈ m(A)

such that m(F) → xm in

(Y2 , s2 ) ∀m ∈ M .

With the help of 1) we get for all m ∈ M that m(y) = xm and hence y∈

m∈M

m−1 (X2 ) = X1 .

On complete, precompact and compact sets 3) The completeness of m(A − y) implies together with 1) that

(X2 ,t2 )

107

and the uniform continuity of m|X1

m(F − y) → 0 in (X2 , t2 ) ∀m ∈ M . Hence for all m ∈ M and for all zero neighborhoods U in (X2 , t2 ) we get the existence of G ∈ F such that G − y ⊂ (m|X1 )−1 (U ). Therefore y ∈ G − (m|X1 )−1 (U ) for all G ∈ F, m ∈ M , and all zero neighborhoods U in (X2 , t2 ). 4) Let W be an arbitrary zero neighborhood in (X1 , t1 ). Then there is a V ∈ V such that V ⊂ W . Choosing G ∈ F with G − G ⊂ V we get with the help of 3): G − y ⊂ G − G + (m|X1 )−1 (U ) ⊂ V + (m|X1 )−1 (U ) for all m ∈ M and all zero neighborhoods U ∈ U0 (X2 ). Hence G−y ⊂V and G − y ⊂ W. (X1 ,t1 )

Because F was arbitrary we get that A is complete. β) i) Let F be a ﬁlter in B. Then there is a ﬁlter G ⊃ F in B which is Cauchy in (Y1 , s1 ) because B is precompact in (Y1 , s1 ). We prove that G is also Cauchy in (X1 , t1 ) (this implies that B is precompact in (X1 , t1 )): Let m ∈ M be arbitrary, and let H be the ﬁlter on B − B having

G − G: G ∈ G

as a basis. Then m(H) → 0 in (Y2 , s2 ).

(1)

On the other hand the compactness of m(B − B) in (X2 , t2 ) guarantees the existence of a cluster point y ∈ m(B − B) of m(H). With the help of (1) we obtain y = 0. Hence, 0 is the only cluster point of m(H) in the compact space m(B − B) and therefore m(H) → 0 in (X2 , t2 ).

108

Frerick

Because m was arbitrary we get that H → 0 in (X1 , t1 ) and G is a Cauchy ﬁlter in (X1 , t1 ). ii) “⇒” is clear. “⇐” In view of i) we only have to show that B is complete. Let F be a Cauchy ﬁlter in B. The compactness of B in (Y1 , s1 ) implies the existence of an x ∈ B such that F → x in (Y1 , s1 ) and m(F − x) → 0 in (Y2 , s2 ), ∀ m ∈ M .

(2)

On the other hand the compactness of all m(B − B) in (X2 , t2 ) (m ∈ M ) guarantees the existence of cluster points in m(B − B) for each one of the ﬁlters m(F − x) (m ∈ M ) . As above we conclude with the help of (2) that these cluster points are all 0 and that m ∈ M m(F − x) → 0 in (X2 , t2 ), ∀ m ∈ M . This means F → x in (X1 , t1 ). Remark 2. and if

α) If W is the collection of all the closed zero neighborhoods in (X2 , t2 ) V := {x1 ∈ X1 : m|X1 (x1 ) ∈ W }: m ∈ M, W ∈ W

is a zero basis in (X1 , t1 ), then automatically for all V ∈ V and all z ∈ V there are m ∈ M and U ∈ U0 (X2 ) such that V ∩ (z − (m|X1 )−1 (U )) = ∅. β) Assuming in addition that the completion of X2 is continuously included in Y2 we can use in the Proposition 1 β), part i) the weaker condition “m(B − B) is precompact for all m ∈ M ” instead of “m(B − B) is relatively compact for all m ∈ M ”. Proposition 1 α) contains a wellknown result due to W. Robertson (see [7]): Corollary 3 Given a topological vector space (E, T1 ) and a Hausdorﬀ vector space topology T2 coarser than T1 on E such that (E, T1 ) has a zero basis V consisting of T2 -closed sets. If for a subset A of E is (A, T2 ∩ A) complete, then (A, T1 ∩ A) is also complete.

On complete, precompact and compact sets

109

Proof. Choose (X1 , t1 ) := (E, T1 ), (X2 , t2 ) := (Y1 , s1 ) := (Y2 , s2 ) := (E, T2 ) and M := {id}. We give applications of our result to vector valued sequence spaces: Definition 4. A locally convex space, which is continuously included in KN and which contains K(N) is a normal locally convex sequence space, if it has a zero basis V satisfying the following (∗) condition: For every V ∈ V, (αn )n∈N ∈ V and (βn )n∈N ∈ KN verifying |βn | ≤ |αn | for every n ∈ N, we have (βn )n∈N ∈ V . Remark 5. i) Every normal locally convex sequence space is normal in the sense of K¨othe (see [5, p. 405ﬀ]). ii) Every normal Banach sequence space (see [1]) is a normal locally convex sequence space. iii) Let (λn )n∈N be a sequence of normal locally convex sequence spaces such that for every n ∈ N the space λn is continuously included in λn+1 . Then the inductive limit indn→ λn is also a normal locally convex sequence space. To prove this, let (Uln )l∈Ln be a basis of the zero neighborhood ﬁlter of λn satisfying the

j ∞ Uln : l ∈ Ln of the zero neighborhood condition (∗). Then the basis j=1 n=1

ﬁlter of indn→ λn satisﬁes the condition (∗) as can easily be seen. iv) Let (λn )n∈N be a sequence of normal locally convex sequence spaces such that for every n ∈ N the space λn+1 is continuously included in λn . Then it is easy to check that the projective limit proj←n λn is also a normal locally convex sequence space. Now we introduce a special kind of vector valued sequence spaces: In the following let E = {0} always denote a Hausdorﬀ locally convex space and let cs (E) be the set of the continuous seminorms on E. If λ is a normal locally convex sequence space we may introduce: Definition 6. Let λ(E) := (xn )n∈N ∈ E N : ∀p∈cs(E) (p(xn ))n∈N ∈ λ be provided with the Hausdorﬀ locally convex topology admitting V := {(xn )n∈N ∈ λ(E): (p(xn ))n∈N ∈ V }: p ∈cs (E), V = V ∈ U0 (λ)

110

Frerick

as a zero basis. For p ∈cs (E) let: Rp : E N → KN (xn )n∈N → (p(xn ))n∈N . Remark 7. i) In the cases λ = c0 or λ = lp , (p ∈ [1, ∞]) the completeness of the spaces λ(E) is well investigated. ii) If λ is a normal Banach sequence space and E is a Banach space then P . Dierolf has proved the completeness of λ(E) in [1]. We deal with the general situation: Theorem 8 Let λ be a normal locally convex sequence space and E be a Hausdorﬀ locally convex space. α) If λ and E are complete, then λ(E) is complete. β) Let B be an absolutely convex subset of λ(E). i) B is precompact if B is precompact in E N and Rp (B) is relatively compact in λ for all p ∈cs (E). ii) B is compact if and only if B is compact in E N and Rp (B) is compact in λ for all p ∈cs (E). Proof. Using the notations of Proposition 1 we choose: (X1 , t1 ) := λ(E), (X2 , t2 ) := λ, (Y1 , s1 ) := E N , (Y2 , s2 ) := KN , M := {Rp : p ∈cs (E)} and V as described in Deﬁnition 6. It is easy to check that the conditions of Proposition 1 are fulﬁlled to get the assertion. Remark 9. i) It is possible to replace in part α “complete” by “quasicomplete”, “p-complete” or “sequentially complete”. ii) The absolute convexity of the set B is essential as the sequence of unit vectors in l∞ (= l∞ (K)) shows. Proposition 1 can also be applied to vector valued sequence spaces of more complicated type. We will treat two concepts: spaces of sequences of bounded variation (see e.g. [3, p. 90ﬀ]) and James’ space (see e.g. [4]). Let ∞ N bv (E) := (xn )n∈N ∈ E : ∀p∈cs(E) p(x1 ) + p(xn − xn+1 ) < ∞ n=1

equipped with the Hausdorﬀ locally convex topology induced by the basis ∞ V := (xn )n∈N ∈bv (E): p(x1 ) + p(xn − xn+1 ) ≤ 1 : p ∈cs (E) . n=1

We get with the help of Proposition 1:

On complete, precompact and compact sets

111

Corollary 10 If E is complete, then bv (E) is complete. Proof. Using the notation of Proposition 1 let: (X1 , t1 ) :=bv (E), (X2 , t2 ) := l1 , (Y1 , s1 ); = E N , (Y2 , s2 ) := KN , M := {mp : p ∈cs (E)} deﬁned by mp : E N → KN , (xn )n∈N → (p(x1 ), (p(xn − xn+1 ))n∈N ), and let V be as described above. Then we can apply Proposition 1 α) and get the assertion. Another result shows the completeness of a vector valued James space: Let F := {1 ≤ n1 < n2 < . . . < nm : m, nk ∈ N} and for q ∈ [1, ∞) let

f

J q (E) := (xn )n∈N

f n −1 −1

m q 1/q k+1 N ∈ E : ∀p∈cs(E) sup p xi <∞

f ∈F

k=1

i=nfk

be equipped with the Hausdorﬀ locally convex topology induced by the basis f nfk+1 −1 −1

m q 1/q q p xi ≤ 1 : p ∈cs (E) . V := (xn )n∈N ∈ J (E): sup f ∈F

k=1

i=nfk

We get with the help of Proposition 1: Corollary 11 If E is complete, then J q (E) is complete. Proof. Using the notation of Proposition 1 let: (X1 , t1 ) := J q (E), (X2 , t2 ) := lF∞ (= {(αf )f ∈F ∈ KF : sup |αf | < ∞}), (Y1 , s1 ) := E N , (Y2 , s2 ) := KF , M := f ∈F

{mp : p ∈cs (E)} deﬁned by

mp : E N → KF

(xn )n∈N →

f

f n −1 −1

m q 1/q k+1 p xi

k=1

i=nfk

f ∈F

and let V as described above. Proposition 1 α) implies the assertion. The next part of this paper contains an application of our result to spaces of vector valued continuous functions. (I do thank Prof. Dr. W.M. Ruess for helpful comments on this topic.) Let X be a completely regular topological space, let again E = {0} be a Hausdorﬀ locally convex space and let C(X, E) be the space of the continuous functions from X into E.

112

Frerick

Let M be a subset of the powerset P (X) of X fulﬁll the conditions:

1) M = X. 2) For all L ∈ M and all f ∈ C(X) is f (L) bounded. 3) For all L1 , L2 ∈ M there is L3 ∈ M with L1 ∪ L2 ⊂ L3 . Then the topology TM of uniform convergence on the elements of M is a Hausdorﬀ locally convex topology on C(X) := C(X, K). For every p ∈cs (E), we introduce the mapping Rp : E X → KX (ex )x∈X → (p(ex ))x∈X and equip C(X, E) with the Hausdorﬀ locally convex topology TM admitting

V := {f ∈ C(X, E): Rp (f ) ∈ U }: p ∈cs (E), U = U ∈ U0 (C(X), TM )

as a zero basis. We get a result similar to some results due to Ruess and Summers (cf. [8]): Theorem 12 If B is an absolutely convex subset of C(X, E) then i) B is precompact if B is precompact in E X and Rp (B) is precompact in C(X) for all p ∈cs (E). ii) B is compact if and only if B is compact in E X and Rp (B) is compact in C(X) for all p ∈cs (E). Proof. Using the notation of Proposition 1 we choose: (X1 , t1 ) := (C(X, E), TM ), (X2 , t2 ) := (C(X), TM ), (Y1 , s1 ) := E X , (Y2 , s2 ) := KX , M := {Rp : p ∈cs (E)} and V as described above. Proposition 1 and Remark 2 implies the assertion, if one notes that the completion of C(X) is continuously included in KX . We denote by L(E, F ) the space of the linear and continuous functions from the Hausdorﬀ locally convex space E into the Hausdorﬀ locally convex space F and we prove by use of Proposition 1 a result of S. Dierolf (see [2, (1.8)]), which generalizes a classical theorem of Grothendieck.

On complete, precompact and compact sets

113

Definition 13 (cf. [2]) 1) Let A be a functor from the category of the Hausdorﬀ locally convex spaces and continuous linear maps into the category of sets and maps, which satisﬁes the following two conditions: i) For every Hausdorﬀ locally convex space E the set A(E) is a subset of the power set P (E). ii) Whenever E, F are Hausdorﬀ locally convex spaces and T ∈ L(E, F ), then A(T ): A(E) → A(F ) satisﬁes A(T )(A) = T (A) for all A ∈ A(E) 2) Let A be a functor as described in 1). We call a Hausdorﬀ locally convex space E A-complete, if A is complete for every A ∈ A(E). In [2] there is given a list of completeness properties which are of type Acomplete: complete: put A(E) := P (E) quasicomplete: put A(E) := {A ⊂ E: A is bounded } p-complete: put A(E) := {A ⊂ E: A is precompact } sequentially complete: put A(E) := {{xn : n ∈ N}: (xn )n∈N ∈ E N is a Cauchy sequence in E} convex compactness property: put A(E) := {Γ(A): A ⊂ E is compact } metric convex compactness property: put A(E): = {Γ(A): A ⊂ E is compact and pseudometrizable } locally complete: put A(E) := {{xn : n ∈ N}: (xn )n∈N ∈ E N is a local Cauchy sequence in E} Let X, Y = {0} be Hausdorﬀ locally convex spaces and let M be a subset of the set of all bounded subsets of X which satisﬁes the conditions 1) and 3). Then we denote by LM (X, Y ) the Hausdorﬀ locally convex space L(X, Y ) equipped with the topology of uniform convergence on the elements of M. If Y = K this space is . With this notation the following theorem holds: called XM Theorem 14 (S. Dierolf, [2, (1.8)]) Let A be a functor as described in Deﬁnition 13. Let L(X, Y ) = L(X, (Y, σ(Y, Y )). If XM and Y are A-complete, then LM (X, Y ) is A-complete. Remark 15. Grothendieck has proved the previous result in the case A(E) := P (E), (see e.g. [6, p. 143])

114

Frerick

Proof. Using the notation of Proposition 1 we choose: (X1 , t1 ) := LM (X, Y ), (X2 , t2 ) := XM , (Y1 , s1 ) := L(X, Y ) (the space of the linear mappings from X into Y equipped with the topology induced by Y X ), (Y2 , s2 ) := KX , M := {Rf : f ∈ Y }, for f ∈ Y we put Rf : L(X, Y ) → KX T → f ◦ T . Moreover let

V := {T ∈ L(X, Y ): T (L) ⊂ U }: L ∈ M, U = Γ(U ) ∈ U0 (Y ) .

Then X2 and Y1 are A-complete. Moreover, from L(X, Y ) = L(X, (Y, σ(Y, Y )), we get X1 = L(X, Y ) = Rf−1 (XM )= m−1 (X2 ) . f ∈Y

m∈M

Let L ∈ M, let U be a closed absolutely convex zero neighborhood in Y and let T0 ∈ L(X, Y ) with T0 (L) ⊂ U . Using the separation theorem for compact convex sets (see [5, p. 243f]), we get the existence of f ∈ Y and W ∈ U0 (XM ) such that −1 (T0 − T )(L) ⊂ U for all T ∈ Rf (W ) ∩ L(X, Y ). Hence, we get for all V ∈ V and all T0 ∈ V an m ∈ M and W ∈ U0 (X2 ) such that V ∩ (T0 − (m|X1 )−1 (W )) = ∅. Now it is easy to derive the theorem from Proposition 1. I thank warmly Prof. Dr. J. Bonet, Prof. Dr. S. Dierolf and especially Prof. Dr. J. Schmets for many helpful hinds and comments.

References 1. J. Bonet, S. Dierolf, Fr´echet spaces of Moscatelli type, Revista Math. de la Universidad Complutense Madrid 2 (1989). 2. S. Dierolf, On spaces of continuous linear mappings between locally convex spaces, Note di Matematica 5 (1985). 3. H. Heuser, Funktionalanalysis, Teubner, Stuttgart, 1986. 4. R.C. James, A separable somewhat reflexive Banach space with non-separable dual, Bull. Amer. Math. Soc. 80 Num. 4, (1974). 5. G. K¨othe, Topological vector Spaces I, Springer, Berlin Heidelberg New York, 1969. 6. G. K¨othe, Topological Vector Spaces II, Springer, Berlin Heidelberg New York, 1979. 7. W. Robertson, Completions of topological vector spaces, Proc. London Math. Soc. III, ser 8, (1958). 8. W.M. Ruess, W.H. Summers, Compactness in spaces of vector valued continuous functions and asymptotic almost periodicity, Math. Nachr. 135 (1988).

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 115–123 c 1994 Universitat de Barcelona

On reverse Hardy’s inequality

Alejandro Garc´ıa del Amo Dpto. An´alisis Matem´atico, Facultad de Ciencias Matem´aticas, Universidad Complutense, 28040-Madrid (Spain)

Abstract In this note, several properties of the class C(p, n) introduced by Neugebauer in [10] are given. In particular we characterize the weight pairs w, v for which ∞ ∞ x p 1 f (x)p w(x)dx ≤ C f v(x)dx x 0 0 0 for nondecreasing functions f and 1 ≤ p < ∞.

Boundedness results for the Hardy-Littlewood maximal operator in the classical Lorentz spaces p (w) have been obtained by Ari˜ no and Muckenhoupt in [2]. They observed that this leads to the study of the inequality ∞ ∞ p Af (x) w(x)dx ≤ C f (x)p v(x)dx 0

0

with w = v for nonincreasing functions f , where A is the averaging operator Af (x) = x 1 f . They solved the problem by introducing the condition Bp . x 0

Later Sawyer [11] characterized the inequality 1/q ∞ 1/p ∞ q p Af (x) w(x)dx ≤C f (x) v(x)dx 0

(∗)

0

Research supported in part by DGICYT grant PB88-0141. AMS subject classification: 26D15, 42B25. Key words and phrases: Weighted inequalities, Hardy’s inequality, Riesz convexity theorem.

115

116

Garc´ıa del Amo

for nonincreasing functions f . Without the restriction to nonincreasing functions f , the inequality (∗) has attracted a great deal of attention (see [1], [4] and [9] for the case 1 < p ≤ q < ∞ and [8, page 47] for the case 1 < q < p < ∞). Recently Neugebauer in [10] has studied the related problem of the converse inequality ∞

∞

f (x)p w(x)dx ≤ C

0

Af (x)p v(x)dx

(∗∗)

0

for nondecreasing functions f . It has been characterized in [10] the weight pairs (w, v) for which (∗∗) holds for nondecreasing functions f when p is a positive integer. Here, we prove, by using an extension of the Riesz convexity theorem given in [6], the general case of considering real p ≥ 1, answering a question of ([10], page 439). In fact, if 1 ≤ p < ∞ and n is a positive integer, then

∞

f (x) w(x)dx ≤ C pn

0

∞

Af (x)pn v(x)dx

0

holds for all nondecreasing functions f if and only if (w, v) ∈ C(p, n). This allows us to ﬁnd somewhat unexpected relationships among the sets C(p, n). For instance it holds that C(p, n) ⊂ C(q, m) (resp. C(p, n) ⊂ C(q, m)) if and only if pn ≥ qm (resp. pn > qm).

=

Throughout the paper we shall use the notation f ↓ (resp. f ↑) to indicate that f : R+ → R+ is nonincreasing (resp. nondecreasing). For 0 < r < ∞, let χr (x) = χ[0,r] (x) and χr (x) = χ[r,∞) (x). By a weight w we mean any nonnegative measurable function deﬁned on R+ , and p will be the conjugate exponent deﬁned by p1 + p1 = 1. The reader is referred to [3] for standard deﬁnitions and terminology. + + We shall denote by M + (µ) (resp. Sdec (µ), Sinc (µ)) the set of all nonnegative (resp. nonincreasing, nondecreasing) µ-measurable (resp. µ-simple) functions. An operator T is called semi-linear if T (f + g) = T (f ) + T (g) and T (af ) = aT (f ) for all a ≥ 0.

Assume that I is a ﬁnite interval, µ is a σ-ﬁnite measure deﬁned on a σ-algebra which contains the Borel sets of I and ν is a σ-ﬁnite measure. Let C be the set of µ-characteristic functions. Then

On reverse Hardy’s inequality

117

+ Definition 1. Let 1 ≤ p, q ≤ ∞. A semi-linear operator T deﬁned on Sdec (µ) ∪ C + + (resp. Sinc (µ) ∪ C) and taking values in M (ν) is said to be of nonincreasing (resp. nondecreasing) strong-type (p, q) if there exists a constant M such that

T f Lq (ν) ≤ M f Lp (µ) , + + (µ) (resp. Sinc (µ)). The least constant M for which the previous for all f in Sdec inequality holds is called the nonincreasing (resp. nondecreasing) strong-type (p, q) norm of T .

To prove Theorem 5 we need the following extension of the Riesz convexity theorem (see [6]): Theorem 2 Suppose 1 ≤ pk ≤ qk ≤ ∞, (k = 0, 1), and let T be a semi-linear operator of nonincreasing (resp. nondecreasing) strong-types (pk , qk ) with nonincreasing (resp. nondecreasing) strong-type norms Mk , (k = 0, 1). Suppose 0 ≤ θ ≤ 1 and deﬁne p and q by 1−θ 1−θ θ 1 θ 1 = = + , + . p p0 p1 q q0 q1 Then T is of nonincreasing (resp. nondecreasing) strong-type (p, q) and its nonincreasing (resp. nondecreasing) strong-type (p, q) norm Mθ satisﬁes Mθ ≤ M01−θ M1θ . If p ≥ 1, we shall denote by Lp (up ) the space of all measurable functions f such that

∞

f Lp (up ) =

1/p |f (u)| d(u ) p

< ∞.

p

0

Applying Fubini’s theorem, we obtain: Lemma 3 If p is a positive integer, then x2 x xp−1 ... 0

=

0

1 (p − 1)!

0

x1

f (u)dudx1 . . . dxp−2 dxp−1

0

x

f (u)(x − u)p−1 du = 0

for all measurable functions f : [0, x] → [0, ∞].

1 p!

x

f (x − u)d(up ) 0

118

Garc´ıa del Amo

The next result will be the key to prove Theorem 5. Lemma 4 If p is a positive integer, then

x

p

x

f (x − u)d(u ) ≤ p

f (u)

0

1/p

du

0

for all nonnegative nondecreasing functions f on [0, x]. Proof. If m ≥ 0 then

m

xm

≤

f (u)du 0

m

xm

f (u)

m/(m+1)

du

f (xm )m/(m+1)

0

so that

xm+1

0

m

xm

f (u)du dxm m xm+1 xm ≤ f (u)m/(m+1) du f (xm )m/(m+1) dxm 0

0

1 = m+1

0 xm+1

m+1 f (u)

m/(m+1)

du

0

and the conclusion follows from Lemma 3. By [x] we denote the integer part of a positive real number x. Theorem 5 If p ≥ 1, then f Lp (up ) ≤ Cp f L1 (u) holds for all functions f ↓, where Cp < 2. Proof. (i) Assume ﬁrst that p is a positive integer. Let A be the set of all nonnegative nonincreasing functions on [0, x]. It will be enough to show that

1/p

x p

p

f (u) d(u ) 0

≤

x

f (u)du 0

On reverse Hardy’s inequality for all f in A. By Lemma 4, we obtain x p f (x − u)d(u ) ≤ 0

119

p

x

f (u)

1/p

du

0

for all nonnegative nondecreasing functions f on [0, x] and, since x x f (x − u)1/p du = f (u)1/p du 0

0

holds for every measurable function f : [0, x] → [0, ∞], this case is proved. (ii) Let us assume now p ∈ [1, ∞) \ N. Again, it will be enough to show that x 1/p x p p f (u) d(u ) ≤ Cp f (u)du 0

0

for all f in A. For pk = [p] + k, k ∈ N ∪ {0}, put 1 1 − θk θk = + . p p0 pk By (i), we obtain x x 1/pk 1/pk pk −1 1/pk pk pk f (u)u u du f (u) d(u ) = pk ≤ 0

0

x

f (u)du,

0

for all f in A. Then we may write 1/pk x pk ≤ ak f (u)u dν 0

x

f (u)du,

0

−1 p

where ak = pk k and ν is the measure on [0, x] deﬁned by ν(E) = u−1 du. E

Thus, using Theorem 2, we get that x 1/p x 1/p x p 1 1−θk θk p p f (u)u dν f (u) d(u ) = ≤ a0 a1 f (u)du. p1/p 0 0 0 It follows, after some easy computations, that 1/p p 1−θk θk 1/p = <2 lim a0 ak p k→∞ p0 and the conclusion follows.

120

Garc´ıa del Amo

Corollary 6 If p ≥ 1, then

x

f (x − u)d(u ) ≤ p

Cp

0

p

x

f (u)

1/p

du

0

holds for all nonnegative nondecreasing functions f on [0, x]. The following deﬁnition extends the class C(p, n) introduced by Neugebauer ([10] page 438). Definition

7. Let 1 ≤ pj < ∞, j = 1, 2, . . . , n. We say that the pair (w, v) ∈ n C (pj )j=1 (or simply (w, v) ∈ C(pj )) if and only if there exists 0 < C < ∞ such that for every choice 0 < r1 , r2 , . . . , rn < ∞,

∞

n

0

χ (x) w(x)dx ≤ C rj

χrj (x)

0

1

x − r pj

n ∞ j=1

j

v(x)dx.

x

Remark 8. (i) In the case pj = p, j = 1, 2, . . . , n, we have C(pj ) = C(p, n). (ii) The smallest C in the above deﬁnition will be referred to as the C(pj )constant of (w, v). Theorem 9 Let fj ↑, 1 ≤ pj < ∞ and aj a positive integer, j = 1, 2, . . . , n. Then 0

n ∞

fj (x) w(x)dx ≤ C 0

1

n ∞ 1

1 xpj

aj

x

fj (x − u)d(upj )

v(x)dx

0

if and only if (w, v) ∈ C(aj pj ) with C equal to the C(aj pj )-constant of (w, v). Proof. The necessary condition is proved easily by taking fj = χrj . We do the converse for n = 2; the general case is obtained by repeating the argument. We suppose a1 = a2 = 1. Let ϕj ↑, ϕj (0) = 0, and let rj = ϕj (yj ), j = 1, 2, where 0 < y1 , y2 < ∞. By integrating the condition C(pj ) over {(y1 , y2 ): y1 , y2 > 0} we obtain ∞ ∞ ∞ χϕ1 (y1 ) (x)χϕ2 (y2 ) (x)w(x)dxdy1 dy2 L≡ 0 0 0 ∞ ∞ ∞ ≤C ψ1 (x, y1 )ψ2 (x, y2 )v(x)dxdy1 dy2 ≡ R , 0

0

0

On reverse Hardy’s inequality

121

x−ϕj (y) pj where ψj (x, y) = χϕj (y) (x) . By Lemma 4.2 of [10] we have x ∞ ∞ ∞ −1 ϕ2 (y2 ) L= ϕ−1 (x)χ (x)w(x)dxdy = ϕ−1 2 1 1 (x)ϕ2 (x)w(x)dx. 0

0

0

We repeat the argument to get x ∞ ∞

1 −1 p1 ϕ (x − u)d(u ) ψ2 (x, y2 )v(x)dxdy2 R= xp1 0 1 0 0 x ∞

1 x

1 −1 −1 p1 p2 = ϕ (x − u)d(u ) ϕ (x − u)d(u ) v(x)dx. xp1 0 1 xp2 0 2 0 Thus we obtain ∞

−1 ϕ−1 1 (x)ϕ2 (x)w(x)dx 0 ∞ x

1 x

1 −1 −1 p1 p2 ≤C ϕ (x − u)d(u ) ϕ (x − u)d(u ) v(x)dx, 1 2 xp1 0 xp2 0 0 where C is the C(pj )-constant of (w, v). If fj (0) = 0 we ﬁnish by taking ϕ−1 j = fj . Otherwise, as in Theorem 4.3 of [10], we 1 1 simple take ϕ−1 = ε f , where ε n j n (x) = nx, if 0 ≤ x ≤ n , and εn (x) = 1, if x > n , j,n and we get the result by letting n → ∞.

Corollary 10 Let f ↑, 1 ≤ p < ∞ and n a positive integer. Then ∞ ∞ x n 1 n p f (x) w(x)dx ≤ C f (x − u)d(u ) v(x)dx xp 0 0 0 if and only if (w, v) ∈ C(p, n) with C equal to the C(p, n)-constant of (w, v). Proof. If (w, v) ∈ C(p, n), then the inequality from Theorem 9 by letting n follows rj f1 = f2 = . . . = fn . Conversely, let f = 1 χ . Then f = f n and by H¨ older’s inequality x n n x

1/n χrj (x − u) d(up ) ≤ χrj (x − u)pup−1 du . 0

1

1

0

So the conclusion follows. Corollary 11 Let fj ↑, 1 ≤ pj < ∞ and aj a positive integer, j = 1, 2, . . . , n. Then ∞ ∞ n n x

aj pj 1 pj w(x)dx ≤ BC v(x)dx fj (x) fj x 0 0 0 1 1 if and only if (w, v) ∈ C(aj pj ) with C equal to the C(aj pj )-constant of (w, v) and Σa p B = Cp j j .

122

Garc´ıa del Amo

Proof. The necessary condition follows by taking fj = χrj . For the suﬃciency, we use the above theorem ∞ x ∞ n n aj

1 pj pj pj fj (x) f (x − u) d(u ) w(x)dx ≤ C v(x)dx. j xpj 0 0 0 1 1 Now we complete the proof by Corollary 6. The next corollary is an extension of Corollary 4.4 in [10] and answers a question in [10] (page 439, remark). (Very recently Lai [7] has also answered this question by using somewhat diﬀerent techniques.) Corollary 12 Let f ↑, 1 ≤ p < ∞ and n a positive integer. Then ∞ ∞ x pn 1 pn pn f (x) w(x)dx ≤ Cp C f v(x)dx x 0 0 0 if and only if (w, v) ∈ C(p, n) with C equal to the C(p, n)-constant of (w, v). Proof. If (w, v) ∈ C(p, n), then the inequality follows Corollary 11 by letting nfrom rj f1 = f2 = . . . = fn . For the converse we take f = 1 χ . Then f = f pn and, by H¨older’s inequality, we conclude n n x

pn p 1 x 1 χrj ≤ χrj . x 0 x 0 1 1 Remark 13. If 1 ≤ p, q < ∞ and n, m are positive integers with pn = qm, then C(p, n) = C(q, m). This provides motivation for studying the relationships among the sets C(p, n). Proposition 14 Let 1 ≤ q < p < ∞ and n a positive integer. Then C(p, n) ⊂ C(q, n). =

Proof. Let r > 0 and m0 a positive integer with 2−m0 ≤ r. For m ≥ m0 we let r−2−(m+1) x − (r − 2−m ) qn Am = dx. x r−2−m Now let us consider the weights w, v deﬁned by ∞ x − (r − 2−m ) qn χ(r−2−m ,r−2−(m+1) ) (x), w(x) = x m=m 0

v(x) = χ(r−2−m0 ,r) (x) =

∞

χ(r−2−m ,r−2−(m+1) ) (x) .

m=m0

We can show that (w, v) ∈ C(q, n) \ C(p, n).

On reverse Hardy’s inequality

123

We give some consequences: Corollary 15 (i) C(p, n) ⊂ C(q, m) if and only if pn ≥ qm. (ii) C(p, n) ⊂ C(q, m) if and only if pn > qm. =

(iii) C(p, n) = C(q, m) if and only if pn = qm. Proof. It is clearly enough to prove only the suﬃciency n

of (i) and (ii). Suppose pn ≥ qm. Then, by Remark 13, C(p, n) = C p m , m ⊂ C(q, m). Similarly, if pn > qm, then, by Proposition 14, C(p, n) ⊂ C(q, m). =

Finally, we also comment a factorization of w ∈ Bp similar to the factorization of w ∈ Ap given in the P. Jones’ factorization theorem (see [5]): Given 1 ≤ p,pq < ∞, it holds that w ∈ Bq if and only if w(x) = u(x) · xq−p with p u(x q ) · x(p−1)(1− q ) ∈ Bp . p−q In the special case of 1 < p < q < ∞, if w ∈ Bq then w = u · x1 , with 1 u, x ∈ Bp .

References 1. K.F. Andersen and B. Muckenhoupt, Weighted weak type Hardy inequalities with applications to Hilbert transforms and maximal functions, Studia Math. 72 (1982), 9–26. 2. M. Ari˜no and B. Muckenhoupt, Maximal functions on classical Lorentz spaces and Hardy’s inequality with weights for nonincreasing functions, Trans. Amer. Math. Soc. 320 (1990), 727–735. 3. C. Bennett and R. Sharpley, Interpolation of Operators, Math. 129, Academic Press, Orlando, 1988. 4. J. Bradley, Hardy inequalities with mixed norms, Canad. Math. Bull. 21 (1978), 405–408. 5. J. Garc´ıa-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North Holland Math. Studies 116, North Holland 1985. 6. A. Garc´ıa del Amo, Weighted inequalities for monotone functions, preprint. 7. S. Lai, Weighted norm inequalities for general operators on monotone functions, Trans. Amer. Math. Soc. (to appear). 8. V. G. Maz’ya, Sobolev Spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, Heidelberg, New York, Tokyo 1985. 9. B. Muckenhoupt, Hardy’s inequality with weights, Studia Math. 44 (1972), 31–38. 10. C.J. Neugebauer, Weighted norm inequalities for averaging operators of monotone functions, Publ. Mat. 35 (1991), 429–447. 11. E. Sawyer, Boundedness of classical operators on classical Lorentz spaces, Studia Math. 96 (1990), 145–158.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 125–127 c 1994 Universitat de Barcelona

A Fourier inequality with Ap and weak-L1 weight

H.P. Heinig Department of Mathematics and Statistics, Mc Master University Hamilton, Ontario, Canada L8S4K1

Abstract The object of this note is to generalize some Fourier inequalities.

The following weighted Fourier norm inequality is known: Theorem A ([1], [2]). Suppose w is a radial weight function on Rn and as radial function nondecreasing on (0, ∞). Let 1 < p ≤ q ≤ p < ∞, then there is a constant C > 0 such that 1/p 1 q/p 1/q q −n(1−q/p ) p ˆ |f (x)| |x| w dx ≤C |f (x)| w(x)dx (1) |x| Rn Rn holds, if and only if w ∈ Ap . Here fˆ denotes the Fourier transform of f , deﬁned by ˆ e−ixy f (y)dy, x ∈ Rn f (x) = Rn

whenever the integral converges. The Muckenhoupt weight class Ap consists of all non-negative measurable functions w for which p−1 1 1 1−p sup w(x)dx w(x) dx < ∞, |Q| Q Q⊂Rn |Q| Q 125

126

Heinig

where Q denotes a cube in Rn with sides parallel to the coordinate axes, |Q| its p is the conjugate index of p. Lebesgue measure and p = p−1 A function ϕ belongs to weak L1 (i.e. ϕ ∈ L1weak ) if there is a constant C > 0 such that for all λ > 0, λm({x ∈ Rn : |ϕ(x)| > λ}) ≤ C, or equivalently yϕ∗ (y) ≤ C, y > 0, where ϕ∗ (y) = inf{λ > 0: m({x ∈ Rn : |ϕ(x)| > λ}) ≤ y} is the equimeasurable decreasing rearrangement of ϕ. Since |x|−n ∈ L1weak one might expect that the term |x|−n occurring in (1) can be replaced by any ϕ ∈ L1weak . The object of this note it to prove this is indeed the case. Theorem 1 Suppose w is a radial weight function in Ap and as radial function nondecreasing in (0, ∞). If 1 < p ≤ q ≤ p < ∞ and ϕ ∈ L1weak , then there is a constant C > 0, such that Rn

|fˆ(x)|q w

1/q 1/p 1 q/p 1−q/p p ϕ(x) dx ≤C |f (x)| w(x)dx . |x| Rn

(2)

Note that the case q = p may be found in [2] while the case w(x) = 1 yields Corollary 1.6 of [4]. Proof. The hypotheses of Theorem 1 imply that inequality (1) holds. Writing 1 1 n( 1 − 1 ) 1 p ) and v(x) = w(x) p then u and v are radial and as radial u(x) = |x| p q w( |x| functions decreasing on (0, ∞). Hence with this change (1) implies by [3, Theorem 3.1] that 1/q κ 1/p κ q n−1 −p n−1 sup u(t) t dt v(t) t dt <∞ s>0

0

0 −1 n

−1 n

where t = |x|, κ = s−2 θn , κ = s2 θn , and θn is the measure of the unit n-sphere. Writing w(t) = w( 1t ) the supremum takes the form

κ

sup s>0

t

n([1/p −1/q]q+1)−1

1/q q/p

w(t)

dt

0

0 1

κ

1/p 1 p /p n−1 (t) t dt <∞ w

−1

and the change of variable t = y n θnn shows that this implies sup s>0

0

s−2n

w(y 1/n θn−1/n )q/p y q/p −1 dy

1/q 0

s2n

1/p 1 1/n −1/n p /p (y θn ) dy < ∞. w (3)

A Fourier inequality

127

But w and w1 are decreasing as radial functions and so equal to their radially decreasing rearrangements. Now the equimeasurable rearrangement of a function g, deﬁned by

g ∗ (y) = inf λ > 0: m({x: |g(x)| > λ}) ≤ y , −1

is related to its radially decreasing rearrangement g ⊗ by g ∗ (y) = g ⊗ (y n θnn ) (cf. [3]). Hence, with λ = s2n , (3) takes the form 1

1/λ

sup

q/p q/p −1

∗

w (y)

λ>0

y

1 ∗ w

0

1/λ

sup

λ

dy

0

But since ϕ ∈ L1weak , ϕ∗ (y) ≤

λ>0

1/q

∗

q/p

w (y)

∗

C y ,y

(y)

1/p < ∞.

dy

> 0, so this implies

1−q/p

ϕ (y)

p /p

1/q dy

0

0

λ

1 ∗ w

p −1

(y)

1/p dy

< ∞.

Since powers and rearrangements commute, i.e. (g α )∗ = (g ∗ )α and since for any h and g, h∗ (y)g ∗ (y) ≥ (hg)∗ (2y), then after a change of variable the last supremum inequality implies

(w

sup λ>0

1/λ q/p

ϕ

1−q/p ∗

1/q

λ

) (y)dy

0

0

1 ∗ w

p −1

(y)

1/p dy

< ∞.

But this (cf. [5]) implies the inequality (2). It is a pleasure to express my appreciation to Professor Raymond Johnson for some fruitful conversations on this topic.

References 1. J.J. Benedetto, H.P. Heinig and R. Johnson, “Fourier inequalities with Ap -weights”, Proc. Conf. Oberwolfach 1986, General Inequalities 5. Internat. Series Numerical Math. 80 Birkh¨auser, Basel (1987), 217–232. 2. H.P. Heinig and G.J. Sinnamon, “Fourier inequalities and integral representations of functions in weighted Bergman Spaces over tube domains”, Indiana Univ. Math. J. 38(3) (1989), 603–628. 3. H.P. Heinig, “Weighted norm inequalities for classes of operators”, Indiana Univ. Math. J. 33(4) (1984), 573–582. 4. L. H¨ormander, “Estimates for translation invariant operators in Lp -spaces”, Acta Math. 104 (1960), 93–140. 5. B. Muckenhoupt, “A note on two weight function conditions for a Fourier transform norms inequality”, Proc. Amer. Math. Soc. 88(1) (1983), 97–100.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 129–134 c 1994 Universitat de Barcelona

Every nonreflexive Banach lattice has the packing constant equal to 1/2

H. Hudzik Institute of Mathematics, A.Mickiewicz University, Matejki 48/49 Pozna´n, 60-769 Poland

Abstract Kottman [9] has proved that any P -convex Banach space X is reflexive. In the case when X is a Banach lattice our result says more. It says that any Banach lattice X with Λ(X) < 1/2 is reflexive. This result generalizes the results of Berezhnoi [2] who proved that Λ(Λ(ϕ)) = Λ(M (ϕ)) = 1/2 for nonreflexive Lorentz space Λ(ϕ) and Marcinkiewicz space M (ϕ). It is proved also that for any Banach lattice X such that its subspace Xa of order continuous elements is nontrivial we have Λ(X) = Λ(Xa ). It is noted also that Orlicz sequence space lΦ is reflexive iff Λ(lΦ ) < 1/2.

In the sequel N, R, R+ denote respectively the sets of natural numbers, of reals and of nonnegative reals. If X is a real Banach space, S(X) and B(X) denote its unit sphere and unit ball, respectively. The packing constant of X is deﬁned by the formula (see [15] and [16]): Λ(X) = sup r > 0: ∃(xn )∞ n=1 in X s.t. xn ≤ 1−r, and xm −xn ≥ 2r if m = n . Kottman [9] has proved that for any inﬁnite dimensional Banach space X, we have Λ(X) = D(X)/(2 + D(X)), 1991 Mathematics Subject Classification. Primary 46B20, 52A45; Secondary 46B30, 46E30. Keywords: Banach lattice, packing constant, P -convexity, reflexivity, Orlicz sequence space.

129

130

Hudzik

where D(X) = sup

inf xm − xn : (xn )∞ n=1 contained in S(X) .

m=n

Recall that a Banach space X is said to be P -convex (see [10]) if P (n, X) < 1/2 for some n ∈ N, n ≥ 2 , where P (n, X) = sup r > 0: ∃(xi )ni=1 s.t. xi ≤ 1 − r and xi − xj ≥ 2r for i = j . Kottman [10] has proved that any P -convex Banach space is reﬂexive. A map Φ: R → R+ is said to be an Orlicz function if Φ(0) = 0, Φ is even, convex, 0 and Φ(u) → +∞. By l we denote the space of all real sequences. Given any Orlicz 0 function Φ deﬁne on l a convex functional IΦ (xλ) ≤ 1 by IΦ (x) =

∞

Φ(x(i))

∀x = (x(i))∞ i=1 ∈ l

0

.

i=1 0

The Orlicz sequence space lΦ is then deﬁned to be the set of these x in l for which IΦ (λx) < +∞ for some λ > 0 depending on x. The space lΦ equipped with the Luxemburg norm ∀x ∈ lΦ xΦ = inf λ > 0: IΦ (x/λ) ≤ 1 is a Banach space (see [9], [11], [12], [13] and [14]). Recall that an Orlicz function Φ satisﬁes the ∆2 -condition at 0 if there exist K > 0 and u0 > 0 such that 0 < Φ(u0 ) < + ∞ and Φ(2u) ≤ KΦ(u) whenever |u| ≤ u0 . Let X, Y be Banach spaces. We say that X contains an isomorphic (almost isometric) copy of Y if for some (for every) ε > 0 there exists a linear operator P : Y → X such that the inequality (∗)

yY ≤ P yX ≤ (1 + ε)yY

holds for every y ∈ Y . For the theory of Banach lattices see [1], [4] and [9]. We start with the following result. Proposition 1 Let X and Y be Banach spaces and assume that X contains an almost isometric copy of Y . Then Λ(X) ≥ Λ(Y ).

Every nonreflexive Banach lattice has the packing constant equal to 1/2

131

Proof. Let ε > 0 be arbitrary and take an arbitrary sequence (yn ) in S(Y ). Let P be a linear operator from Y into X satisfying condition (∗). Deﬁne the sequence xn = P yn /P yn X in X. In virtue of (∗), we have 1 P ym − P yn X P ym X 1 1 − + P yn X P ym X P yn X 1 1 − ≤ (1 + ε)ym − yn Y + (1 + ε) P ym X P yn X ≤ (1 + ε)ym − yn Y + ε.

xm − xn X ≤

(1)

On the other hand,

(2)

y yn ym yn

m − P − ≤ P

P ym X P yn X Y P ym X P yn X X

and

(3)

1 ε 1 1 = < ε. − ≤1− P ym X P yn X 1+ε 1+ε

Applying (3), we get (4)

y −y

ym yn 1 1

m n

− −

≥

− P ym X P yn X Y P ym X Y P ym X P yn X 1 ym − yn Y − ε . > 1+ε

Combining (2) and (4), we obtain xm − xn X ≥

(5)

1 ym − yn Y − ε. 1+ε

Inequalities (1) and (5) imply our result. Theorem 1 Every nonreﬂexive Banach lattice X has the packing constant equal to 1/2.

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Hudzik 1

Proof. By the assumption, X contains an isomorphic copy of c0 or l (see [1], [4] and [9]). So, by the James theorem (see [8]) X contains an almost isometric copy 1 1 of c0 or l , respectively. By Proposition 1, we get Λ(X) ≥ Λ(c0 ) or Λ(X) ≥ Λ(l ), 1 respectively. However, Λ(c0 ) = Λ(l ) = 1/2 (see [7], [16] and [17]), whence it follows that Λ(X) ≥ 1/2. Since, the inequality Λ(X) ≤ 1/2 is always true, we get Λ(X) = 1/2. Remark 1. Since Λ(X) ≤ P (n, X) for any n ∈ N(n ≥ 2), Theorem 1 gives stronger result for Banach lattices than Kottman theorem which says that any P -convex Banach space is reﬂexive. Remark 2. There exists a reﬂexive Banach lattice X with Λ(X) = 1/2.

pi In fact, deﬁne X to be the Hilbertian direct sum l , where 1 < pi 1. pi Every l , i = 1, 2 . . . , is isometrically embedded into X, so we get for any i ∈ N: pi

Λ(X) ≥ Λ(l ) = 1/(2 + 21−1/pi ) 1/2 (for the inside equality see [16] and [17]). Since we always have Λ(X) ≤ 1/2, it follows that Λ(X) = 1/2. Clearly, X is a Banach lattice and as the Hilbertian direct pi sum of reﬂexive Banach lattices l (1 < pi < + ∞), X is reﬂexive. Theorem 2 Let X be an arbitrary Banach lattice and Xa be its subspace of order continuous elements. If Xa = {0}, then Λ(Xa ) = Λ(X). Proof. Assume ﬁrst that X has an order continuous norm, i.e. Xa = X. Then the equality is obvious. Assume now that Xa = X and Xa = {0}. Then Xa contains an isomorphic copy of c0 (see [1], [4] and [9]), and so, by the James theorem (see [7]), Xa contains almost isometric copy of c0 . Hence, Λ(X) ≥ Λ(Xa ) = 1/2, and consequently Λ(X) = Λ(Xa ) = 1/2. Yining Ye, He Miaohong and Ryszard PHluciennik [18] have proved that Orlicz function space LΦ as well as Orlicz sequence space lΦ equipped with the Luxemburg norm is P -convex iﬀ it is reﬂexive, i.e. both Φ and Φ∗ (the complementary function to Φ in the sense of Young) satisfy the suitable ∆2 -condition (i.e. the ∆2 -condition at zero in the sequence case). We will prove now an analogous result for lΦ in terms of Λ(lΦ ). Theorem 3 An Orlicz sequence space lΦ is reﬂexive if and only if Λ(lΦ ) < 1/2.

Every nonreflexive Banach lattice has the packing constant equal to 1/2

133

Proof. If Λ(lΦ ) < 1/2 then lΦ is reﬂexive by Theorem 1. Assume that lΦ is reﬂexive, i.e. Φ and Φ∗ satisfy the ∆2 -condition at zero. Then (see [15] and [17]), (6)

D(lΦ ) = sup

cx > 0: IΦ (x/cx ) =

x∈S(lΦ )

1 . 2

To get this formula only the ∆2 -condition at zero for Φ is important. Since Φ∗ also satisﬁes the ∆2 -condition at zero, we have for a > 0 such that Φ(a) = 1: (7)

∃p > 1 ∀λ ∈ (0, 1) ∀|u| ≤ a: Φ(λu) ≤ λp Φ(u).

Hence, taking into account that, in view of the ∆2 -condition at zero for Φ, the equality IΦ (x) = 1 holds for any x ∈ S(lΦ ), we get for any x ∈ S(lΦ ): IΦ

x 1 1 ≤ IΦ (x) = . 1/p 2 2 2

Hence, it follows that cx ≤ 21/p for any x ∈ S(lΦ ). Therefore, D(lΦ ) ≤ 21/p , and consequently Λ(lΦ ) ≤ 1/(2 + 21−1/p ) < 1/2.

References 1. C.D. Aliprantis and O. Burkinshaw, Positive Operators, Pure and Applied Math., Academic Press, Inc. 1985. 2. E.I. Berezhnoi, Packing constant of the unit ball of Lorentz and Marcinkiewicz spaces, Qualitative and Approximate Methods for the Investigation of Operator Equations, Yaroslavl 1984, 24–28. 3. Ch.C. Cleaver, Packing spheres in Orlicz spaces, Pacific J. Math. 65 (2) (1976), 325–335. 4. J. Diestel, Sequences and Series in Banach Spaces, Lecture Notes in Math., Springer-Verlag 1984. (1) 5. H. Hudzik, Uniformly non-ln Orlicz spaces with Luxemburg norm, Studia Math. 81 (3) (1985), 271–284. 6. H. Hudzik and T. Landes, Packing constant in Orlicz spaces equipped with the Luxemburg norm, to appear in Bolletino della Unione Matematica Italiana. 7. H. Hudzik, Congxin Wu and Yining Ye, Packing constant in Musielak-Orlicz sequence spaces equipped with the Luxemburg norm, to appear in Revista Matem´atica de la Universidad Complutense de Madrid 7(1) (1994). 8. R.C. James, Uniformly non-square Banach spaces, Annals of Math. 80 (1964), 542–550. 9. L. V. Kantoroviˇc and G.P. Akilov, Functional Analysis, Moscow 1977 (in Russian). 10. C.A. Kottman, Packing and reflexivity in Banach spaces, Trans. Amer. Math. Soc. 150 (1970), 565–576.

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11. M.A. Krasnoselskiˇi and Ya. B. Rutickiˇi, Convex functions and Orlicz spaces, P. Noordhoof Ltd., Groningen 1961. 12. W.A.J. Luxemburg, Banach Function Spaces, Thesis, Delft 1955. 13. J. Musielak, Orlicz Spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer-Verlag 1983. 14. M.M. Rao and Z.D. Ren, Theory of Orlicz spaces, Marcel Dekker Inc. New York-Basel-Hong Kong 1991. 15. T. Wang and Y. Liu, Packing constant of a type of sequence spaces, Comment. Math. (Prace Mat.) 30 (1) (1990), 197–203. 16. J.H. Wells and L.R. Williams, Embedding and Extension Problems in Analysis, Lecture Notes in Math., Springer-Verlag 1975. 17. Yining Ye, Packing spheres in Orlicz sequence spaces, Chinese Ann. Math. 4A (4) (1983), 487–493. 18. Yining Ye, He Miaohong and Ryszard Pluciennik, P -convexity and reflexivity of Orlicz spaces, Comment. Math. (Prace Mat.) 31 (1991), 203–216.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 135–146 c 1994 Universitat de Barcelona

On extreme points of Orlicz spaces with Orlicz norm

Henryk Hudzik and Marek Wisla Institute of Mathematics, A. Mickiewicz University, Matejki 48/49 Pozna´n, 60-769 Poland

Abstract In the paper we consider a class of Orlicz spaces equipped with the Orlicz norm over a non-negative, complete and σ -finite measure space (T, Σ, µ), which covers, among others, Orlicz spaces isomorphic to L∞ and the interpolation space L1 + L∞ . We give some necessary conditions for a point x from the unit sphere to be extreme. Applying this characterization, in the case of an atomless measure µ, we find a description of the set of extreme points of L1 + L∞ which corresponds with the result obtained by R.Grz¸as´lewicz and H.Schaefer [3] and H.Schaefer [13].

The aim of this paper is to extend some known descriptions of the set of extreme points of Orlicz spaces yielded with the Orlicz norm (cf., e.g., [7], [15], [6]) to the case that covers classical Banach spaces like L∞ and the interpolation space L1 +L∞ with the norm xL1 +L∞ = inf y1 + z∞ : y + z = x, y ∈ L1 , z ∈ L∞ . The point is that in the previous papers on this subject the authors have assumed that the function Φ generating the Orlicz space LΦ is an N-function, i.e., Φ : R → [0, ∞) is even, convex, continuous, vanishing at 0 function satisfying Φ(u)/u → 0 as u → 0 and Φ(u)/u → ∞ as u → ∞. Keywords: extreme point, Orlicz space, space L1 + L∞ . AMS Subject Classification: 46B20.

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In this paper we will take into consideration a more general class of the functions Φ. We shall assume that Φ : R → [0, ∞] (so Φ can take inﬁnity value), Φ vanishes at 0, it is even, convex, left-continuous on [0, ∞), nonidentically equal to 0 and such that 0 ≤ Φ(u) < ∞ for some u > 0. To motivate this sort of conditions, let us consider the following function Φ : R → [0, ∞): Φ(u) =

0

if |u| ≤ 1

|u| − 1

otherwise.

(1)

Then, an easy calculation shows that the space L1 + L∞ is equal (as a set) to Φ L of all those measurable functions x : T → R for whichΦ IΦ (λx) := the space Φ λx(t) dµ < ∞ for some λ > 0 (depending on x). The space L , and thus T 1 L + L∞ , is, in fact, an Orlicz space generated by the function Φ (cf. [8], [12], [4], [11]). It occurs that the norm · L1 +L∞ can be described by means of the function Φ as well, namely · L1 +L∞ is equal to the Orlicz norm · 0Φ given by x0Φ

|x(t)y(t)| dµ : y ∈ L

= sup

Φ∗

, IΦ∗ (y) ≤ 1

,

(2)

T

where Φ∗ denotes the complementary function to Φ in the Young sense, i.e., Φ∗ (u) = sup uv − Φ(v) : v ≥ 0 (cf. [12]) and Φ is deﬁned by (1). It is easy to show that , if Φ is given by (1) then Φ∗ (u) =

|u|

if |u| ≤ 1

+∞ otherwise. ∗

1 ∞ Moreover, LΦ = (as sets) and the classical norm · L1 ∩L∞ = L ∩ L ∗ max · 1 , · ∞ coincides with the Luxemburg norm · Φ∗ on LΦ deﬁned by

· Φ∗ = inf{λ > 0 : IΦ x(t)/λ ≤ 1}. Thus (2) follows from the well–known formula xL1 +L∞ = sup T

(cf. [1], [5]).

∞

|x(t)y(t)| dµ : y ∈ L ∩ L , yL1 ∩L∞ ≤ 1 1

On extreme points of Orlicz spaces with Orlicz norm

137

If we consider the function Φ given by (1) then, obviously, Φ(u)/u → 1 as u → ∞, thus the above mentioned results cannot be applied. In fact, a description which covers all the cases of Orlicz functions is not known yet. (Let us mention that the similar problem concerning the description of the set of extreme points of Orlicz spaces yielded with the Luxemburg norm and Lorentz spaces have been already solved – cf. [2], [14]). The Orlicz norm given by (2) is not easy to deal with. It is far more convenient to make use of the Amemiya formula: x0Φ =

1 1 + IΦ (kx) 0
(3)

(cf. [9], [10]). The set of all k’s at which the inﬁmum is attained (for a ﬁxed x ∈ LΦ ) will be denoted by K(x). In particular, the set K(x) can be empty. To simplify the notation, by < a, b > we shall denote the interval with the endpoints a and b, i.e., < a, b >= {λa + (1 − λ)b : 0 ≤ λ ≤ 1}. In the following, the set of all extreme points of the unit ball B(X) will be denoted by Ext B(X). Theorem 1 Let Φ be an Orlicz function and let µ be an arbitrary non–negative complete and σ–ﬁnite measure (not necessarily atomless). If z ∈ Ext B(LΦ , · 0Φ ) and supp z does not reduce to an atom, then the set K(z) consists of exactly one element. First, we prove an auxiliary lemma. Lemma 1 Under the assumptions of Theorem 1, the set K(z) is not empty. = ∞ and let z ∈ LΦ \ {0}. Then there exists Proof. a) Assume that limu→∞ Φ(u) u ε > 0 such that µ(Aε ) > 0, where Aε = {t ∈ T : |z(t)| > ε}. Thus

so

1 1 1 −− −→ ∞ , IΦ (kz) ≥ IΦ (kzχAε ) ≥ Φ(kε)µ(Aε ) − k→∞ k k k 1 1 + IΦ (kz) ≤ 2z0Φ < ∞ . k2 := max k ∈ (0, ∞) : k

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Since

1 1 1 + IΦ (kz) ≥ > 2z0Φ k k

provided k < k1 := (2z0Φ )−1 , K(z) ⊂ [k1 , k2 ]. Since, moreover, the function k →

1 1 + IΦ (kz) k

is continuous on [k1 , k2 ], we infer that K(z) = ∅. b) Let g = limu→∞ Φ(u)/u and let us assume that 0 < g < ∞. Since supp z does not reduce to an atom, there exists ε > 0 such that the set C = {t ∈ T : |z(t)| > ε} also does not reduce to an atom. Let A,B be disjoint subsets of C such that 0 < µ(A), µ(B) < ∞. Without loss of generality we can assume that A |z(t)|dµ ≤ |z(t)|dµ. Let λ ∈ (0, 1] be a number such that A |z(t)|dµ = λ B |z(t)|dµ and B deﬁne x = z + zχA − λzχB , y = z − zχA + λzχB . Obviously, x = y and (x + y)/2 = z . Let un = nε for n ∈ N. Applying the monotonicity of u → Φ(u)/u and the convexity of Φ we have Φ(u) ≤ g|u| for every u ∈ R and Φ(un ) |u| ≤ Φ(u) un

for every |u| ≥ un .

Thus gn 2 A |z(t)|dµ + (1 − λ) B |z(t)|dµ IΦ (2nzχA) + IΦ (n(1 − λ)zχB ) ≤ Φ(un ) IΦ (nzχA∪B ) un · n A∪B |z(t)|dµ gun = Φ(un ) and

gn(1 + λ) B |z(t)|dµ IΦ (n(1 + λ)zχB ) gun ≤ Φ(u ) . = n IΦ (nzχA∪B ) Φ(u n) un n A∪B |z(t)|dµ Now, suppose that K(z) = ∅. Then 1 1 1 1 + IΦ (kz) = lim 1 + IΦ (kz) = lim IΦ (kz). k>0 k k→∞ k k→∞ k

1 = z0Φ = inf

On extreme points of Orlicz spaces with Orlicz norm

139

Thus 1 1 IΦ (nx) = IΦ (2nzχA) + IΦ (n(1 − λ)zχB ) + IΦ (nzχT \(A∪B)) n n 1 gun IΦ (nzχA∪B ) + IΦ (nzχT \(A∪B)) ≤ nΦ(un ) n gun −−− −→ 1, IΦ (nz) n→∞ ≤ nΦ(un ) so x0Φ ≤ 1. Analogously, 1 1 IΦ (ny) = IΦ (n(1 + λ)zχB ) + IΦ (nzχT \(A∪B)) n n gun −−− −→ 1, ≤ IΦ (nz) n→∞ nΦ(un ) so y0Φ ≤ 1 as well. Thus z is not an extreme point of B(LΦ , · 0Φ ) - and we arrived at a contradiction which ends the proof. The assumption “supp z does not reduce to an atom” cannot be omitted. Indeed, consider the sequence space 1 and the sequence z = (1, 0, . . .). Obviously, z is an extreme point of B(1 ). Since ∞ 1 1 1+ |kzi | = + 1 > 1 k k i=1

for every k = 0 the set K(z) is empty. Proof of Theorem 1. Suppose that K(z) is not a one element set and let k2 > k1 be such that k1 , k2 ∈ K(z). We have 2k1 k2 k1 + k2 1 + IΦ z 2k1 k2 k1 + k2 k2 k1 k1 + k2 1 + IΦ k1 z + k2 z = 2k1 k2 k1 + k2 k1 + k2 1 1 1 ≤ 1 + IΦ (k1 z) + 1 + IΦ (k2 z) = z 0Φ . 2 k1 k2

z0Φ ≤

1 k2 Thus the numbers k1 z(t), k2k z(t), k2 z(t) belong to the same interval on which Φ 1 +k2 is aﬃne and this fact holds true for µ–a.e. t in T .

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1 k2 In order to simplify put k0 = k2k and denote by SCΦ the set 1 +k2 the notation, of all u ∈ R for which u, Φ(u) is a point of strict convexity of the epigraph of Φ. Then there exist sequences (an ), (bn ) of numbers, bn > an for every n ∈ N, such that < k1 z(t), k2 z(t) > ⊂ R \ SCΦ = (an , bn ) (4)

n

for µ–a.e. t in T (Φ is aﬃne on each interval [an , bn ]). Therefore µ {t ∈ T :< k1 z(t), k2 z(t) >⊂ [an , bn ]} > 0 for some, ﬁxed from now on, n ∈ N. Denote C = {t ∈ T :< k1 z(t), k2 z(t) >⊂ [an , bn ]}. If C reduces to an atom then, by assumptions and by (4), there exists p = n such that µ(D) > 0, where D = {t ∈ T :< k1 z(t), k2 z(t) >⊂ [ap , bp ]}. Let us deﬁne the sets A1 , A2 and the numbers α1 , β1 , α2 , β2 in the following manner: – if C reduces to an atom: put A1 = C, α1 = an , β1 = bn , α2 = ap , β2 = bp and take A2 ⊂ D with 0 < µ(A2 ) < ∞; – in the other case: let A1 , A2 ⊂ C be disjoint sets such that 0 < µ(A1 ), µ(A2 ) < ∞ and put α1 = α2 = an , β1 = β2 = bn . Since Φ is aﬃne on the intervals [αi , βi ], Φ(u) = mi u + pi for every u ∈ [αi , βi ] and some mi , pi ∈ R (i = 1, 2). Our ﬁrst claim is that both m1 and m2 are diﬀerent from zero. Suppose m1 is −k1 equal to zero. Take λ = k22k and put 2 x = z + λzχA1 ,

y = z − λzχA1 .

Obviously, x = y and (x + y)/2 = z. Further, since k2 > k1 ,

2k1 k2 k2 − k1 k2 − k1 · max 1 − , 1+ k0 · max{1 − λ, 1 + λ} ≤ k1 + k2 2k2 2k1 = max{k1 , k2 } = k2 . Moreover, IΦ (k2 x(t)χA1 ) = 0, so IΦ (k0 x) = IΦ (k0 zχT \A1 ) + IΦ (k0 (1 + λ)zχA1 ) ≤ IΦ (k0 zχT \A1 ) + IΦ (k2 z(t)χA1 ) = IΦ (k0 zχT \A1 ) ≤ IΦ (k0 z).

On extreme points of Orlicz spaces with Orlicz norm

141

Thus x0Φ ≤ 1 and, analogously, y0Φ ≤ 1 – a contradiction. Therefore m1 = 0 and m2 = 0. Note that z(t)mi > 0 for every t ∈ Ai (i = 1, 2). −k1 Let λ1 , λ2 ∈ 0, k22k be numbers such that 2 λ1 m1

z(t) dµ = λ2 m2

A1

z(t) dµ. A2

Observe that k1 + k2 k2 − k1 ≤ k0 (1 − λi ) ≤ k0 (1 + λi ) = k0 1 − 2k2 2k2 k1 + k2 k2 − k1 k2 − k1 ≤ k0 1 + = k2 ≤ k0 1 + = k0 2k2 2k1 2k1

k1 = k0

for i = 1, 2. Now, deﬁne x = z + λ1 zχA1 − λ2 zχA2 ,

y = z − λ1 zχA1 + λ2 zχA2 .

Plainly, x = y and (x + y)/2 = z. Moreover, IΦ (k0 x) = IΦ (k0 zχT \(A1 ∪A2 )) z(t) dµ + p1 µ(A1 ) + m2 k0 (1 − λ2 ) + m1 k0 (1 + λ1 ) A1 × z(t) dµ + p2 µ(A2 ) A2 = IΦ (k0 zχT \(A1 ∪A2 )) + m1 k0 z(t) + p1 dµ A1 + m2 k0 z(t) + p2 dµ = IΦ (k0 z). A2

Thus x0Φ ≤ 1 and, analogously, y0Φ ≤ 1. This contradiction proves that the strong inequality k2 > k1 is false, i.e., K(z) is a one–point set. Theorem 2 Let Φ be an Orlicz function and let µ be an atomless measure. If z is an extreme point of B(LΦ , · 0Φ ) then (i) the of one element, set K(z) consists (ii) kz(t), Φ(kz(t)) are points of strict convexity of the epigraph of Φ for k ∈ K(z) and µ–a.e. t in T .

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Proof. Condition (i) follows immediately from Theorem 1. Suppose that (ii) is not satisﬁed and let k ∈ K(z). Then, there exist numbers a, b ∈ R and ε > 0 with a < b and ε < (b − a)/2k such that Φ is aﬃne on the interval (a, b), i.e., Φ(u) = mu + p for some m, p ∈ R and every u ∈ (a, b), and, moreover, kz(t) ∈ (a + kε, b − kε) on a set A of positive measure. Let B, C be two disjoint subsets of A with 0 < µ(B) = µ(C) < ∞. Deﬁne x = (z − ε)χB + (z + ε)χC + zχT \(B∪C),

y = (z + ε)χB + (z − ε)χC + zχT \(B∪C).

Then, obviously, x = y and (x + y)/2 = z. Moreover,

mk(z(t) − ε) + p dµ

IΦ (kx) =

B

+

C

= B∪C

mk(z(t) + ε) + p dµ + IΦ (zχT \(B∪C))

mkz(t) + p dµ + IΦ (zχT \(B∪C)) = IΦ (kz),

so x0Φ ≤ z0Φ = 1. Similarly, y0Φ ≤ 1. Thus z is not extreme – a contradiction. If the space L∞ is included in LΦ it is interesting to establish when the extreme points of B(L∞ ) are extreme in LΦ as well. Theorem 3 Let µ be an atomless measure with µ(T ) > 1 and let us assume that a point z ∈ LΦ satisﬁes the following conditions: (i) z ∈ Ext B(L∞ ) ∩ B(LΦ ), (ii) K(z) is a one element set, (iii) (k, Φ(k)) is a point of strict convexity of the epigraph of Φ, where k ∈ K(z), (iv) there exists 0 < ε < 2 such that Φ(u) > u − 1 for every u > 2 − ε. Then z is an extreme point of B(LΦ ). Proof. Let z be an extreme point of B(L∞ ). It is well–known that the absolute value of z(t) must be equal to 1 for µ–a.e. t in T . Suppose that z is not an extreme point of B(LΦ ), i.e., z = (x + y)/2 for some x, y ∈ B(LΦ ) with x = y. We shall consider three cases.

On extreme points of Orlicz spaces with Orlicz norm

143

10 . K(x) = ∅ and K(y) = ∅. Let a ∈ K(x) and b ∈ K(y). Then 1 1 1 1 = z0Φ = x0Φ + y0Φ = 1 + IΦ (ax) + 1 + IΦ (by) 2a 2b 2 b a a+b 1+ IΦ (ax) + IΦ (by) = 2ab a+b a+b 2ab x + y a+b ≥ 1 + IΦ 2ab a+b 2 2ab a+b = 1 + IΦ z ≥ z0Φ = 1, 2ab a+b so all the inequalities in the above formulae are, in fact, equalities. Therefore a+b ∈ K(z) = {k}, 2ab Φ is aﬃne on the intervals < ax(t), by(t) > and, moreover, kz(t) ∈< ax(t), b(y(t) > for µ − a.e. t in T . Since x = y and (x + y)/20Φ = 1, ax(t) = by(t) on a set of positive measure. Thus the epigraph of Φ is not strictly convex at k|z(t)| = k and we arrive at a contradiction. 20 . K(x) = ∅ and K(y) = ∅. Then, by the Amemiya formula (3), 1 IΦ (nx) n→∞ n

1 = x0Φ = lim and, similarly, limn→∞ n1 IΦ (ny) = 1. Thus 1=

z0Φ

1 1 lim ≤ lim IΦ (nz) ≤ n→∞ n 2 n→∞

On the other hand, by (iv), 1 1 IΦ (nz) ≥ lim (n − 1)µ(T ) > 1 n→∞ n n→∞ n lim

1 1 IΦ (nx) + IΦ (ny) n n

= 1.

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Hudzik and Wisla

– a contradiction. 30 . K(x) = ∅ and K(y) = ∅. Then limn→∞ n1 IΦ (nx) = 1 and 1b 1 + IΦ (by) = 1 for some 1 ≤ b < ∞. For every n ∈ N suﬃciently large we have 2nb 2nb n+b n+b 1+ − 1 µ(T ) ≤ 1+Φ µ(T ) 2< nb n+b nb n+b 2nb x + y n+b 1 + IΦ = nb n+b 2 n+b b n ≤ 1+ IΦ (nx) + IΦ (by) nb n+b n+b 1 1 −−− −→ 2 1 + IΦ (nx) + 1 + IΦ (by) n→∞ = n b and this contradiction ends the proof. Now, we can apply the obtained results to the space L1 + L∞ . Theorem 4 (R. Grz¸a´slewicz and H. Schaefer [3], H. Schaefer [13]) Let µ be an atomless measure. A point z of the unit sphere of the space L1 +L∞ is extreme if and only if (i) |z(t)| ≡ 1 for µ–a.e. t in T , (ii) µ(T ) > 1. In other words: the set of extreme points of the unit ball of L1 + L∞ is either empty (if µ(T ) ≤ 1) or it coincides with the set Ext B(L∞ ) (provided µ(T ) > 1). Proof. Suﬃciency. Let z ∗ be the rearrangement function of |z|. Then z ∗ (t) ≡ 1, so

z

L1 +L∞

=

1

z ∗ (t) dµ = 1,

0

i.e, z belongs to the unit sphere of L + L∞ . Now, let Φ be the function deﬁned by (1). Then the set K(z) consists exactly of one element and, moreover, K(z) = {1} – this is an easy consequence of the assumption µ(T ) > 1 and the following equality  1   ∈ [1, ∞) if 0 < k ≤ 1,  1 k 1 + IΦ (kz) = 1 1  k  µ(T ) if 1 < k < ∞.  + 1− k k 1

Obviously (1, 0) is a point of strictly convexity of Φ. Finally, it is evident that condition (iv) of Theorem 3 is satisﬁed as well. Thus z is an extreme point of B(L1 + L∞ ).

On extreme points of Orlicz spaces with Orlicz norm

145

Necessity. Let us note that if µ(T ) ≤ 1 then the space L1 + L∞ is isometric to L . Indeed, it is obvious that for any ﬁnite measure µ, L∞ ⊂ L1 , so L1 + L∞ = L1 . Thus any x ∈ L1 + L∞ admits a decomposition x = x + 0, hence 1

xL1 +L∞ ≤ x1 . On the other hand, for any y ∈ L∞ we have y1 ≤ y∞ µ(T ) ≤ y∞ . Thus, considering any of the decompositions x = y + z of x, where y ∈ L1 and z ∈ L∞ , we have y1 + z∞ ≥ y1 + z1 ≥ y + z1 = x1 . Hence, passing to inﬁmum, we obtain xL1 +L∞ ≥ x1 , i.e., L1 + L∞ is isometric to L1 . Assume that z ∈ Ext B(L1 +L∞ ) and let the function Φ be deﬁned by (1). Then, by Theorem 2, K(z) = {k} for some 0 < k < ∞ and, moreover, k|z(t)| = 1 for µ–a.e. t in T . Similarly as in the proof of Theorem 1, one can show that zL1 +L∞ = 1/k. Thus k = 1 and (i) is proved. Since, by the assumption, the set of extreme points is not empty, the space L1 + L∞ can not be isometric to L1 , so the measure of T must be grater than one. Remark. Theorem 4 was given in [3] and [13] for the inﬁnite Lebesgue measure space only.

References 1. C. Bennett, R. Sharpley, Interpolation of Operators, Academic Press Inc. 1988. 2. R. Grz¸as´lewicz, H. Hudzik, W. Kurc, Extreme and exposed points in Orlicz spaces, Canad. J. Math. 44 (1992), 505–515. 3. R. Grz¸as´lewicz, H. H. Schaefer, On the isometries of L1 ∩ L∞ [0, ∞) and L1 + L∞ [0, ∞), Indagationes Math. NS 3 (2) (1992), 173–178. 4. A. Kami´nska, Extreme points in Orlicz-Lorentz spaces, Arch. Math. 52 (1990), 1–8.

146

Hudzik and Wisla

5. M. A. Krasnosel’skii, Y. B. Rutickii, Convex Functions and Orlicz Spaces, P. Nordhoff Ltd, Groningen 1961. 6. S. G. Krein, Yu. I. Petunin, E. M. Semenov, Interpolation of Linear Operators, Moskow 1978 (in Russian). 7. W. Kurc, Extreme points of the unit ball in Orlicz spaces of vector–valued functions with the Amemiya norm, Mathematica Japonica 38 (2) (1993), 277–288. 8. Bing Yuan Lao, Xiping Zhu, Extreme points of Orlicz spaces, J. Zhongshan University 2 (1983), 27–36 (in Chinese). 9. J. Musielak, Orlicz spaces and Modular Spaces, Lecture Notes in Math. 1034, Springer Verlag 1983. 10. H. Nakano, Generalized modular spaces, Studia Math. 31 (1968), 439–449. 11. H. Nakano, Modulared Semi–Ordered Linear Spaces, Maruzem, Tokyo 1950. ¨ 12. W. Orlicz, Uber eine gewisse Klasse von R¨aumen vom Typus B, Bull. Intern. Acad. Pol. S´erie A, Krak´ow 0 (1932), 207–220. 13. M. M. Rao, Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker Inc., New York 1991. 14. H. Schaefer, On convex hulls, Arch. Math. 58 (1992), 160–163. 15. Congxin Wu, Shutao Chen, Extreme points and rotundity of Musielak-Orlicz sequence spaces, J. Math. Res. Exposition 2 (1988), 195–200. 16. Zhuogiang Wang, Extreme points of Orlicz sequence spaces, J. Daging Oil College 1 (1983), 112–121 (in Chinese).

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 147–154 c 1994 Universitat de Barcelona

Higher order Hardy inequalities

Alois Kufner Math. Institute, Czech. Acad. Sci., Zitna 25, 11567 Praha 1, Czech Republic

Abstract This note deals with the inequality

1/q

b

|u(x)|q w0 (x)dx

≤C

a

1/p

b

|u(k) (x)|p wk (x)dx

,

(1)

a

more precisely, with conditions on the parameters p > 1, q > 0 and on the weight functions w0 , wk (measurable and positive almost everywhere) which ensure that (1) holds for all functions u from a certain class K with a constant C > 0 independent of u.

Here −∞ ≤ a < b ≤ ∞ and k ∈ N and we will consider classes K of functions u = u(x) deﬁned on (a, b) whose derivatives of order k − 1 are absolutely continuous and which satisfy the “boundary conditions” u(i) (a) = 0 u

(j)

(b) = 0

for i ∈ M0 , for j ∈ M1

(2)

where M0 , M1 are subsets of the set M = {0, 1, . . . , k − 1}; we will suppose that the number of conditions in (2) is exactly k. This class will be denoted by AC (k−1) (a, b; M0 , M1 ).

(3)

The conditions (2) are reasonable since they allow to exclude functions like polynomials of order ≤ k − 1 for which the right hand side in (1) is zero while the left hand side is positive. 147

148

Kufner

Let us start with some remarks. (i) We will concentrate on the case k>1

(4)

since for k = 1 the problem is completely solved: see, e.g., the book Opic, Kufner [4], Chapter 1. Some particular results concerning the case k = 2, k = 3 and - for a special choice of the sets M0 , M1 - also higher values of k can be found in the paper Kufner, Wannebo [3]. (ii) For (a, b) = (0, ∞), k ∈ N arbitrary and M0 = M, M1 = ∅ or M0 = {0, 1, . . . , m − 1}, M1 = M \ M0 , 0 < m < k, the problem is also solved: see Stepanov [5] or Kufner, Heinig [2], respectively. These results cover all reasonable cases when the interval (a, b) is inﬁnite. Therefore, we will concentrate on the case of a ﬁnite interval (a, b). Without loss of generality it can be assumed that (a, b) = (0, 1). (5) In the sequel, we will make substantial use of some functions and constants. For r = 1, we will denote r =

1 1 + =1. r r

r , i.e. r−1

Further, let us denote for i = 1, 2 W0i (t) = w0 (t)tαi q (1 − t)βi q ,

(6)

Wki (t) = wk1−p (t)tγi p (1 − t)δi p ,

where w0 (t), wk (t) are the weight functions appearing in (1) and αi , βi , γi , δi (i = 1, 2) are certain nonnegative integers, and let us introduce functions

1/q

1

B1 (x) =

W01 (t)dt

x

Wk1 (t)dt 0

1/q

x

B2 (x) =

1/p

x

(7)

1/p

1

W02 (t)dt

,

Wk2 (t)dt

0

(8)

x

and constants

1

r/q

1

A1 =

W01 (t)dt 0

x 1

A2 =

Wk1 (t)dt

0

Wk1 (x)dx

,

(9)

,

(10)

0

r/q

x

Wk2 (t)dt x

1/r

r/q

1

W02 (t)dt 0

1/r

r/q

x

Wk2 (x)dx

Higher order Hardy inequalities

149

where 1 1 1 = − . r q p

(11)

We suppose that all expressions appearing in formulas (7) - (10) are well deﬁned. Of course, it also depends on the values αi , βi , γi , δi which have not yet been determined. Later, we will show how these integers can be determined by the sets M0 and M1 which appear in the conditions (2). If we suppose for a moment that these integers are known, then the main result can be formulated as follows: Proposition 1 Let M0 , M1 be two nonempty subsets of the set {0, 1, . . . , k − 1} containing together k elements. Let αi , βi , γi , δi , i = 1, 2, be nonnegative integers corresponding to the pair M0 , M1 . Let w0 (t), wk (t) be weight functions deﬁned on (0, 1) and let 1 < p < ∞, 0 < q < ∞, q = 1. Then the (HARDY) inequality

1/q

1

|u(t)| w0 (t)dt q

≤C

0

1/p

1

|u

(k)

p

(t)| wk (t)dt

(12)

0

holds for every function u ∈ AC (k−1) (0, 1) satisfying the conditions u(i) (0) = 0

for i ∈ M0 ,

u(j) (1) = 0

for j ∈ M1

(13)

if and only if sup Bi (x) = Bi < ∞,

i = 1, 2

(14)

0<x<1

in the case p ≤ q , and Ai < ∞,

i = 1, 2

in the case p > q , where Bi (x) and Ai are given by formulas (7) - (11).

(15)

150

Kufner

Determination of the integers α1 , . . . , δ2 . Let us consider a very simple boundary value problem u(k) = f

in

(0, 1),

u (0) = 0

for i ∈ M0 ,

u(j) (1) = 0

for j ∈ M1

(i)

(16)

where f does not change the sign in (0, 1) and M0 , M1 are the subsets of M = {0, 1, ..., k − 1} mentioned in Proposition 1. Suppose that the solution u can be expressed uniquely in the form 1 x K1 (x, t)f (t)dt + K2 (x, t)f (t)dt. (17) u(x) = 0

x

The kernels K1 (x, t), K2 (x, t) are then polynomials. We will write Ki (x, t) ≈ xαi (1 − x)βi tγi (1 − t)δi

(18)

if there exist positive constants c1 , c2 such that the estimates c1 ≤

xαi (1

Ki (x, t) ≤ c2 − x)βi tγi (1 − t)δi

hold for 0 < t < x < 1 (i = 1) and 0 < x < t < 1 (i = 2), respectively. Now, we will show under what conditions (18) is fulﬁlled. For this purpose, let us split the set M = {0, 1, . . . , k − 1} into s successive groups G1 , G2 , . . . , Gs (s ≥ 2) according to the following scheme: G1 = {0, 1, . . . , m − 1}

(k1 elements,

k1 = m),

G2 = {m, m + 1, . . . , n − 1}

(k2 elements,

k2 = n − m),

G3 = {n, n + 1, . . . , r − 1}

(k3 elements,

k3 = r − n),

(19)

. . Gs = {h, h + 1, . . . , k − 1}

(ks elements,

ks = k − h),

(i.e. Gi has ki elements, ki > 0, i = 1, 2, . . . , s, and k1 + k2 + . . . + ks = k), and suppose that the sets M0 and M1 appearing in the boundary conditions in (16) are deﬁned as follows: M0 = G1 ∪ G2 ∪ . . . ∪ Gs−1 , M0 = G1 ∪ G2 ∪ . . . ∪ Gs , Then we have

M1 = G2 ∪ G4 ∪ . . . ∪ Gs for s even.

(20)

M1 = G2 ∪ G4 ∪ . . . ∪ Gs−1 for s odd.

(21)

Higher order Hardy inequalities

151

Proposition 2 If the set M = {0, 1, . . . , k − 1} is splitted into s groups according to (19), the sets M0 and M1 are deﬁned by (20) and (21) and the solution u to the boundary value problem (16) can be expressed in the form (17), then K1 (x, t) ≈ xk1 −1 tk2 , K2 (x, t) ≈ xk1 tk2 −1 for s = 2,

(22)

Ki (x, t) ≈ xk1 (1 − t)ks , i = 1, 2, for s odd, Ki (x, t) ≈ xk1 tks , i = 1, 2, for s > 2 even. Remarks. (i) The proof of Proposition 2 is elementary but cumbersome. It is based on the fact that the solution u to the boundary value problem (16) can be expressed in the form

x

k1 −1

1

(x − t1 )

u(x) = co 0

k2 −1

(t2 − t1 )

t2

(t2 − t3 )k3 −1 . . .

0

t1

. . . F (ts−1 )dts−1 . . . dt2 dt1 where c0 = [(k1 − 1)!(k2 − 1)! . . . (ks − 1)!]−1 and F (ts−1 ) is either

1

(ts − ts−1 )ks −1 f (ts )dts

for s even

(ts−1 − ts )ks −1 f (ts )dts

for s odd .

ts−1

or

ts−1

0

For s = 2, it can be found in the paper [3], for s > 2 in the preprint [1]. (ii) In (20), (21) we have always assumed that the ﬁrst group G1 belongs to M0 so that we start with the boundary condition u(0) = 0, 0 ∈ M0 . If we suppose that 0 ∈ M1 , i.e. that the boundary condition u(1) = 0 appears in (16), and have M0 = G2 ∪ G4 ∪ . . . ,

M1 = G1 ∪ G3 ∪ . . . ,

then we simply exchange the role of the sets M0 and M1 , i.e. of the endpoints x = 0 and x = 1, and a corresponding assertion holds again, if we replace in (22) x by 1 − x and t by 1 − t. (iii) In the foregoing cases, we have assumed that M0 ∪ M1 = M, i.e.

M0 ∩ M1 = ∅.

152

Kufner

If the sets M0 and M1 again have together k elements, but have a nonempty intersection, then the method described above cannot be used. Nonetheless, many examples allow to expect that - provided there is a unique representation of the solution u of (16) in the form (17) - the kernels Ki (x, t) again behave according to (18). Therefore, let us formulate the following conjecture: Suppose that M0 ∩ M1 = ∅. 1 by (a) Deﬁne M 1 = M \ M0 . M 1 satisﬁes the conditions of either Proposition 2 (if G1 ⊂ M0 ) Then the pair M0 , M 1 ), and consequently, the kernels or of part (ii) of this Remark (if G1 ⊂ M (a) Ki (x, t) corresponding to the pair M0 , M1 satisfy (18): There are positive integers αi (a), βi (a), γi (a), δi (a) such that (a)

Ki (x, t) ≈ xαi (a) (1 − x)βi (a) tγi (a) (1 − t)δi (a) , 0 by (b) Deﬁne M

i = 1, 2.

0 = M \ M1 . M

0 , M1 again satisﬁes the conditions which allow to state that for the Then the pair M (b) corresponding kernels Ki (x, t) we have (b)

Ki (x, t) ≈ xαi (b) (1 − x)βi (b) tγi (b) (1 − t)δi (b) ,

i = 1, 2.

(c) For the kernels Ki (x, t) corresponding to the initial pair M0 , M1 we have (18) with αi = αi (a), βi = βi (b), γi = γi (a), δi = δi (b). Idea of the proof of Proposition 1 We consider the Hardy inequality (12) on the class AC (k−1) (0, 1; M0 , M1 ), i.e., for functions u satisfying (13). Therefore, let us consider the boundary value problem (16) and denote by T the operator deﬁned by formula (17): 1 x K1 (x, t)f (t)dt + K2 (x, t)f (t)dt. (T f )(x) = 0

x

Since the function u = T f satisﬁes conditions (13) and u(k) = f , we can instead of the inequality (12) investigate the inequality

1/q

1

|(T f )(x)| w0 (x)dx q

0

for functions f ≥ 0.

≤C

1/p

1 p

f (x)wk (x)dx 0

(23)

Higher order Hardy inequalities

153

Now, it can be shown that the validity of (23) for f ≥ 0 is equivalent to the validity of the inequalities

1

1/q |(Ji f )(x)| w0 (x)dx ≤ Ci

1

q

0

1/p f (x)wk (x)dx , p

i = 1, 2,

(24)

0

where

x

(J1 f )(x) =

K1 (x, t)f (t)dt,

1

(J2 f )(x) =

K2 (x, t)f (t)dt.

0

x

But due to (18), the inequalities (24) are equivalent to the inequalities

1

x

xα1 (1 − x)β1

0

1/q q tγ1 (1 − t)δ1 f (t)dt w0 (x)dx

0

1 ≤ C

1/p

1 p

f (x)wk (x)dx 0

and

1

x (1 − x) α2

0

1/q

q

1

t (1 − t) f (t)dt w0 (x)dx

β2

γ2

x

2 ≤ C

δ2

1/p

1 p

f (x)wk (x)dx 0

respectively, and these last two inequalities can be easily rewritten into the form

1/q

1

|(Hg)(x)| w(x)dx q

≤C

0

1/p

1 p

g (x)v(x)dx

,

(25)

0

where H is the Hardy operator, x g(t)dt (Hg)(x) = 0

or

1

(Hg)(x) =

g(t)dt . x

Finally, necessary and suﬃcient conditions for the validity of (25) (see, e.g., [4]) lead to the conditions (14) (if p ≤ q) or (15) (if p > q). Consequently, the integers α1 , . . . , δ2 which appear in (6) can be determined from the behavior of the kernels K1 (x, t), K2 (x, t) described by (18).

154

Kufner References

1. A. Kufner, Higher order Hardy inequalities, Preprint No. 672, Universit¨at Heidelberg, Sonderforschungsbereich 123, 1992. 2. A. Kufner, H.P. Heinig, Hardy’s inequality for higher order derivatives, Trudy Mat. Inst. Steklov 192 (1990), 105–113, (Russian). 3. A. Kufner, A. Wannebo, Some remarks on the Hardy inequality for higher order derivatives. In: General Inequalities VI. International Series of Numerical Mathematics 103. Birkh¨auser Verlag, Basel 1992, 33–48. 4. B. Opic, A. Kufner, Hardy-type inequalities, Pitman Research Notes in Mathematics Series 219. Longman Scientific and Technical, Harlow 1990. 5. V.D. Stepanov, Two-weighted estimates for Riemann-Liouville integrals. Preprint No. 39, Math. Inst. Czech. Acad. Sci., Prague 1988.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 155–165 c 1994 Universitat de Barcelona

A dual property to uniform monotonicity in Banach lattices

W. Kurc Institute of Mathematics, A. Mickiewicz University, Pozna´n, Poland

Abstract For Banach lattices X with strictly or uniformly monotone lattice norm dual, properties (o)-smoothness and (o)-uniform smoothness are introduced. Lindenstrauss type duality formulas are proved and duality theorems are derived. It is observed that (o)-uniformly smooth Banach lattices X are order dense in X ∗∗ . An application to an optimization problem is given.

1. Introduction Let X be a Banach lattice with the dual X ∗ and let · stands for the corresponding dual (monotone) norms. X is said to be strictly monotone (STM) (we will often write X ∈ STM etc.) if x − y < x whenever 0 < y ≤ x. The strongest property in this direction is the uniform monotonicity (UM) of X which means that δX () > 0 for all ∈ (0, 1] where δX () = inf 1− x − y : 0 ≤ y ≤ x, x = 1, y ≥ ( ∈ [0, 1]). In [2] (Chap. XV) this is called a “UMB” space. It is worth noticing the following fact (cf. [5] and [4], p. 124). Lemma For ∈ [0, 1) the following formula holds true: δX () =

σX () 1 + σX ()

155

156

Kurc

where σX () is a modulus of the uniform monotonicity deﬁned by

σX () = inf

x + y −1 : x, y ≥ 0, x = 1, y ≥

( ∈ [0, 1]).

Proof. It suﬃces to apply the following identity, with u = 1, u+z u + z −1 z =1− − u+z u + z u + z and pass to the inﬁmum over the set U = {(u, z) : u, z ≥ 0, u = 1, z ≥ }. Let us point out that the indicated correspondence of the diﬀerent deﬁnitions of UM spaces is not longer true for local properties (eg. LUM, cf. [4]). The UM and STM can be viewed as restrictions of the uniform rotundity (UR) and the strict convexity (R) to the positive cone X+ , respectively ([4], Proposition 1.2 and 1.3). Thus UR ⇒ UM and R ⇒ STM. We will call X order smooth, in abbreviation (o)-Sm, if for each x ∈ S(X+ ) (the positive part of the unit sphere in X) and each order interval [u∗ , v ∗ ] ⊂ ∂+ x ∗ there holds u∗ = v ∗ , where ∂+ x ={x∗ ∈ S(X+ ) : < x, x∗ >= x }. The strongest notion of smoothness of X is the order uniform smoothness, in abbreviation (o)-USm. We say X to be (o)-USm if ρX (τ )/τ → 0, whenever τ 0, where the modulus of smoothness ρX (τ ) is deﬁned as follows: ρX (τ ) = sup

x ∨ τ y −1 : 0 ≤ x, y, x = 1, y = 1

(τ ∈ [0, 1]) .

Lemma For all , τ ∈ [0, 1] the following inequalities hold true (i)

0 ≤ αX () ≤ δX () ≤ ,

(ii)

0 ≤ ρX (τ )/τ ≤ βX (τ )/τ ≤ 1.

By αX () and βX (τ ) we mean the modulus of the uniform rotundity (cf. [4], Proposition 1.2) and the modulus of smoothness (0 ≤ , τ ): αX () = inf{1− x ± y : x = 1, y ≥ } βX (τ ) = sup

x + τy + x − τy − 1 : x = 1, y = 1 . 2

A dual property to uniform monotonicity in Banach lattices

157

Corollary (a) If X is UR (resp. USm) then X is UM (resp. (o)-USm). (b) If X is an R (i.e. rotund) space (resp. Sm space , i.e. smooth) then X is a STM space (resp. (o)-Sm space). Recall (cf. [4]) that any UM Banach lattice X is a KB space (i.e. the norm is order continuous and X is monotonically complete). Example: It follows easily from the deﬁnitions that δL1 () ≡ , ρL∞ (τ ) ≡ 0. However δL∞ () ≡ 0 but ρL1 (τ ) ≡ τ . Roughly speaking the space L1 is the best (worst) UM (resp. (o)-USm ) space and the space L∞ is the best (worst) (o)-USm (resp. UM ) space since the respective modules attain their bounds.

2. (o)-Smoothness and strict monotonicity The following theorem is true also for normed lattices. Theorem 1 Let X be a Banach lattice with the dual X ∗ . Then (a) if X ∗ is a STM space then X is (o)-Sm space, (b) if X ∗ is (o)-Sm space then X is a STM space, If moreover X is reﬂexive then the converse implications are also true. Proof. (a) If X is not (o)-Sm then there exists a proper (order) interval [u∗ , v ∗ ] ⊂ ∂+ x . Hence in particular 0 < u∗ < v ∗ and [u∗ , v ∗ ] ⊂ S(X+ ) i.e. X ∗ is not STM space which proves (a). (b). Let X ∗ be (o)-Sm space but X is not STM, i.e. x = x − y for some y and x ∈ S(X+ ) such that 0 < y < x. There exists a positive functional x∗ ∈ X ∗ satisfying < x − y, x∗ >= x − y . Hence we conclude that also < x, x∗ >= x . Let u = x − y. Denoting the canonical injections of x and u into X ∗∗ by x ˆ and u ˆ, respectively, we obtain ﬁnally that the proper interval [ˆ u,ˆ x] ⊂ ∂+ x∗ , a contradiction with the (o)-Sm of X ∗ . The converse implications for X reﬂexive are now clear. In the following we will try to explain the meaning of the (o)-Sm by means of the behavior of the function t → η(t) (t > 0), where η(t) =

x ∨ ty − x t

(x, y ≥ 0, t > 0).

Lemma The function t → x ∨ ty is convex and the function η(t) is nonnegative and nondecreasing for t > 0.

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Proof. Applying the formula x ∨ ty = 12 (x + ty+ | x − ty |) the convexity of the function x ∨ ty easily follows. Now the standard reasoning yields the second assertion. As a corollary it follows that η = limt0 η(t) = inf t>0 η(t) exists and the limit η is ﬁnite and nonnegative. Now we will prove the basic duality formula relating the notion of the (o)smoothness with the behavior of divided diﬀerence of special kind. Theorem 2 Let x, y be arbitrary in S(X+ ). The following duality formula holds true: inf

t>0

x ∨ ty − x = sup (< y, x∗ − y ∗ >) t ∗ ∗ ∗ ∗ x ,y ∈∂x,0≤y ≤x

(1)

where the “sup” on the right side is attained. Proof. First we will prove the inequality “≤”. Let x, y ∈ S(X+ ) be arbitrary. In virtue of Lemma above the function t → η(t) is nondecreasing and nonnegative. Next, for the function η(t) we have (cf. [1] pp. 55 and 175): 1 η(t) = sup < y, x∗ − y ∗ > + (< x, y ∗ > −1) , sup ∗ ) (x∗ ≥y ∗ ≥0) t x∗ ∈S(X+ and η = limt0 η(t). Hence there exist nets (tα ) , (x∗α ), (yα∗ ) such that tα 0, ∗ x∗α ∈ S(X+ ), 0 ≤ yα∗ ≤ x∗α and 1 (< x, yα∗ > −1) −→ η. tα Since the ﬁrst term is bounded and tα 0 we conclude that < y, yα∗ >→ 1 and ∗ ∗ therefore < x, yα∗ >→ 1. Since S(X+ ) is weakly∗ compact there exist x∗ ∈ S(X+ ) ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ and y with 0 ≤ y ≤ x such that xβ → x and yβ → y weakly for a subnet (β). Hence < x, x∗ >=< x, y ∗ >= x and consequently x∗ , y ∗ ∈ ∂ x , 0 ≤ y ∗ ≤ x∗ . Passing now to the limit above we see that with these x∗ and y ∗ there holds x ∨ ty − x inf =< y, x∗ − y ∗ >, t>0 t and the inequality “≤” follows. To prove the inequality “≥” let us conﬁne with the supremum in the formula for η(t) to x∗ , y ∗ ∈ ∂ x such that x∗ ≥ y ∗ ≥ 0. Then < x, y ∗ >= 1 and the desired inequality follows which concludes the proof. < y, x∗α − yα∗ > +

We will relate the smoothness with the (o)-smoothness. Let f (x) = x and f+ (x, y) be the directional derivative of f at x in the direction y. It is a well known fact in convex analysis that f+ (x, y) = max {< y, x∗ >: x∗ ∈ ∂ x }. For the left directional derivative we have −f+ (x, −y) = min {< y, x∗ >: x∗ ∈ ∂ x }.

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Corollary If x in S(X+ ) is a smooth point then it is an (o)-smooth point. More precisely if x ∈ S(X+ ) and y ≥ 0 then f+ (x, y) + f+ (x, −y) ≥

max

x∗ ,y ∗ ∈∂x,0≤y ∗ ≤x∗

(< y, x∗ − y ∗ >) ≥ 0.

Moreover x ∈ S(X+ ) is an (o)-smooth point if and only if X+ ⊥ (∂+ x −∂+ x ), where y ⊥ x∗ means that < y, x∗ >= 0. 2 Example: Any point x ∈ S(l∞ ), x ≥ 0, is an (o)-smooth point (in fact this space is (o)-USm). Indeed, it suﬃces to consider the extreme point x = (1, 1) only. In this case ∂+ x can be identiﬁed with the positive part of the unit sphere in l12 which does not contain any order interval (the coordinatewise ordering is considered).

On the other hand the space l12 is not (o)-smooth. Indeed, a point x = (0, 1) has ∂+ x containing an order interval [y ∗ , x∗ ] (x∗ = (1, 1), y ∗ = (0, 1)) which is the largest possible.

3. Uniform properties and duality In this paragraph Lindenstrauss type duality formulas relating the modulus of uniform monotonicity δX () and the modulus of (o)-uniform smoothness ((o)-USm) ρX (τ ) are proved and the main duality theorem is derived. Let us ﬁrst observe that ρX (τ ) ≤ ρX ∗∗ (τ ) and δX () ≥ δX ∗∗ () (, τ ∈ [0, 1]). Theorem 3 Let x, y be arbitrary in S(X+ ). The following duality formulas hold true: (a) ρX (τ ) = ρX ∗∗ (τ ), (b) δX () = δX ∗∗ () and (c) ρX ∗ (τ ) = sup0≤≤1 τ − δX () , (d) δX () = sup0≤τ ≤1 (τ − ρX ∗ (τ )) where , τ ∈ [0, 1].

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∗ Proof. (c). Let x∗ , y ∗ ∈ S(X+ ) be arbitrary but ﬁxed, τ ∈ [0, 1] and x ∈ S(X+ ). Then

< x, x∗ ∨ τ y ∗ > −1 = sup (< x − u, x∗ > +τ < u, y ∗ >) − 1 x≥u≥0

≤ sup ( x − u +τ y ∗ u ) − 1 x≥u≥0

≤

( x − u −1 + τ )

sup

sup

(0≤≤1) (0≤u≤x,x=1,u≥)

= sup 0≤≤1

τ − δX () .

∗ Now, passing to the “sup” over x ∈ S(X+ ) and then over x∗ , y ∗ ∈ S(X+ ) we get

ρX ∗ (τ ) ≤ sup

0≤≤0

τ − δX () .

(2)

Now, let ∈ [0, 1], and ﬁx x ∈ S(X+ ) and u such that 0 ≤ u ≤ x. Then there ∗ ) such that < x − u, x∗ >= x − u and < u, y ∗ >= u . Hence exist x∗ , y ∗ ∈ S(X+ for τ ∈ [0, 1] ρX ∗ (τ ) ≥ x∗ ∨ τ y ∗ −1 ≥ < x, x∗ ∨ τ y ∗ > −1 = sup < x − y, x∗ > +τ < y ∗ , y ∗ > − 1 0≤y≤x

≥ x − u +τ u −1 ≥ τ − (1− x − u ). Now, passing to the supremum over x and u indicated and then over ∈ [0, 1], we obtain (3) ρX ∗ (τ ) ≥ sup τ − δX () . 0≤≤1

Collecting (2) and (3) the property (c) follows. To prove (b) we will estimate δX () from below. First in virtue of (c) δX () ≥ sup

0≤τ ≤1

τ −

sup (x∗ ,y ∗ )∈S(τ )

( x∗ ∨ y ∗ −1)

(4)

for all ∈ [0, 1], where S(τ ) = {(x∗ , y ∗ ) : x∗ , y ∗ ≥ 0, x∗ = 1, y ∗ ≤ τ }. Let , η, τ ∈ (0, 1] be arbitrary. For each (x∗ , y ∗ ) ∈ S(τ ) there exists x ∈ S(X+ ) such that (5) x∗ ∨ y ∗ ≤ < x, x∗ ∨ y ∗ > +η.

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Denote A = {(x, y) : 0 ≤ y ≤ x, y ≤ } and B = {(x, y) : 0 ≤ y ≤ x, y ≤ }. Given x∗ , y ∗ ∈ S(τ ) we have < x, x∗ ∨ y ∗ > −1 = sup (< x − y, x∗ > + < y, y ∗ >) − 1 0≤y≤x

≤ max{sup ( x − y −1 + τ y ), A

sup ( x − y −1 + τ y )} B

≤ max {τ , −δX () + τ } ≤ τ − δX () + τ. Hence with x∗ , y ∗ and x as above, from (4) and (5) it follows that δX () ≥ sup {τ − ρX ∗ (τ )} 0≤τ ≤1

≥ τ − (< x, x∗ ∨ y ∗ > −1 + η) = δX () − τ − η. Since η and τ were arbitrary in (0, 1] we get the equality in (4) for each ∈ (0, 1] as desired. The case = 0 is obvious in virtue of ρ( (τ )X ∗ ) ≤ τ and δX (0) = 0. To prove (a) it suﬃces to prove that ρX (τ ) ≥ ρX ∗∗ (τ ). For this let x∗ , y ∗ ∈ X+ be such that x∗ = 1, 0 ≤ y ∗ ≤ x∗ , 0 = y ∗ and let η ∈ (0, 1]. Then there exist x, y ∈ S(X+ ) such that x∗ − y ∗ ≤ < x, x∗ − y ∗ > + η and y ∗ ≤ < y, y ∗ > + η. With these x, y, x∗ , y ∗ and η we have ρX (τ ) ≥ x ∨ τ y −1 ≥ < x ∨ τ y, x∗ > −1 = sup < x, x∗ − y ∗ > + τ < y, y ∗ > − 1 0≤y ∗ ≤x∗

≥ < x, x∗ − y ∗ > + τ < y, y ∗ > −1 ≥ x∗ − y ∗ − η + τ ( y ∗ −η) − 1. Taking the supremum over x∗ , y ∗ indicated we get ρX (τ ) ≥ sup

0≤≤1

sup (x∗ =1,0≤y ∗ ≤x∗ ,y ∗ =)

(τ + x∗ − y ∗ −1) − 2η

= sup (τ − δX ∗ ()) − 2η = ρX ∗∗ (τ ) − 2η. 0≤≤1

(6)

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Since η ∈ (0, 1] was arbitrary we get the desired inequality and hence (a) follows. Finally to prove (b) it suﬃces to apply (a) and (d) respectively: δX ∗∗ () = sup (τ − ρX ∗∗∗ (τ )) 0≤τ ≤1

= sup (τ − ρX ∗ (τ )) 0≤τ ≤1

= δX (). In the Proposition below we collect basic properties of the modules δX () and ρX ∗ (τ ). Proposition 4 The following properties hold true. (a) δX () ≡ 0 (resp. ) if and only if ρX ∗ (τ ) ≡ τ (resp. 0). (b) τ ≤ δX () + ρX ∗ (τ ) ≤ + τ. Moreover, given , τ ∈ [0, 1] the equality on the right is attained if and only if δX () = and ρX ∗ (τ ) = τ . (c) The functions δX (), ρX ∗ (τ ) are convex (nonnegative) and continuous on the interval [0, 1] with δX (0) = ρ∗X (0) = 0 and therefore nondecreasing. Proof. (a) In virtue of Theorem 3(d), δX () = 0 for all ∈ [0, 1] implies that τ ≤ ρX ∗ (τ ) ≤ τ for all ∈ [0, 1]. Hence ρX ∗ (τ ) ≡ 0. To prove the converse implication we put in Theorem 3(d) ρX ∗ (τ ) ≡ τ . Hence δX () ≡ 0. The remaining cases follow in the same way so we omit their proofs. (b) It was already stated that 0 ≤ δX () ≤ and 0 ≤ ρX ∗ (τ ) ≤ τ . Hence and from (d) in Theorem 3, (b) follows. (c) From (c) and (d) in Theorem 3 it follows that the functions δX (), ρX ∗ (τ ) are pointwise suprema of families of aﬃne functions on the interval (0, 1). Therefore they are lsc and convex on (0, 1) and hence continuous and nondecreasing. From the deﬁnitions it follows that δX (0) = ρX ∗ (0) = 0 and consequently they are continuous at zero from the right. Since δX (1) ≥ δX () = sup0≤τ ≤1 (τ − ρX ∗ (τ ) − τ (1 − )) ≥ δX (1) + (1 − ), we conclude that δX () is left continuous at 1. The same reasoning applies to ρX ∗ (τ ) so the proof is ﬁnished. As a consequence of Theorem 3 we get the following duality theorem. Theorem 5 Let X be a Banach lattice. Then (a) X is UM (resp. (o)-USm) if and only if X ∗∗ is UM (resp. (o)-USm). (b) X is UM if and only if X ∗ is (o)-USm. (c) X ∗ is UM if and only if X is (o)-USm.

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Proof. (a) This follows immediately from Theorem 3 ((a),(b)). (b) If X ∗ is not an (o)-USm space then inf τ >0 ρX ∗ (τ )/τ > α for some α > 0, because the function τ → ρX ∗ (τ )/τ is nondecreasing and continuous on (0, 1] (apply the property of the function η(t) from Sec. 2). Therefore ρX ∗ (τ ) δX () = sup τ − ≤ sup τ ( − α) = 0 τ 0≤τ ≤1 0≤τ ≤1 whenever 0 < ≤ α, i.e. X is not UM. To prove the converse implication let X be not UM, i.e. δX (0 ) = 0 for some 0 ∈ (0, 1). Then ρX ∗ (τ ) δX () = sup − ≥ 0 for τ ∈ (0, 1], τ τ 0≤≤1 i.e. X ∗ is not (o)-USm. Collecting these all (b) follows. Now let X ∗ be UM. From (b) X ∗∗ is then (o)-USm. Since X embeds as a closed sublattice (isometrically) into X ∗∗ we conclude that X is (o)-USm. To prove the converse let X be (o)-USm. Then in virtue of (a) X ∗∗ is (o)-USm and hence (using (b)) X ∗ is UM as desired. Thus (c) holds true and the proof is ﬁnished. As a corollary we get the following applications of the notion of UM and (o)USm spaces. Theorem 6 Let X be a Banach lattice. Then (a) If X is UM then X is a KB -space. (b) If X is (o)-USm then X ∗∗ is the band generated by X in X ∗∗ . Proof. (a) This is a known fact (cf. [2], Chap. XV, Theorem 21) so we omit the proof. (b) In virtue of Theorem 5 if X is (o)-USm then X ∗ is a UM-space. Now applying Theorem 2.4.14 from [8] we conclude that X ∗∗ is the band generated by X in X ∗∗ . Applying results from this section and characterizations of STM and UM Orlicz spaces for Luxemburg and Orlicz (in the Amemiya form, cf. [3], [7]) norm (see [4], [5], [3]), we derive in [7] characterizations of (o)-Sm and (o)-USm Orlicz spaces as well we obtain estimations for the modules δX (), ρX ∗ (τ ).

164

Kurc 4. An application to optimization

Let x be arbitrary but ﬁxed on S(X+ ). Deﬁne a functional fx (y) = x ∨ y ,

where y ∈ X and y ≥ 0.

Clearly fx (y) ≥ fx (0) and fx (y) ≥ fx (x). Therefore the (order) interval [0, x] ⊂ Pfx = {y ∈ X : y ≥ 0, fx (0) = x ∨ y }. Definition. Let x be ﬁxed as above. We say that Pfx is a set of solutions of the optimization problem: fx (y) −→ min y ≥ 0. As a corollary from Theorem 2 we get a criterion for potential members of Pfx . Theorem 7 A necessary condition for u ∈ Pfx is max

x∗ ,y ∗ ∈∂x,0≤y ∗ ≤x∗

< u, x∗ − y ∗ > = 0.

∗ contains This condition is trivially satisﬁed if x is an (o)-Sm point, i.e. ∂ x ∩X+ no proper order interval.

Example: Let us consider the space l1 and let x = (xn ) be in S(l1 ) with xn ≥ 0.

Let x∗ ≥ y ∗ ≥ 0 where x∗ = (αn ), y ∗ = (βn ) are from ∂ x . Thus n αn = 1

and n βn = 1 with αn ≥ βn ≥ 0. Hence, in particular, xn (αn − βn ) = 0 for all n. Let y = (yn ) be nonnegative such that < y, x∗ − y ∗ >= 0. Hence if follows yn (αn − βn ) = 0 for all n. Consequently the supports of x∗ and y ∗ are the same: suppn (xn ) = suppn (yn ). In fact we have a little more. Namely y = (yn ) is in Pfx if yn ∈ [0, xn ] for all n, since x ∨ y = (xn ∨ yn ) and x ∨ y = x . In the theorem below full characterization of elements y ∈ Pfx is given. Theorem 8 Let x ∈ S(X+ ) be ﬁxed and let y ≥ 0. The following statements are equivalent. (a) y ∈ Pfx . ∗ such that (b) There exists x∗ ∈ X+ ∗ (i) x = 1 and < x, x∗ >= x , (ii) < x ∨ y, x∗ >= x ∨ y , (iii) ∀(0≤y∗ ≤x∗ ) < y − x, y ∗ > ≤ 0.

A dual property to uniform monotonicity in Banach lattices

165

∗ Proof. (a)⇒(b). There exists x∗ ∈ S(X+ ) such that < x, x∗ >= x . Now, applying (a), we obtain

x = x ∨ y ≥< x ∨ y, x∗ > =

sup (< x, x∗ − y ∗ > + < y, y ∗ >)

0≤y ∗ ≤x

=< x, x∗ > + sup < y − x, y ∗ > 0≤y ∗ ≤x

= x + sup < y − x, y ∗ >. 0≤y ∗ ≤x

Hence (b)(ii)-(iii) follow. (a)⇐(b). We have to prove that for y ≥ 0 satisfying (b) there holds x ∨ y = x . In virtue of (b) x ∨ y =< x ∨ y, x∗ > =

sup (< x, x∗ − y ∗ > + < y, y ∗ >)

0≤y ∗ ≤x

=< x, x∗ > + sup < y − x, y ∗ > = x 0≤y ∗ ≤x

which ﬁnishes the proof.

References 1. C. Aliprantis, O. Burkinshaw, Positive Operators, Academic Press Inc., 1985. 2. G. Birkhoff, Lattice Theory, Providence, RI, 1967. 3. H. Hudzik, W. Kurc, Monotonicity of Musielak-Orlicz spaces equipped with the Orlicz norm (to appear). 4. W. Kurc, Strictly and uniformly monotone Musielak-Orlicz spaces and applicatios to best approximation, Journal of Approximation Theory 69(2), (1962), 173–187. 5. W. Kurc, Strictly and uniformly monotone sequential Musielak-Orlicz spaces (to appear). 6. W. Kurc, Extreme points of the unit ball in Orlicz spaces of vector-valued functions with the Amemiya norm, Mathematica Japonica 38(2), (1993), 277–288. 7. W. Kurc, Musielak-Orlicz spaces with order smooth norms, (to appear). 8. P. Nieberg, Banach Lattices, Springer Verlag, 1991.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 181–199 c 1994 Universitat de Barcelona

Inequalities and interpolation

L. Maligranda and L.E. Persson Department of Mathematics, Lule˚a University, S-971 87 Lule˚a, Sweden1

Abstract Some examples of the close interaction between inequalities and interpolation are presented and discussed. An interpolation technique to prove generalized Clarkson type inequalities is pointed out. We also discuss and apply to the theory of interpolation the recently found facts that the Gustavsson-Peetre class P +− can be described by one Carlson type inequality and that the wider class P0 can be characterized by another Carlson type inequality with “blocks”.

0. Introduction The ﬁrst interpolation proof of an inequality (Hausdorﬀ-Young’s inequality) was given already in 1926 by M. Riesz [56]. He wanted to ﬁnd a simple proof of the Hausdorﬀ-Young inequality and this was the main reason to prove the convexity theorem of Riesz. Nowadays it is well-known that also most of the other classical inequalities (e.g. those by Paley, Young, H¨ older, Minkowski, Beckenbach-Dresher, Clarkson, Carlson, Grothendieck etc.) can easily be proved by using such interpolation results. On the other hand, inequalities have been used to develop the theory of interpolation and its applications in various ways. In this paper we will present, discuss and complement some recently obtained examples of such interactions between inequalities and interpolation in both directions. Some new examples, proofs and results are also included. 1

This research was partly supported by a grant of the Swedish Natural Science Council (Contract F-FU 8685-300).

181

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This paper contains the following contributions: For the reader’s convenience and as an introduction of some ideas we use Section 1 to present and discuss four examples of well-known or “folklore” interpolation proofs of classical inequalities. In Section 2 we present an elementary interpolation technique to create Clarkson type inequalities and we also give some examples of results obtained in this way (see Theorems 1 and 2). In Section 3 we discuss some recently found results concerning generalized Carlson type inequalities (see [33]) namely that the Gustavsson-Peetre class P +− can be exactly described by one Carlson type inequality (see Theorem 3) and that the wider class P0 can be exactly characterized by another Carlson type inequality with “blocks” (see Th. 4). This exact information about inequalities gives us new information in the theory of interpolation e.g. concerning the +− method by Gustavsson-Peetre (see [21]) and that the Peetre interpolation functor (see [45]) on a couple of Banach lattices can be characterized by the Calder´onLozanovskii construction for every ϕ ∈ P0 . Ovchinnikov [43] was the ﬁrst who used the famous Grothendieck inequality to prove that the Gagliardo completion ϕ(·)c of the Calder´ on-Lozanovskii construction is an interpolation functor on a class of Banach function spaces. Finally, Section 4 is reserved for some concluding remarks and additional examples. The fundamental interpolation theorems, e.g. the Riesz-Thorin and Marcinkiewicz interpolation theorems, and the basic results of the real interpolation method of Lions-Peetre and the complex interpolation method of Calder´ on can be found in the books of Bennett-Sharpley [6], Bergh-L¨ ofstr¨om [8], Brudnyi-Krugljak [11], KreinPetunin-Semenov [32] and Triebel [64]. Conventions. For 0 < p ≤ ∞, p is deﬁned by p1 + p1 = 1 (p = ∞ for p = 1 and p = 1 for p = ∞). Let f ∗ denote the nonincreasing rearrangement of a measurable function |f | on a measure space (Ω, µ). The Lorentz Lp,q -spaces (0 < p < ∞, 0 < q ≤ ∞) are deﬁned by using the quasinorm (with the usual supremum interpretation when q = ∞) ∞ 1/q ∗ ∗ 1/p q dt (f (t)t ) . f Lp,q = t 0

1. Interpolation proofs of some classical inequalities Example 1 (Hausdorﬀ-Young’s and Paley’s inequalities): Consider the Fourier transform e−ixy f (y)dy. F f (x) = Rn

Inequalities and interpolation

183

Then we have boundedness F : L1 → L∞ with the norm M1 ≤ 1, and n F : L2 → L2 with the norm M2 = (2π) 2 (the Parseval equality). 10 . By using complex interpolation we obtain boundedness F : Lp ≡ [L1 , L2 ]θ → [L∞ , L2 ]θ ≡ Lq , where

1 p

=

1−θ 1

+ θn 2

θ 2

and

1 q

=

n p

1−θ ∞

+

θ 2

(which gives q = p ) with the norm Mθ ≤

M11−θ M2θ ≤ (2π) = (2π) . Equivalently, we can formulate this as the HausdorﬀYoung inequality:

F f Lp ≤ (2π)n/p f Lp ∀f ∈ Lp , 1 ≤ p ≤ 2.

(1.1)

20 . By using real interpolation we obtain boundedness F : Lp,q = (L1 , L2 )θ,q → (L∞ , L2 )θ,q = Lp ,q , θn

where 1 < p ≤ 2, 0 < q ≤ ∞, with the norm Mθ ≤ CM11−θ M2θ ≤ C(2π) 2 . Thus, in particular, we have proved the following version of the Paley inequality (sometimes also called the Hardy-Littlewood inequality): F f Lp ,p ≤ Cf Lp ∀f ∈ Lp , 1 < p ≤ 2.

(1.2)

n

Remark 1. The best constant in (1.1) is not (2π) p . In fact, Babenko and Beckner n (cf. [2]) proved that the best constant is equal to Cp,n = (Ap )n (2π) p , where Ap = 1 1 1 [p p /p p ] 2 . The best constant C = Cp,n in (1.2) is not known for p = 2. Remark 2. Some generalizations of the Hausdorﬀ-Young inequality for the Fourier transform on Orlicz spaces are done by Luxemburg [35] and Jodeit-Torchinsky [27], and on the rearrangement-invariant spaces by Bennett [4], [5]. Moreover, Russo [58] has obtained some generalizations of the Hausdorﬀ-Young theorem on integral operators. Remark 3. According to 10 and 20 we see that, for 1 < p ≤ 2, f ∈ Lp ⇒ F f ∈ Lp

(∗)

f ∈ Lp ⇒ F f ∈ Lp ,p ,

(∗∗)

and

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Maligranda and Persson

respectively. Moreover, since Lp ,p is continuously and properly embedded in Lp , we see that (∗∗) is a sharper criterion than (∗). Moreover, it can be confusing to compare these criterions with the following criterion (see [47]): (2−)p/(p−1) ∞ 1 p |F f | h max |F f |, dx < ∞, (∗ ∗ ∗) f ∈ Lp ⇒ |F f | 0 for some function h ≥ 1, 1/th(t) ∈ L1 (1, ∞) such that h(x)xa is a decreasing or increasing function of x for some real number a (1 < p ≤ 2). It is possible to prove directly that (∗∗) and (∗ ∗ ∗) are, in a way, equivalent (see [48]). Another way to understand this fact is to use interpolation in the following way: it is well-known that the Lorentz space Lp ,p coincides with the interpolation space (Lp0 , Lp1 )η,p , 0 < η < 1, 1/p = (1 − η)/p0 + η/p1 . Moreover, by restricting the general descriptions of real interpolation spaces in oﬀ-diagonal cases obtained in [48] to this case we ﬁnd that (Lp0 , Lp1 )η,p also coincides with the spaces described by the right hand side of (∗ ∗ ∗) (see [49], Corollary 3.3 and cf. also [38]). Remark 4. This interpolation proof shows that both (1.1) and (1.2) are true not only for the Fourier operator but also for any operators bounded from L1 into L∞ and from L2 into L2 . Example 2 (Young’s inequality): Here we consider the convolution operator k(x − y)f (y)dy = k ∗ f (x), with k ∈ Lq (Rn ), 1 ≤ q ≤ ∞. T f (x) = Rn

older Then we have boundedness T : Lq → L∞ with the norm ≤ kLq (by the H¨ inequality), and T : L1 → Lq with the norm ≤ kLq (by a generalized Minkowski inequality). By using complex interpolation we obtain boundedness T : Lp ≡ [Lq , L1 ]θ → [L∞ , Lq ]θ ≡ Lr , θ 1 1−θ θ 1 1 1 where p1 = 1−θ q + 1 and r = ∞ + q (which gives 1 ≤ p ≤ q and r = p + q − 1) with the norm ≤ kLq . Thus we have proved the Young inequality: if 1 ≤ q ≤ ∞, 1 ≤ p ≤ q and 1r = 1 1 p + q − 1, then (1.3) k ∗ f Lr ≤ kLq f Lp .

By instead using real interpolation we obtain the inequality k ∗ f Lr,s ≤ CkLq f Lp,s , 1 < q < ∞, 1 < p < q , 1r =

1 p

+

1 q

− 1 and s > 0.

(1.4)

Inequalities and interpolation

185

Remark 5. Beckner [2] found the following sharp form of the Young inequality (1.3): k ∗ f Lr ≤ (Ap Aq Ar )n kLq f Lp ∀f ∈ Lp (Rn ) ∀k ∈ Lq (Rn ), 1

where As = [s s /s s ] 2 and where (Ap Aq Ar )n is the best constant. The best constant C = Cp,q,r,n in (1.4) is not known when either r = p or r = q. 1

1

Remark 6. The inequality (1.4) is still true even if we only assume that k ∈ Lq,∞ (Rn ). This sharper result is due to O’Neil [42]. All such inequalities are very useful for numerous applications, e.g. in fractional diﬀerentiation, imbeddings between Sobolev spaces etc. Remark 7. A bilinear interpolation theorem for the general K-method of interpolation (generated by the Banach sequence lattices Φ, not only by lp (2−nθ )) is equivalent to the boundedness of the convolution operator from Φ0 × Φ1 into Φ (see [1] and [36]). In this connection we remark that Cwikel and Kerman [18] recently have used interpolation to prove some new interesting inequalities of Young type to hold for the more general case with positive multilinear operators acting on weighted Lp -spaces. Example 3 (H¨ older’s inequality): We note that the multiplication operator T (f, g) = f g is a bilinear bounded operator from L∞ × L1 into L1 and from L1 × L∞ into L1 , and that T (f, g)L1 ≤ f L∞ gL1 ∀f ∈ L∞ ∀g ∈ L1 , T (f, gL1 ≤ f L1 gL∞ ∀f ∈ L1 ∀g ∈ L∞ . Using the interpolation theorem for bilinear operators in complex spaces (Calder´ on theorem [12]) we ﬁnd that T : [L1 , L∞ ]θ × [L∞ , L1 ]θ → [L1 , L1 ]θ with the norm ≤ 1. Since [L1 , L∞ ]θ ≡ Lp (p = θ1 ), [L∞ , L1 ]θ ≡ [L1 , L∞ ]1−θ ≡ Lp older inequality [L1 , L1 ]θ ≡ L1 we obtain the H¨

(p =

f gL1 = T (f, g)L1 ≤ f Lp gLp ∀f ∈ Lp ∀g ∈ Lp .

1 1−θ )

and (1.5)

Remark 8. It is well-known that the Minkowski and Beckenbach-Dresher inequalities follow from the H¨ older inequality (1.5). Here we remark that also the Carlson inequality (see [14]) ∞ n=1

√ an ≤ π

∞ n=1

1/4 a2n

∞

n=1

1/4 n2 a2n

,

(1.6)

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where an are positive numbers, follows from the H¨ older inequality in the following way (cf. Hardy [22]): ∞ ∞ Let α = n2 a2n and β = a2n . Then n=1

∞

n=1

2 an

=

n=1

∞

an

α + βn2 α + βn2

n=1

≤

∞

2

1

2

∞

1 α + βn2 n=1

a2n (α

+ βn )

1 dx = π αβ = π α + βx2

n=1

≤ 2αβ 0

∞

∞

1/2 a2n

n=1

∞

1/2 n2 a2n

.

n=1

(Carlson in his original paper [14] suggested that (1.6) does not follow from the H¨older inequality). It is obviously easy to generalize this proof in various directions. In our Section 3 we discuss some new precise forms of Carlson type inequalities, which recently have been proved partly guided by the Hardy idea presented above (see [33]). (2)

Example 4 (Clarkson’s inequalities): Let lp be the 2-dimensional complex lp (2) space. Clearly, lp can be identiﬁed with C2 endowed with the p-norm (a, b)p = (|a|p + |b|p )1/p . (2)

(2)

We consider an elementary operator T : lp → lq

given by

T (a, b) = (a + b, a − b). (2)

(2)

By the triangle inequality, T : l1 → l∞ has norm 1 and, by the parallelogram law, 1 (2) (2) T : l1 → l2 has norm 2 2 . Therefore, by using complex interpolation, we ﬁnd that, 1 (2) (2) for 1 ≤ p ≤ 2, T : lp → lp has norm ≤ 2 p , i.e., for 1 ≤ p ≤ 2 and all a, b ∈ C,

(|a + b|p + |a − b|p )1/p ≤ 21/p (|a|p + |b|p )1/p .

(1.7)

By integrating (1.7) and using standard arguments we obtain the Clarkson inequalities:

((x + yLp )p + (x − yLp )p )1/p ≤ 21/p ((xLp )p + (yLp )p )1/p , 1 ≤ p ≤ 2,

Inequalities and interpolation

187

((x + yLp )p + (x − yLp )p )1/p ≤ 21/p (xLp )p + (yLp )p )1/p , p ≥ 2. 1

1

Remark 9. Consider Ap = (|a|p + |b|p ) p and Bp = (|a + b|p + |a − b|p ) p . It is well-known that Ap , Bp are nonincreasing in p and that 2−1/p Ap , 2−1/p Bp are nondecreasing in p. By using these facts together with the (Clarkson-HausdorﬀYoung) estimate (1.7) we obtain the following inequality: Let r ∈ R, r = 0, s > 0, q = min(2, s) if r ≤ 2 and q = min(r , s) if r > 2. Then, for a, b ∈ C (for the case r ≤ 0 we put by deﬁnition 0r = 0), (|a + b|r + |a − b|r )1/r ≤ 2γ (|a|s + |b|s )1/s , γ =

1 1 1 − − . r s q

This estimate was proved in [39]. The case r > 0 with a diﬀerent proof is due to Koskela [31].

2. An interpolation technique to obtain Clarkson type inequalities The idea to prove Clarkson type inequalities presented in Example 4 is easy to generalize in various directions, e.g. to more dimensions. Here we present some results which can be obtained in this way. (n) (m) Let m, n ∈ N and consider any linear bounded operator T : l1 → l∞ with the (n) (m) norm M1 which is also bounded T : l2 → l2 with the norm M2 . By using complex interpolation we obtain the (Hausdorﬀ-Young type) estimate 1−1/p

T (a)l(m) ≤ M1 p

2/p

M2

al(n) , 1 ≤ p ≤ 2,

(2.1)

p

which may be seen as a genuine generalization of (1.7). By using this interpolated estimate with diﬀerent operators and using the monotonicity arguments discussed in Remark 9 we can prove the following Clarkson type inequality, which, in particular, uniﬁes and generalizes some earlier results of this kind (see e.g. [15], [29], [31], [39], [51] and [66]). Theorem 1 (n) (m) Let T be any linear bounded operator T : l1 → l∞ with the norm M1 (n)

1− 2

(m)

2

and T : l2 → l2 with the norm M2 . Put Mu = M1 u M2u and assume that r ∈ R, r = 0, s > 0, u = max(r, 2), q = min(s, 2) if r ≤ 2 and q = min(r , s) if r > 2. Then, for any complex numbers a = (a1 , a2 , . . . , an ), m

1

|(T (a))i |r

1/r

≤ Mu m1/r−1/u n1/q−1/s

n

1

|ai |s

1/s .

(2.2)

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Maligranda and Persson

Proof. We put m

Ar =

|(T (a))i |r

1/r and Br =

n

1

|ai |r

1/r

1

and note that, by (2.1), Ap ≤ Mp Bp . Moreover, by using this estimate together 1 1 with the well-known facts that Ar , Br are nonincreasing in r and Ar m− r , Br n− r are nondecreasing in r (see e.g. [39, Lemma 1]), we obtain the following estimates: r ≤ 2,s ≤ 2: Ar ≤ m1/r−1/2 A2 ≤ M2 m1/r−1/2 B2 ≤ M2 m1/r−1/2 Bs , r ≤ 2,s ≤ 2: Ar ≤ m1/r−1/2 A2 ≤ M2 m1/r−1/2 B2 ≤ M2 m1/r−1/2 n1/2−1/s Bs , r ≤ 2,s ≤ 2: Ar ≤ Mr Br ≤ Mr Bs ,

r ≤ 2,s ≤ 2: Ar ≤ Mr Br ≤ Mr n1/r −1/s Bs . The proof of (2.2) follows by combining these inequalities. Example 5 (see [39]): Consider the operator T : a →

n

ai , . . . ,

1

n

εi ai , . . . ,

1

n

−ai ,

1

where εi = ±1, 1 ≤ i ≤ n (each coordinate of the vector T (a) ∈ Rm , m = 2n , is n n equal to a sum of the type εi ai ). It is easy to see that here M1 = 1 and M2 = 2 2 1

and, thus, according to Theorem 1, we obtain that

2

n r ε a i i

−n

εi =±1

1/r ≤n

1/q−1/s

1

n

1/s |ai |

s

,

1

for every r ∈ R, r = 0, s > 0, q = min(2, s) for r ≤ 2 and q = min(r , s) for r > 2. For the case r > 0 this inequality can be rewritten as 0

1

n r ϕ (t)a i i dt 1

1/r ≤n

1/q−1/s

n

1/s |ai |

s

,

1

where ϕi (t) =sign (sin(2i πt)) are the usual Rademacher functions. For the case n = 2 the last estimate coincides with a result of Koskela [31, Th. 1] and for the case s = r , r > 2 and s = r ≤ 2 another proof has been done by Williams-Wells [66, formulas (26)-(27)].

Inequalities and interpolation

189

Example 6: Consider the Littlewood-Walsh matrices A2n = (εij ), 1 ≤ i, j ≤ 2n , deﬁned recursively in the following way: A21 =

1 1

1 −1

,...,A

2n

=

A2n−1 A2n−1

A2n−1 −A2n−1

,

n = 2, 3, . . . .

By applying Theorem 1 to the operator 2n 2n 2n n n T:a → ε1j aj , ε2j aj , . . . , ε2n j aj from R2 to R2 , we ﬁnd that 1

1

1

 n 1/r 2n 1/s 2 2n r n(1/r−1/s+1/q) s  εij aj  ≤2 |aj | , 1

1

j=1

for every r ∈ R, r = 0, s > 0, q = min(2, s) if r ≤ 2 and q = min(r , s) if r > 2. For some special cases this estimate was also proved by Pietsch [51, p. 15] (see also Kato [29, p. 164]). Remark 10. Gurarii-Kadec-Macaev [20] used also an interpolation proof in the estimate of the Littlewood-Walsh matrix operator between lp -spaces. Their interest was to ﬁnd the order of the Banach-Mazur distance between the spaces lpn and lqn . Example 7: We note that for the operator T:a →

n

ai , a1 − a2 , a1 − a3 , a1 − a4 , . . . , a1 − an , . . . , an−1 − an

1 1

+ 1, we have M1 = 1 and M2 = n 2 and, thus, by from Rn to Rm , m = n(n−1) 2 Theorem 1, it holds that if r ∈ R, r = 0 and s > 0, then  n r  ai + 1

1/r 1/s n s r ai − aj  ai ≤C ,

1

1≤i<j≤n

where C = m r − 2 n 2 + q − s , q = min(2, s) for r ≤ 2 and C = n r + q − s , q = min(s, r ) for r > 2. In particular, for the case r = s = p ≥ 2, this estimate reads 1

1

1

1

1

n p ai + 1

1

1≤i<j≤n

|ai − aj |p ≤ np−1

n 1

1

|ai |p .

1

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Maligranda and Persson

For the case n = 3 Shapiro proved (by interpolation) this and some similar estimates already 1973 in a talk at the meeting of the Swedish Mathematical Society (see [61]). It is well-known that local versions of the Clarkson type inequalities always imply some corresponding global versions yielding e.g. for Lp -spaces or more general function spaces. Here we consider the following situation: Let T denote a linear class of functions f = f (t) deﬁned on a non-empty set E. Moreover, we let B: Γ → R, denote an isotone sublinear functional (“isotone” means that for every f, g ∈ L such that |f (t)| ≥ |g(t)| on E it holds that L(f ) ≥ L(g)). We also say that f ∈ Bp if 1 Bp (f ) = (B(|f |p )) p < ∞, 0 < p < ∞. Our global version of Theorem 1 reads Theorem 2 (n)

Let T be any linear operator such that T : l1 (n)

(m)

→ l∞ , with the norm M1 1− 2

(m)

2

and T : l2 → l2 , with the norm M2 . Put Mu = M1 u M2u and assume that r ∈ R, r = 0, p, s > 0, u = max(r, 2) and q = min(p, p , r , s) with the conventions that p is omitted if 0 < p ≤ 1 and r is omitted if r ≤ 1. Then, for any isotone linear functional B and any f1 , f2 , . . . , fn ∈ Bp , it holds that m

1/r Bpr ((T (f ))i )

≤ Mu m1/r−1/u n1/q−1/s

m

1

1/s Bps (fi )

.

1

By applying Theorem 1 to various operators, e.g. those considered in Examples 5-7, we obtain Clarkson type inequalities, for example those obtained in [29], [31], [39], [50]. Here we only give the following example: Example 8 (see [39, Theorem 4.1]): We apply Theorem 2 with the operator considered in Example 6 and obtain the following result: Let 0 < p, s < ∞ and r ∈ R, r = 0. Then, for any n ∈ Z+ and any f1 , f2 , . . . , f2n ∈ Bp , 2n 1

Bpr

2n 1

1/r εij fj

≤ 2n(1/r−1/s+1/q)

2n

1/s Bps (fj )

,

1

where q = min(p, p , s, r ) with the convention that p is omitted if 0 < p ≤ 1 and r is omitted if r ≤ 1. Remark 11. We can e.g. use Example 8 with Bp (f ) = f X p (X is any Banach function space) and, thus, in particular, with B(f ) = Ω |f (t)|dµ(t) (f is a measurable function on a measure space (Ω, Σ, µ)) so that Bp (f ) = f Lp . Therefore

Inequalities and interpolation

191

Example 8 generalizes some previous results by Kato [29, Th. 1], Koskela [31, Th. 2] and Persson [50, Th. 5.1]. Proof of Theorem 2. 10 . Let 0 < s ≤ p ≤ r < ∞. According to the well-known facts that B pr is superadditive and B ps is subadditive (see e.g. [46, Lemma 1.2]) we ﬁnd that m m m r r r (2.3) Bp ((T (f ))i ) = Bp/r (|T (f ))i | ) ≤ Bp/r |T (f ))i | 1

and

1

i

 r/s  n r/s n s s  = Bp/s Bp/r  |fi | |fi | 1

1

≤

n 1

r/s Bp/s (|fi |s )

=

n

(2.4)

r/s (Bp (fi ))s

.

1

In view of (2.3) and (2.4) we can use Theorem 1 and the isotonity of the functional B to obtain that 1/r n 1/s m Bpr (T (f ))i ≤ Mu m1/r−1/u n1/q−1/s Bps (fi ) , (2.5) 1

1

where q = s for 0 < r ≤ 2 and q = min(s, r ) for r ≥ 2. 20 . Let s, r ≤ p. We use (2.5) for the case 10 with r = p and also the well-known m r1 r − r1 Bp (T (f ))i is nondecreasing in r to see that in this case (2.5) fact that m 1

holds with q = s for 0 < r ≤ 2 and q = min(p, r ) for r ≥ 2. 30 . Let p ≤ s, r. Here we use (2.5) for the case 10 with s = p and ﬁnd as above that in this case (2.5) holds with q = p for 0 < r ≤ 2 and q = min(p, r ) for r ≥ 2. The proof is complete.

3. Some exact Carlson type inequalities and interpolation Let P denote the set of all positive concave functions ϕ: R+ → R+ (R+ = (0, ∞)). It is easy to see that ϕ(t) is nondecreasing and that ϕ(t)/t is nonincreasing. For ϕ ∈ P we consider

ϕ(st) sϕ (t) = sup , 00

192

Maligranda and Persson

We also deﬁne the following subsets of P : ϕ(t) =0 , P0 = ϕ ∈ P | lim ϕ(t) = lim t→∞ t t→0+ sϕ (t) =0 . P +− = ϕ ∈ P | lim sϕ (t) = lim t→∞ t→0+ t For our discussions later on it is important to note that P +− is a genuine subset of 1 P0 . For example, the functions ϕ1 (t) = min(1, t) and ϕ2 (t) = min(1, t 2 ) belong to P0 but not to P +− . Moreover, in the sequel we let ϕ(s, t) denote a 1-homogeneous function of two variables deﬁned as ϕ(s, t) = sϕ( st ) if s, t > 0 and ϕ(s, t) = 0 if s = 0 or t = 0. Next, we remark that some straightforward calculations show that the Carlson type inequality 1/4 ∞ 1/4 ∞ ∞ 2 2 2 an ≤ C an n an n=1

n=1

n=1

can equivalently be rewritten as

a 2n a n n {an }l1 ≤ Cϕ , , n n ϕ(2 ) lp ϕ(2 ) lq

(3.1)

1

where ϕ(t) = t 2 and p = q = 2. In 1977 Gustavsson and Peetre [21] proved (in connection to some problems in the theory of interpolation between Orlicz spaces) that (3.1) holds for any ϕ ∈ P +− and for all p, q > 1 (cf. also [37, pp. 143-145]). The following recent result by Krugljak-Maligranda-Persson [33] shows that, in fact, the (Carslon-GustavssonPeetre) inequality (3.1) is exactly equipped with the class P +− in the following way: Theorem 3 Let ϕ ∈ P0 and let 1 < min(p, q) ≤ ∞. The following statements are equivalent: (i) ϕ ∈ P +− . (ii) The following inequality holds:

a 2n a n n {an }l1 ≤ Cϕ . , ϕ(2n ) lp ϕ(2n ) lq (iii) The following inequality holds: {ϕ(an , bn )}l1 ≤ Cϕ

ak , lp

k:(ak ,bk )∈Sn

where Sn , n = 1, 2, . . . , are the areas in y = 2n+1 x.

R2+

k:(ak ,bk )∈Sn

bk

,

lq

between the lines y = 2n x and

Inequalities and interpolation

193

→

For any couple of Banach lattices X= (X0 , X1 ) we consider the Calder´on→

Lozanovskii interpolation space ϕ(X) equipped with the norm f

→

ϕ(X)

= inf max(f0 X0 , f1 X1 ),

where the inﬁmum is taken over all representations of |f | in the form |f | = ϕ(|f0 |, |f1 |), fi ∈ Xi , i = 0, 1. Remark 12. By using Theorem 3 it is possible to prove that, for any ϕ ∈ P +− , the →

→

Peetre interpolation functor G0ϕ (X) (see [44]) and its Gagliardo closure [G0ϕ (X)]c can be identiﬁed with a Calder´ on-Lozanovskii interpolation space (cf. also Remark 14). In order to be able to generalize the statements in Theorem 3 and Remark 12 to the “ﬁnal” case ϕ ∈ P0 , Krugljak-Maligranda-Persson [33] considered a tricky increasing sequence {tn } constructed by Brudnyi-Krugljak already in 1981 (in connection to their ﬁnal solution of the K-divisibility problem in interpolation theory, cf. [11]) and having the useful properties that, for ϕ ∈ P0 , ϕ(t) ≈

ϕ(t2n+1 ) min 1,

t t2n+1

and ϕ(t) ≈ max ϕ(t2n+1 ) min 1, n

t t2n+1

,

where the constants of equivalences are independent of ϕ and t. More exactly, the following Carlson type inequality with “blocks” was proved in [33]: Theorem 4 Assume that 1 < p, q ≤ ∞, ϕ ∈ P0 and let χn = [t2n , t2n+2 ). Then, for any positive sequence {an }, it holds that

{an }l1 ≤ Cϕ

k:2k ∈Xn

ak , ϕ(2k ) lp

k:2k ∈Xn

2k ak , ϕ(2k ) lq

(3.2)

with the constant C not depending on {an }. Moreover, the inequality (3.2) is equivalent to the following inequality:

ak , bk , (3.3) {ϕ(an , bn )}l1 ≤ Cϕ k:(an ,bk )∈Tn

lp

k:(ak ,bk )∈Tn

lq

where Tn , n = 1, 2, . . . , are the areas in R2+ between the lines y = t2n x and y = t2n+2 x.

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Maligranda and Persson

Our proof of Theorem 4 shows that the constants in the inequalities (3.2) and (3.3) √ √ can be estimated by (1 + 2)2 and 2(1 + 2)2 , respectively. Remark 13. By using Theorem 4 it is possible to prove the following generalization of an interpolation result by Ovchinnikov [43] and Nilsson [41] (see [33]): If ϕ ∈ P0 , →

→

→

→

→

then, for any couple of Banach lattices X, G0ϕ (X) = ϕ(X)0 , and [G0ϕ (X)]c = [ϕ(X)]c and the constants in the equivalence of the norms in these equalities do neither →

depend on the couple X nor on the function ϕ (here, as usual, for any intermediate space A, A0 denotes the closure of A0 ∩A1 in A and Ac denotes the Gagliardo closure with respect to A0 + A1 ). →

Remark 14. The main interest to consider the Peetre functor G0ϕ (X) is inspired by →

the fact that if ϕ(t) = tθ (0 < θ < 1), then, on couples of Banach lattices X, and their retracts, this functor coincides with the complex method (see [62], [41], [44], [11]) and, thus, it may be regarded as a “real version” of the complex method of interpolation.

4. Concluding remarks on [13] generalized the classical Marcinkiewicz interpolation theorem. 10 . Calder´ He even pointed out the maximal operator (1 ≤ p0 < p1 ≤ ∞, 1 ≤ q0 = q1 ≤ ∞)

∞

Sf (t) =

min(t−1/q0 s1/p0 , t−1/q1 s1/p1 )f (s)

0

ds , s

which is bounded from Lpi ,1 to Lqi ,∞ and for any operator acting from Lpi ,1 to Lqi ,∞ we have the estimate (T f )∗ (x) ≤ CSf ∗ (x), where C depends only on pi and on operator can be written as qi (i = 0, 1). The Calder´ −1/q0

s1/p0 f (s)

Sf (t) = t

0

where m =

1 q0 1 p0

− q1

1

− p1

.

tm

ds + t−1/q1 s

∞

s1/p1 f (s)

tm

ds , s

Therefore, in order to obtain generalized forms of the

1

Marcinkiewicz interpolation theorem, we only need to investigate the boundedness of the Hardy operator H and its dual H ∗ , deﬁned by x ∞ g(s)ds and H ∗ g(x) = x−1 g(s)ds, Hg(x) = x−1 0

x

Inequalities and interpolation

195

in the function spaces we are interested in. These types of the Hardy inequalities can be found in many places (cf. [32], [10], [7] and the literature given there). 20 . Interpolation theorems can also be used to obtain estimates of operators between vector-valued spaces (see e.g. [19]). For example, if T : Lp → Lp , is bounded then the natural vector-valued extension TE : Lp (Lp ) → Lp (Lp ) given by (TE f )(x, y) = T f (·, y)(x) has the same norm. On the other hand, according to the Marcinkiewicz-Zygmund theorem, TE has also the same norm on Lp (L2 ). Thus, by using interpolation, we ﬁnd that TE has the same norm on Lp (Lq ) for all q satisfying min(p, 2) ≤ q ≤ max(p, 2). 30 . Jameson [23] proved the famous Grothendieck inequality by using an interpolation technique. Moreover, Pisier [54] used the Riesz-Thorin interpolation theorem to obtain the following estimate of the complex Grothendieck constant: C KG ≤ e1−γ = 1.527 . . ., where γ is the Euler constant. 40 . Interpolation techniques can also be used to ﬁnd the “regularity” of the solutions of boundary-initial value problems. For example, if we consider the KleinGordon equation (KG) in R3 given by utt − ∆x u + u = 0, u(x, 0) = 0, ut (x, 0) = f (x), and the operator Tt f = u(·, t), then Tt : Lp → Lp is bounded if and only if 4/3 ≤ p ≤ 2 (see [40]). This statement in one direction can easily be proved by interpolation (for p = 2 we use the energetic identity and for p = 4/3 we use the fact that the KG equation is connected with the wave equation utt = ∆x u and in this case it is well-known that we have boundedness of Tt ). Some similar results connected to the Korteweg-De Vries equation and other equations can be found in [9], [34] and [63]. onig (1978) and Pietsch (1980) 50 . Karadzov (1973), Birman-Solomjak (1977), K¨ have used interpolation techniques to improve s-number estimates or estimates of entropy numbers or eigenvalues of operators in Banach spaces. For all these results and related references we refer to the books of K¨onig [30] and Pietsch [52], [53]. 60 . Rochberg-Weiss [57], Jawerth-Rochberg-Weiss [26] and Kalton [28] used the complex and real methods of interpolation to obtain nonlinear inequalities of the form (1 ≤ p0 < p < p1 < ∞) T (f log |f |) − T f log |T f |Lp ≤ Cp f Lp , for operators T which maps Lpi boundedly into Lpi for i = 0, 1.

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70 . Wells-Williams [65] used a generalized Riesz-Thorin interpolation theorem to obtain the exact value of so-called packing constant for the space Lp (µ). 80 . Interpolation results for vector-valued spaces give us information about the type and cotype of concrete spaces because a Banach space X has Rademacher type p, 1 ≤ p ≤ 2, if the operator T : lp (X) → Lp (X) given by T ((xk )) = xk rk (t) is bounded. This possibility to use interpolation was investigated by Cobos [16], [17]. 90 . Pisier [55] used the complex method of interpolation to prove the inverse BrunnMinkowski inequality due to V.D. Milman. 100 . In connection to the real method of interpolation there is also an extrapolation theory giving some old and new inequalities like those by Yano (1951), Stein (1969), Moser (1971) etc. (cf. Jawerth-Milman [25]). 110 . Another example of a new interesting inequality where interpolation is used in a crucial way can be found in Semenov’s paper [60] in this volume. 120 . For concrete operators, e.g. the average operator, the Hardy maximal operator, the conjugate operator, the Hilbert transform, the singular integral operators, etc., we can use Riesz-Thorin or Marcinkiewicz interpolation theorems to prove their boundedness between Lp -spaces and more general function spaces. This boundedness can be interpreted as an inequality. Such simple interpolation techniques to prove inequalities can be avoided in many concrete cases but to the cost of some more complicated computations.

References 1. S.V. Astashkin, On interpolation of bilinear operators by the real method, Mat. Zametki 52 (1992), 15–24 (Russian). 2. W. Beckner, Inequalities in Fourier analysis, Ann. of Math. 102 (1975), 159–182. 3. E.F. Beckenbach and R. Bellman, Inequalities, Springer-Verlag, 1983. 4. C. Bennett, A Hausdorff-Young theorem for rearrangement-invariant spaces, Pacific J. Math. 47 (1973), 311–328. 5. C. Bennett, Banach function spaces and interpolation methods. III. Hausdorff-Young estimates, J. Approx. Theory 13 (1975), 267–275. 6. C. Bennett and R. Sharpley, Interpolation of Operators, Pure and Applied Mathematics, Academic Press, Boston 1988. 7. J. Bergh, V. Burenkov and L.E. Persson, Best constants in reversed Hardy’s inequalities for quasi-monotone functions, Acta Sci. Math (Szeged), to appear. 8. J. Bergh and J. L¨ofstr¨om, Interpolation Spaces, Springer Verlag, 1976. 9. J. Bona and R. Scott, Solutions of the Korteweg-De Vries equation in fractional order Sobolev spaces, Duke Math. J. 43 (1976), 87–99.

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10. Ju. A. Brudnyi, S.G. Krein and E.M. Semenov, Interpolation of Linear Operators, Itogi Nauki i Techniki, T. Mat. Analiz, Moscow 1986, 3-163; English transl. in Journal of Soviet Math. 42 (1988), 2009–2113. 11. Ju. A. Brudnyi and N. Ja. Krugljak, Interpolation Functors and Interpolation Spaces I, NorthHolland, 1991. 12. A.P. Calder´on, Intermediate spaces and interpolation, the complex method, Studia Math. 24 (1964), 133–190. 13. A.P. Calder´on, Spaces between L1 and L∞ and the theorems of Marcinkiewicz, Studia Math. 26 (1966), 273–299. 14. F. Carlson, Une in´egalit´e, Arkiv Mat., Astr. och Fysik 25 B (1934), 1–5. 15. J.A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc. 40 (1936), 396–414. 16. F. Cobos, On the type of interpolation spaces and Sp,q , Math. Nachr. 113 (1983), 59–64. 17. F. Cobos, Propiedades Geom´etricas de los Espacios de Interpolaci´on, Seminars in Math. 4, Campinas 1989. 18. M. Cwikel and R. Kerman, Positive multilinear operators acting on weighted Lp -spaces, J. Funct. Anal. 106 (1992), 130–144. 19. J. Gasch and L. Maligranda, On vector-valued inequalities of the Marcinkiewicz-Zygmund, Herz and Krivine type, Math. Nachr. 166 (1994), to appear. 20. V.I. Gurarii, M.I. Kadec and V.I. Macaev, On the distance between finite-dimensional Lp -spaces, Mat. Sb. 70 (1966), 481–489 (Russian). 21. J. Gustavsson and J. Peetre, Interpolation of Orlicz spaces, Studia Math. 60 (1977), 33–59. 22. G.H. Hardy, A note on two inequalities, J. London Math. Soc. 11 (1936), 167–170. 23. G.J.O. Jameson, The interpolation proof of Grothendieck’s inequality, Proc. Edinburgh Math. Soc. 28 (1985), 217–223. 24. S. Janson, Minimal and maximal methods of interpolation, J. Functional Anal. 44 (1981), 50–73. 25. B. Jawerth and M. Milman, Extrapolation theory with applications, Memoirs of Amer. Math. Soc. 89, No. 440, Providence 1991. 26. B. Jawerth, R. Rochberg and G. Weiss, Commutator and other second order estimates in real interpolation theory, Ark. Mat. 24 (1986), 191–219. 27. M.A. Jodeit, Jr. and A. Torchinsky, Inequalities for Fourier transforms, Studia Math. 37 (1971), 245–276. 28. N.J. Kalton, Nonlinear commutators in interpolation theory, Memoirs of Amer. Math. Soc. 73, No. 385, Providence 1988. 29. M. Kato, Generalized Clarkson’s inequalities and the norms of the Littlewood matrices, Math. Nachr. 114 (1983), 163–170. 30. H. K¨onig, Eigenvalue Distribution of Compact Operators, Birkh¨auser Verlag, Basel 1986. 31. M. Koskela, Some generalizations of Clarkson’s inequalities, Univ. Beograd. Publ. Elektrotechn. Fak. Ser. Mat. Fiz. 634–677 (1979), 89–93. 32. S.G. Krein, Ju.I. Petunin and E.M. Semenov, Interpolation of Linear Operators, Nauka, Moscow 1978; English Transl., Amer. Math. Soc., Providence, 1982. 33. N.J. Krugljak, L. Maligranda and L.E. Persson, A Carlson type inequality with blocks and interpolation, Studia Math. 104 (1993), 161–180.

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34. J.L. Lions, Quelques M´ethodes de R´esolution de Probl´emes aux Limites Non Lin´eaires, Dunood Gauthier Villars, Paris 1969. 35. W.A.J. Luxemburg, The Hausdorff-Young-Riesz theorem in Orlicz spaces, Report of the Summer Research Institute of the Canadian Math. Congress 1957, 14–15. 36. L. Maligranda, Interpolation of some spaces of Orlicz type II. Bilinear interpolation, Bull. Polish Acad. Sci. Math. 37 (1989), 453–457. 37. L. Maligranda, Orlicz Spaces and Interpolation, Seminars in Math. 5, Campinas 1989. 38. L. Maligranda and L.E. Persson, Real interpolation of weighted Lp -spaces and Lorenz spaces, Bull. Polish Acad. Sci. Math. 35 (1987), 765–778. 39. L. Maligranda and L.E. Persson, On Clarkson’s inequalities and interpolation, Math. Nachr. 155 (1992), 187–197. 40. B. Marshall, W. Strauss and S. Wainger, Lp − Lq estimates for the Klein-Gordon equation, J. Math. Pures Appl. 59 (1980), 417–440. 41. P. Nilsson, Interpolation of Banach lattices, Studia Math. 82 (1985), 133–154. 42. R. O’Neil, Convolution operators and L(p, q) spaces, Duke Math. J. 30 (1963), 129–142. 43. V.I. Ovchinnikov, Interpolation theorems resulting from Grothendieck’s inequality, Functional. Anal. Prilozen. 10 (1976), 45–54 (Russian). 44. V.I. Ovchinnikov, The Methods of Orbits in Interpolation Theory, Math. Reports Vol. 1, Part 2, Harwood Academic Publishers 1984. 45. J. Peetre, Sur l’utilisation des suites inconditionellement sommables dans la th´eorie des espaces d’interpolation, Rend. Sem. Math., Univ. Padova 46 (1971), 173–190. 46. J. Peetre and L.E. Persson, A general Beckenbach’s inequality with applications, In: Function Spaces, Differential Operators and Nonlinear Analysis, Pitman Research Notes in Math., Ser. 211 (1989), 125–139. 47. L.E. Persson, An exact description of Lorentz spaces, Acta Sci. Math. 46 (1983), 177–195. 48. L.E. Persson, Descriptions of some interpolation spaces in off-diagonal cases, Lecture Notes in Math. 1070 (1984), 213–231. 49. L.E. Persson, Exact relations between some scales of spaces and interpolation, Teubner Texte zur Math. 103 (1988), 112–122. 50. L.E. Persson, Some elementary inequalities in connection with Xp -spaces, Publishing House of the Bulgarian Academy of Sciences, 1988, 367–376. 51. A. Pietsch, Absolutely-p-summing operators in Lr -spaces 2, Sem. Goulaouic-Schwartz, Paris 1970/71. 52. A. Pietsch, Operator Ideals, North-Holland 1980. 53. A. Pietsch, Eigenvalues and s-Numbers, Akad. Verlag, Leipzig 1987. 54. G. Pisier, Grothendieck’s theorem for noncommutative C ∗ -algebras with an appendix on Grothendieck’s constants, J. Functional Anal. 29 (1978), 397–415. 55. G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge Univ. Press, Cambridge 1989. 56. M. Riesz, Sur les maxima des formes bilinearies et sur les fonctionelles lin´eaires, Acta Math. 49 (1926), 465–497.

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57. R. Rochberg and G. Weiss, Derivatives of analytic families of Banach spaces, Ann. Math. 118 (1983), 315–347. 58. B. Russo, On the Hausdorff-Young theorem for integral operators, Pacific J. Math. 68 (1977), 241–253. 59. Y. Sagher, An application of interpolation theory to Fourier series, Studia Math. 41 (1972), 169–181. 60. E.M. Semenov, Random rearrangements in functional spaces, to appear. 61. H. S. Shapiro, The uses of soft analysis,Research report 2, Dept. of Math., Royal Institute of Technology, Stockholm, 1974. 62. V.A. Shestakov, Complex interpolation in Banach spaces of measurable functions, Vestnik Leningrad Univ. 19 (1974), 64–68 [Russian]. 63. L. Tartar, Interpolation non lin´eaire et r´egularit´e, J. Functional Anal. 9 (1972), 469–489. 64. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, VEB, Berlin 1978. 65. J.H. Wells and L.R. Williams, Imbeddings and Extensions in Analysis, Springer-Verlag 1975. 66. L.R. Williams and J.H. Wells, Lp -inequalities, J. Math. Anal. Appl. 64 (1978), 518–529.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 201–210 c 1994 Universitat de Barcelona

A commutator theorem with applications

Mario Milman∗ Department of Mathematics, Florida Atlantic University, Boca Raton, Florida 33431, U.S.A.

Abstract We give an extension of the commutator theorems of Jawerth, Rochberg and Weiss [9] for the real method of interpolation. The results are motivated by recent work by Iwaniec and Sbordone [6] on generalized Hodge decompositions. The main estimates of these authors are based on a commutator theorem for a specific operator acting on Lp spaces and through the use of the complex method of interpolation. In this note we give an extension of the Iwaniec-Sbordone theorem to general real interpolation scales.

1. Introduction In [13] and [9] Jawerth, Rochberg and Weiss initiated the study of second order and abstract commutator theorems for scales of interpolation spaces. Recall that given a compatible pair of Banach spaces the classical constructions of interpolation theory provide methods to obtain parameterized families of spaces with the interpolation property. That is if an operator T is bounded from a compatible pair A to another compatible pair B then T will also be bounded on the corresponding interpolation spaces. Jawerth, Rochberg, and Weiss (cf. [13] and [9]) have shown that associated with the classical methods of interpolation are certain operators, Ω, generally unbounded and non-linear, which can be obtained by diﬀerentiation with respect to certain parameters used in the speciﬁc method. These operators have the property that their commutator with a bounded operator T in the scale, ∗

Supported in part by NSF grant DMS–9100383

201

202

Milman

[Ω, T ], is also bounded in the scale. This was shown in [13] for the complex method, and in [9] for the real methods. For example, in the case of Lp spaces one can use the operators Ωf = f log |f |/f p . These results have interesting applications in analysis. We refer to these papers, and also to [2] and the survey [3] for a detailed account. We should also point out an interesting connection of the subject under consideration and the theory of logarithmic Sobolev inequalities (cf. [3]), in fact some of the basic ideas of the theory, for the complex method and in the Lp setting, are already implicit in Feissner’s [5] study of higher order logarithmic Sobolev inequalities. In the setting of lattices these results have been considerably extended by Kalton (cf. [10] and the papers quoted therein) who has exhibited a large class of operators “Ω” which commute with bounded operators in an interpolation scale. The methods developed in Kalton’s papers are very interesting and his results have many new applications. However, it is not yet clear how Kalton’s methods can be incorporated in the general theory. In [12] a new approach to the abstract commutator theorems for the real method was given, showing, in particular, commutation relations with certain non-linear operators. We also mention [11] where a connection to the functional calculus for positive operators in Banach spaces is developed. The connections of this subject with “extrapolation theory” are also explored in [7] and [11]. A general uniﬁed approach to commutator theorems for the real and complex methods has been obtained in the forthcoming paper [4]. Recently in their study of minimizers for variational problems Iwaniec and Sbordone [6] have obtained and used the following commutator theorem using the complex method of interpolation. Theorem 1 p r2

Let T be an operator T : Lp → Lp , p ∈ [r1 , r2 ], where 1 ≤ r1 < r2 < ∞, and let − 1 ≤ ε ≤ rp1 − 1. Deﬁne Ωε (f ) =

|f | f p

ε f.

Then, [T, Ωε ]p/(1+ε) ≤ cp |ε| f p where cp =

2p(r2 − r1 ) sup T s . (p − r1 )(r2 − p) r1 ≤s≤r2

(1)

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203

This is a useful variant of the commutator theorem of [13] and can be obtained by the complex method. One of the main points in the applications of (1) is the fact that ε can be negative. Let us also point out that, as was observed in [6], letting ε → 0 in (1) we obtain the Rochberg-Weiss [13] theorem in the context of Lp spaces [T, Ω]p ≤ cp f p

(2)

where Ω f = f log |f |/f p . The purpose of this note is to point out that the Iwaniec-Sbordone result can be incorporated to the general theory of commutator inequalities for real method of interpolation. Thus, we exhibit a general class of operators Ωε which commute, with bounded operators T acting on the initial pairs, in the sense that an estimate of the type (1) holds for [Ωε , T ] inside the real interpolation scale. When specialized to the Lp setting our results give the Iwaniec-Sbordone theorem with a less precise constant. We assume that the reader is familiar with the basic results of interpolation theory as developed in [1], where we refer for background information. In order to make the paper self contained we have included a brief summary of the necessary deﬁnitions concerning the theory of real interpolation commutators.

2. Quasi-logarithmic operators associated to real interpolation In this section we brieﬂy review the relevant deﬁnitions from interpolation theory and introduce the relevant operators that we shall study in this paper. We refer to [1], [9] and [3] for more details. Let A = (A0 , A1 ) be a Banach pair, a ∈ Σ(A) = A0 + A1 , and recall that the K functional of a is deﬁned, for t > 0, by K(t, a; A) = inf{a0 A0 + ta1 A1 : a = a0 + a1 } . The interpolation spaces Aθ,q;K , 0 < θ < 1, 0 < q ≤ ∞, are deﬁned by Aθ,q;K = a : aAθ,q;K =

0

∞

[t−θ K(t, a, A)]q

dt 1/q <∞ . t

(3)

We shall be concerned with the process of computing these interpolation norms. We say that the decomposition a = a0 (t) + a1 (t) is almost optimal for the K method if a0 (t)A0 + ta1 (t)A1 ≤ cK(t, a; A)

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where c is a constant ﬁxed before hand, say c = 2. We then write DK (t; A)a = DK (t)a = a0 (t). The operator Ω associated with this decomposition is deﬁned by ΩA;K a =

0

1

dt DK (t)a − t

∞

1

dt I − DK (t) a . t

(4)

Similarly, we can deﬁne the corresponding operators Ω associated with the J and E methods. Recall that given a Banach pair A the spaces Aθ,q;J , 0 < θ < 1, 0 < q ≤ ∞, are deﬁned using the quasi-norms ∞ ∞ −θ q ds 1/q ds aAθ,q;J = inf :a = u(s) J s, u(s); A s s s 0 0 where u: (0, ∞) → ∆(A), and the J functional is deﬁned for h ∈ ∆(A), t > 0, by J(t, h; A) = max{hA0 , thA1 }. We shall say that u(t) is an almost optimal decomposition of a for the J method, and write DJ (t, A)a = DJ (t)a = u(t), if a= 0

∞

ds u(s) , aAa,q;J ≈ s

∞

[J(s, u(s); A)s−θ ]q

0

The corresponding ΩJ operator is deﬁned by ∞ dt DJ (t)a log t . ΩJ a = t 0

ds 1/q . s

(5)

(6)

For the E method we have a similar deﬁnition. Recall that E(t, a; A) =

inf

{a0 A0 : a = a0 + a1 }.

a1 A1 ≤t

The corresponding interpolation spaces Aθ,q;E , 0 < θ < ∞, 0 < q ≤ ∞, are deﬁned using the quasi-norms 1/q ∞ θ q dt f Aθ,q;E = [t E(t, f, A)] . (7) t 0 Let DE (t; A) = DE (t)a = a0 (t), for an almost optimal decomposition, that is such that E(t, a; A) ≈ DE (t)aA0 . (8)

A commutator theorem with applications

205

Then, the corresponding Ω s are deﬁned by

1

ΩE a =

DE (t)a 0

dt − t

∞

dt I − DE (t) a . t

1

(9)

The main result of [9] is that if T is a bounded operator T : A → B, and F denotes any of these methods of interpolation, then there exists a constant c(F ) such that if we let [ΩF , T ] = ΩF (B) T − T ΩF (A) , then [ΩF , T ] f F (B) ≤ cf F (A) . This result also holds for the complex method (cf. [13]). In the next sections we consider variants of these operators and commutator theorems for them.

3. A commutator theorem for the E method We consider ﬁrst variants of the Ω operators associated with the E method since it is the method that will provide us with an appropriate generalization of Theorem 1. Let α ∈ (−1, 1), α = 0, and deﬁne

∞

ΩE,α a = Ωα a = α

α dt

DE (t)at 1

t

1

−

dt . I − DE (t) atα t

0

Theorem 2 Let A and B be a Banach pairs, T : A → B be a bounded operator, then there exists a constant c > 0 such that if θ + α > 0, [Ωα , T ]f (B0 ,B1 )θ/(α+1),q;E ≤

c |α|(2cα )θ/(α+1) (α + 1)1/q f (A0 ,A1 )θ+α,q;E . θ

Proof. It is easy to see that according to our deﬁnitions for any Banach pair H, and for t > 0, we have

Ωα,H a + aϕα (t) = α where ϕα (t) = 1 − tα .

t

∞

α ds

DE,H (s)as

s

− 0

t

ds I − DE,H (s) asα s

(10)

206

Milman Let a1 (t) =

t 1 α ( 0 (I

− DE (s))asα ds s ), then

a1 (t)H1 ≤ |α|

0

t

ds |α| α+1 ≤ t I − DE (s) aH1 sα . s (α + 1)

(11)

Thus, letting cα = |α|(α + 1)−1 and combining (10), (11), and (8), we get

E cα tα+1 , Ωα a + ϕα (t)a; H ≤ |α|

∞

E(s, a, H)sα

t

ds . s

(12)

Therefore if T : A → B, then we can estimate E(2cα tα+1 , Ωα,B T a − T Ωα,A a; B) as less than or equal to

E cα tα+1 , Ωα,B T a + ϕα (t)T a; B + E cα tα+1 , T Ωα,A a + ϕα (t)a ; B . Using the fact that T is bounded, and applying (12) to each of these terms we get E(2cα t

α+1

, Ωα,B T a − T Ωα,A a; B) ≤ c|α|

∞

t

α ds

E(s, a, A)s

s

where c depends only on the norm of T on the initial pair. An application of Hardy’s inequality (cf. [14]) now yields 0

∞

dt 1/q [tθ E(2cα tα+1 , Ωα,B T a − T Ωα,A a; B)]q t ∞ ds 1/q c|α| ≤ [E(s, a, A)sα+θ ]q θ s 0

and therefore we ﬁnally get [Ωα , T ]a(B0 ,B1 )θ/(α+1),q;E ≤

c|α| (2cα )θ/(α+1) (α + 1)1/q a(A0 ,A1 )θ+α,q;E . θ

We consider now in detail the special case of Lp spaces. Although the calculation of the interpolation spaces in this case is well known we include the details for the sake of completeness and the reader’s convenience. The E functional for the pair (L1 , L∞ ) is well known and easy to compute (cf. [1], [8]) E(t, f, L1 , L∞ ) =

∞

λf (s)ds t

(13)

A commutator theorem with applications

207

and an approximate optimal decomposition is given by f = f χ{|f |>t} +f χ{|f |≤t} . (In fact an optimal decomposition is given by f = (f −t)+ +tχ{|f |≤t} ). The interpolation spaces for the E method can be determined using this formula. In fact if we recall the formula ∞ 1/p f p = p λf (s)sp−1 ds 0

we see, using (13) and integration by parts that f (L1 ,L∞ )p−1,1;E = [(p − 1)p]−1 f pp . A calculation using (9) gives Ωα f = f |f |α − f. Let us set Sα f = f |f |α , then we clearly have [T, Ωα ] = [T, Sα ]. Now to apply Theorem 2 we let θ r = − 1, α+1 1+α

then θ + α = r − 1

and the previous discussion gives T Sα f −

r/1+α Sα T f r/1+α

≤ c2

(r−1−α)/(α+1)

|α| (α + 1)

r/α+1

Raising both members of the previous inequality to the power of Iwaniec-Sbordone type,

1 f rr . (r − 1)

1+α r

gives an estimate

α (α+1)/r α 1 |α| T |f | f − |T f | T f ≤c f r . f α f α (α + 1) (r − 1) r r r/1+α

(14)

In order to obtain a version of Theorem 1 we argue that α α α |f | f |T f |α T f T |f | f − |T f | T f − ≤ T α α α f α T f f f r r r r r/1+α r/1+α α |T f |α T f T f |T f | + = I + II, say. f α − T f α r

r

I is controlled by (14) while II can be readily computed T f r α II = T f r − 1. f r

r/1+α

208

Milman

Let x = T f r , y = f r , u = y/x, ϕ(u) = uα+1 − u, and assume, as we may, that T r→r ≤ 1, then u ∈ [0, 1], and we have reduced everything to prove that there exists c > 0, such that ∀u ∈ [0, 1] |ϕ(u)| ≤ c|α|.

(15) 1+α

1 ) α . We We study ϕ using calculus and we see that (15) holds with c = ( 1+α conclude the analysis by observing that the factor 1/(1+α) is under control by r2 /r. By collecting estimates we see that we have thus obtained an end point version of Theorem 1 by real methods with a somewhat worst constant, but with the right control when α → 0. By reiteration we may obtain the full result. In a similar fashion we can deal with the family of error functionals Eβ introduced in [9], this is particularly useful when dealing with pairs of weighted Lp spaces. As an example when dealing with the pair (Lp0 (w0 (x)dx), Lp1 (w1 (s)dx)) the corresponding Ω s can be chosen to be of the form ε w0 Ωf = f − f. w1

For brevity sake we refer to [9] for other possible applications of Theorem 2, and where similar calculations are performed. Using these methods one can also deal with operators T that are not necessarily linear (cf. [12] for a detailed treatment of non-linear operators in the context of the K method).

4. Remarks on the K and J methods There are many variants of the results of the previous section. We can consider the K and J methods, or consider variants of the E method (as in [9]), etc. However, since the analysis of these methods is similar to the one we developed in detail in the previous section we shall be rather brief here. In fact in the case of the K method the analysis follows closely the one given in [12]. We consider operators deﬁned by

1

Ωα a = α

α ds

DK (s)as 0

s

−

∞

1

ds . I − DK (s) asα s

Then, as before we see that Ωα a − ϕα (t)a = α

0

t

DK (s)asα

ds − s

t

∞

ds . I − DK (s) asα s

A commutator theorem with applications

209

This leads to the estimate

K t, Ωα a + ϕα (t)a, A ≤ c|α| 0

∞

t ds K(s, a; A)sα . min 1, s s

(16)

Thus, if T : A → B, we see, using the cancellation property for commutators in the usual fashion, the estimate (16), and Hardy’s inequality, that [T, Ωα ]f B θ,q;K ≤ c(θ, q)|α| f Aθ−α,q;K . We formally state this result as, Theorem 3 Let A, B, be Banach pairs, let T : A → B, be a bounded operator, and let α ∈ (−1, 1) \ {0}, 0 < θ < 1, 0 < q ≤ ∞, and suppose that 0 < θ − α < 1. Then there exists a constant c = c(θ, q) such that [T, Ωα ]f B θ,q;K ≤ c|α| f Aθ−α,q;K . Let us remark that the operators Ωα for this method are diﬀerent than those for the E method (cf. [9]). In the familiar examples of the theory they can be easily calculated by trivial modiﬁcations to the calculations of Ω in [9] and [3]. For example, for the pair (L1 , L∞ ) a possible choice of Ωα is given by Ωα f = f (rf )α − f , where rf is the “rank function” of f deﬁned by rf (x) = |{y: |f (y)| > |f (x)| or |f (y)| = |f (x)| and y ≤ x}| (cf. [3]). The J method admits a similar treatment and analogous results. For example a class of operators that can be treated by these methods is given by (cf. [3]) ∞ dt tα DJ (t)a . Ωα a = t 0 The relationship to the corresponding Ωα;K operators is, as usual, given by the fact that the fundamental lemma of interpolation theory implies that we can take d DK (t)a. We also point out that the resulting theory is closely related to DJ (t)a = t dt the functional calculus associated with positive operators in Banach spaces (cf. [11] and the references therein) and Zafran’s work [15]. We shall deal elsewhere with the complex method and with applications to weighted norm inequalities for classical operators (cf. also [9], [12], [13]).

210

Milman References

1. J. Bergh and J. L¨ofstr¨om, Interpolation Spaces: An introduction, Springer-Verlag, Berlin, Heidelberg and New York, 1976. 2. M. Cwikel, B. Jawerth, and M. Milman, The domain spaces of quasilogarithmic operators, Trans. Amer. Math. Soc. 317 (1990), 599–609. 3. M. Cwikel, B. Jawerth, M. Milman, and R. Rochberg, Differential estimates and commutators in interpolation theory, “Analysis at Urbana II”, London Math. Soc., Cambridge Univ. Press 1989, pp. 170–220. 4. M. Cwikel, N. Kalton, M. Milman, and R. Rochberg, in preparation. 5. F. Feissner, Hypercontractive semigroups and Sobolev’s inequality, Trans. Amer. Math. Soc. 210 (1975), 51–62. 6. T. Iwaniec, and C. Sbordone, Weak minima of variational integrals, University of Naples, preprint 1992. 7. B. Jawerth and M. Milman, New results and applications of extrapolation theory, Interpolation Spaces and Related Topics, M. Cwikel et al editors, Israel Math. Conf. Proc. 5 (1992), 81–105. 8. B. Jawerth and M. Milman, Interpolation of weak type spaces, Math. Z. 201 (1989), 509–519. 9. B. Jawerth, R. Rochberg, and G. Weiss, Commutator and other second order estimates in real interpolation theory, Ark. Mat. 24 (1986), 191–219. 10. N. Kalton, Differentials of complex interpolation process for K¨othe function spaces, to appear. 11. M. Milman, Extrapolation and Optimal Decompositions with Applications to Analysis, Lecture Notes in Math., Springer, 1994. 12. M. Milman and T. Schonbek, Second order estimates in interpolation theory and applications, Proc. Amer. Math. Soc. 110 (1990), 961–969. 13. R. Rochberg and G. Weiss, Derivatives of analytic families of Banach spaces, Ann. Math. 118 (1983), 315–347. 14. E.M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, 1970. 15. M. Zafran, Spectral theory and interpolation of operators, J. Funct. Anal. 36 (1980), 185–204.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 211–215 c 1994 Universitat de Barcelona

Boundary spaces for inclusion map between rearrangement invariant spaces

S. Ya. Novikov Samara State University, 443023 Samara 23, Russia

Abstract Let E([0, 1]; m) be a rearrangement invariant space (RIS) on [0, 1] with Lebesgue measure m. That is, E is a Banach lattice and if m(t: |x(t)| > τ ) = m(t: |y(t)| > τ )∀τ , then xE = yE . For each of this kind of spaces we have inclusions C ⊂ L∞ ⊂ E ⊂ L1 and canonical inclusion maps I(C, E) or I(E1 , E2 ). The aim of this paper is to represent a number of RIS, which are boundary for various properties of canonical inclusion maps. There are still some unsolved problem in this area.

1. Strict singularity An operator T ∈ L(X, Y ) between two Banach spaces (BS) X and Y is called strictly singular if there is no inﬁnite dimensional subspace Z of X such that the restriction T |Z is an isomorphism. The set of this kind of operators will be denoted σ(X, Y ). It is an ideal in the Pietsch sense. According to a well-known Grothendieck’s theorem I(L∞ , Lp )∈ σ, 1 ≤ p < ∞ (see, for example, the text book of W. Rudin). A more general fact seems to be true: Theorem 1 Let E be a RIS and E = L∞ . Then I(L∞ , E) is strictly singular. 211

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Proof. The function ϕE (t) := 1[0,t] E , the so-called fundamental function of the space We may deﬁne Lorentz space Λ(ϕE ) := 1 E∗ is a quasi-concave function. ∗ {f : 0 f dϕE < ∞}, where f is the decreasing rearrangement of |f |, besides we have another inclusion: E ⊃ Λ(ϕE ), E = L∞ . It’s known, that if E = L∞ , then the function ϕE is continuous at zero and the space Λ(ϕE ) is weakly sequential complete. From this we deduce that the p-convexiﬁcation Λp (ϕE ) := {f : |f |p ∈ Λ(ϕE )} is reﬂexive for 1 < p < ∞. So, we have: E ⊃ Λ(ϕE ) ⊃ Λp (ϕE ) ⊃ L∞ . As L∞ has the Dunford-Pettis property (i.e., ∀Y, ∀ weakly compact T ∈ L(L∞ , Y ), ∀ convex weakly compact K ⊂ L∞ , T (K) is compact in Y ), we have that the unit ball BH of each subspace H ⊂ E, such that H ⊂ L∞ , is compact in E. In spite of the fact that Theorem 1 solves the problem of strict singularity of the inclusion map I(L∞ , E), there are still left a lot of problems concerning inclusion maps between general RIS E1 ⊂ E2 . For example, there is no full description of the set of such RIS E, for which I(E, L1 ) ∈ σ. In this direction we know only a partial answer: Theorem 2 If RIS E ⊂ L2 , then I(E, L1 ) ∈ σ iﬀ E ⊃ G, where G is the closure of C[0, 1] 2 in the Orlicz space LN , N (u) = eu − 1. Proof. If E ⊃ G, then according to the classical result of Rodin-Semenov ([6], [2]), E contains an inﬁnite dimensional subspace R closed in L1 . Now suppose that I(E, L1 ) is not strictly singular. It means that E contains an inﬁnite dimensional subspace H, closed in L1 . This subspace is closed in L2 also (cf. condition). Let {fi } be a sequence of elements of H, equivalent to the unit basis of l2 and fi L2 = 1, i = 1, 2, . . .. We can assume that fi → 0 weakly in L2 and lim inf fi L1 > 0; this may be done by choosing subsequences. The last inequality ensures the existence of a function 0 ≤ g ∈ L1 with m(supp g) > 0 such that fi2 → g weakly in L1 . Now we will use the following theorem of V. Gaposhkin ([1], Th. 1.5.1): If {fk } is a sequence of functions such that: 1) fk L2 = 1 ∀k; 2) fk → 0 weakly in L2 ; 3) ∃g ∈ L+ 1 , gL1 = 1 such that fk → g weakly in L1 ; then it’s possible to choose a subsequence {fki } such that the next equality, like in central limit theorem, takes place 1 ∞ m −u2 1 1 du . fki (t) ≥ s = dt exp lim m t: √ √ m→∞ 2π 0 2 m i=1 s/ g(t)

Boundary spaces for inclusion map

213

Using this fact it’s not diﬃcult to see that the function (ln 1t ) 2 ∈ E , where E is the K¨othe dual of E. The last condition is known to be equivalent to the inclusion E ⊃ G. 1

2. Absolutely summing properties Definition. An operator T is called (q, p) – absolutely summing (T ∈ Πq,p (X, Y ) if ∃ C > 0: ∀{x1 , x2 , . . . , xn } ∈ X i

T xi q

1/q

≤ C sup

|F (xi )|p

1/p

: F X ∗ ≤ 1 .

i

This deﬁnition makes sense only if 0 < p ≤ q < ∞; if p > q then only 0 – operator is (q, p) – absolutely summing. For p = q we use the notations Πp and “p – absolutely summing”. Theorem 3 Let E1 ⊂ E2 and p ≥ 1. The inclusion map I(E1 , E2 ) ∈ Πp iﬀ E1 = L∞ , E2 ⊃ Lp . Proof. Suﬃciency is obvious. Now assume that I(E1 , E2 ) is p – absolutely summing. Then each weak convergent sequence in E1 is convergent in norm in E2 . Repeating the proof of Theorem 1 we deduce that E1 = L∞ . From classical factorization theorem of Pietsch we have: ∃ probability measure ν on [0, 1] such that 1/p f E2 ≤ πp (I) |f (s)|p dν(s) , f ∈ C[0, 1]. Now let t ∈ [0, 1] and ft (s) := f (t + s), addition by mod 1. We have: p ft E2 ≤ πp (I) |ft (s)|p dν(s) , t ∈ [0, 1]. Integrating this inequality by Lebesgue measure, we have f E2 ≤ πp (I)f Lp , f ∈ C[0, 1].

In order to give the analogous fact for (q, p) – absolutely summing operators, we 1 again return to Lorentz spaces Lq,1 := Λ(ϕq ), where ϕq (t) = t q . Another description ∞ 1 of its norm is as following: f = 0 (m(|f | > t)) q dt .

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Theorem 4 Let 1 ≤ p < q < ∞. The following assertions are equivalent: 1) I(C[0, 1]; E[0, 1]) ∈ Πq,p ; 1 2) ∃K > 0: ϕE (t) ≤ Kt q , 0 ≤ t ≤ 1; 3) E ⊃ Lq,1 . Proof. This theorem may be easily deduced from the recent factorization theorem of G. Pisier [5], but we prefer the direct way from the rather old paper of I. Novikov [3]. 1) ⇒ 2). Let I(C, E) ∈ Πq,p . Then, as is known from the results of B. Maurey I ∈ Πq,1 , that is ∃K > 0: ∀{x1 , . . . , xn } ∈ C[0, 1],

xi q

1/q

≤ K |xi | .

This inequality may be continued on {x1 , . . . , xn } ⊂ L∞ . If we set xi = 1[ i−1 , i ] , n n then nϕq ( n1 ) ≤ K, n = 1, 2, . . .; that is equivalent to 2). 2)⇒3) is well-known ([7]). 3)⇒1). Simple calculations (cf. [8] for q = 2) show that I(C, Lq,1 ) is (q, 1) – absolutely summing. There are some open problems in this area. As far as I know, there is not a single result concerning the (q, p) – absolutely summing property of inclusion map I(E1 , E2 ) for another RIS besides Lp -spaces.

3. Another ideal properties Definition. An operator T ∈ L(X, Y ) is of gaussian cotype q if for some C > 0

1 and all sequences (xi ) of X, we have ( T xi q ) q ≤ CEgi xi , where (gi ) denotes a sequence of independent normalized N (0, 1) –gaussian random variables. The set (g) of all operators of such kind forms an ideal and will be denoted by Cq . Not long ago M. Talagrand (preprint) and S. Montgomery-Smith (dissertation) found boundary spaces for the gaussian cotype 2 – property of inclusion map. Their result is the following Theorem 5 I(C, E) is of gaussian cotype 2 iﬀ E ⊃ LΦ,2 , where Φ(t) = t2 log t. The space LΦ,2 is deﬁned by the following norm: 1/2 2 f = θ m(|f | ≥ t) dt2 , where θ(t) = t ln . t

Boundary spaces for inclusion map

215 (g)

It’s not diﬃcult to show that LΦ,2 ⊇ L2,1 and so I(C; L2,1 ) ∈ Π2,1 \ C2 , i.e. (g) we have a nice counterexample to the conjecture C2 = Π2,1 . Thus, the space L2,1 is still a rich source of counterexamples. Another example of this statement is the following. Let E ⊆ L2 . The following conjecture was made by M. Braverman, N. Carothers and others. If (f1 ) ⊂ E and (fi ) are independent, identically distributed random variables such that Efi = 0, then [span (fi )]E is isomorphic to l2 . But this conjecture is not true. As shown in [4] the following equality is valid:

A(L2,1 ) := {(ai ) ∈ R∞ : ai fi converges for each sequence of i.i.d. {fi } : fi = 0, f1 ∈ L2,1 } = l2,1 . If the conjecture were true, we would have to have that A(L2,1 ) = l2 . The Theorems 1–5 give the basis for the following Conjecture. For each ideal U of operators there exists a boundary RIS EU such that I(C, E) ∈ U iﬀ E ⊃ EU , where the inclusion in the right hand may be strict or unstrict in dependence of the ideal U. As far as I know there is no answer to the question about the boundary space for the ideal of Rademacher cotype q – property.

References 1. V.F. Gaposhkin, Lacunary series and independent functions, Uspechy mat. nauk. 21, no 6 (1966), 3–83 (Russian). 2. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Function Spaces, Ergebnisse der Math., v. 97, Springer Verlag 1979. 3. I. Ya. Novikov, (p, q) – concavity of canonical inclusion map, Scientific Student Conference, Novosibirsk, 18 (1980), 36–42. 4. S. Ya. Novikov, The classes of coefficients for convergent random series in Lp,q -spaces, Theory of operators in function spaces (red. S. Gindikin), Saratov Univ. (1989), 180–192. English translation: Transl. Amer. Math. Soc. 153 (1992). 5. G. Pisier, Factorization of operators through Lp,∞ or Lp,1 , Math. Ann. 276 (1986), 105–136. 6. V.A. Rodin and E.M. Semenov, Rademacher series in symmetric spaces, Anal. Math. 1, no 3 (1975), 207–222. 7. E.M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton Univ. Press, 1971. 8. M. Talagrand, The canonical injection from C[0, 1] into L2,1 is not of cotype 2, Contemp. Math.: Banach Space Theory, 85 (1989), 513–521.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 217–236 c 1994 Universitat de Barcelona

Order continuous seminorms and weak compactness in Orlicz spaces

Marian Nowak Institute of Mathematics, Adam Mickiewicz University, Matejki 48/49, 60-769 Pozna´n, Poland

Abstract ϕ

Let L be an Orlicz space defined by a Young function ϕ over a σ -finite measure space, and let ϕ∗ denote the complementary function in the sense of Young. We ∗ give a characterization of the Mackey topology τ (L∗ , Lϕ ) in terms of some family of norms defined by some regular Young functions. Next, we describe order continuous (= absolutely continuous) Riesz seminorms on Lϕ , and obtain ∗ a criterion for relative σ(Lϕ , Lϕ )-compactness in Lϕ . As an application we get a representation of Lϕ as the union of some family of other Orlicz spaces. Finally, we apply the above results to the theory of Lebesgue spaces.

0. Introduction and preliminaries In 1915 de la Vall´ee Poussin (see [12]) showed that a set Z of L1 for a ﬁnite measure space (Ω, Σ, µ) has uniformly absolutely continuous L1 -norms (i.e., lim (sup |x(t)|dµ) = 0) iﬀ there exists a Young function ψ such that µ(E)→0 x∈Z E lim ψ(u)/u = ∞ in terms of which sup ψ(|x(t)|)dµ < ∞. u→∞

x∈Z Ω

On the other hand, in view of the Dunford-Pettis criterion (on relatively compact sets in L1 )(see [3, p. 294]) the set Z ⊂ L1 has uniformly absolutely continuous L1 -norms iﬀ it is relatively σ(L1 , L∞ )-compact. Thus we have the following criterion for relative weak compactness in L1 (for ﬁnite measures): a set Z of L1 is relatively σ(L1 , L∞)-compact iﬀ there exists a Young function ψ such that lim ψ(u)/u = ∞ and sup ψ(|x(t)|)dµ < ∞. u→∞

x∈Z Ω

217

218

Nowak ∗

In 1962 T. Ando [2, Theorem 2] found similar criterion for relative σ(Lϕ , Lϕ )compactness in Lϕ for ϕ being an N -function and a ﬁnite measure. This criterion was extended by the present author to the case of σ-ﬁnite measures ([15, Theorem 1.2]). In this paper, using a diﬀerent method, we extend the Ando’s criterion to the case of ϕ belonging to a much wider class of Young functions and σ-ﬁnite measures. We can include Lϕ being equal to L1 + L∞ (so L1 if µ(Ω) < ∞), L1 + Lp , Lp + L∞ (p > 1). In section 1, making use of the author’s results concerning the so-called modular topology Tϕ∧ on Lϕ (see [13], [14], [18], [19]), we obtain a characterization of the ∗ Mackey topology τ (Lϕ , Lϕ ) in terms of some family of norms deﬁned by some regular Young functions, dependent on ϕ (see Theorem 1.5). As an application we have a description of absolutely continuous (= order continuous) Riesz seminorms on Lϕ (see Corollary 1.6). ∗ In section 2, in view of the close connection between relative σ(Lϕ , Lϕ )∗ compactness in Lϕ and the absolute continuity of some seminorm in Lϕ , we can ∗ describe relatively σ(Lϕ , Lϕ )-compact sets in Lϕ as norm bounded subsets of an Orlicz space Lψ for some regular Young function ψ (see Theorem 2.4). As an application we get a representation of the Orlicz space Lϕ as the union of some family of ∗ other Orlicz spaces. At last, we examine the absolute weak topology |σ|(Lϕ , Lϕ ). In section 3 we apply the results of sections 1 and 2 to the theory of Lebesgue spaces. For notation and terminology concerning Riesz spaces we refer to [1], [21]. As usual, N stands for the set of all natural numbers. Let (Ω, Σ, µ) be σ-ﬁnite measure space, and let L0 denote the set of equivalence classes of all real valued measurable functions deﬁned and a.e. ﬁnite on Ω. Then L0 is a super Dedekind complete Riesz space under the ordering x ≤ y whenever x(t) ≤ y(t) a.e. on Ω. The Riesz F -norm x0 = Ω

|x(t)| f (t)dµ 1 + |x(t)|

where f : Ω → (0, ∞) is measurable and

Ω

for x ∈ L0 ,

f (t)dµ = 1, determines the Lebesgue

topology T0 on L0 , which generates the convergence in measure on subsets of Ω of ﬁnite measure. For a sequence (xn ) in L0 we will write xn → x(µ) whenever xn − x0 → 0. For a subset A of Ω and x ∈ L0 we will write xA = x · χA , where χA stands for the characteristic function of A. We will write En ∅ if (En ) is a decreasing

Order continuous seminorms and weak compactness in Orlicz spaces

219

sequence of measurable subsets of Ω such that µ(En ∩ E) → 0 for every set E ⊂ Ω of ﬁnite measure. Now we recall some notation and terminology concerning Orlicz spaces (see [6], [8], [10], [20] for more details). By an Orlicz function we mean a function ϕ: [0, ∞) → [0, ∞] which is nondecreasing, left continuous, continuous at 0 with ϕ(0) = 0, not identically equal to 0. An Orlicz function ϕ is called convex, whenever ϕ(αu + βv) ≤ αϕ(u) + βϕ(v) for α, β ≥ 0, α + β = 1 and u, v ≥ 0. A convex Orlicz function is usually called a Young function. For a Young function ϕ we denote by ϕ∗ the function complementary to ϕ in the sense of Young, i.e., for v ≥ 0. ϕ∗ (v) = sup uv − ϕ(u): u ≥ 0 For a set Ψ of Young functions we will write Ψ∗ = {ψ ∗ : ψ ∈ Ψ} . We shall say that an Orlicz function ψ is completely weaker than another ϕ a s for all u (resp. for small u; resp. for large u), in symbols ψ ϕ (resp. ψ ϕ; l

resp. ψ ϕ), if for an arbitrary c > 1 there exists a constant d > 0 such that ψ(cu) ≤ dϕ(u) for u ≥ 0 (resp. for 0 ≤ u ≤ u0 ; resp. for u ≥ u0 ≥ 0). (See [2], [20, Ch. II]). It is seen that ϕ satisﬁes the so called ∆2 -condition for all u (resp. for small u; a

s

l

resp. for large u) if and only if ϕ ϕ (resp. ϕ ϕ; resp. ϕ ϕ). We shall say that an Orlicz function ϕ increases more rapidly than another ψ a s for all u (resp. for small u; resp. for large u) in symbols ψ ≺ − ϕ (resp. ψ ≺ − ϕ; resp. l

ψ≺ − ϕ), if for an arbitrary c > 0 there exists d > 0 such that cψ(u) ≤ d1 ϕ(du) for all u ≥ 0 (resp. for 0 ≤ u ≤ u0 ; resp. for u ≥ uo ≥ 0). Note that ϕ satisﬁes the so called ∇2 -condition for all u (resp. for small u; resp. a

s

l

for large u) if and only if ϕ ≺ − ϕ (resp. ϕ ≺ − ϕ; resp. ϕ ≺ − ϕ). a One can verify that for given Young functions ψ and ϕ the relation ψ ϕ (resp. s

l

a

s

l

ψ ϕ; resp. ψ ϕ) holds iﬀ ϕ∗ ≺ − ψ ∗ (resp. ϕ∗ ≺ − ϕ∗ , resp. ϕ∗ ≺ − ψ ∗ ) holds (see [2], [20, Proposition 2.2.4]).

220

Nowak An Orlicz function ϕ determines the functional mϕ : L0 → [0, ∞] by ϕ(|x(t)|)dµ. mϕ (x) = Ω

The Orlicz space generated by ϕ is the ideal of L0 deﬁned by Lϕ = {x ∈ L0 : mϕ (λx) < ∞ for some λ > 0}. The functional mϕ restricted to Lϕ is an orthogonally additive semimodular (see [10], [11]). Lϕ can be equipped with the complete metrizable linear topology Tϕ of the Riesz F -norm x

≤λ . x ϕ = inf λ > 0: mϕ λ Moreover, when ϕ is a Young function, the topology Tϕ can be generated by two Riesz norms (called the Orlicz and the Luxemburg norms resp.) deﬁned as follows: ∗ xϕ = sup |x(t)y(t)|dµ: y ∈ Lϕ , mϕ∗ (y) ≤ 1 Ω x |||x|||ϕ = inf λ > 0: mϕ ( ) ≤ 1 . λ For an Orlicz function ϕ let E ϕ = {x ∈ L0 : mϕ (λx) < ∞ for all λ > 0} and ϕ Lϕ a = {x ∈ L : xEn ϕ → 0

as En ∅}.

It is well known that for ϕ taking only ﬁnite values these spaces coincide, i.e., E ϕ = Lϕ a.

∗

1. The Mackey topology τ (Lϕ , Lϕ ) First we recall the deﬁnition and the basic properties of the so-called modular topology on Orlicz spaces (see [13], [14]). Let ϕ be an Orlicz function vanishing only at 0. For given ε > 0, let Uϕ (ε) = {x ∈ Lϕ : mϕ (x) ≤ ε}. Then the family of all sets of the form ∞ N

Uϕ (εn ) N =1

n=1

where (εn ) is a sequence of positive numbers, forms a base of neighborhoods of 0 for a linear topology on Lϕ , that will be called the modular topology on Lϕ and will be denoted by Tϕ∧ . The basic properties of Tϕ∧ are included in the following theorem (see [14, Theorem 1.1], [18, Theorem 2.2], [19. Theorem 4.2]):

Order continuous seminorms and weak compactness in Orlicz spaces

221

Theorem 1.1 Let ϕ be an Orlicz function vanishing only at 0. Then the following statements hold: (i) Tϕ∧ is the ﬁnest σ-Lebesgue topology on Lϕ . (ii) Tϕ∧ ⊂ Tϕ and the equality Tϕ∧ = Tϕ holds whenever ϕ ∈ ∆2 . ∗ (iii) Tϕ∧ coincides with the Mackey topology τ (Lϕ , Lϕ ), whenever ϕ is a Young function. To present the crucial for this paper characterization of the modular topology we will distinguish some classes of Orlicz functions. An Orlicz function ϕ continuous for all u ≥ 0, taking only ﬁnite values, vanishing only at zero, and such that ϕ(u) → ∞ as u → ∞ is usually called a ϕ-function (see [10]). We will denote by Φ the collection of all ϕ-functions. A Young function ϕ vanishing only at 0 and taking only ﬁnite values is called an N -function whenever lim ϕ(u)/u = 0 and lim ϕ(u)/u = ∞ (see [6], [10]). We

Tϕ∧

u→∞

u→0

will denote by ΦN the collection of all N -functions. Let Φ0 be the collection of all Orlicz functions ϕ vanishing only at 0 and such that ϕ(u) → ∞ as u → ∞. Let Φ01 = {ϕ ∈ Φ0 : ϕ(u) < ∞ for u ≥ 0} Φ02 = {ϕ ∈ Φ0 : ϕ jumps to ∞, i.e., ϕ(u) = ∞ for u > u0 > 0}. The following characterizations of the modular topology Tϕ∧ will be crucial for this paper (see [13, Theorem 2.1], [14, Theorem 1.2]). Theorem 1.2 Let ϕ ∈ Φ0i (i = 1, 2). Then the modular topology Tϕ∧ is generated by the family of F -norms: { · ψ|Lϕ : ψ ∈ Ψϕ 0i } where a

Ψϕ 01 = {ψ ∈ Ψ: ψ ϕ},

s

Ψϕ 02 = {ψ ∈ Φ: ψ ϕ}.

Now, for ϕ being a Young function we are going to apply Theorem 1.2 to obtain ∗ a description of the Mackey topology τ (Lϕ , Lϕ ) in terms of some family of norms deﬁned by some regular Young functions. For this purpose we distinguish some classes of Young functions.

222

Nowak

Let Φc0 be the collection of all Young functions ϕ vanishing only at 0 and such that lim ϕ(u)/u = ∞ . u→∞

Let Φc01 = {ϕ ∈ Φc0 : ϕ(u) < ∞ for all u ≥ 0 and lim

u→0

Φc02 = {ϕ ∈ Φc0 : ϕ jumps to ∞

and lim

u→0

ϕ(u) u

u→0

4

Then Φc0 = that Φc01 =

and lim

u→0

ϕ(u) u

= 0},

= 0},

Φc03 = {ϕ ∈ Φc0 : ϕ(u) < ∞ for all u ≥ 0 and lim Φc04 = {ϕ ∈ Φc0 : ϕ jumps to ∞

ϕ(u) u

ϕ(u) u

> 0},

> 0}.

Φc0i , and the sets Φc0i (i = 1, 2, 3, 4) are pairwise disjoint. It is seen

i=1 ΦN .

Denote by a

c Ψϕ 01 (c) = {ψ ∈ ΦN : ψ ϕ}, whenever ϕ ∈ Φ01 , s

c Ψϕ 02 (c) = {ψ ∈ ΦN : ψ ϕ}, whenever ϕ ∈ Φ02 , l

c Ψϕ 03 (c) = {ψ ∈ Φ03 : ψ ϕ}, whenever ϕ ∈ Φ03 , c c Ψϕ 04 (c) = Φ03 , whenever ϕ ∈ Φ04 .

The following two lemmas will be needed. Lemma 1.3 a

Let ϕ ∈ Φc0i (i = 1, 2) and let ψ be a ϕ-function such that ψ ϕ for i = 1 (resp. s ψ ϕ for i = 2). Then there exists ψ0 ∈ Ψϕ 0i (c) such that ψ(u) ≤ ψ0 (2u) for u ≥ 0.

a

s

Proof. Take an arbitrary N -function ψ1 such that ψ1 ϕ for i = 1 (resp. ψ1 ϕ for i = 2). Let us set ψ2 (u) = max ψ(u), ψ1 (u) for u ≥ 0. Let us put

0

p(s) = sup0
for s = 0, ψ2 (t) t

for s > 0,

Order continuous seminorms and weak compactness in Orlicz spaces

and let ψ0 (u) =

223

u

p(s)ds. 0

To show that ψ0 is an N -function we have to check that lim p(u) = 0 and lim p(u) = u→∞ u→0 ∞. a Indeed, since ψ ϕ we get that ψ(u) ≤ aϕ(u) for some a > 0 and u ≥ 0. Hence ψ2 (t) ψ(t) ψ1 (t) ≤ sup + sup t t t 0
p(u) = sup

ϕ(t) ψ1 (t) ϕ(u) ψ1 (t) + sup =a + . t t u u 0
≤ a sup

Thus lim p(u) = 0, because lim ϕ(u)/u = 0 and lim ψ1 (u)/u = 0. Moreover, u→∞

u→0

u→0

we have: p(u) = sup ψ2 (t)/t ≥ sup ψ1 (t)/t = ψ1 (u)/u, because ψ1 is a Young 0
0
function. Hence lim p(u) = ∞, because lim ψ1 (u)/u = ∞. u→∞

u→∞

a

s

Now we shall show that ψ0 ϕ if i = 1 (resp. ψ ϕ if i = 2). Indeed, given c > 0 there exist d > 0 such that u

for u ≥ 0. ψ2 (u) = ψ(u) ∨ ψ1 (u) ≤ dϕ c Hence

dϕ( ct ) ψ2 (t) dϕ(u) ≤ sup = for u ≥ 0, t t cu 0
p(cu) = sup so

ψ0 (cu) ≤ p(cu) · cu ≤ dϕ(u) for u ≥ 0. s

Similarly we can show that ψ0 ϕ if i = 2. At last we will show that ψ(u) ≤ ψ0 (2u) for u ≥ 0. Indeed, we have ψ0 (2u) ≥ p(u) · u and ψ2 (t) ψ(t) ψ(u) ≥ sup ≥ for u ≥ 0. t u 0
p(u) = sup Thus

ψ0 (2u) ≥ ψ(u) for u ≥ 0.

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Lemma 1.4 a

Let ϕ ∈ Φc0i (i = 3, 4) and let ψ be a ϕ-function such that ψ ϕ for i = 3 (resp. s ψ ϕ for i = 4). Then there exists a Young function ψ0 ∈ Ψϕ 0i (c) such that ψ(u) ≤ ψ0 (2u) for u ≥ 0.

a

Proof. Take an arbitrary N -function ψ1 such that ψ1 (u) ≤ ϕ(u) and ψ1 ϕ. Let a > 1 be such that aψ1 (1) = ϕ(1). Let us set ψ2 (u) =

max ψ(u), ϕ(u) max ψ(u), aψ1 (u)

Let

0

for u ≥ 1. for s = 0,

p(s) = sup0
and let

for 0 ≤ u ≤ 1,

ψ2 (t) t

for s > 0,

u

p(s)ds for u ≥ 0.

ψ0 (u) = 0

We shall show that ψ0 ∈ Ψc03 , i.e., that lim ψ0 (u)/u > 0 and lim ψ0 (u)/u = ∞. u→∞

u→0

Indeed, for 0 ≤ u ≤ 1 we have

ψ2 (t) ϕ(t) ϕ(u) ≥ sup = , t u 0
p(u) = sup so

lim p(u) ≥ lim

u→0

u→0

ϕ(u) > 0. u

Since ψ0 (u) ≥ p( u2 ) · u2 , we get lim ψ0 (u)/u > 0. u→0

To show that lim ψ0 (u)/u = ∞ it is enough to show that lim p(u) = ∞. u→∞

u→∞

Indeed, let u0 > 1 be such that aψ1 (u)/u ≥ K = sup ψ2 (t)/t for u ≥ u0 . Then for u ≥ u0 we have:

0
ψ2 (t) ψ2 (t)

ψ1 (t)

= max K, sup ≥ max K, sup t t t 0
ψ1 (u) aψ1 (u) = max K, sup = . u u 1≤t≤u

p(u) = sup

Order continuous seminorms and weak compactness in Orlicz spaces

225

Thus lim p(u) = ∞, because lim ψ1 (u)/u = ∞. u→∞

u→∞

l

Now, for i = 3 we shall show that ψ0 ϕ. Indeed, given c > 1 there exists d > 1 such that for u ≥ 0 ψ(u) ≤ dϕ

u

c

and aψ1 (u) ≤ dϕ

u

c

.

Let u0 > 0 be such that dϕ(u0 )/u0 ≥ K = sup ψ2 (t)/t. Then for u ≥ uo we get 0
ψ2 (t) ψ2 (t) ψ2 (t)

= max sup , sup t t t 0

dϕ( c ) dϕ(u) dϕ(u) = max K, = . ≤ max K, sup t cu cu 1≤t≤cu

p(cu) = sup

Thus for u ≥ u0 ψ0 (cu) ≤ p(cu) · cu ≤ dϕ(u) , l

i.e. ψ0 ϕ. At last, we shall show that ψ(u) ≤ ψ0 (2u) for u ≥ 0 (i = 3, 4). Indeed, we have ψ0 (2u) ≥ p(u) · u and ψ2 (t) ψ(u) ≥ sup t 0
p(u) = sup

for u ≥ 0.

Thus ψ(u) ≤ ψ0 (2u) for u ≥ 0. We are now in position to present a description of the Mackey topology ∗ τ (Lϕ , Lϕ ) in terms of some family of norms deﬁned by some regular Young functions. Theorem 1.5 ∗

Let ϕ ∈ Φc0i (i = 1, 2, 3, 4). Then the Mackey topology τ (Lϕ , Lϕ ) is generated by the family of norms: {||| · |||ψ|Lϕ : ψ ∈ Ψϕ 0i (c)}.

226

Nowak ∗

Proof. In view of Theorem 1.1 the equality Tϕ∧ = τ (Lϕ , Lϕ ) holds. Let ϕ ∈ Φc0i (i = 1, 2, 3, 4). Then ϕ ∈ Φ01 for i = 1, 3, and ϕ ∈ Φ02 for i = 2, 4. Thus, ∗ according to Theorem 1.2 the Mackey topology τ (Lϕ , Lϕ ) is generated by the family ϕ { · ψ|Lϕ : ψ ∈ Ψϕ 01 } for i = 1, 3, and by the family { · ψ|Lϕ : ψ ∈ Ψ01 } for i = 2, 4. ϕ Now let ψ ∈ Ψϕ 01 (resp. ψ ∈ Ψ02 ), and let r > 0 be given. In view of Lemma ϕ 1.3 (resp. Lemma 1.4) there exists ψ0 ∈ Ψϕ 0i (c) for i = 1, 3 (resp. ψ0 ∈ Ψ0i (c) for i = 2, 4) such that ψ(u) ≤ ψ0 (2u) for u ≥ 0. Hence x Since the F -norms · such that

ψ

ψ0

≤ 2x

ψ0

for all x ∈ Lψ0 .

(1)

and ||| · |||ψ0 are equivalent on Lψ0 , there exists r1 > 0 B(ψ0 ) (r1 ) ⊂ Bψ0 (r),

(2)

where Bψ0 (r) = {x ∈ Lψ0 : x

ψ0

≤ r}

and B(ψ0 ) (r1 ) = {x ∈ Lψ0 : |||x|||ψ0 ≤ r1 }.

We shall show that B(ψo ) ( r21 ) ∩ Lϕ ⊂ Bψ (r). Indeed, let x ∈ B(ψ0 ) ( r21 ) ∩ Lϕ . Then |||2x|||ψ0 ≤ r1 ; hence by (2), 2x ψ0 ≤ r. Next, by (1) we get that x ψ ≤ r. Thus we proved that the topology τϕ∗ generated by the family of norms ϕ ϕ∗ {||| · |||ψ : ψ ∈ Ψϕ 0i (c)} is ﬁner than τ (L , L ). On the other hand, since for ψ ∈ Ψϕ 0i (c) the F -norms · ψ and ||| · |||ψ are equiv∗ alent on Lϕ , we get that τ (Lϕ , Lϕ ) is ﬁner than τϕ∗ . Thus the proof is completed. As an application of Theorem 1.5 we obtain a characterization of absolutely continuous (order continuous) seminorms on Lϕ (see [2, Theorem 3]). Corollary 1.6 Let ϕ ∈ Φc0i (i = 1, 2, 3, 4). Then for a Riesz seminorm p on Lϕ the following statements are equivalent: (i) p is order continuous (i.e., p(xn ) ↓ 0 whenever xn ↓ 0 in Lϕ ). (ii) p is absolutely continuous (i.e. p(xEn ) → 0 whenever En ∅ and x ∈ Lϕ ). (iii) There exists ψ ∈ Ψϕ 0i (c) and a number a > 0 such that p(x) ≤ a|||x|||ψ

for x ∈ Lϕ .

Order continuous seminorms and weak compactness in Orlicz spaces Proof. (i)⇔(ii)

227

See [9, Theorem 2.1]. ∗

(i)⇒(iii) Let ϕ ∈ Φc0i (i = 1, 2, 3, 4). Since τ (Lϕ , Lϕ ) is the ﬁnest σ-Lebesgue topology on Lϕ (see Theorem 1.1) in view of Theorem 1.5 and [5, Ch. 4, § 18, (4)] there exist ψ1 , . . . , ψn ∈ Ψϕ 0i (c) and a number a > 0 such that p(x) ≤ a max{|||x|||ψ1 , . . . , |||x|||ψn } Let us put

ψ(u) = max ψ1 (u), . . . , ψn (u)

for all x ∈ Lϕ .

for u ≥ 0.

ϕ ϕ Then ψ ∈ Ψϕ 0i (c) and |||x|||ψi ≤ |||x|||ψ for x ∈ L , so p(x) ≤ a|||x|||ψ for x ∈ L . ∗

(iii)⇒(i) Since τ (Lϕ , Lϕ ) is a σ-Lebesgue topology, by Theorem 1.5, for each ϕ ψ ∈ Ψϕ 0i (c) the norm ||| · |||ψ is order continuous on L ; so p is also order continuous ϕ on L .

2. Weak compactness in Orlicz spaces Throughout this section we assume that (Ω, Σ, µ) is a σ-ﬁnite measure space. ∗ For any Young function ϕ the following criterion for relative σ(Lϕ , Lϕ )compactness is well known (see [11, §28], [8, Ch. I, §3, Theorem 5], [20, Corollary 4.5.2]): Theorem 2.1 Let ϕ be a Young function. For a subset Z of Lϕ the following statements are equivalent: ∗ (i) Z is relatively σ(Lϕ , Lϕ )-compact. ∗ ∗ (ii) Z is σ(Lϕ , Lϕ )-bounded and for each y ∈ Lϕ |x(t)y(t)|dµ = 0

lim sup

n→∞ x∈Z

whenever En ∅.

En

∗

The next theorem presents conditions for relative σ(Lϕ , Lϕ )-compact embeddings of Orlicz spaces. This theorem was proved in a diﬀerent way in [20, Theorem 5.3.3] for ϕ being an N -function.

228

Nowak

Theorem 2.2 Let ϕ and ψ be Young functions. a

l

s

10 . If ϕ ≺ − ψ (resp. ϕ ≺ − ψ if µ(Ω) < ∞, resp. ϕ ≺ − ψ if µ is the counting measure on N), then the embedding i: Lψ (→ Lϕ ∗

is relatively σ(Lϕ , Lϕ )-compact (i.e., every norm bounded subset of Lϕ is relatively ∗ σ(Lϕ , Lϕ )-compact). 2 0 . Let Lψ ⊂ Lϕ with lim ψ(u)/u = ∞, and let the measure space (Ω, Σ, µ) be u→∞

inﬁnite and atomless (resp. ﬁnite and atomless; resp. Ω = N with µ being the counting measure). If the embedding i: Lψ (→ Lϕ a

∗

l

s

is relatively σ(Lϕ , Lϕ )-compact, then ϕ ≺ − ψ (resp. ϕ ≺ − ψ; resp. ϕ ≺ − ψ). Proof. 10 . We have Lψ ⊂ Lϕ and the Young function ψ ∗ is ﬁnite valued because lim ψ(u)/u = ∞. Let the set Z ⊂ Lψ be norm bounded, i.e., sup{|||x|||ψ : x ∈ Z} < u→∞

∗

∞. For y ∈ Lϕ let us put

|x(t)y(t)|dµ: x ∈ Z .

pZ (y) = sup Ω

In view of Theorem 2.1 we have to show that the seminorm pZ is absolutely contin∗ ∗ ∗ uous on Lϕ , i.e., pZ (yEn ) → 0, as En ∅ for y ∈ Lϕ . Indeed, let y ∈ Lϕ and a

l

s

a

En ∅. Since ϕ ≺ − ψ (resp. ϕ ≺ − ψ; resp. ϕ ≺ − ψ) we get that ψ ∗ ϕ∗ (resp. l

s

∗

∗

∗

ψ ∗ ϕ∗ ; resp. ψ ∗ ϕ∗ ). Hence Lψ ⊂ E ψ = Lψ a (see [20, Theorem 5.3.1]). By applying H¨ older’s inequality (see [20, Ch. III, §3]) we get pZ (yEn ) = sup |x(t)yEn (t)|dµ: x ∈ Z Ω ≤ yEn ψ∗ · sup |||x|||ψ : x ∈ Z . ∗

Thus pZ (yEn ) → 0, because y ∈ Lψ a . ∗ ∗ 20 . Since Lψ ⊂ Lϕ we have Lϕ ⊂ Lψ , and ψ ∗ is ﬁnite valued. To prove that a

s

l

∗

∗

− ψ (resp. ϕ ≺ − ψ, resp. ϕ ≺ − ψ) it is enough to show that Lϕ ⊂ E ψ , because ϕ≺ a

s

l

this inclusion implies that ψ ∗ ϕ∗ (resp. ψ ∗ ϕ∗ ; resp. ψ ∗ ϕ∗ ) (see [20, Theorem 5.3.1]).

Order continuous seminorms and weak compactness in Orlicz spaces

229

∗

Indeed, let y ∈ Lϕ . Since the unit ball Bψ (1) = {x ∈ Lψ : |||x|||ψ ≤ 1} ⊂ Lϕ is ∗ ϕ σ(L , Lϕ )-compact, in view of Theorem 2.1 we get that yEn ψ∗ = sup |x(t)yEn (t)|dµ: x ∈ Lψ , x ∈ Bψ (1) → 0 as En ∅. Ω ∗

∗

ψ This means that y ∈ Lψ . a =E

Corollary 2.3 Let ϕ be a Young function, and let the measure space (Ω, Σ, µ) be inﬁnite and atomless (resp. ﬁnite and atomless; resp. Ω = N with µ being the counting measure). Then the following statements are equivalent: (i) ϕ satisﬁes the 2 -condition for all u (resp. for large u; resp. for small u). ∗ (ii) Every norm bounded subset of Lϕ is relatively σ(Lϕ , Lϕ )-compact. ∗

The main aim of this section is to show that a relatively σ(Lϕ , Lϕ )-compact subset of Lϕ (for ϕ being a ﬁnite valued Young function) is norm bounded in Lψ for some regular Young function ψ dependent on ϕ. This result extends the well-known Ando’s criterion for relative weak compactness in Lϕ obtained for ϕ being an N -function and ﬁnite measures (see [2, Theorem 2]). For this purpose we distinguish some classes of Young functions. Let Φc1 be the collection of Young functions taking only ﬁnite values and such that lim ϕ(u)/u = 0. u→0

Let

Then Φc1 =

Φc11 = {ϕ ∈ Φc1 : ϕ(u) > 0

for u > 0, and lim

ϕ(u) u

= ∞} ,

Φc12 = {ϕ ∈ Φc1 : ϕ(u) > 0

for u > 0, and lim

ϕ(u) u

< ∞} ,

Φc13 = {ϕ ∈ Φc1 : ϕ(u) = 0

near u > 0, and lim

ϕ(u) u

= ∞} ,

Φc14 = {ϕ ∈ Φc1 : ϕ(u) = 0

near u > 0, and lim

ϕ(u) u

< ∞} .

∞ i=1

u→∞ u→∞

u→∞ u→∞

Φc1i , and the sets Φc1i are pairwise disjoint. It is seen that Φc11 = ΦN .

Denote by a

whenever ϕ ∈ Φc11 ,

s

whenever ϕ ∈ Φc12 ,

l

whenever ϕ ∈ Φc13 ,

Ψϕ − ψ}, 11 (c) = {ψ ∈ ΦN : ϕ ≺ − ψ}, Ψϕ 12 (c) = {ψ ∈ ΦN : ϕ ≺ c − ψ}, Ψϕ 13 (c) = {ψ ∈ Φ13 : ϕ ≺ c Ψϕ 14 (c) = Φ13 ,

whenever ϕ ∈ Φc14 .

The next important lemma shows the relation between the sets Φc0i and Φc1i , ϕ and the sets Ψϕ 0i (c) and Ψ1i (c) (i = 1, 2, 3, 4).

230

Nowak

Lemma 2.4 10 .

Let ϕ ∈ Φc0i (i = 1, 2, 3, 4). Then ϕ∗ ∈ Φc1i and ϕ ∗ ∗ Ψ0i (c) = Ψϕ 1i (c).

2 0.

Let ϕ ∈ Φc1i (i = 1, 2, 3, 4). Then ϕ∗ ∈ Φc0i and ϕ ∗ ∗ Ψ1i (c) = Ψϕ 0i (c).

Proof. In view of [17, Lemma 3.1] ϕ∗ ∈ Φc1i whenever ϕ ∈ Φc0i , and ϕ∗ ∈ Φc0i whenever ϕ ∈ Φc1i (i = 1, 2, 3, 4). a But it is known that for Young functions ψ and ϕ the relation ψ ϕ (resp. s

a

l

s

ψ ϕ; resp. ψ ϕ) holds if and only if the relation ϕ∗ ≺ − ψ ∗ (resp. ϕ∗ ≺ − ψ ∗ ; resp. l

ϕ∗ ≺ − ψ ∗ ) holds (see [20, proposition 2.2.4]). ∗

Now we are ready to obtain our desired description of relatively σ(Lϕ , Lϕ )compact sets in Lϕ . Theorem 2.5 Let ϕ ∈ Φc1i (i = 1, 2, 3, 4). For a subset Z of Lϕ the following statements are equivalent: ∗ (i) Z is relatively σ(Lϕ , Lϕ )-compact. ψ (ii) There exists ψ ∈ Ψϕ 1i (c) such that Z ⊂ L and sup xψ : x ∈ Z < ∞ .

∗

Proof. (i)⇒(ii) Since the set Z ⊂ Lϕ is relatively σ(Lϕ , Lϕ )-compact, in view of Theorem 2.1 the seminorm pZ (y) = sup{ |x(t)y(t)|dµ: x ∈ Z} is absolutely Ω

∗

∗

∗

ϕ ⊂ E ψ0 ) continuous on Lϕ . Hence by Corollary 1.6 there exist ψ0 ∈ Ψϕ 0i (c) (so L and a number a > 0 such that

pZ (y) ≤ a|||y|||ψ0 ∗

∗

for y ∈ Lϕ .

We shall show that Z ⊂ Lψ0 and sup{xψ0∗ : x ∈ Z} ≤ a.

(1)

Order continuous seminorms and weak compactness in Orlicz spaces

231

∗

Indeed, let x ∈ Z. Then by (1), for y ∈ Lϕ , |||y|||ψ0 ≤ 1 we get that |x(t)y(t)|dµ ≤ a.

(2)

Ω

Since the measure space (Ω, Σ, µ) is σ-ﬁnite, there exists a sequence (Ωn ) of measur∞ able subsets of Ω such that Ωn ↑, Ω = Ωn , µ(Ωn ) < ∞. Let z ∈ Lψ0 and z = 0. n=1

For n = 1, 2, . . . denote by z Then |x(t)z

(n)

(t) =

z(t)

if |z(t)| ≤ n and t ∈ Ωn ,

0

elsewhere.

(t)| ↑n |x(t)z(t)| on Ω, so by Fatou’s lemma and (2) we obtain 1 |x(t)z(t)|dµ ≤ sup |x(t)z (n) (t)|dµ |||z|||ψ0 n Ω Ω ∗ ≤ sup |x(t)y(t)|dµ: y ∈ Lϕ , |||y|||ψ0 ≤ 1 ≤ a.

(n)

1 |||z|||ψ0

Ω ψ0 ×

ψ0∗

ψ0 ×

Hence x ∈ (L ) = L , where (L ) denotes the K¨othe dual of Lψ0 . Moreover, since ∗ xψ0 = sup x(t)z(t)dµ: z ∈ Lψ0 , |||z|||ψ0 ≤ 1 Ω

we get that x ≤ a. Putting ψ = ψ0∗ and using Lemma 2.4 we get that ψ ∈ Ψϕ 1i (c) and Z ⊂ Lψ with sup{xψ : x ∈ Z} ≤ a. (ii)⇒(i) It follows from Theorem 2.2. ψ0∗

As an application of Theorem 2.5 we obtain a representation of Lϕ as the union of some family of other Orlicz spaces. Corollary 2.6 Let ϕ ∈ Φc1i (i = 1, 2, 3, 4). Then the following equality holds: Lϕ = {Lψ : ψ ∈ Ψϕ 1i (c)}. Proof. From Theorem 2.5 we obtain that Lϕ ⊂ hand, Lψ ⊂ Lϕ for each ψ ∈ Ψϕ 1i (c).

{Lψ : ψ ∈ Ψϕ 1i (c)}. On the other

Remark. The equality from Corollary 2.6 for i = 1, 2 was obtained in a diﬀerent way in [14, Theorem 2.6]. At last, we apply Theorem 2.5 to examination of the absolute weak topology ∗ |σ|(Lϕ , Lϕ ) (see [8, Deﬁnition 2, p. 27]).

232

Nowak

Theorem 2.7 Let ϕ ∈ Φc1i (i = 1, 2, 3, 4). For a sequence (xn ) in Lϕ the following statements are equivalent: ∗ (i) xn → 0 for |σ|(Lϕ , Lϕ ). ∗ (ii) xn → 0(µ) and the set {xn } is relatively σ(Lϕ , Lϕ )-compact. (iii) xn → 0(µ) and supn xn ψ < ∞ for some Young function ψ ∈ Ψϕ 1i (c). Proof. (i)⇔(ii) (ii)⇔(iii)

See [16, Theorem 2.1]. It follows from Theorem 2.5.

Theorem 2.8 ψ ϕ Let ϕ ∈ Φc1i (i = 1, 2, 3, 4). If ψ ∈ Ψϕ 1i (c) and Z ⊂ L ⊂ L and sup{xψ : x ∈ ∗ Z} < ∞, then the topologies T0 and |σ|(Lϕ , Lϕ ) coincide on Z, i.e., ∗

T0|Z = |σ|(Lϕ , Lϕ )|Z .

∗

Proof. It is well known that T0|Z ⊂ |σ|(Lϕ , Lϕ )|Z (see [7, Ch. X, §5, Lemma 1]). Since T0 is a linear metrizable topology from Theorem 2.7 it follows that ∗ |σ|(Lϕ , Lϕ )|Z ⊂ T0|Z . Theorem 2.9 Let ϕ ∈ Φc1i (i = 1, 2, 3, 4). For a subset Z of Lϕ the following statements are equivalent: ∗ (i) Z is relatively compact for |σ|(Lϕ , Lϕ ). (ii) Z is relatively compact for T0 , and there exists a Young function ψ ∈ Ψϕ 1i (c) such that Z ⊂ Lψ and sup{xψ : x ∈ Z} < ∞. Proof. It follows from Theorem 2.5 and [8, Ch. I, §3, Corollary of Lemma 11].

3. Applications to the theory of Lebesgue spaces In this section we will apply Theorem 1.5, Corollary 1.6, Theorem 2.5 and Corollary 2.6 to the theory of Lebesgue spaces. We will assume that (Ω, Σ, µ) is a σ-ﬁnite measure space. A. Let ϕ(u) = χ1 (u) ∨ χ∞ (u) for u ≥ 0,

Order continuous seminorms and weak compactness in Orlicz spaces where

χ1 (u) = u

for u ≥ 0,

χ∞ =

0

233

for 0 ≤ u ≤ 1,

∞ for u > 1.

Then ϕ ∈ Φc04 and Lϕ = L1 ∩ L∞ . Moreover, by Lemma 2.4, ϕ∗ ∈ Φc14 and ∗ a ϕ∗ = (χ1 ∨ χ∞ )∗ ∼ χ∗1 ∨ χ∗∞ = χ∞ ∨ χ1 , so Lϕ = L1 + L∞ (see [4, Theorem 3]). Theorem 3.1 The following statements hold: 10 . The Mackey topology τ (L1 ∩ L∞ , L1 + L∞ ) is generated by the family of norms: {||| · |||ψ|L1 ∩L∞ : ψ ∈ Φc03 }. 2 0 . For a Riesz seminorm p on L1 ∩ L∞ the following statements are equivalent: (i) p is order continuous. (ii) There exist ψ ∈ Φc03 and a number a > 0 such that p(x) ≤ a|||x|||ψ

for x ∈ L1 ∩ L∞ .

3 0 . For a subset Z of L1 + L∞ the following statements are equivalent (i) Z is relatively σ(L1 + L∞ , L1 ∩ L∞ )-compact. (ii) There exists ψ ∈ Φc13 such that Z ⊂ Lψ and sup{|||x|||ψ : x ∈ Z} < ∞. 4 0 . The following equality holds L1 + L∞ =

B. Let p > 1, q > 1 and

1 p

+

1 q

{Lψ : ψ ∈ Φc13 }.

= 1. Let

ϕ(u) = χp (u) ∨ χ∞ (u)

for u ≥ 0,

where χp (u) = up for u ≥ 0. Then ϕ ∈ Φc02 and Lϕ = Lp ∩ L∞ . Moreover, by ∗ a Lemma 2.4, ϕ∗ ∈ Φc12 and ϕ∗ = (χp ∨ χ∞ )∗ ∼ χ∗p ∧ χ∗∞ = χq ∧ χ1 ; so Lϕ = L1 + Lq .

234

Nowak

Theorem 3.2. The following statements hold: 1 . The Mackey topology τ (Lp ∩ L∞ , Lq + L1 ) is generated by the family of norms: 0

||| · |||ψ|Lp ∩L∞ : ψ ∈ ΦN

and lim sup u→0

ψ(u) < ∞ . up

2 0 . For a Riesz seminorm p on Lp ∩ L∞ the following statements are equivalent: (i) p is order continuous. (ii) There exist an N -function ψ with lim sup ψ(u) up < ∞ and a number a > 0 u→0

such that for x ∈ Lp ∩ L∞ .

p(x) ≤ a|||x|||ψ

3 0 . For a subset Z of Lq + L1 the following statements are equivalent: (i) Z is relatively σ(Lq + L1 , Lp ∩ L∞ )-compact. s

− ψ such that Z ⊂ Lψ and (ii) There exists an N -function ψ with χ1 ≺ sup{|||x|||ψ : x ∈ Z} < ∞. 4 0 . The following equality holds: Lq + L1 =

C. Let p > 1, q > 1 and

1 p

+

{Lψ : ψ ∈ ΦN

1 q

s

and χq ≺ − ψ}.

= 1. Let

ϕ(u) = χ1 (u) ∨ χp (u)

for u ≥ 0.

Then ϕ ∈ Φc03 and Lϕ = L1 ∩ Lp . Moreover, by Lemma 2.4, ϕ∗ ∈ Φc13 , and ∗ a ϕ∗ = (χ1 ∨ χp )∗ ∼ χ∗1 ∧ χ∗p = χ∞ ∧ χq . Hence Lϕ = Lq + L∞ . Theorem 3.3 The following statements hold: 1 . The Mackey topology τ (L1 ∩ Lp , Lq + L∞ ) is generated by the family of norms: 0

||| · |||ψ|L1 ∩Lp : ψ ∈ Φc03

and lim sup u→∞

ψ(u) < ∞ . up

2 0 . For a Riesz seminorm p on L1 ∩ Lp the following statements are equivalent: (i) p is order continuous on L1 ∩ Lp .

Order continuous seminorms and weak compactness in Orlicz spaces

235

(ii) There exist ψ ∈ Φc03 with lim sup ψ(u) up < ∞ and a number a > 0 such that u→∞

p(x) ≤ a|||x|||ψ

for x ∈ L1 ∩ Lp .

3 0 . For a subset Z of Lq + L∞ the following statements are equivalent: (i) Z is relatively σ(Lq + L∞ , L1 ∩ Lp )-compact. l

(ii) There exists ψ ∈ Φc13 with χq ≺ − ψ such that Z ⊂ Lψ and sup{|||x|||ψ : x ∈ Z} < ∞. 4 0 . The following equality holds: Lq + L∞ =

{Lψ : ψ ∈ Φc13

l

and χq ≺ − ψ}.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

C.D. Aliprantis, 0. Burkinshaw, Locally solid Riesz spaces, Academic Press, New York, 1978. T. Ando, Weakly compact sets in Orlicz spaces, Canadian J. Math. 14 (1962), 170–176. N. Dunford, J. Schwartz, Linear operators I, Interscience Publishers, New York, 1958. H. Hudzik, Intersections and algebraic sums of Musielak-Orlicz spaces, Portugaliae Math. 40 (1985), 287–296. G. K¨othe, Topological vector spaces I, Springer-Verlag, Berlin, Heidelberg, New York, 1983. M. Krasnoselskii, Ya. B. Rutickii, Convex functions and Orlicz spaces, P. Noordhoof Ltd., Groningen, 1961. L.V. Kantorovitch, G.P. Akilov, Functional Analysis, Nauka, Moskow 1984 (in Russian). W.A. Luxemburg, Banach function spaces, Delft, 1955. W.A. Luxemburg, A.C. Zaanen, Compactness of Integral Operators in Banach function spaces, Math. Ann. 149(2), (1963), 150–180. J. Musielak, Orlicz spaces and modular spaces, Lectures Notes in Math. 1034, Springer-Verlag, New York, 1983. H. Nakano, Modulared semi-ordered linear spaces, Maruzen Co., Ltd. Tokyo, 1950. I.P. Natanson, Theory of functions of a real variable, Frederic Unger Publishing Co., New York, 1961. M. Nowak, On the finest of all linear topologies on Orlicz spaces for which ϕ-modular convergence implies convergence in these topologies, Bull. Acad. Polon. Sci. 32 (1984), 439–445. M. Nowak, On the modular topology on Orlicz spaces, Bull. Acad. Polon. Sci. 36 (1988), 41–50.

236

Nowak ∗

15. M. Nowak, A characterization of the Mackey topology τ (Lϕ , Lϕ ) on Orlicz spaces, Bull. Acad. Polon. Sci. 34 (1986), 576–583. 16. M. Nowak, Some remarks on absolutely weak topologies on Orlicz spaces, Bull. Acad. Polon. Sci. 34 (1986), 569–575. 17. M. Nowak, Some equalities among Orlicz spaces II, Bull. Acad. Polon. Sci. 34 (1986), 675–687. 18. M. Nowak, Orlicz lattices with modular topology I, Comment. Math. Universitatis Carolinae 30, No. 1 (1989), 261–270. 19. M. Nowak, Orlicz lattices with modular topology II, Comment. Math. Universitatis Carolinae 30, No. 1 (1989), 271–279. 20. M.M. Rao, Z.D. Ren, Theory of Orlicz spaces, Marcel Dekker, New York, Besel, Hong Kong, 1991. 21. A.C. Zaanen, Riesz spaces II, North. Holland Publ. Comp., Amsterdam, New York, Oxford, 1983.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 237–246 c 1994 Universitat de Barcelona

Properties of some bivariate approximants

Paulina Pych–Taberska Institute of Mathematics, Adam Mickiewicz University, Matejki 48/49, 60-769 Pozna´n, Poland

Abstract The smoothness and approximation properties of certain discrete operators for bivariate functions are examined.

1. Preliminaries Let I be a ﬁnite or inﬁnite interval and let Q be the square I × I. A bivariate (complex-valued) function f deﬁned on Q is said to be B¨ ogel-continuous, in symbols f ∈ BC(Q), if for every (x, y) ∈ Q there holds lim (u,v)→(x,y)

∆u,v f (x, y) = 0,

where (u, v) ∈ Q, ∆u,v f (x, y) := f (u, v) − f (u, y) − f (x, v) + f (x, y). The mixed modulus of continuity ω(f ; δ, η)Q of a function f on Q is deﬁned for δ ≥ 0, η ≥ 0 as the supremum of |∆u,v f (x, y)| extended over all (x, y) ∈ Q, (u, v) ∈ Q such that |u − x| ≤ δ, |v − y| ≤ η. As is known, if f is uniformly B¨ ogel-continuous on Q then lim (δ,η)→(0,0)

ω(f ; δ, η)Q = 0.

In particular, this relation holds if f ∈ BC(Q) on a compact square Q. Some other properties of the B¨ ogel-continuous functions and their mixed moduli of continuity can be found e.g. in [2] and [3]. 237

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Let ϕ be a positive bivariate function on the square (0, 1]×(0, 1], non-decreasing in each variable, with ϕ(1, 1) ≤ 1 and ϕ(s, t) → 0 as (s, t) → (0, 0). Take a positive ϕ (Q) the class of all functions f ∈ BC(Q) for which number A and denote by HA ω(f ; δ, η)Q ≤ Aϕ(δ, η)

if

0 < δ ≤ 1, 0 < η ≤ 1.

α,β ϕ (Q) instead of HA (Q) when ϕ(s, t) = sα tβ (α ≥ 0, β ≥ 0). Write HA Given a rectangle P and a bivariate (complex-valued) function f deﬁned on P we introduce the quantities

f P := sup |f (x, y)| and f P ;ϕ := f P + sup

|∆u,v f (x, y)| , ϕ(|u − x|, |v − y|)

where the ﬁrst supremum is taken over all (x, y) ∈ P and the second one is extended over all (x, y) ∈ P, (u, v) ∈ P such that 0 < |u − x| ≤ 1, 0 < |v − y| ≤ 1. Clearly, if ϕ HA (Q) then f P ;ϕ is ﬁnite for every rectangle P ⊆ Q on which f ∈ H ϕ (Q) := A>0

f is bounded. This non-negative number is called the H¨ older-type norm of f on P . Consider now a sequence J1 , J2 , . . . of some index sets contained in Z := {0, ±1, ±2, . . .}, choose real numbers ξj,k ∈ I and real-valued functions pj,k continuous on an interval I ⊆ I and write, formally, Lk g(t) :=

k ∈ N) g(ξj,k )pj,k (t) (t ∈ I,

(1)

j∈Jk

for univariate (complex-valued) functions g deﬁned on I. For bivariate (complexvalued) functions f deﬁned on Q we introduce, also formally, the Boolean sums Lm,n (m, n ∈ N) of parametric extensions of operators Lm and Ln , i.e. Lm,n f := (L·m + L∗n − L·m ◦ L∗n )f,

(2)

where L·m f (x, y) := Lm f y (x), L∗n f (x, y) := Ln f x (y) := I × I. and f y (x) = f x (y) = f (x, y) for (x, y) ∈ Q Clearly, if k ∈ N, |pj,k (t)| ≤ c1 for all t ∈ I, j∈Jk

(3)

Properties of some bivariate approximants

239

with a positive constant c1 , then all Lm,n f are well-deﬁned for every function f bounded on Q. In the case of unbounded Q, under the additional assumption |µ2,k |(t) :=

k ∈ N, (ξj,k − t)2 |pj,k (t)| < ∞ for all t ∈ I,

j∈Jk

Lm,n f are meaningful also for unbounded functions f such that f (x, y) = O((1 + x2 )(1 + y 2 )) uniformly in (x, y) ∈ Q. In Section 2 of this paper we examine the relations between the mixed moduli of continuity of functions f and Lm,n f satisfying some appropriate conditions. With the help of the results obtained here we estimate, in Section 3, the degree of older-type one. approximation of f by Lm,n f in the supremum norm and in the H¨ Analogous problems concerning the rate of convergence of univariate operators (1) were discussed in [5].

2. Smoothness properties Let {Jk , pj,k ; j ∈ Jk , k ∈ N} be a system satisfying (3) and let

pj,k (t) = sk

for all

k ∈ N, t ∈ I,

(4)

j∈Jk

Suppose, moreover, that ξj,k ∈ I where sk are real numbers independent of t ∈ I. and that pj,k have continuous derivatives pj,k such that

|(ξj,k − t)pj,k (t)| ≤ c2

for all

k ∈ N, t ∈ Int I,

(5)

j∈Jk

c2 being a positive constant. Then the ordinary moduli of continuity of univariate functions g on I and Lk g on I satisfy ω(Lk g; δ)I ≤ 2(c1 + c2 )ω(g; δ)I was proved recently by W. Kratz and U. for all δ ≥ 0, k ∈ N. This fact, when I = I, Stadtm¨ uller [4]. Under some additional assumptions, the same inequality with an improved constant was derived in [1]. Corresponding result for the mixed moduli of continuity of bivariate functions f and Lm,n f can be stated as follows.

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Theorem 1 Suppose that f ∈ BC(Q)∩ Dom (Lm,n ) (m, n ∈ N) and that conditions (3)-(5) are fulﬁlled. Then, for all positive numbers δ, η, ω(Lm,n f ; δ, η)Q ≤ c3 ω(f ; δ, η)Q , with c3 = 4(c1 + c2 )(1 + c1 + c2 ). (u, v) ∈ Q, 0 < u − x ≤ δ, 0 < v − y ≤ η and let x0 := Proof. Let (x, y) ∈ Q, (x + u)/2, y0 := (y + v)/2. By the deﬁnition,

(L·m ◦ L∗n )f (x, y) =

f (ξi,m , ξj,n )pi,m (x)pj,n (y).

i∈Jm j∈Jn

Hence, in view of (4), ∆u,v (L·m ◦ L∗n )f (x, y) = f (ξi,m , ξj,n ){pi,m (u) − pi,m (x)}{pj,n (v) − pj,n (y)} i∈Jm j∈Jn

=

∆x0 ,y0 f (ξi,m , ξj,n ){pi,m (u) − pi,m (x)}{pj,n (v) − pj,n (y)}.

i∈Jm j∈Jn

Applying the known property of the mixed modulus of continuity ([2], Lemma 2.1) we get |∆u,v (L·m ◦ L∗n )f (x, y)| ≤ Am (x, u; δ)An (y, v; η)ω(f ; δ, η)Q , where − x0 |

|pi,m (u) − pi,m (x)| δ i∈Jm 1 u ≤ |pi,m (u) − pi,m (x)| + |ξi,m − x0 ||pi,m (t)|dt. δ x

Am (x, u; δ) :=

i∈Jm

1+

|ξ

i,m

|ξi,m −x0 |≥δ

Observing that |ξi,m − x0 | ≤ 2|ξi,m − t| if x < t < u, |ξi,m − x0 | ≥ u − x and using (3), (5) we easily verify that Am (x, u; δ) ≤ 2(c1 + c2 )

Properties of some bivariate approximants

241

δ > 0 (see [4], p. 330). Consequently, for all m ∈ N, u, x ∈ I,

2 ω (L·m ◦ L∗n )f ; δ, η Q ≤ 4(c1 + c2 ) ω(f ; δ, η)Q . Analogously, one can get inequalities for the mixed moduli of continuity of the remaining terms of the Boolean sum (2). Namely, we have |∆u,v L·m f (x, y)|

= ∆x0 ,v f (ξi,m , y){pi,m (u) − pi,m (x)} i∈Jm

≤ Am (x, u; δ)ω(f ; δ, η)Q , which implies ω(L·m f ; δ, η)Q ≤ 2(c1 + c2 )ω(f ; δ, η)Q . By symmetry, the same inequality remains also valid for the mixed modulus of continuity of L∗n f . These results together with (2) lead to the desired inequality. For many well-known operators the “weights” pj,k (j ∈ Jk , k ∈ N) satisfy the assumptions pj,k (t) ≥ 0, pj,k (t) = 1 for all t ∈ I j∈Jk

and |µ2,k |(t) > 0, |µ2,k |(t)pj,k (t) = (ξj,k − t)pj,k (t)

for all t ∈ Int I.

Hence, in these cases, c1 = c2 = 1 and the constant c3 in Theorem 1 equals 24. Inequalities obtained in the proof of Theorem 1 yield the implication ϕ f ∈ HA (Q)∩ Dom (Lm,n ) ⇒ ϕ ϕ ϕ ⇒ L·m f ∈ HB (Q), L∗n f ∈ HB (Q), (L·m ◦ L∗n )f ∈ HM (Q) ,

where B = 2(c1 + c2 )A, M = 4(c1 + c2 )2 A. This means that, under assumptions of Theorem 1, the terms of the Boolean sum (2) have the property of preserving the H¨older class with the same order that f but with the diﬀerent constants. In the case ϕ(s, t) = sα tβ (0 < α ≤ 1, 0 < β ≤ 1) we will indicate a wide class of operators for which the order (α, β) as well as the H¨older constant A are retained. To this end, let us introduce a sequence (ψk )∞ 1 of continuous functions on I0 = [0, ∞), with values ψk (0) = 1, satisfying for some positive numbers q = q(k) and a certain interval I0 ⊆ I0 (such that 0 ∈ I0 ) the following conditions

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Pych–Taberska

10 (−1)j Dqj ψk (t) ≥ 0 whenever t ∈ I0 , j ∈ N0 := {0, 1, . . .}, where Dq0 ψk := ψk , Dq1 ψk (t) := (ψk (t + q) − ψk (t))/q and Dqj ψk (t) := Dq1 (Dqj−1 ψk )(t) if j > 1; 20 under the restriction t, x ∈ Io , ψk (t) =

∞ (t − x)(j,q) j=0

j!

Dqj ψk (x),

where h(0,ρ) := 1, h(j,ρ) := h(h − ρ) . . . (h − (j − 1)ρ)

if j ≥ 1 (h, ρ ∈ R).

Consider, as in [6], the class of linear operators Vk = Lk deﬁned by (1) for univariate functions g on I0 , with Jk = N0 , ξj,k = j/k and pj,k (t) = (−1)j

t(j,−q) j Dq ψk (t) j!

(t ∈ I0 ).

We note occasionally that from some operators of this class (with q independent of k) the classical Bernstein polynomials, the Sz´ asz-Mirakyan operators or the Baskakov operators can be obtained by letting q → 0+. Theorem 2 Suppose that Vk e1 (t) = γk t for all t ∈ I0 , k ∈ N, where e1 (τ ) = τ (τ ≥ 0) and γk are some constants from [0, 1]. Denote by Vm,n (m, n ∈ N) the Boolean sum of 0 = parametric extensions of univariate operators Vm and Vn . Put Q0 = I0 × I0 , Q α,β I0 × I0 . Then if f ∈ HA (Q0 )∩ Dom (Vm,n ), with 0 < α, β ≤ 1, the functions α,β Vm· f, Vn∗ f and (Vm· ◦ Vn∗ )f are in the class HA (Q0 ). α,β Q0 ∩ Dom (Vm,n ) and let δ, η be arbitrary positive numbers. Proof. Let f ∈ HA 0 such that 0 < u − x ≤ δ, 0 < v − y ≤ η. Consider (x, y) ∈ Q0 , (u, v) ∈ Q The argumentation similar to that of the proof of Theorem 2.1 of [6] yields the identities

Vm· f (u, t) = Vm· f (x, t) =

∞ ∞ i=0 l=0 ∞ ∞ i=0 l=0

(−1)i+l

i + l

x(i,−q) (u − x)(l,−q) i+l Dq ψm (u)f ,t , i!l! m

(−1)i+l

i

x(i,−q) (u − x)(l,−q) i+l Dq ψm (u)f ,t , i!l! m

Properties of some bivariate approximants

243

for every t ∈ I0 . Consequently, |∆u,v Vm· f (x, y)| ∞ ∞ i x(i,−q) (u − x)(l,−q) i+l = Dq ψm (u)∆ i+l ,v f ,y (−1)i+l m i!l! m i=0 l=0

≤ Aη β

∞ ∞

(−1)i+l

i=0 l=0

l α x(i,−q) (u − x)(l,−q) i+l Dq ψm (u) i!l! m

= Aη Vm gα (u − x), β

where gα (τ ) = τ α (τ ≥ 0). Since Vk gα (t) ≤ tα

for all t ∈ I0 , k ∈ N

(see [6], p. 128), we have |∆u,v Vm· f (x, y)| ≤ Aη β (u − x)α . This implies the inequality α β ω(Vm· f ; δ, η)Q ≤ Aδ η . 0

Analogously, α β ω(Vn∗ f ; δ, η)Q ≤ Aδ η . 0

Considering the superposition

Vm·

◦

Vn∗

we easily observe that

|∆u,v (Vm· ◦ Vn∗ )f (x, y)| ∞ ∞ ∞ ∞ x(i,−q) (u − x)(l,−q) = (−1)i+l+j+r i!l! i=0 j=0 r=0 l=0

i j (v − y)(r,−q) i+l Dq ψm (u)Dqj+r ψn (v)∆(i+l)/m,(j+r)/n f , j!r! m n ≤ AVm gα (u − x)Vn gβ (v − y). ×

y

(j,−q)

Applying inequality (6) we get

α β ω (Vm· ◦ Vn∗ )f ; δ, η Q ≤ Aδ η , 0

and this completes the proof.

(6)

244

Pych–Taberska 3. Approximation properties

Let us return to the general operators Lm,n given by (2) for functions f deﬁned on Q. Make the standing assumption k ∈ N. pj,k (t) = 1 for all t ∈ I, j∈Jk

In this case, for any f ∈Dom (Lm,n ) and all (x, y) ∈ Q, f (x, y) − Lm,n f (x, y) =

∆x,y f (ξi,m , ξj,n )pi,m (x)pj,n (y).

i∈Jm j∈Jn

By a small modiﬁcation of the proof of Theorem 2.2 in [2] one can get Theorem 3 Let condition (3) be satisﬁed and let for a certain interval Y ⊆ I, sup |µ2,k |(t) ≤ λd2k ,

(7)

t∈Y

where λ is a positive constant and (dk )∞ 1 is a sequence of positive numbers not greater than 1. Suppose that f belongs to the class BC(Q)∩ Dom (Lm,n ) and that P := Y × Y . Then f − Lm,n f P ≤ (c1 + λ)2 ω(f ; dm , dn )Q . older-type norm it is In order to estimate the deviation Lm,n f from f in H¨ convenient to apply the following Lemma Suppose that f ∈ Dom (Lm,n ) and that 0 < δm ≤ 1, 0 < ηn ≤ 1. Then, for every rectangle P ⊆ Q,

4 f − Lm,n f P ϕ(δm , ηn ) 1 {ω(f ; δ, η)P + ω(Lm,n f ; δ, η)P }, + sup ϕ(δ, η)

f − Lm,n f P ;ϕ ≤ 1 +

the supremum being taken over all pairs (δ, η) belonging to the set R(δm , ηn ) := (0, 1] × (0, 1] \ (δm , 1] × (ηn , 1].

Properties of some bivariate approximants

245

The above inequality follows at once from the two obvious facts: (i) if (x, y) ∈ P, (u, v) ∈ P, δm ≤ |u − x| ≤ 1, ηn ≤ |v − y| ≤ 1, then |∆u,v (f − Lm,n f )(x, y)| 4 ≤ f − Lm,n f P ; ϕ(|u − x|, |v − y|) ϕ(δm , ηn ) (ii) if 0 < |u − x| ≤ δm , 0 < |v − y| ≤ 1 or if δm ≤ |u − x| ≤ 1, 0 < |v − y| ≤ ηn , then |∆u,v (f − Lm,n f )(x, y)| ϕ(|u − x|, |v − y|) ω(f ; |u − x|, |v − y|)P + ω(Lm,n f ; |u − x|, |v − y|)P ≤ . ϕ(|u − x|, |v − y|) Combining Theorems 1, 3 and Lemma we obtain Theorem 4 Suppose that conditions (3), (5) and (7) are fulﬁlled. Then if f ∈ H ϕ (Q)∩ Dom (Lm,n ) and P := Y × Y , we have f − Lm,n f P ;ϕ ≤ c4 sup

ω(f ; δ, η) Q

ϕ(δ, η)

,

where c4 = 5(c1 + λ)2 + c3 + 1 and the supremum is taken over all (δ, η) ∈ R(dm , dn ). Corollary α,β (Q)∩ Dom (Lm,n ) and let 0 < a < α ≤ 1, 0 < b < β ≤ 1. Then, in Let f ∈ HA a b case ϕ(s, t) = s t (0 < s, t ≤ 1),

f − Lm,n f P ;ϕ ≤ Ac4 (dα−a + dβ−b m n ) whenever assumptions (3), (5), (7) hold. α,β (Q0 ) Remark. For operators Vm,n considered in Theorem 2 and functions f ∈ HA 2 2 (0 < α, β ≤ 1) satisfying the condition f (x, y) = O((1 + x )(1 + y )) uniformly in α,β (x, y) ∈ Q0 , the relation Vm,n f ∈ H3A (Q0 ) is valid. Further, by Theorem 3,

f − Vm,n f P ≤ (1 + λ)2 ω(f ; dm , dn )Q0 , where P = Y × Y , λ and dk are determined via condition (7). Applying Lemma we get, for all m, n ∈ N, the estimate of f − Vm,n f P ;ϕ as in Corollary, with c4 replaced by 5(1 + λ)2 + 4.

246

Pych–Taberska References

1. G.A. Anastassiou, C. Cottin, H.H. Gonska, Global smoothness of approximating functions, Analysis 11 (1991), 43–57. 2. C. Badea, I. Badea, C. Cottin, H.H. Gonska, Notes on degree of approximation of B -continuous and B -differentiable functions, Approx. Theory and its Appl. 4 (1988), 95–108. 3. C. Cottin, Approximation by bounded pseudo-polynomials, in “Function Spaces” (ed. J. Musielak et al.), Teubner-Texte zur Mathematik 120 (1991), 152-160. 4. W. Kratz, U. Stadtm¨uller, On the uniform modulus of continuity of certain discrete approximation operators, J. Approx. Theory 54 (1988), 326–337. 5. M. Powierska, P. Pych-Taberska, Approximation of continuous functions by certain discrete operators in H¨older’s norms, Functiones et Approximatio, Comment. Math. 21 (1992), 75–83. 6. B.D. Vecchia, On the preservation of Lipschitz constants for some linear operators, Bolletino U.M.I. (7) 3-B (1989), 125–136.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 247–260 c 1994 Universitat de Barcelona

On complemented subspaces of rearrangement invariant function spaces

Yves Raynaud Equipe d’Analyse (CNRS), Universit´e Paris 6, 4, place Jussieu, 75252-PARIS-Cedex 05, France

Abstract A necessary and sufficient condition is given for a r.i. function space to contain a complemented isomorphic copy of 1 (2 ).

1. Introduction In the paper [3] was investigated the existence of complemented copies of the space 2 in rearrangement invariant (“r.i”) function spaces. We showed in particular that if the r.i. space X does not contain a subspace isomorphic to c0 , then it contains a complemented copy of 2 iﬀ either it contains a complemented sublattice isomorphic to 2 or X and its K¨ othe dual X both contain a Gaussian variable. In the same paper was also investigated the existence of an isomorphism between X and its Hilbert-valued extension X(2 ) (which is in fact equivalent to the existence of a complemented copy of X(2 ) in X), in the case where X is a q-concave (q < 2) r.i. function space over I = [0, 1]. In this case, a necessary and suﬃcient condition is that the multiplication operator MG : L0 (I) → L0 (I × I), f → f ⊗ G operates from X (I) into X (I × I) (where G is a normal Gaussian variable and f ⊗ G(s, t) = f (s)G(t)). Here we are interested in the existence of a complemented copy of the space 1 (2 ) in X. When X contains itself 1 as complemented sublattice (which is the case in particular if simple integrable functions are not dense in X ), it is clear that this question is intermediate between the two preceding; thus the criterion we ﬁnd is naturally intermediate between the two criterions given above. In the case where X has ﬁnite upper Boyd index, and 1 (2 ) does not embed in X as complemented 247

248

Raynaud

sublattice, the criterion reduces to the fact that the domain of MG in X is not included in the closure of simple integrable functions. We state now our main results. Proposition 1 Let X be a rearrangement invariant function space over (Ω, A, µ), not containing c0 . Suppose that X does not contain 1 (2 ) as complemented sublattice. Then X contains 1 (2 ) as complemented subspace iﬀ there exist disjoint functions Ai , i ≥ 1 othe dual X (Ω), such that, denoting by in X+ (Ω) and an element B ≥ 0 in the K¨ G a normal Gaussian variable deﬁned on the auxiliary probability space (S, Σ, σ), i) ∀ ≥ 1, Ai ⊗ G X = 1 = Ai , B and ii) B ⊗ G belongs to X (Ω × S). Corollary 2 Suppose that X satisﬁes the hypotheses of Proposition 1 and, moreover, has ﬁnite upper Boyd index. Then a necessary and suﬃcient condition for X to contain 1 (2 ) as complemented subspace is the existence of an element B of X which is not in the closure of the space of simple integrable functions but such that B ⊗ G still belongs to X (Ω × S). We give now some deﬁnitions. If X is a r.i. space over I = [0, m] and (Ω, A, µ) is a measure space with µ(Ω) = m (possibly inﬁnite), we denote by X(Ω, A, µ) the space of measurable functions over (Ω, A, µ) whose non-increasing rearrangement is in X = X(I). We say that a bounded sequence (xn )n in the r.i. space X is X-equiintegrable if the following conditions are satisﬁed: i)

lim sup 1A xn X = 0

µ(A)→0 n

ii)

inf

sup 1Ac xn X = 0

µ(A)<∞ n

where Ac denotes the complementary set of A. We say that a sequence (xn )n converges weakly conditionally in distribution (in short “wcd”) if there exists a measurable function Y ∈ L0 (Ω × S), deﬁned on a superspace of measure (Ω × S, A ⊗ Σ, µ ⊗ σ) (where σ is a probability measure) such that for every µ-integrable subset U of Ω, and every bounded continuous function ϕ(Y )dµdσ. ϕ on R, U ϕ(xn )dµ −−−→ n→∞

U ×S

We say that Y is conditionally Gaussian (r.r. to the ﬁrst variable) iﬀ for µa.e. ω ∈ Ω, the partial function Yω = Y (ω, ·) has Gaussian probability distribution (hence is equimeasurable with A(ω)G(·), where G is a normal gaussian variable).

On complemented subspaces of rearrangement invariant function spaces

249

The main tool used here (as in [3]) is the following: for every 2 -basic sequence (xn )n in L1 (Ω), there exists a sequence of successive normalized blocks (yn ) build on the xn which converges wcd to a conditionally Gaussian variable. This is for instance a consequence of [1] and [4], as noticed in [5]. In section 2 below, we prepare the proof of Proposition 1 by several technical lemmas. The proof of Proposition 1 itself and of its corollary are given in section 3. Unexplained notions or facts about r.i. spaces can be found in [2], which we follow in particular for the precise deﬁnition of r.i. spaces ([2], 2a1).

2. Some technical lemmas The ﬁrst lemma is a reﬁnement of Lemma 10 of [3]: Lemma 3 Let X be a r.i. function space not containing c0 and (xj,n )j,n∈N be a system of elements of X such that for each j ∈ N, the sequence (xj,n )n is X-equiintegrable and converges wcd to a conditionally Gaussian variable. Then for each j there is a subsequence (xj,n(j) ) such that for every ﬁnite system (λj, ) of reals: 1/2 2 λj, xj,n(j) ∼ |λj, | xj,n(j) (1) .

j,

1+ε

j

Moreover we can choose these subsequences such that each Fj = span[xj,n(j) ] has X-equiintegrable unit ball, and every weakly null subsequence of Fj converges wcd to a conditionally Gaussian variable. wcd

Proof. We have xj,n −−−→ Aj ⊗ Gj ∈ X(Ω × S), where we may suppose the Gj n→∞

to be independent. We suppose that L0 (S) contains a sequence (Gj ) of Gaussian variables which are independent and independent of the Gj . We ﬁx a sequence of (j) εj . Suppose we have chosen the n , with j, ≥ 1 and positive reals εj with ε = j

j + ≤ m, verifying:  For every system (λj , ) with j ≥ 1, ≥ 1 and j + ≤ m     and every sequence (ρj ), j ≤ m:    m   λ x (j) + ρj Aj ⊗ Gj j, j,n Hm j=1 j,≥1   j+≥m  m m−j  1/2  m   2 2  ≤ |λj, | + |ρj | Aj ⊗ G j εj . −  j=1

=1

j=1

250

Raynaud Then we have for every systems (λj, )j,≥1,j+≤m+1 and (ρj )j≤m+1 :

un1 ,n2 ,...,nm (λj, ), (ρj ) :=

λj, xj,n(j) +

j,≥1 j+≤m

+

m+1

m

λj,m+1−j xj,nj

j=1

ρ j Aj ⊗ G j

j=1

wcd

−→

nm →∞;nm−1 →∞;...n1 →∞

λj, xj,n(j) +

j,≥1 j+≤m

+

m

m+1

λj,m+1−j Aj ⊗ Gj

j=1

ρ j Aj ⊗ G j

j=1 dist

∼

λj, xj,n(j) +

j,≥1 j+≤m

m+1

(|λj,m+1−j |2 + |ρj |2 )1/2 Aj ⊗ Gj

j=1

=: u∞ (λj, ), (ρj ) .

Hence we deduce the convergence a.e. of the rearrangements:

∗

∗ un1 ,n2 ,...,nm (λj, ), (ρj ) → u∞ (λj, ), (ρj ) .

(2)

As in the proof of Lemma 10 in [3], using the order-continuity of X, we deduce the convergence of: m

Fn1 ,n2 ,...,nm (λj, ), (ρj ) := λj, xj,n(j) + λj,m+1−j xj,nj

j,≥1 j+≤m

+

m+1 j=1

j=1

ρ j Aj ⊗ G j

X

to F∞

m+1 (λj, ), (ρj ) := λ x + (|λj,m+1−j |2 (j) j, j,n

j,≥1 j+≤m

+ |ρj | )

2 1/2

j=1

Aj ⊗ G j

X

On complemented subspaces of rearrangement invariant function spaces

251

and by Ascoli’s theorem, this convergence is uniform on each set m+1 m+1−j j=1 (1)

|λj, | + |ρj | 2

1/2

≤K .

=1 (2)

(m)

Hence we can choose nm , nm−1 , . . . , n1 (2) Fn(1) m ,n

2

such that:

(λj, ), (ρj ) − F∞ (λj, ), (ρj ) ≤ εm+1

(m) ,...,n1 m−1

uniformly on the set

m+1 m+1−j 2 2 1/2 (λj, ), (ρj ) : (|λj,m+1−j | + |ρj | ) Aj ⊗ Gj j=1

≤1 .

X

=1

Together with (Hm ) we obtain (Hm+1 ). The subsequences (xj,n(j) ) we obtain satisfy

then the equivalence (1) (take m suﬃciently large and ρj = 0 in (Hm )). Finally the assertions about equiintegrability and wcd convergence of blocks are also a consequence of the convergence of rearrangements (2) (see [3] for more details). Lemma 4 Let X be a r.i. space over (Ω, A, µ), not containing c0 . If X contains 1 (2 ) as complemented subspace, but not as complemented sublattice, then X(Ω × [0, 1]) contains a complemented subspace with a 1 (2 )-basis of the form (Aj ⊗ Gnj )j,n≥1 , n where Aj ∈ L+ 0 (Ω) and the Gj ∈ L0 ([0, 1]) are independent Gaussian variables. Proof. A) Let E be a complemented subspace of X, isomorphic to 1 (2 ); write E = ⊕j Ej , where the “ﬁbers” Ej are isomorphic to 2 , and the direct sum is a 1 sum: for every ﬁnite sequence (xj )j , xj ∈ Ej , we have xj ∼ xj . We j

j

remark ﬁrst that for all but a ﬁnite numbers of indices j, there exists a subset Uj of Ω, of ﬁnite µ-measure, such that the X-norm and the L1 (Uj )-norm are equivalent on Ej (these equivalence need not be uniform with respect to j). For, if not we have an inﬁnite subset J of N, such that for each j ∈ J, and every µ-ﬁnite subset U of Ω, the L1 (U )-norm and the X-norm are not equivalent on Ej . It is then easy to ﬁnd, for each j, a normalized sequence (fj,n )n in Ej which is weakly null and converges to 0 locally in measure. It follows that this sequence is quasidisjoint for both the lattice structures of X and of Ej (this last one being

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Raynaud

given by the isomorphism with 2 ), i.e. fj,n − gj,n −−−→ 0 and fj,n − hj,n −−−→ 0, n→∞

n→∞

where (gj,n )n is disjoint in X and (hj,n )n is disjoint in Ej w.r. to the 2 -basis. Then (hj,n )j∈J,n∈N is a 1 (2 )-basic sequence, spanning a complemented subspace of X. If nski we suppose, as we may, that: hj,n − gj,n X ≤ ε2−(j+n) , then by Bessaga-PeClczy´ perturbation principle the same holds for the doubly indexed sequence (gj,n )j∈J,n∈N , providing a complemented sublattice of X isomorphic to 1 (2 ), a contradiction. B) From now on we suppose that J = N. There exists for each j a normalized sequence (xj,n )n in Ej which converges wcd on Uj to a conditionally Gaussian variable. This can be done in fact on every U ⊃ Uj (since the L1 (U ) and the X-norm are still equivalent on Ej ), hence by a diagonal argument we can obtain this wcd convergence on the whole of Ω. Now the subspace Fj = span[xj,n ] is C-complemented in the hilbertian space Ej (with C independent from j), hence F = ⊕Fj is complemented in E. Thus we suppose from now on that E has a 1 (2 )-basis (xj,n )j,n such that for every j, the sequence (xj,n )n converges wcd to a conditionally gaussian variable. Now using a “subsequence splitting lemma” (see [6] for instance), after extraction, we may decompose: xj,n = xj,n + xj,n , where xj,n ⊥ xj,n , the sequence (xj,n )n is X-equiintegrable and the sequence (xj,n )n is disjoint. We have for all ﬁxed j two operators Sj and Sj : Ej → X, such that S xj,n = xj,n and S xj,n = xj,n and which are uniformly bounded (w.r. to j). Since E = ⊕Ej is a 1 -direct sum (up to isomorphism), we deduce the existence of two bounded operators S , S : E → X, whose restriction to each subspace Ej are respectively Sj and Sj . C) Let P be a projection from X onto E. For each j, n, we denote by Ej,n the closed span of (xj,m )m>n . We claim that for all but a ﬁnite number of indices j, there exist a positive real σj and an integer Nj such that: ∀y ∈ Ej,Nj , P S y ≥ σj y . For, if not, there exist an inﬁnite subset J of N, and for all j ∈ J a sequence yj,n in Ej , such that: uj,n = 1,

yj,n −−−→ 0 and P S yj,n −−−→ 0 . w

n→∞

n→∞

We can suppose that P S yj,n ≤ ε2−(j+n) . After extraction, the doubly indexed sequence (yj,n )j,n is equivalent to the 1 (2 ) basis and spans a 2-complemented subspace F of E. Let Q be a projection from E onto F with Q ≤ 2. Since

On complemented subspaces of rearrangement invariant function spaces

253

yj,n − QP S yj,n = QP Sj,n , we see that J = QP S is an isomorphism of F ; S F is isomorphic to F and complemented in X (by S J −1 QP ). Thus (S yj,n )j,n spans )n is a disjoint sequence. a complemented 1 (2 ) subspace of X. For each j, (yj,n Using the order continuity of X and the Bessaga-PeClczy´ nski perturbation principle, we deduce that 1 (2 ) embeds as complemented sublattice in X, a contradiction.

D) We want now to extract subsequences (xj,n(j) )∞ =1 such that for some δ > 0

and every element y of the closed span F of (xj,n(j) )j, , one has S y ≥ δ y .

(It will be also useful for the sequel that the unit ball of each closed subspace Fj generated by the sequence (x (j) ) is X-equiintegrable, and that every weakly null j,n

sequence in Fj converges wcd to a conditionally Gaussian variable). Using Lemma 3, we can extract subsequences (xj,n(j) )∞ =1 such that:

xj,n(j)

j,

1/2 ∼ |λj,l |2 xj,n(j)

X 1+ε

j

.

(3)

X

(j)

Relabeling, in order to simplify notations, we can suppose n = (∀j, ). If is not equivalent to the 1 -basis, there exist 1 -normalized blocks yp = (xj,1 )∞ j=1 αj xj,1 , ( |αj | = 1) such that yp −−−→ 0. Supposing ∀p, yp ≤ p→∞ j∈Jp j∈Jp ε αj xj,1 , and setting: zp, = αj xj, , we obtain for every ﬁnite sysj∈Jp

j∈Jp

tem of scalars (λp, ): 1/2 λp, P S zp, ≤ (1 + ε) P |λp, |2 yp p

p,

≤ ε(1 + ε) P

p

|λp, |2

1/2 αj xj,1 j∈Jp

≤ Cε(1 + ε) P λ z p, p, p,

where C is the equivalence constant of (xj, ) with the basis of 1 (2 ). Hence for small ε, the operator (I − P S ) = P S is an isomorphism from Z = span[zp, ] onto its image. In particular, S Z and P S Z are isomorphic to 1 (2 ). The subspace P S Z, being a copy of 1 (2 ) in the space E, which is itself isomorphic to 1 (2 ), contains a further subspace G, which is isomorphic to 1 (2 ), build on a subset of the 1 (2 )-basis of P S Z, and complemented in E. Let Z1 = (P S )−1 (G), and Q be a projection of E onto G. Let J = P|S Z . Then J −1 QP is a projection from 1 X onto S Z1 , which proves that X contains a complemented 1 (2 ) sublattice, a contradiction.

254

Raynaud

Hence in fact the sequence (xj,1 )j is equivalent to the 1 -basis, which implies by (3) that (xj, ) is equivalent to the 1 (2 )-basis (and S is an isomorphism on its range). E) From now on we suppose that S y ≥ δ y for y ∈ E, and we prove now the existence of a closed subspace F of E, generated by a system of block sequences (yj, ) of the (xj, ) (j ∈ N), on which P S is an isomorphism, i.e. ∀y ∈ E, P S y ≥ ρ y . Since the subspace P S Ej is hilbertian by the point C) above, so is the subspace S P S Ej (S being an isomorphism), and we can ﬁnd appropriate normalized successive blocks (yj, ) on each sequence (xj, ) such that the sequence (S P S yj, ) converges wcd to a (nonzero) conditionally Gaussian variable. By Lemma 3, we can suppose that: 1/2 ∼ 2 λj, S P S yj, |λj, | S P S yj,1 1+ε

j,

j

hence, using again the fact that S is an isomorphism, we have: 1/2 λj, P S yj, ∼ |λj, |2 P S yj,1 1+ε

j,

j

and, reasoning as in the point D above (using P S = I − P S on E), this implies that (P S yj,1 )j is equivalent to the 1 -basis, hence: 1/2 λj, S P S yj, ∼ |λj, |2 . j,

j

F) Let Y = span[yj, ]j, . The subspace P S (Y ), being a 1 (2 )-subspace of the 1 (2 )-subspace E, contains a subspace G which is isomorphic to 1 (2 ), complemented in E and spanned by a subset of the P S yj, . Set Z = (P S )−1 (G): then S Z is a subspace of X isomorphic to 1 (2 ) and complemented in X. Note that the basis zi,m of S Z is a subset of the basis of S Y : zi,m = S yj(i),(i,m) , hence each sequence (zi,m )m is X-equiintegrable and converges wcd to a conditionally Gaussian variable. G) We have thus reduced the situation to the case where the elements (xj,n )j,n of a 1 (2 ) basis (spanning a complemented closed subspace of X) are, for each ﬁxed j, X-equiintegrable and converging wcd to a conditionally Gaussian variable. Now we apply the ultrapower procedure of the §3 in [3]. We have for every doubly indexed ﬁnite set of natural numbers S = (n(j, ))1≤j,l≤k a projection πS : X →

On complemented subspaces of rearrangement invariant function spaces

255

span [xj,n(j,) ]1≤j,≤k with norm bounded by a constant K (independent of S). Let (of X) relative to the U be a free ultraﬁlter over N. Passing to the ultrapower X iterated limit: lim . . . lim lim lim . . . lim k,U n(k,k),U

n(k,1),U n(k−1,k),U

n(1,1),U

which is equivalent to the we obtain a doubly indexed sequence (ξj,n )j,n≥1 in X, → span[ξj,n ]j,n . Moreover the ξj,n lie in 1 (2 )-basis, and a bounded projection π: X eq fact in X , the subspace of X whose elements can be deﬁned by X-equiintegrable families of elements of X, since for each j, the sequence (xj,n )n is in fact X eq identiﬁes to a space X(Ω, A, µ equiintegrable. It is known that X ), the measure A, µ space (Ω, A, µ) identifying to that generated by a sub-σ-algebra in (Ω, ) ([6]). The conditional distribution of the elements ξj,n , w.r. to the initial σ-ﬁeld A is the same as the limit conditional distribution of the xj,n , i.e. that of a sequence Aj ⊗ Gnj in X(Ω × [0, 1]), where the Aj are nonnegative, A-measurable, and the Gnj are independent normal Gaussian variables in [0, 1]. µ The last point is to use a transformation of the measure algebra (A, ) conserving the measure, leaving the elements of A invariant and carrying each ξj,n on Aj ⊗ Gnj , where Aj is A-measurable, and Gnj are normal Gaussian variables indeµ pendent of A. That such a transformation exists, at least after enlarging (A, ) is an easy measure-theoretic exercise. Lemma 5 Let X be a r.i. space not containing c0 and (xn )n a basic sequence in X , equivalent to the 1 -basis. There exists a sequence of 1 -normalized successive blocks fi build on the sequence (xn ) , which is quasidisjoint in X (i.e. ∀i, |fi | ∧ |fj | X −−−→ 0). j→∞

Proof. Passing to a subsequence, we may suppose (by the subsequence splitting lemma) that we have a decomposition: xn = xn + xn , into a X-equiintegrable part (xn )n and a disjoint part (xn )n , with ∀n, xn ⊥ xn . Let F = span[xn ], F = span[xn ], F = span[xn ] and S : F → F , (resp. S : F → F ) be the bounded linear operator such that S xn = xn , (resp. S xn = xn ). If there are no n0 ∈ N and δ > 0 such that S x ≥ δ x for every x ∈ Fn0 = span[xn ]n≥no , then there is a sequence (yn )n of successive normalized blocks on the basis of F such that yn − S yn → 0, hence (yn ) is quasidisjoint. If at the contrary S x ≥ δ x for every x ∈ Fn0 , then (xn )n≥n0 is equivalent to the 1 -basis. This implies that the X-norm and the L1 (U )-norm are equivalent on Fn 0 for no integrable subset U of Ω (since a 1 -basic

256

Raynaud

sequence in L1 (U ) cannot be equiintegrable for the norm of L1 (U ), and a fortiori for that of X). Hence there exists a sequence of normalized successive blocks on the xn which converges to 0 in measure, hence is quasidisjoint (by order continuity). The homologous blocks build on the xn are also quasidisjoint. Lemma 6 Let X be a r.i. space not containing c0 nor 1 (2 ) as a complemented sublattice. If X contains 1 (2 ) as a complemented subspace, then there exists in the extended r.i. space X(Ω × S, A⊗Σ, µ ⊗ σ) a 1 (2 )-basis (spanning a complemented closed subspace too) having the form (Aj ⊗ Gnj )j,n , where the Gnj ∈ L0 (S) are independent normal Gaussian variables, and the functions Aj ∈ L+ 0 (Ω) have disjoint supports. Proof. We start from the elements (Aj ⊗ Gnj ) given by Lemma 4. By applying Lemma 5 to the 1 -basic sequence (Aj ⊗ G1j )j , we obtain a quasidisjoint sequence αj Aj ⊗ G1j ; due to the symmetry of of successive 1 -normalized blocks y = j∈J

G1j ,

the variables we may suppose in fact that the αj are nonnegative. We set now: αj Aj ⊗Gnj . The doubly indexed sequences (yn ),n and (B ⊗Gn ),n , where yn = j∈J (αj Aj )2 )1/2 , are equivalent in distribution (in fact, conditionally w.r. to B = ( j∈J

the ﬁrst coordinate). It follows that F = span[B ⊗ Gn ] is complemented in X(Ω × S). For, we may suppose that the variables (Gn ) generate the σ-algebra Σ. There exists a measure presenting set transformation T deﬁned on the A ⊗ Σ, with µ ⊗ σ-measurable values, α A ⊗ whose associated isometry T: X(Ω⊗S) → X(Ω×S) maps φ⊗Gn onto Bφ j∈J

for all φ ∈ L∞ (Ω): then T(B ⊗ Gn ) = yn . The subspace span[yn ] = TF is complemented in E = span[Aj ⊗ Gn ] by the projection Q: λj,n Aj ⊗ Gj,n → jn ( λj,n )y,n . Since E is itself complemented in X(Ω × S) by a projection P, F Gn ,

,n j∈J

is also complemented by the projection T−1 QP T. On the other hand, the sequence (z ) := (B ⊗ G1 ) is quasidisjoint in X (since it is equimeasurable with (y1 ) ). This implies that (z ) converges to zero locally in measure. Hence (B ) itself goes to zero locally in measure; hence (B ∧ Bp ) ⊗ G −−−→ 0 by ordercontinuity, and we can ﬁnd a disjoint sequence (C ) with p→∞

(B − C ) ⊗ G −−−→ 0; equivalently, B ⊗ G1 − C ⊗ G1 → 0, and by a standard l→∞

reasoning, we deduce that for some subsequence (D ) of (C ) , the double sequence

On complemented subspaces of rearrangement invariant function spaces

257

(D ⊗ Gn ),n is equivalent to the 1 (2 )-basis and spans a closed subspace which is complemented.

3. Proof of the main results Proof of Proposition 1. A) We prove ﬁrst the necessity of the given conditions. Suppose that X contains 1 (2 ) as subspace, and let (Aj ⊗ Gnj )j,n be a special (1 (2 ))-basis given by Lemma 6. If P is a projection from X(Ω × S) onto E = span[Aj ⊗ Gnj ]j,n , we can replace it by another projection Q conserving the support Uj of each function Aj ; we set simply: Qf =

1Uj P 1Uj f .

j

This series is norm-convergent in X, and Qf ≤ P f . For, denote by Sj the isometry of X deﬁned by: Sj f = 1Uj f − 1Ujc f , and deﬁne recursively a sequence (Pn ) of projections by: P0 = P, Pj+1 = 12 (Pj + Sj Pj Sj ). Then Pj ≤ P and n Pn f = 1Uj P 1Uj f + Rn f , with Rn f = 1n U c P 1n U c f . Then Rn f ≤ 1n

j=1

j=1

j=1

Ujc f

j

j=1

j

which goes to zero as n → ∞ when f is supported by the union of

the Uj , by order-continuity of X, and the sequence (Pn f )n is stationary when f is supported in a ﬁnite union of the Uj . Denoting by π the band projection in X deﬁned by the union of the Uj , this shows that Pn π −−−→ Q (in strong operator n→∞

topology). Now let Qj = 1Uj Q1Uj , which acts as a projection from Xj = X(Uj × S) onto Ej = span[Aj × Gnj ]n≥1 . By [3] §2, there is another projection Rj : Xj → Ej , having the form: Rj f =

f, Bj , ⊗Gnj Aj ⊗ Gnj n

with Bj ⊗ G ∈ X , Aj , Bj = 1 and Rj ≤ Qj . More precisely, the proof in [3] shows that: ∀f ∈ X, Rj 1Uj f + 1Ujc f ≤ Qj 1Uj f + 1Ujc f hence:

n Rj 1Uj f + 1n j=1

j=1

n f ≤ Qj 1Uj f + 1n Uj j=1

j=1

f Uj .

258

Raynaud

Note that we may suppose that Bj is supported by Uj , and that Bj ≥ 0 (replacing ∞ |Bj | if necessary Bj by Aj ,|B ). Thus Rf := j=1 Rj 1Uj f converges in norm and j | Rf ≤ Qf . We have then: Rf =

f, Bj ⊗ Gnj Aj ⊗ Gnj . j,n

Since (Bj ⊗ Gnj )j,n is biorthogonal with (Aj ⊗ Gnj )j,n , which spans a complemented 1 (2 ) subspace, it is a c0 (2 )-basis, and in particular: n Bj ⊗ G1j

X

j=1

n 1 ≤ P B ⊗ G j j

E∗

j=1

n Bj ⊗G1j but since the Bj are disjoint, j=1

and consequently B ⊗ G ∈ X , where B =

≤ C P

n = Bj ⊗G

X ∞

j=1

X

(same distribution),

Bj .

j=1

B) Now we prove the suﬃciency of the conditions. Let Ui be the support of Ai and Bi = 1Ui B. Then in X , the sequence (Bi ⊗G)i is equivalent to the c0 basis, since Bi ⊗ G ≥ Bi ⊗ G, Ai ⊗ G = Bi , Ai = 1, n and ∀n, Bi ⊗ G ≤ B ⊗ G ; the biorthogonal sequence (Ai ⊗ G)i in X is thus i=1

then F = span[Ai ⊗G]i is complemented by the projection equivalent to the 1 -basis; R deﬁned by Rf = f, Bi ⊗ GAi ⊗ G, since: i

Rf ≤

| f, Bi ⊗ G| ≤

i

f, Bi ⊗ |G| ≤ f,

i

Bi ⊗ |G|

i

= f, B ⊗ |G| ≤ f X B ⊗ G X . We remark now that the same happens when we replace the Ai ⊗ G by elements Ai ⊗ Gi , where (Gi )i is a sequence of independent normal Gaussian variables. For, we may suppose that Gi = Zi G, where Zi is an onto isometry of X(S) induced by a measure preserving set transformation. Let Zi = id ⊗ Zi be the natural extensions = Zi fi . of these isometries to X(Ω × S). If f ∈ X(Ω × S), set fi = 1Ui f and Zf i We deﬁne Rf = f, Bi ⊗ Gi Ai ⊗ Gi . We have then: i

Rf =

−1 fi , Bi ⊗ G

fi , Bi ⊗ Gi Ai ⊗ Gi =

Z i i

=

i

i

−1

Z

f, Bi ⊗ GAi ⊗ Gi

On complemented subspaces of rearrangement invariant function spaces which is equimeasurable with

259

−1 −1 f ); hence R ≤

Z f, Bi ⊗ GAi ⊗ G = R(Z i

RZ−1 = R .

Finally let (Gi,n )i,n be a doubly indexed sequence of independent normal Gaussian variables. Then (Ai ⊗ Gi,n ) is equivalent to the 1 (2 )-basis. Let us show that E = span[Ai ⊗ Gi,n ]i,n is complemented. We set: Rf =

f, Bi ⊗ Gi,n Ai ⊗ Gi,n . i,n

We have: Rf =

1/2

Ai ⊗ G i

n

i

=

| f, Bi ⊗ Gi,n |2

λi,n f, Bi ⊗ Gi,n Ai ⊗ Gi

n

i

(for some sequence (λi,n ) with |λi,n |2 = 1) n = f, Bi ⊗ λi,n Gi,n Ai ⊗ Gi n

i

=

f, Bi ⊗ Γi Ai ⊗ Gi i

where the Γi =

λi,n Gi,n are independent normal Gaussian variables. By the

n

preceding, we deduce: Rf ≤ f X B ⊗ G X R is the desired projection. Proof of Corollary 2. If X contains 1 (2 ) as a complemented subspace, let Ai , B be given by Proposition 1, Ui be the support of Ai , and Bi = 1Ui B. Then B ∈ X (Ω) and Ai X ≤ (E|G|)−1 Ai ⊗ G = C, hence Bi ≥ C1 Bi , Ai ≥ C1 . The order ideal generated by B in X is thus not order continuous, while the closure SX , of the space of simple integrable functions in X is order continuous (since X = L∞ ). Hence B ∈ SX . Conversely if B ≥ 0 veriﬁes: B ⊗ G ∈ X , B ∈ SX , there exists a disjoint sequence (Bi ) in X+ , which is bounded from below (say Bi ≥ 1), and such that B = Bi . Let Ai ∈ X+ verify 1 ≤ Ai ≤ (1 + ε) Ai , Bi , with support included in i

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that of Bi . Since X has ﬁnite upper Boyd index, we have f ⊗ G X ≤ C f X for every f ∈ X; in particular: Ai ⊗ G ≤ C Ai , Bi = C Ai , B . Since Ai ⊗ G ≥ (E|G|) Ai = E|G|, we see that up to a constant factor the conditions i) and ii) of Proposition 1 are fulﬁlled.

References 1. V.F. Gaposhkin, Lacunary series and independent functions, Usp. Math. Nauk (Engl. transl.: Russian Math. Surveys) 21 no. 6 (1966), 33–82. 2. J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces II, Function Spaces, Ergebnisse der Math. 97 (Springer Verlag), 1979. 3. Y. Raynaud, Complemented hilbertian subspaces of rearrangement invariant function spaces, Illinois. J. Math., to appear. 4. H.P. Rosenthal, On subspaces of Lp , Ann. Math. 97, no. 2 (1969), 344–373. 5. E.V. Tokarev, Subspaces of symmetric spaces of functions, Funct. Anal. Appl. 13, no. 2 (1979), 93–94. 6. L. Weiss, Banach lattices with the subsequence splitting property, Proc. Amer. Math. Soc. 105, no. 1 (1989), 87–96.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 261–268 c 1994 Universitat de Barcelona

Random rearrangements in functional spaces

E.M. Semenov Department of Mathematics, Voronezh State University, Voronezh 394693, Russia

Abstract We give an operator approach to several inequalities of S. Kwapien and C. Sch¨utt, which allows us to obtain more general results.

Section 0 Let n be an integer, x = {xij }, 1 ≤ i, j ≤ n, Π = Πn be the set of rearrangements of {1, 2, . . . , n}. Denote by s1 , s2 , . . . , sn2 the rearrangement of |xij | in the decreasing order. S. Kwapien and C. Sch¨ utt proved the following statements. Theorem A ([3]) The inequalities n n 1 1 1 sk ≤ max |xiπ(i) | ≤ sk 1≤i≤n 2n n! n k=1

π∈Π

k=1

are valid. Theorem B ([5]) If 1 ≤ p ≤ q < ∞, then 261

262

Semenov n n2 1 1 p 1/p 1 q 1/q sk + sk 10 n n k=1 k=n+1 n p/q 1/p 1 |xxπ(i) |q ≤ n! i=1 π∈Π

n n2 1/p 1 1/q 1 p ≤ sk + sqk . n n k=1

k=n+1

The operator approach to such problems is presented in this article. It allows to obtain more general results.

Section 1 If x(t) is a measurable function on [0, 1], we denote by x∗ (t) the decreasing rearrangement of |x(t)|. A Banach functional space E on [0, 1] with the Lebesgue measure m is said to be rearrangement invariant (r.i.) if it satisﬁes the following condition: if y ∈ E and x∗ (t) ≤ y ∗ (t) for all t ∈ [0, 1], then x ∈ E and xE ≤ yE . Let τ > 0. The compression operators t x( τ ), 0 ≤ t ≤ min(τ, 1) στ x(t) = 0, min(τ, 1) < t ≤ 1 act in every r.i. space. The numbers ln στ E , τ →0 ln τ

αE = lim

ln στ E τ →∞ ln τ

βE = lim

are named Boyd indexes of the r.i. space E. It’s known that 0 ≤ αE ≤ βE ≤ 1. Let x, y ∈ L1 . We denote x ≺ y if τ τ ∗ x (t)dt ≤ y ∗ (t)dt 0

0

for each τ ∈ [0, 1]. If a r.i. space E is separable or isometric to the conjugate of some separable r.i. space, then x ≺ y implies xE ≤ yE . For simplicity we shall assume that a r.i. space E satisﬁes this assumption. The Hardy operator 1 x(s) ds Hx(t) = s t is bounded in a r.i. space E iﬀ αE > 0. Without loss of generality 1E = 1.

Random rearrangements in functional spaces

263

Orlicz, Lorentz, Marcinkiewicz spaces are r.i. ones. If a function M (u) is even, convex, increasing on [0, ∞) and lim

u→∞

then

M (u) = ∞, u

M (0) = 0 1

xLM = inf λ: λ > 0, 0

x(t) dt ≤ 1 . M λ

Let ϕ(t) be an increasing concave function on [0, 1] s.t. ϕ(0) = 0. Then 1 x∗ (t)dϕ(t). xΓ(ϕ) = 0

All the above mentioned properties of r.i. spaces can be found in [1,2,4].

Section 2 Let us ﬁx some one-to-one correspondence of Π into {1, 2, . . . , n!}. Let 1 ≤ q ≤ ∞ and x is an n-square matrix. We deﬁne the quasi-linear operator Tq x(t) =

n

|xiπ(i) |q

1/q

, t∈

i=1

(π) − 1 (π) , n! n!

with usual modiﬁcation for q = ∞. It’s evident that Tq xE does not depend on if E is a r.i. space. Deﬁne the operator k − 1 k Sx(t) = sk , t ∈ , , 1 ≤ k ≤ n. n n The following statement generalizes Theorem A. Theorem 1 Let E be a r.i. space. Then 1 SxE ≤ T∞ xE ≤ SxE . 2 Proof. For simplicity we assume that s1 > s2 > . . . > sn > 0. Then m{t: Sx(t) ≥ sk } = k/n for each k = 1, 2, . . . , n . As m{t: T∞ x(t) = sk } ≤

1 (n − 1)! = , k = 1, 2, . . . , n n! n

264

Semenov

then m{t: T∞ x(t) ≥ sk } ≤

k . n

Hence (T∞ x)∗ (t) ≤ Sx(t) and T∞ xE ≤ SxE . To prove the left side of the inequality, we ﬁx k ∈ {1, 2, . . . , n} and construct the matrix |xij |, |xij | ≥ sk yij = . 0, |xij | < sk Applying Theorem A to matrix Y = {yij } we have k 1 1 si ≤ max yiπ(i) . 1≤i≤n 2n 1 n!

(1)

π∈Π

Denote ek = {t: T∞ y(t) = 0}. Then mek ≤ k/n and 1 max yiπ(i) = T∞ y(t)dt 1≤i≤n 2n e k π∈Π T∞ x(t)dt ≤ ≤ ek

As

1 si = n i=1 n

k/n

(T∞ x)∗ (t)dt .

0

k/n

Sx(t)dt 0

then (1) and (2) imply the inequalities k/n 1 k/n Sx(t)dt ≤ (T∞ x)∗ (t)dt. 2 0 0 The function τ (T∞ x)∗ (t)dt

0

is concave on [0, 1] and the function 1 τ Sx(t)dt 2 0 k is linear on each interval [ k−1 n , n ], 1 ≤ k ≤ n. Therefore the inequality τ τ 1 Sx(t)dt ≤ (T∞ x)∗ (t)dt 2 0 0 is valid for each τ ∈ [0, 1]. Hence 12 Sx ≺ T∞ x and 1 SxE ≤ T∞ xE . 2

Usually the Orlicz space LM where M (u) = e|u| − 1 is denoted by exp L.

(2)

Random rearrangements in functional spaces

265

Theorem 2 There exists a constant C > 0 such that

n 1 max |xij | + |xij | . 1≤i,j≤n n i,j=1

T1 xexp L ≤ C

(3)

Proof. It is well known that the extremal points of the convex set max |xij | ≤ 1

n

and

1≤i,j≤n

|xij | ≤ n

i,j=1

are matrices such that some n elements xij (1 ≤ i, j ≤ n) are equal to ±1 and n2 − n elements are equal to 0. It is suﬃcient to prove inequality (3) only for such matrices. Let matrix z belong to this set and z ≥ 0, 1 ≤ j ≤ n. We have

(n − j)! 1 = . m t: T1 z(t) = j ≤ Cnj n! j! Therefore

1

(e

T1 z(t) λ

− 1)dt ≤

0

∞ ej/λ j=1

This means that C in (3) may be chosen as

j!

1/λ

− 1 = ee

1 ln ln 3 .

− 2.

Lemma 3 If x is an n × n

matrix and {(i, j): xij = 0} ≤ n

then T1 x ≺ 8HSx . Proof. First we consider the case:

si =

1,

1≤i≤k

0, k < i ≤ n

for some k ≤ n. Given 1 ≤ j ≤ k we denote n k xiπ(i) = j , Qj = Rm Rj = π: π ∈ Π, i=1

m=j

(4)

266

Semenov

and τj =

|Qj | n! .

It is clear that

τj ≤

2Ckj (n − j)! 2k!(n − j)! 2(k − j + 1) . . . k 2k = = ≤ . n! j!(k − j)!n! j!(n − j + 1) . . . n j!n

As

HSx(t) =

k ln nt , 0
0,

k n

then m{t: HSx(t) ≥ j} =

k n

≤t≤1 k −j e . n

Therefore 2e2 k −j e ≤ 8m{t: HSx(t) ≥ j}. 2n

m{t: T1 x(t) ≥ j} = τj ≤ So

T1 x ≺ 8HSx. Let us consider the general case. There exist ak ≥ 0, n-square matrices zk (1 ≤ k ≤ n) such that some k elements of zk are equal to 1 and the other n2 − k elements are equal to 0, {(i, j): (zk )ij = 1} ⊂ {(i, j): (zk+1 )ij = 1} for each k = 1, 2, . . . , n − 1 and x=

n

ak zk .

k=1

Then

τ

(T1 x)∗ (t)dt ≤

0

n

τ

ak

k=1 n

≤8

k=1

(T1 zk )∗ (t)dt

0

ak

τ

0

τ

HSx(t)dt.

HSzk (t)dt = 8 0

Theorem 4 Let 1 ≤ q < ∞, E be a r.i. space, αE > 0. Then n2 1/q 1 q Tq xE ≤ C SxE + sk n k=n+1

where C depends only on E.

(5)

Random rearrangements in functional spaces Proof. Let

267

2

SxE ≤ 1,

n 1 q sk ≤ 1. n

(6)

k=n+1

We ﬁnd n-square matrices y and z such that their supports are disjoint, x = y + z, | supp y| ≤ n and Sx = Sy. By Lemma 3, we have Tq yE ≤ 8HSyE ≤ 8HE SyE = 8HE SxE . Denote |zij |q = uij , 1 ≤ i, j ≤ n. Then 1/q

Tq zE = (T1 u)1/q E ≤ T1 uE . Assumptions (6) imply that 0 ≤ uij ≤ 1, 1 ≤ i, j ≤ n,

n

uij ≤ n.

i,j=1

Applying Theorem 2 we have 1 . ln ln 3 > 0 implies E ⊃ Lτ for some τ < ∞. So

T1 uexp L ≤

It is well known that the assumption αE E ⊃ exp L and xE ≤ C1 xexp L

for some C1 > 0 and every x ∈ exp L. Therefore C 1/q 1 Tq zE ≤ ln ln 3 and Tq xE ≤ Tq yE + Tq zE C 1/q 1 ≤ 8HE + . ln ln 3 The assumption αE > 0 in Theorem 4 is essential, however it is not necessary. In fact, the function Tq In (t) takes the value n1/q on some interval of length 1/n!. Hence lim Tq In L∞ = ∞ . n→∞

On the other hand, SIn (t) = 1 and sk = 0 for n < k ≤ n2 . The inequality inverse to (5) is true without any restrictions.

268

Semenov

Theorem 5 Let E be a r.i. space and 1 ≤ q < ∞. Then n2 1 1/q 1 q SxE + sk ≤ Tq xE . 12 n k=n+1

Proof. By Theorem 3, SxE ≤ 2T∞ xE ≤ 2Tq xE . A space E is embedded into L1 with constant 1 ([2], II.4.1). Applying Theorem B with p = 1 we have 1/q n2 1 q sk ≤ 10Tq xL1 ≤ 10Tq xE . n k=n+1

From the above given inequality we obtain the needed one. Corollary 6 If M ∈ ∆2 , 1 ≤ q < ∞, then

Tq xLM ≈ SxLM +

2

n 1 q sk n

1/q .

k=n+1

Corollary 7 1 ≤ p, q < ∞ then Tq xLp ≈

1 p sk n n

1/p

+

k=1

2

n 1 q sk n

1/q .

k=n+1

Corollary 7 states that the restriction p ≤ q in Theorem B is superﬂuous.

References 1. C. Bennett, R. Sharpley, Interpolation of Operators. Academic Press, London, 1988. 2. S.G. Krein, Ju.I. Petunin, E.M. Semenov, Interpolation of Linear Operators. Transl. Math. Monogr., Amer. Math. Soc., Providence, 1982. 3. S. Kwapien, C. Sch¨utt, Some combinatorial and probabilistic inequalities and their applications to Banach space theory, Studia Math. 82 (1985), 91–106. 4. J. Lindenstrauss, L. Tzafriri, Classical Banach Spaces. II. Springer Verlag 1979. 5. C. Sch¨utt, Lorentz spaces that are isomorphic to subspace L1 . Trans Amer. Math. Soc. 89, No 2 (1985), 583–595.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 269–277 c 1994 Universitat de Barcelona

Besov spaces and function series on Lie groups II

Leszek Skrzypczak Institute of Mathematics, A. Mickiewicz University, Pozna´n, Poland

Abstract In the paper we investigate the absolute convergence in the sup-norm of two-sided Harish-Chandra’s Fourier series of functions belonging to Zygmund-H¨older spaces defined on non-compact connected Lie groups.

Let G be an n-dimensional connected unimodular Lie group countable at inﬁnity and let K be a k-dimensional connected compact subgroup of G. Let Σ(K) denote the set of all equivalence classes of ﬁnite-dimensional irreducible representation of K. For any σ ∈ Σ(K) let χδ be a character of the class δ and d(δ) its degree. We put ¯δ . (1) αδ = d(δ)χ Let L(x) denote the left regular representation of G on C ∞ (G) (or C0∞ (G)) i.e. (L(x)f )(y) = f (x−1 y) and R(x) be the right regular representation of G on the same spaces i.e. (R(x)f )(y) = f (yx). If f is a suitable function on G then αδ (y)f (y −1 x)dy, x ∈ G, (2) (αδ ∗ f )(x) = K

and

αδ (y −1 )f (xy)dy,

(f ∗ αδ )(x) =

x ∈ G,

(3)

K

are called a δ-Fourier component of the function f with respect to the representation L(x) and R(x) respectively, dy being the normalized Haar measure on K. The group G is countable at inﬁnity therefore the space of smooth function C ∞ (G) and the space of smooth functions with compact support C0∞ (G) taken with their usual 269

270

Skrzypczak

topologies are locally convex complete and metrizable vector topological spaces. Let D (G) be the continuous dual of C0∞ (G). We call the elements of D (G) distributions on G. Identifying αδ with an element of the space of Radon measures with compact support on G we can regard (2) and (3) as the convolutions on G. Due to this identiﬁcation, we can deﬁne the δ-Fourier component with respect to L(x) and R(x) of every distribution T ∈ D (G) by αδ ∗ T,

T ∗ αδ

(convolutions of distributions).

Theorem 1 (cf. [2]) Let f ∈ C ∞ (G) (f ∈ C0∞ (G)) then the Fourier series

αδ ∗ f and

δ∈Σ(K)

f ∗ αδ

δ∈Σ(K)

converge absolutely to f in C ∞ (G) (C0∞ (G)) . Corollary 1 (cf. [14] §4.4.3) The Fourier series of the distribution T converges to T in D (G) equipped with the topology of uniform convergence on bounded subsets. Note that L(x) and R(y) (x, y ∈ G) commute and hence α ∗ (f ∗ β) = (α ∗ f ) ∗ β

α, β ∈ C(K) .

We may therefore simply write α ∗ f ∗ β. In the present paper we will regard also the following series αδ1 ∗ f ∗ αδ2 . (4) δ1 ,δ2 ∈Σ(K)

Generally the above series does not coincide with the series deﬁned in (2) and (3). If the group G is abelian then the last series coincides with the previous ones because αδ ∗ αδ = αδ and αδ1 ∗ αδ2 = 0 if δ1 = δ2 . Proposition 1 (cf. [2]) Let E denote either one of the spaces C ∞ (G) or C0∞ (G). Then for any f ∈ E the series (4) converges absolutely to f in E. On the Lie group G we can deﬁne a left-invariant Riemannian metric tensor g and a right-invariant Riemannian metric tensor g as well (cf. [3]). The Riemannian

Besov spaces and function series on Lie groups II

271

manifolds (G, g) and (G, g) are both connected complete Riemannian manifolds with a positive injectivity and bounded geometry. Therefore we can deﬁne the two scales s s of Besov spaces on G Bp,q (G) and B p,q (G), −∞ < s < ∞, 0 < p ≤ ∞, 0 < q ≤ ∞. The ﬁrst scale corresponds to the Riemannian manifold (G, g) the second one to (G, g) in the sense of the Triebel deﬁnition (cf. [10], [12]). Generally these two scales of function spaces do not coincide. Let R be the Lie algebra of K. Since K is compact we can choose a positivedeﬁned quadratic form Q on R invariant with respect of the adjoint representation Adk . Let X1 , . . . , Xk be a basis of R orthonormal with respect to Q, then the diﬀerential operator Ω = I − (X12 + · · · + Xk2 ) (5) commutes with both left and right translation of K. It is well known that the functions αδ , δ ∈ Σ(K), are eigenvectors of Ω with eigenvalues c(δ) ≥ 1, and that d(δ)2 c(δ)−m < ∞ δ∈Σ(K)

for a suﬃciently large positive number m (cf. [2]), d(δ) being the degree of the class δ. Thus for every r, 0 < r ≤ 2, there is the smallest number mr such that sup d(δ)r c(δ)−m < ∞,

for every m > mr .

(6)

δ∈Σ(K)

Let C(G) denote the Banach space of bounded continuous functions on G with the standard norm. In [7] we proved the following theorem Theorem 2 Let 1 ≤ p ≤ ∞, 1 ≤ q ≤ ∞ and s > s (f ∈ Bp,q (G)). Then the Fourier series δ∈Σ(K)

αδ ∗ f

n p

s

+ 2m1 + max(0, k2 − kp ). Let f ∈ B p,q (G)

f ∗ αδ

δ∈Σ(K)

converges absolutely in C(G) to the function f . Moreover, there is a constant C such that s s αδ ∗ f ∞ ≤ C f |B p,q (G) , f ∗ αδ ∞ ≤ C f |Bp,q (G) . δ∈Σ(K)

δ∈Σ(K)

272

Skrzypczak 2. The absolute convergence of the series Σαδ1 ∗ f ∗ αδ2

The main result of the paper reads as follows. Theorem 3 s

s (G) ∩ B ∞,∞ (G) the series Let s > 4m1 + k. Then for every f ∈ B∞,∞

αδ1 ∗ f ∗ αδ2

δ1 ,δ2 ∈Σ(K)

converges absolutely to f in C(G) and

s s αδ1 ∗ f ∗ αδ2 ∞ ≤ C max f |B∞,∞ (G) , f |B ∞,∞ (G) .

δ1 ,δ2 ∈Σ(K)

Proof. We divide the proof into several steps. = G×G and K = K ×K, where × denotes the cartesian product Step 1. Let G is a 2n-dimensional connected Lie of groups and manifolds as well. The group G is isomorphic to the group, and K is its compact subgroup. The Lie algebra of G direct sum G ⊕ G of the Lie algebra G of G. In this step we describe a Riemannian needed later on. structure on G onto the corresponding factor of the Let πi , i = 1, 2, denote a projection of G as a cartesian product g = g × g product. We deﬁne the Riemannian metric g on G of the Riemannian metric g and g i.e. g(x,y) (X, Y ) = g x (d(x,y) π1 X, d(x,y) π1 Y ) + gy (d(x,y) π2 X, d(x,y) π2 Y ), X, Y ∈ T(x,y) G. The manifold (G, g˜) is a connected homogeneous (x, y) ∈ G, Riemannian manifold. The mappings (x, y) → (xa, b−1 y) ∈ G, a, b ∈ G Φ(a,b) : G The transitivity of the action is form a group of isometries acting transitively on G. obvious. Since Φ(a,b) = Φ(a,e) ◦ Φ(e,b) , it is suﬃcient to prove that Φ(a,e) and Φ(e,b) are isometries. To prove that Φ(a,e) is an isometry we ought to show that g(x,y) (X, Y ) = g (xa,y) (d(x,y) Φ(a,e) X, d(x,y) Φ(a,e) Y ),

(7)

Besov spaces and function series on Lie groups II

273

X, Y ∈ T(x,y) G. Using the product structure of G it is for every (x, y) ∈ G, not diﬃcult to see that d(xa,y) π1 ◦ d(x,y) Φ(a,e) = dy ra ◦ d(x,y) π1 , and d(xa,y) π2 ◦ d(x,y) Φ(a,e) = d(x,y) π2 , where ra : G x → xa ∈ G. These identities and the fact that ra is an isometry of (G, g) imply (7). The proof for Φ(e,b) is the same. g) is a Riemannian manifold with positive injectivity radius and bounded Thus (G, s are well deﬁned on G. (G) geometry and the spaces Bp,p of (G, g), of The following relation between the covariant differentiation (G, g) and of (G, g) is well known: (X ,X ) (Y1 , Y2 ) = ( Y1 , X Y1 ), where X1 , X2 , Y1 , Y2 are vector ﬁelds on G. X1 2 1 2 The last identity makes it obvious that exp(x,y) X = (expx d(x,y) π1 X, expy d(x,y) π2 X), X ∈ T(x,y) G. be the injectivity radius of (G, g), (G, g) and (G, g), Let i(G), i(G) and i(G) ) G . Then there are positive numbers α and respectively. Let ε < min i(G),i(G),i( 8 β, 0 < α < β < ε, and sequences of points {xi }, {yi } ⊂ G such that the family of geodesic balls {B(xi , β)}, ({B(yi , β)}) forms a uniformly locally ﬁnite covering of (G, g) (and (G, g), respectively), and the balls B(xi , α) (B(yi , α)) are pairwise disjoint. The sets √ B(yi , β)×B(xj , β) are also pairwise disjoint. The geodesic balls Bij = It is not diﬃcult to see that this covering B((yi , xj ), 2β) form a covering of G. has bounded geometry therefore is uniformly locally ﬁnite. In fact, the manifold G √ there are constants C1 , C2 > 0 such that vol(B(x, √ 3 2β)) < C1 and √ vol(B(x, α)) < C2 for every x ∈ G. Let Jij = {(k, l): B((yk , xl ), 2β)∩B((yi , xj ), 2β) = ∅}. Then √ vol B(yk , xl ), α > C2 |Jij | C1 > vol B(yi , xj ), 3 2β > k,l∈Jij

(cf. [1] Lemma 2.25 and 2.26, [8]). s if Step 2. Let f(y, x) = f (yx), x, y ∈ G. We prove that f ∈ B ∞,∞ (G) s s f ∈ B∞,∞ (G) ∩ B ∞,∞ (G). We will need the following lemma, which is a direct consequence of Theorem 2.5.13 in [13]. Lemma 1 Let 1 ≤ p ≤ ∞ and s > 0. Then s s f |Bp,p (Rn+m ) ∼ f (·, y)|Bp,p (Rn ) Lp (Rm ) s (Rm ) Lp (Rn ) . + f (x, ·)|Bp,p Lemma 2 Let −∞ < s < ∞, 0 < p ≤ ∞, 0 < q ≤ ∞. Then L(x)(R(x)) is an isomorphism s s (G) (B p,q (G)), and there is a constant C such that L(x) ≤ C, ( R(x) ≤ C) of Bp,q for every x ∈ G.

274

Skrzypczak

Proof of Lemma 2. Let κ and κ0 be a rotation invariant C ∞ functions in Rn such that supp κ ⊆ B(0, 1), and κ(0) = 0, κ ˆ 0 (ξ) = 0 for all ξ ∈ Rn , whereˆ denotes the Fourier −n κN (t−1 exp−1 transform. Let k0,t (x) = κ0 (t−1 exp−1 e x), and kN,t (x) = t e x), where N

n ∂ κN = κ, N = 1, 2, . . . Then for suﬃciently small ε > 0 and r > 0 and ∂x2 j=1

j

N > max(s, 5 + 2 np ) + max(o, n( p1 − 1)) the expression  s (G) 1 = f ∗ k0,ε |Lp (G) +  f |Bp,q

r

1/q t−sq f ∗ kN,t |Lp (G) q

dt  t

0 s is an equivalent norm in Bp,q (G) (cf. [11]). But (L(x)f )∗kN,t = L(x)(f ∗kN,t ). s s f |Bp,q (G) 1 = L(x)f |Bp,q (G) 1 . For right translations the proof is similar.

Thus

Let {ϕi } be the resolution of unity corresponding to the covering {B(xi , β)} and {ψj } the resolution of unity corresponding to the covering {B(yi , B)}. If these resolutions of unity satisfy the assumptions needed to deﬁne the scale of Besov spaces (cf. [10], [12]) then χij (y, x) = ψi (y)ϕj (x) is the resolution of unity corresponding to the covering Bij and satisfying the same assumptions. We have s = sup χij f · exp(y ,x ) |B s (R2n ) f|B∞,∞ (G) ∞,∞ i j i,j

s (Rn ) L∞ (Rn ) ≤ sup ψi (expyi ξ) ϕj (·)f(exp)yi ξ, expxj ·)|B∞,∞ i,j

s (Rn ) L∞ (Rn ) + sup ϕj (expxj ξ) ψi (·)f(expyi ·, expxj ξ)|B∞,∞ i,j

s (Rn ) ≤ sup sup ϕj (·)f(x, expxj ·)|B∞,∞ j

x∈G

s (Rn ) + sup sup ψi (·)f(expyi ·, y)|B∞,∞ i

y∈G

s

s (G) + sup f (·y)|B ∞,∞ (G) ≤ sup f (x·)|B∞,∞ x∈G

s ≤ f |B∞,∞ (G) +

y∈G s f |B ∞,∞ (G) .

The last inequality follows from Lemma 2. Thus s ≤ C max f |B s (G) , f |B s (G) f|B∞,∞ ∞,∞ ∞,∞ (G) .

(8)

Step 3. In the third step we deal with expansions of functions from the spaces s of K needed later on. On the Lie algebra we deﬁne a positive-deﬁned (K) B∞,∞ by quadratic form Q Q(X, Y ) = Q(d(e,e) π1 X, d(e,e) π1 Y ) + Q(d(e,e) π2 X, d(e,e) π2 Y ),

X, Y ∈ ,

Besov spaces and function series on Lie groups II

275

is invariant with respect to where Q is the form on described in §1. The form Q AdK . Let X1 , . . . , Xk be the base in orthonormal with respect to Q. Then the vectors k = (Xk , 0), X k+1 = (0, X1 ), . . . , X 2k = (0, Xk ) 1 = (X1 , 0), . . . , X X orthonormal with respect to Q, and therefore the diﬀerential form a basis of operator 2k 2 X Ω=I− i i=1

The operator Ω is a positivecommutes with both left and right translations of K. so we can deﬁne the abstract Besov spaces deﬁned self-adjoint operator in L2 (K) s Bq (Ω), s > 0, 1 ≤ q ≤ ∞, connected with this operator (cf. [6], §6.2). The ab coincides with the space B 2s (K) deﬁned on K by the stract Besov space Bqs (Ω) 2,q Riemannian approach (cf. [10], [12]). The functions βδ1 ,δ2 (x, y) = αδ1 (x)αδ2 (y) as well as the functions χδ1 ,δ2 (x, y) = with eigenvalues c(δ1 , δ2 ) = c(δ1 ) + c(δ2 ) − χδ1 (x)χδ2 (y) are the eigenfunctions of Ω 1, δ1 , δ2 ∈ Σ(K). The operator Ω has a pure point spectrum and the functions therefore for every r, w ∈ R χδ1 ,δ2 form the orthonormal system of eigenvectors of Ω r such that w + k(1 − 2 ) > 0, there is a positive constant C such that

s r c(δ1 , δ2 )w | < χδ1 ,δ2 , f > |r ≤ C f |B2,r (K)

δ1 ,δ2 ∈Σ(K) s s = 2 w + 2k( 1 − 1 ) (cf. Theorem 6.4.3 in [6]). Here holds for all f ∈ B2,r (K), r r 2 Thus < ·, · > denotes the scalar product in L2 (K). c(δ1 , δ2 )w | < βδ1 ,δ2 , f > |r δ1 ,δ2 ∈Σ(K)

s r ≤ C sup c(δ1 , δ2 )−m d(δ1 )r d(δ2 )r f |B2,r (K) δ1 ,δ2

holds for s = 2 w+m + 2k( 1r − 12 ). But c(δ1 , δ2 ) ≥ max(c(δ1 ), c(δ2 )) therefore the last r inequality and (6) imply s r, c(δ1 , δ2 )w | < βδ1 ,δ2 , f > |r ≤ C f |B2,r (K) (9) δ1 ,δ2 ∈Σ(K)

for s > 2r (w + 2mr ) + 2k( 1r − 12 ).

276

K:

Skrzypczak The following embedding is a consequence of the compactness of the manifold s ⊂ B s0 (K), 1 ≤ r ≤ ∞, −∞ < s0 < s < ∞, cf. [7]. (K) B∞,∞ 2,r

Now if w = 0 and r = 1 then (9) implies that there is a positive constant c > 0 such that s | < βδ1 δ1 , f > | ≤ C f |B∞,∞ (K) (10) δ1 ,δ2 ∈Σ(K) s s > 4m1 + k. holds for every f ∈ B∞,∞ (K),

Step 4. Let f be a suitable function on G. Then (αδ1 ∗ f ∗ αδ2 )(x) = αδ1 (y)αδ2 (z −1 )f (y −1 xz)dzdy K K = αδ1 (y)αδ2 (z)f (yxz)dzdy = βδ1 δ2 (y, z)f (yxz)dydz. K K K We put fx (y, z) = f (yxz), x, y, z ∈ G. Then fx = fe ◦ Φ(e,x−1 ) and (αδ1 ∗ f ∗ αδ2 )(x) = βδ1 ,δ2 (y, z)fx (y, z)dydz =< fx , βδ1 ,δ2 > . K

(11)

x = {(y, z) ∈ G: (y, x−1 z) ∈ K} = Φe,x−1 (K), x ∈ G. Then K x is Let K s s → B a compact submanifold of G. Let Rx : B∞,∞ (G) ∞,∞ (Kx ) be the restriction operator (cf. [8]). We recall that it is a continuous surjective linear operator. It was s x ) can be deﬁned in (K proved in Lemma 1 of [7] that the norms in the spaces B∞,∞ such a way that s = Rx (fe )|B s (K x ) (K) Re (fx )|B∞,∞ ∞,∞

and

Rx ≤ C,

(12)

where C is a constant independent of x. s s s (cf. Step 2). (G) ∩ B ∞,∞ (G), s > 4m1 + k, then fe ∈ B∞,∞ (G) If f ∈ B∞,∞ Now it follows from (8) and (10)-(12) that s s αδ1 ∗ f ∗ αδ2 ∞ ≤ C max f |B∞,∞ (G) , f |B ∞,∞ (G) . δ1 ,δ2 ∈Σ(K)

Thus the series converges absolutely in C(G). But it converges to f in the sense of the strong topology of D (G) so it converges to f also in C(G).

Besov spaces and function series on Lie groups II

277

References 1. T. Aubin, Nonlinear Analysis on Manifolds, Monge-Ampere Equations. Springer Verlag, New York, 1982. 2. Harish-Chandra, Discrete series for semi-simple Lie groups II, Acta Math. 116 (1966), 1–111. 3. J.F. Price, Lie Groups and Compact Groups, University Press, Cambridge 1979. 4. D.W. Robinson, Lie group and Lipschitz spaces,Duke J. Math. 57 (1988), 357–395. 5. D.W. Robinson, Lipschitz operators, J. Funct. Anal. 85 (1989), 179–211. 6. H.J. Schmeisser, H. Triebel, Topics in Fourier Analysis and Function Spaces, Wiley, Chichester 1987. 7. L. Skrzypczak, Besov Spaces and function series on Lie Group, Comment. Math. Univ. Carolinae 34 (1993), 193–147. s s 8. L. Skrzypczak, Traces of function spaces of Fp,q − Bp,q type on submanifolds, Math. Nach. 146 (1990), 137–147. 9. H. Triebel, Diffeomorphism properties and pointwise multiplier for function spaces, In: Function Spaces. Proc. Inter. Conf. Pozna´n 1986, Teubner-Texte, Leipzig 1988, 75–84. 10. H. Triebel, Function spaces on Lie groups, the Riemannian approach, J. London Math. Soc. 35 (1987), 327–338. 11. H. Triebel, How to measure smoothness of distributions on Riemannian symmetric manifolds and Lie groups? Zeitsch. Anal. ihre Anwend. 7 (1988), 471–480. 12. H. Triebel, Spaces of Besov-Hardy Sobolev type on complete Riemannian manifolds, Ark. Mat. 24 (1986), 299–337. 13. H. Triebel, Theory of function spaces, Birkh¨auser, Basel 1983. 14. G. Warner, Harmonic Analysis on Semi-simple Lie groups I, Springer Verlag, Berlin 1972.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 279–299 c 1994 Universitat de Barcelona

On W ∗ U R point and U R point of Orlicz spaces with Orlicz norm∗

Tingfu Wang, Zhongrui Shi and Quandi Wang P.O. Box 610, Math. Dept. Harbin Univ. Sci. Tech., Harbin, Heilongjiang, 150080, P.R. China

Abstract For Orlicz spaces with Orlicz norm, a criterion of W ∗ U R point is given, and previous results about U R points and W U R points are amended.

In the sequel, X denotes a Banach space, S(X) and B(X) denote the unit sphere and ball of X respectively. x ∈ S(X), x is said to be an uniformly round (U R) point, uniformly weak round (W U R) point and uniformly weak star round (W ∗ U R) point w provided that xn ∈ S(X), xn + x → 2 imply xn − x → 0, xn − x −→ 0 and w∗

xn − x −→ 0, respectively. M (u) and N (v) denote a pair of complemented N -functions, p(u) denotes the right-side derivative of M (u). M ∈ ∆2 (M ∈ ∇2 ) denotes that M (u) satisﬁes ∆2 condition (∇2 -condition) for large u. SM denotes the set of all strictly convex points of M (u). {a } (respectively, {b }) denotes the sets of all left-extreme points (resp., right-extreme points) of aﬃne segments of M (u) with p− (a ) = p(a ) (resp., p− (b ) = p(b)), but for {a}, {b} with p− (a) < p(a) and p− (b) < p(b). (G, Σ, µ) denotes a non-atomic ﬁnite measure space, x(t) denotes a measurable real function. We call RM (x) = G M (x(t))dµ a modular of x. By an Orlicz space we shall mean the space LM (G, Σ, µ) = {x(t): for some c > 0, RM (cx) < ∞} equipped with the norm x(t)y(t)dµ. x0 = sup RN (y)≤1

G

Keywords: Orlicz spaces, U R, W U R, W ∗ U R points. Classification: 46B30, 46E30 ∗ Partially supported by NSF of China.

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For x ∈ LM , we denote

x <∞ ξ(x) = inf c > 0: RM c ∗ kx = inf k > 0: RN p(kx) = N p kx(t) dµ ≥ 1 G

kx∗∗ = sup k > 0: RN p(kx) ≤ 1 , K(x) = [kx∗ , kx∗∗ ]

It is well known that k ∈ K(x) if and only if x0 = k1 (1 + RM (kx)). For each x ∈ LM , there is a supporting functional f = y + ϕ, i.e., f (x) = x(t)y(t)dµ + ϕ(x), f = 1 and f (x) = x0 where y ∈ LN and ϕ is singular. G The criteria of U R and W U R points in LM have been discussed in [1-3], but in [1, 2] they are restricted to p(u) continuous, and the condition given in [3] is not necessary. Even without the restriction in that paper, we ﬁrst give the criterion of W ∗ U R point and deduce easily the criteria of W U R and U R points. Lemma 1 For any 0 < λ < 1, 0 < δ < 1 and 0 < ε < 1 there is 0 < δ < 1 such that for u, v, with M (λu + (1 − λ)v) ≤ (1 − δ)(λM (u) + (1 − λ)M (v)) we have that for all λ ∈ [ε, 1 − ε] M λ u + (1 − λ )v ≤ (1 − δ ) λ M (u) + (1 − λ )M (v) . Proof. The proof is easy and is left to the reader. Lemma 2 Let 0 = x ∈ LM . Then f = y + ϕ is a supporting functional of x ⇔ RN (y) + ϕ = 1, ϕ = ϕ(kx), p− (kx(t)) ≤ y(t) ≤ p(kx(t)) µ - a.e., for k ∈ K(x). Proof. Necessity. If f is the supporting functional of x, then y ϕ 1 = f = inf c > 0: RN + ≤1 c c 0 kx = f (kx) = kx(t)y(t)dµ + ϕ(kx) ≤ kx(t)y(t)dµ + ϕ G

G

≤ RM (kx) + RN (y) + ϕ ≤ RM (kx) + 1 = kx0 .

So we get G kx(t)y(t)dµ = RM (kx) + RN (y), thus p− (kx(t)) ≤ y(t) ≤ p(kx(t)). Moreover ϕ = ϕ(kx) and RN (y) + ϕ = 1. The suﬃciency part of the proof is clear. For the convenience of the reader, we split the main result into several lemmas.

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281

Lemma 3 Let x ∈ S(LM ) and k ∈ K(x). If x is a W ∗ U R point then M ∈ ∇2 . Proof. Take yn (t), RN (yn ) = 1 with G x(t)yn (t)dµ > 1 − 1/n. Let d be such that µGd = µ{t ∈ G: |x(t)| ≤ d} > 0. Suppose that M ∈ ∇2 . Then there exists a sequence (vn ) with vn vn Take Gn ⊂ Gd , N (vn )µGn = 1/n and deﬁne zn (t) = ∞, N ( 1− 1 ) > 2nN (vn ). n zn vn χGn (t). Then RN (zn ) = 1/n < 1 and RN ( 1− 1 ) > 2nN (vn )µGn = 2 so we get n 1 ≥ zn N ≥ 1 − 1/n, where yN = inf{c > 0: RN (y/c) ≤ 1} (it is called the Luxemburg norm of y). By [4], there are xn (t) = un χGn (t), xn 0 = 1 such that 1 un vn µGn = xn (t)zn (t)dµ = zn N > 1 − . n Gn Put gn (t) = (1 − n1 )(yn (t)χG\Gn (t) + zn (t)). Then 1 1 1 1 RN (yn ) + RN (zn ) ≤ 1 − 1+ = 1 − 2. RN (gn ) ≤ 1 − n n n n Thus

xn (t) + x(t) gn (t)dµ G 1 = 1− xn (t)yn (t)dµ + x(t)yn (t)dµ n G\Gn G\Gn

+ xn (t)zn (t)dµ + x(t)zn (t)dµ Gn Gn 1 ≥ 1− 0+ x(t)yn (t)dµ − x(t)yn (t)dµ + un vn µGn n G Gn − x(t)zn (t)dµ

xn + x ≥ 0

Gn

1 1 1 1 − − d · χGn 0 + 1 − − dχGn 0 → 2 n n n as n → ∞. Take h ∈ EN , with G x(t)h(t)dµ > 0, then x(t) − xn (t) h(t)dµ ≥ x(t)h(t)dµ − xn 0 · hχGn 0 G G → x(t)h(t)dµ > 0

≥ 1−

G

which contradicts the fact that x is a W ∗ U R point.

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Lemma 4 Let x ∈ S(LM ). If x is a W ∗ U R point then µ{t ∈ G: |kx(t)| ∈ R \ SM ∪ {a } ∪ {b }} = 0 for every k ∈ K(x). Proof. Since a W ∗ U R point is an extreme point, by [4], it follows µ{t ∈ G: |kx(t)| ∈ R \ SM } = 0 . Denote Ga = {t ∈ G: |kx(t)| ∈ {a }}, Gb = {t ∈ G: |kx(t)| ∈ {b }}. Suppose that µGb > 0. Without loss of generality, we can assume that µG0 = µ{t ∈ G: kx(t) = b } > 0 for some b ∈ {b }, b > 0. Deﬁne y(t) = x(t)χG\G0 (t) +

a χG0 (t), k

where a ∈ {a } is such that M is aﬃne on the interval [a , b ]. For any η > 0, RN p((1 − η)ky) ≤ RN p((1 − η)kx) ≤ RN p((1 − η)kx∗∗ x) ≤ 1 and RN (p((1+η)ky)) ≥ 1. Indeed, when RN (p((1+η)kx)) = ∞, we have RN (p((1+ η)ky)) ≥ RN (p((1 + η)kxχG\G0 (t))) = ∞; in the other case, i.e., if RN (p((1 + η)kx)) < ∞ we have N p((1 + η)kx(t)) dµ + N p((1 + η)a ) µG0 RN p((1 + η)ky) = G\G0 ≥ lim N p((1 + s)kx(t)) dµ + N p((1 + s)a ) µG0 s→0

G\G0

= lim RN p((1 + s)kx) ≥ 1 . s→0

Hence k ∈ K(y). Let f = v + ϕ be a supporting functional of x. By Lemma 2, we get that RN (v) + ϕ = 1, ϕ = ϕ(kx) and p− (kx(t)) ≤ v(t) ≤ p(kx(t)). Clearly ϕ(ky) = ϕ(kx) and p− (ky(t)) ≤ v(t) ≤ p(ky(t)), so f is the supporting functional of y too, i.e., f (y) = y0 . Therefore f x+

y f (y) =2 = f (x) + 0 y y0

On W ∗ U R point and U R point of Orlicz spaces with Orlicz norm

283

we get x + y/y0 0 = 2. But x = y/y0 , which contradicts the fact that x is a W ∗ U R point. If we suppose that µGa > 0, without loss of generality, we can assume that for some a ∈ {a }, a > 0 we have µG0 = µ{t ∈ G: kx(t) = a } > 0. Deﬁne y(t) = x(t)χG\G0 (t) +

a + b χG0 (t), 2k

where b ∈ {b } is such that M is aﬃne on the interval [a , b ]. For any η > 0 suﬃciently small, RN (p((1+η)ky)) ≥ RN (p((1+η)kx)) ≥ 1, and RN (p((1−η)ky)) ≤ 1, which shows that k ∈ K(y). Indeed if suppose that N p((1 − η)kx(t)) dµ + N p(a ) µG0 1 < RN p(1 − η)ky) = G\G0 = N p((1 − η)kx(t)) dµ + N p− (a ) µG0 . G\G0

By p− (a ) = p(a ), it follows that there is 0 < ξ < η such that N p((1 − η)kx(t)) dµ + N p((1 − ξ)a ) µG0 > 1 RN p((1 − ξ)kx) ≥ G\G0

which is a contradiction with k ∈ K(x). Finally one can reach a contradiction analogously as in the case µGb > 0. Lemma 5 Let x ∈ S(LM ) and k ∈ K(x). If x is a W ∗ U R point, then (i) µGa > 0 implies RN (p− (kx)) = 1 (ii) µGb > 0 implies RN (p(kx)) = 1 and, for some 0 < τ < 1, kx <∞ p 1−τ

RN

where Ga = {t ∈ G: |kx(t)| ∈ {a}}, Gb = {t ∈ G: |kx(t)| ∈ {b}}. Proof. (i) If µGa > 0, then µG0 = µ{t ∈ G: |kx(t)| = a} > 0 for some a ∈ {a}. We ﬁrst show RN (p(kx)) ≥ 1. Indeed, if RN (p(kx)) < 1, then, for the supporting functional f = v + ϕ of x, we have ϕ = 0. Take λ > 0 with RN p(kx) + λϕ = 1 .

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Put y(t) = x(t)χG\G0 (t)+(a + b)/(2k)χG0 (t). Then RN (p(ky)) = RN (p(kx)) < 1 and RN (p((1 + η)ky)) ≥ RN (p((1 + η)kx)) ≥ 1, so k ∈ K(y). Put v(t) = p(kx)χG\G0 (t) + p(a)χG0 (t). Then f = v + λϕ is the supporting functional of y, since RN (v) + λϕ = 1, λϕ = λϕ = λϕ(kx) = λϕ(ky), and v(t) = p(ky(t)). Further, f is the supporting functional of x too. Therefore f (x + y/y0 ) = 2, x + y/y0 0 = 2. But x = y/y0 , which contradicts the fact that x is a W ∗ U R point. Now suppose that RN (p− (kx)) = 1. Then RN (p− (kx)) < 1. We shall consider two cases. Assume that G\G0 N p− (kx(t)) dµ + N (p(a))µG0 < 1. Since RN (p(kx)) ≥ 1, there exists v(t), such that v(t) = p(a) (t ∈ G0 ); p− (kx(t)) ≤ v(t) ≤ p(kx(t)) (t ∈ G \ G0 ) and RN (v) = 1. Hence v is the supporting functional of x. Deﬁne y(t) = x(t)χG\G0 (t) + (a + b)/(2k)χG0 (t). Since RN p(ky) ≥ RN p(kx) ≥ 1 and RN p((1 − η)ky) ≤ N p− (kx(t)) dµ + N p(a) µG0 < 1 , G\G0

k ∈ K(y) and v is the supporting functional of y too. From G (x(t) + y(t)/y0 )v(t)dµ = 2, we get x + y/y0 0 = 2. But x = y/y0 ∗ which contradicts the fact that x is a W U R point. Now, assume that G\G0 N p− (kx(t)) dµ + N (p(a))µG0 ≥ 1. 0 = 1. Deﬁne 0 ⊂ G0 with N p (kx(t)) dµ + N (p(a))µG Take G − G\G0 v(t) = p− (kx(t))χG\G 0 (t) + p(a)χG 0 (t), then RN (v) = 1 and v is the supporting functional of x. Put y(t) = x(t)χG\G 0 (t) + (a+ b)/(2k)χG 0 (t). Then RN (p(ky)) ≥ RN (p(kx)) ≥ 1 and RN (p((1 − η)ky)) ≤ G\G 0 N (p− (kx(t)))dµ + N (p(a))µ functional of y too. G0 = 1, so k ∈0 K(y). Clearly v is the supporting Hence G (x(t) + y(t)/y )v(t)dµ = 2, so x + y/y0 0 = 2. But x = y/y0 , a contradiction. (ii) If µGb > 0, then µG = µ{t ∈ G: |kx(t)| = b} > 0, for some b ∈ {b}. Supposing that RN (p(kx)) > 1 and applying the fact that RN (p− (kx)) ≤ 1, analogously as in (i) we obtain a contradiction. Thus RN (p(kx)) ≤ 1. kx Now we show that RN (p( 1−τ )) < ∞ for some τ > 0. In fact if we suppose that RN (p((1 + ε)kx)) = ∞ for any ε > 0, then M ∈ / ∆2 , i. e., there exists a sequence un ∞, M (un )/un p(un ) → 0. Since, by Lemma 3, M ∈ ∇2 , there is d > 0 such that up(u) ≤ dN (p(u)) for every large u, so M (un )/N (p(un )) → 0. Take Gn ⊂ G0 satisfying N p(kx(t)) dµ + N p(a) µ(G0 \ Gn ) + N p(un ) µGn = 1 . G\G0

On W ∗ U R point and U R point of Orlicz spaces with Orlicz norm Clearly µGn → 0 (n → ∞). Deﬁne a un χG \G (t) + χGn (t) . k 0 n k

yn (t) = x(t)χG\G0 (t) +

From RN (p(kyn )) = 1, it follows that k ∈ K(yn ), kyn 0 ∈ K(yn /yn 0 ). Hence kn = kyn 0 = 1 + RM (kyn ) =1+ M kx(t) dµ + M (a)µ(G0 \ Gn ) + M (un )µGn G\G0 −→ 1 + M kx(t) dµ + M (a)µG0 = k < k . G\G0

Since k · kn k yn (t) kn kx(t) + kyn (t) x(t) + = k + kn yn 0 k + kn k + kn  kx(t)      kn b + k a = k + kn k + kn    kn k   b+ un k + kn k + kn

(t ∈ G \ G0 ) (t ∈ G0 \ Gn ) (t ∈ Gn ),

we have RN

k · k yn n p x + k + kn yn 0

N p(kx(t)) dµ

< G\G0

+ N p(a) µ(G0 \ Gn ) + N p(un ) µGn = 1

and Rn

Hence

k · kn yn (1 + η) x + k + kn yn 0

k·kn k+kn

>

∈ K(x + y/y0 ). Therefore

G\G0

N p((1 + η)kx(t)) dµ = ∞.

285

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k · k 0 y k + k yn n n n x+ 1 + RM = yn 0 k · kn k + kn yn 0 k k k + kn n b+ a µ(G0 \ Gn ) ≥ 1+ M kx(t) dµ + M k · kn kn + k kn + k G\G0 1 = 1+ M kx(t) dµ + M (b)µ(G0 \ Gn ) k G\G0 1 + 1+ M kx(t) dµ + M (a)µ(G0 \ Gn ) kn G\G0

1 1 −→ 1 + RM kx) + 1 + M kx(t) dµ + M (a)µG0 = 2 . k k G\G0

x +

But k yn (t) − x(t) sign x(t)χ (t)dµ = − 1 |x(t)|dµ G\G0 0 G yn G\G0 kn k −→ −1 |x(t)|dµ > 0 k G\G0 which contradicts the fact that x is a W ∗ U R point. Thus there is 0 < τ < 1, kx such that RN (p( 1−τ )) < ∞. Applying the right-hand-side continuity of p(u), we get RM (kx) ≥ 1. Since we have veriﬁed that RM (p(kx)) ≤ 1, we deduce RM (p(kx)) = 1. Lemma 6 Let us assume that x ∈ S(LM ), k ∈ K(x) and x is a W ∗ U R point with RN (p(kx)) < 1. Then, for any ε, ε > 0, there is δ > 0 such that, for any measurable function u(t) and e ⊂ G satisfying µe < δ and εε ≤ ε u(t) ≤ kx(t) ≤ u(t) for all t ∈ e, if M (u(t)) ≥ εN (p− (u(t))) and M

u(t) + kx(t)

then RM (uχe ) < ε.

2

M u(t) + M kx(t) > (1 − δ) , 2

On W ∗ U R point and U R point of Orlicz spaces with Orlicz norm

287

Proof. Suppose on the contrary, that for some ε > 0, for all n, there exist un (t) and en with µen < 1/n, such that ε2 ≤ εun (t) ≤ kx(t) ≤ un (t) for t ∈ en , M (un (t)) ≥ (kx(t)) εN (p− (un (t))), M ( un (t)+kx(t) ) > (1 − n1 ) M (un (t)+M but RM (un χen ) ≥ ε. 2 2 If RN (p− (kx)) + RN (p− (un χen )) ≤ 1, put En = en . Then RM (un χEn ) ≥ ε. If RN (p− (kx)) + RN (p− (un χen )) > 1, take En ⊂ en , such that RN p− (kx) + RN p− (un χEn ) = 1 . Then RM (un χEn ) ≥ εRN p− (un χEn ) ≥ ε 1 − RN (p− (kx)) = (1 − θ)ε. Deﬁne xn (t) = x(t)χG\En (t) +

1 un (t)χEn (t) k

(n = 1, 2, . . .) .

Since RN (p((1 − η)kxn )) ≤ G\En N (p− (kx(t)))dµ + En N (p− (un (t)))dµ ≤ 1 and RN (p((1 + η)kxn )) ≥ RN (p((1 + η)kx)) ≥ 1, from kx(t) ≤ un (t), we get that k ∈ K(xn ) (n = 1, 2, . . .), kxn 0 ∈ K(xn /xn 0 ). Since kn = kxn 0 = 1 + RM (kxχG\En ) + RM (un χEn ) → k ≥ k + (1 − θ)ε and k · kn k xn (t) kn kx(t) + kxn (t) x(t) + = 0 k + kn xn k + kn k + kn  t ∈ G \ En  kx(t) = kn k  kx(t) + un (t) t ∈ En k + kn k + kn k·kn we have that RN (p((1 + η) k+k (x + xn /xn 0 ))) ≥ RN (p((1 + η)kx)) ≥ 1 and n k·kn (x + xn /xn 0 ))) ≤ RN (p− (kx)χG\En ) + RN (p− (un χEn )) ≤ 1, RN (p((1 − η) k+k n so k · kn /(k + kn ) ∈ K(x + xn /xn 0 ). By Lemma 1, it follows that there exists 1 M (u)+M (v) , then M (λn u + (1 − λn )v) ≥ δn ↓ 0 such that if M ( u+v 2 ) ≥ (1 − n ) 2 1 k , 1+k ], where k = sup kn < ∞ (1 − δn )(λn M (u) + (1 − λn )M (v)) for λn ∈ [ 1+k

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(because of M ∈ ∇2 ). Thus we have k + kn xn 0 1 + = M kx(t) dµ x + 0 xn k · kn G\En k k n M kx(t) + un (t) dµ + k + kn k + kn En k + kn M kx(t) dµ ≥ 1+ k · kn G\En kn k + (1 − δn ) M kx(t) + M un (t) dµ k + kn En k + kn 1 1+ = M kx(t) dµ + (1 − δn ) M kx(t) dµ k G\En En 1 M kx(t) dµ + (1 − δn ) M un (t) dµ → 2 . + 1+ kn G\En En µ

But xn /xn 0 −→ kx/k = x, which contradicts the fact that x is a W ∗ U R point. Lemma 7 If M ∈ ∇2 , 1 = x0 = k1 (1 + RM (kx)), 1 = xn 0 = and

1 1 + RM (kn xn ) , kn

xn + x0 → 2

µ t ∈ G: |kx(t)| ∈ R \ SM ∪ {a } ∪ {b } ∪ {a} ∪ {b} = 0, µ

then kn xn − kx −→ 0. µ

/ 0. Then there are ε, δ > 0 such that Proof. Suppose that kn xn − kx −→

µ t ∈ G: |kn xn (t) − kx(t)| ≥ ε ≥ σ (n = 1, 2, . . .) . Since xn 0 = 1, k = sup kn < ∞. Further, for any D, n

k ≥ kn ≥ RM (kn xn ) ≥ M kn xn (t) dµ ≥ M (D)µ{t ∈ G: |kn xn (t)| > D} . {t:|kn xn (t)|>D}

On W ∗ U R point and U R point of Orlicz spaces with Orlicz norm

289

Thus µ{t: |kx(t)| > D} < σ/4, µ{t: |kn xn (t)| > D} < σ/4 (n = 1, 2, . . .) for some D large enough. Denote all left and right extreme points of aﬃne segments of M (u) as c1 , c2 , . . . Since kx(t) = ci (i = 1, 2, . . .) there are open segments Vi including ci with µ{t: kx(t) ∈ Vi } < σ/(2i · 4) (i = 1, 2, . . .), then ∞ σ Vi ≤ . µ t: kx(t) ∈ 4 i=1

Hence µGn = µ t ∈ G: |kn xn (t) − kx(t)| ≥ ε, |kn xn (t)|, |kx(t)| ≤ D, kx(t) ∈ SM \

σ Vi ≥ . 4 i=1 ∞

For the bounded closed set of three dimension space (u, v, λ): |u − v| ≥ ε, |u|, |v| ≤ D, v ∈ SM

1 k \ Vi , λ ∈ , 1+k 1+k i=1 ∞

there is a common δ, 0 < δ < 1 such that for all (u, v, λ) of the above set M λu + (1 − λ)v ≤ (1 − δ) λM (u) + (1 − λ)M (v) ,

so, for t ∈ Gn ,

M

k k·k kn n xn (t) + x(t) ≤ (1 − δ) M (kn xn (t)) + M (kx(t)) . k + kn k + kn k + kn

Hence

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Wang, Shi and Wang

0 ← xn 0 + x0 − xn + x0 1 1 1 + RM (kn xn ) + 1 + RM (kx) ≥ kn k k · kn k + kn 1 + RM (xn + x) − k · kn k + kn k·k k k + kn kn n xn (t) = M kn xn (t) + M kx(t) − M k · kn G k + kn k + kn k + kn + x(t) dµ k·k k k + kn kn n ≥ M kn xn (t) + M kx(t) − M xn (t) k · kn Gn k + kn k + kn k + kn + x(t) dµ k k + kn kn ≥ δ M kn xn (t) + M kx(t) dµ k · kn k + kn Gn k + kn εδ 1 ≥ δM >0 2 4 k µ

this contradiction shows that kn xn − kx −→ 0. Lemma 8 Let x ∈ S(LM ) and k ∈ K(x). Then for any ε > 0 there is y such that ky x − y0 < ε, k ∈ K(y) and RN (p( 1− ε )) < ∞. 2

Proof. If RN (p((1 + η)kx)) < ∞ for some η > 0, we can take y = x. Assume that RN (p((1 + η)kx)) = ∞ for every η > 0. Let ε > 0. Since RN (p((1 − ε)kx)) ≤ 1 and RN (p((1 + ε)kx)) = ∞, we can ﬁnd Gc = {t ∈ G: |x(t)| > c} with RN p (1 + ε)kx χG\Gc + RN p (1 − ε)kx χGc = 1 . Clearly µGc > 0. Put y(t) = (1 + ε)xχG\Gc + (1 − ε)xχGc . Then y − x0 = ε, k ∈ K(y), RN (p(ky)) = 1 and ky 1+ε 1−ε = RN p kx χG\Gc + RN p kx χGc RN p 1 − 2ε 1 − 2ε 1 − 2ε 1+ε ≤N p µG + RN p− (kx) < ∞ . εc 1− 2

On W ∗ U R point and U R point of Orlicz spaces with Orlicz norm

291

Theorem 1 (1) (2) (3) (4)

Let x ∈ S(LM ) and let k ∈ K(x). A point x is W ∗ U R if and only if M ∈ ∇2 µ{t ∈ G: |kx(t)| ∈ R \ SM ∪ {a } ∪ {b }} = 0 if µGa > 0 then RN (p− (kx)) = 1; kx )) < ∞, for some 0 < τ < 1. if µGb > 0 then RN (p(kx)) = 1 and RN (p 1−τ If RN (p− (kx)) < 1, then for any ε, ε > 0 there is δ > 0 such that for any measurable function u(t) ande ⊂ G satisfying µe < δ and εε ≤ ε u(t) ≤ kx(t) ≤ u(t) for all t ∈ e, if M u(t) ≥ εN p− (u(t)) and u(t) + kx(t) M u(t) + M kx(t) > (1 − δ) , M 2 2 then RM (uχe ) < ε.

Proof. Necessity, see Lemmas 3-6. Suﬃciency. We shall discuss three cases. Let x0 = k1n (1 + RM (kn xn )) = 1, xn + x0 → 2. Without loss of generality we can assume that xn (t) ≥ 0, x(t) ≥ 0 and, by Lemma 8, RN (p(ξn kn xn )) < ∞ for some ξn > 1. (I) µGa = µGb = 0 µ In this case, by (2) and Lemma 7, it follows that kn xn −kx −→ 0. To prove that µ xn −x −→ 0, it is enough to show kn → k. Note that kn −k = RM (kn xn )−RM (kx), so by the Egoroﬀ theorem, it suﬃces to prove lim sup RM (kn xn χe ) = 0. This will µe→0 n

be split also into two cases. (I1) RN (p− (kx)) = 1. Assume that kn xn (t) → kx(t), t ∈ G, µ - a.e., passing to a subsequence if necessary. Since p− (u) is left-continuous and nondecreasing, we get lim inf N (p− (kn xn (t))) ≥ N (p− (kx(t))), t ∈ G, µ - a.e. Since RN (p− (kn xn )) ≤ 1 it n→∞

follows that 1 ≥ lim inf RN (p− (kn xn )) ≥ RN (p− (kx)) = 1. Since µ(G) < ∞ we get n→∞

lim sup RN (p− (kn xn χe )) = 0. From M ∈ ∇2 , M (u) ≤ up− (u) ≤ dN (p− (u)) we

µe→0 n

deduce lim sup RM (kn xn χe ) = 0

µe→0 n

(I2) RN (p− (kx)) < 1. n /2 Let k = sup kn and ε > 0. Then there exists 0 < δ < 1 such that M ( k+k k+kn u) ≤ n

n /2 (1 − δ ) k+k k+kn M (u), for every u ≥ ε. Let 0 < ε < δ /k and let δ > 0 be taken from (4) for those ε and ε .

292

Wang, Shi and Wang For e ⊂ G and each n, denote An = t ∈ e: kn xn (t) < ε or kn xn (t) < kx(t) or M kn xn (t) < εN p− (kn xn (t)) k·k k n Bn = t ∈ e \ An : M xn (t) + x(t) ≤ (1 − δ) M kn xn (t) k + kn k + kn kn M kx(t) or kx(t) < ε kn xn (t) + k + kn Cn = e \ An \ Bn = t ∈ e: εε ≤ ε kn xn (t) ≤ kx(t) ≤ kn xn (t), M kn xn (t) ≥ εN p− (kn xn (t)) , and

M

k·k k kn n xn (t) + x(t) > (1 − δ) M kn xn (t) + M kx(t) . k + kn k + kn k + kn

From RM (kn xn χAn ) ≤ M (ε)µe+ RM (kxχe ) + εRN (p− (kn xn )), combining with RM (kxχe ) → 0 (µe → 0) and RN (p− (kn xn )) ≤ 1 we deduce lim sup RM (kn xn χAn ) = O(ε)

µe→0 n

If kx(t) < ε kn xn (t) then M

k·k k + ε k n n kn xn (t) xn (t) + x(t) ≤ M k + kn k + kn k + ε kn ≤ (1 − δ) M (kn xn (t)) k + kn k + ε kn k = (1 − δ ) M kn xn (t) k k + kn k M kn xn (t) ≤ (1 − δ )(1 + ε k) k + kn k kn 2 M kn xn (t) + M kx(t) . ≤ (1 − δ ) k + kn k + kn

On W ∗ U R point and U R point of Orlicz spaces with Orlicz norm

293

Putting δ = min{δ, δ }, we obtain k k·k kn n M kn xn (t) + M kx(t) xn (t) + x(t) ≤ (1 − δ ) M k + kn k + kn k + kn for t ∈ Bn . So 2

0 ← xn 0 + x0 − xn + x0 k k + kn kn ≥ δ M kn xn (t) + M kx(t) dµ. k · kn Bn k + kn k + kn Thus RM (kn xn χBn ) < ε for n large enough. Hence we have lim sup RM (kn xn χBn ) = O(ε) .

µe→0 n

Finally from (4), we get lim sup RM (kn xn χCn ) ≤ ε. By the arbitrariness of ε, µe→0 n

combining with the above, we obtain lim supn RM (kn xn χe ) = 0. µe→0

(II) µGa > 0 . From 1 = RN (p− (kx)) < RN (p(kx)) , combining with (3), it follows µGb = 0 . µ Now, analogously as in Lemma 7, we deduce kn xn − kx −→ 0 on G \ Ga , so without loss of generality, we can assume that kn xn (t) → kx(t) t ∈ G \ Ga µ-a.e. Since p− (u) is left-continuous and nondecreasing, we get lim inf N (p− (kn xn (t))) ≥ N (p− (kx(t)))

t ∈ G \ Ga

n→∞

µ-a.e.

Hence

lim inf RN (p− (kn xn χG\Ga )) ≥

n→∞

RN (p− (kxχG\Ga )). On the other hand, RN (p− (kn xn )) ≤ 1 = RN (p− (kx)), so we have lim sup RN p− (kn xn χGa ) ≤ RN p− (kxχGa ) . n→∞

Denote Gn = {t ∈ Ga : kn xn (t) ≤ kx(t)}. Applying the fact that a is a left extreme point of an aﬃne segment of M (u), we obtain, like in Lemma 7, µ{t ∈ Ga : kn xn (t) ≤ kx(t) − ε} → 0 for any ε > 0. Since lim sup RN (p− (kn xn χe )) = 0 for all e ⊂ Gn , we deduce µe→0 n lim sup RN p− (kn xn χGn ) = lim sup RN p− (kxχGn ) , n→∞

so

n→∞

lim sup RN p− (kn xn χGa \Gn ) ≤ lim sup RN p− (kxχGa \Gn ) .

n→∞

n→∞

From kn xn (t) > kx(t) (t ∈ Ga \ Gn ) we derive N p− (kn xn (t)) − N p− (kx(t)) dµ = 0. lim sup n→∞

Ga \Gn

By p− (a) < p(a) (a ∈ {a}), µ{t ∈ Ga \ Gn : kn xn (t) ≥ kx(t) + ε} −→ 0 .

294

Wang, Shi and Wang µ

Combining the above we deduce kn xn − kx −→ 0 on G, like (I1), which follows that µ lim sup RN (p− (kn xn χe )) = 0 and lim sup RM (kn xn χe ) = 0, hence xn − x −→ 0. µe→0 n

µe→0 n

(III) µGb > 0. kx )) < In this case, 1 = RN (p(kx)) > RN (p− (kx)), so µGa = 0. From RN (p( 1−τ kx ∞, so RM ( 1−τ ) < ∞, it yields that lim sup RM (kn xn χe ) = 0. Indeed if we suppose µe→0 n

that for some ε0 > 0, there exist en ⊂ G and xn , if necessary pass to a subsequence, such that RM (kn xn χen ) ≥ ε0 , then k·k k + kn n 1 + RM 2← (xn + x) k · kn k + kn k·k k + kn n 1 + RM = (xn + x)χG\en k · kn k + kn k(1 + τ ) k x kn (1 − τ ) kx n n n + χ en + RM k + kn 1 + τn k + kn 1 − τ k·k k + kn n 1 + RM ≤ (xn + x)χG\en k · kn k + kn k x k (1 − τ ) kx k(1 + τn ) n n n χe + RM χe + RM k + kn 1 + τn n k + kn 1−τ n k·k k + kn n ≤ (xn + x)χG\en 1 + RM k · kn k + kn kx k(1 + τn ) RM (kn xn χe ) kn (1 − τ ) χe + (1 − δ) + RM k + kn 1 + τn k + kn 1−τ n k + kn k kn ≤ 1+ RM (kn xn χG\en ) + RM (kxχG\en ) k · kn k + kn k + kn k kn kn RM (kn xn χen ) + RM (kxχen ) − RM (kxχen ) k + kn k + kn k + kn kx kn (1 − τ ) kδ χe RM (kn xn χen ) + RM − k + kn k + kn 1−τ n kx δ 1 1−τ RM χen ≤ 2 − RM (kn xn χen ) − RM (kxχen ) + k k 1−τ k δε0 −→ 2 − k +

On W ∗ U R point and U R point of Orlicz spaces with Orlicz norm

295

(where τn > 0, kτn = kn τ, τ, δ > 0 satisfying M ( 1+τu /k ) ≤ (1−δ) 1+τ1 /k M (u) u ≥ ε0 ) - a contradiction. Since u p(t)dµ ≥ p (1 − θ)u θu M (u) > (1−θ)u

=

θ θ (1 − θ)u · p (1 − θ)u ≥ N p((1 − θ)u) 1−θ 1−θ

we have that for θ, 0 < θ < 1, lim sup RN p((1 − θ)kn xn χe ) = 0 .

µe→0 n

For θ small enough, if kn xn (t) ≤ (1 + θ) and if kn xn (t) >

then

k · kn kx(t) kx(t) ≤ xn (t) + x(t) ≤ (1 + θ) k + kn 1 − τ /2 1−τ

kx(t) 1−τ /2

(1 + θ)

kx(t) 1−τ /2

(∗)

then

τ k · kn xn (t) + x(t) ≤ (1 + θ) 1 − kn xn (t) k + kn 2(1 + k) = (1 − θ )kn xn (t) .

kx )) < ∞ and (∗), we have Combining RN (p( 1−τ

k · kn lim sup RN p (1 + θ) (xn + x)χe = 0 . µe→0 n k + kn

(∗∗)

In the following we shall show that for any η > 0 k · kn (xn + x) ≥ 1 . lim sup RN p (1 + η) n→∞ n k + kn

(∗ ∗ ∗)

Indeed if we suppose that for some η0 > 0, θ0 > 0 and for all n (if necessary passing to a subsequence) k · kn RN p (1 + η0 ) (xn + x) ≤ 1 − θ0 . k + kn

296

Wang, Shi and Wang For hn =

0←

k·kn k+kn ,

we have shown

1 hn (1 + RM (hn (xn

+ x))) − xn + x0 → 0, thus

1 1 1 + RM h(xn + x) 1 + RM hn (xn + x) − inf h>0 h hn 1 1 1 + RM hn (xn + x) − 1 + RM (1 + η0 )hn (xn + x) ≥ hn (1 + η0 )hn 1 + η0 η0 RM (1 + η0 )hn (xn + x) − RM hn (xn + x) 1− = (1 + η0 )hn η0 + RM (1 + η0 )hn (xn + x) η0 ≥ (1 + η0 )hn 1 + η 0 × 1− η0 hn xn (t) + x(t) p (1 + η0 )hn xn (t) + x(t) dµ η0 G + RM (1 + η0 )hn (xn + x) η0 η0 θ0 1 − RN (p((1 + η0 )hn (xn + x))) ≥ (1 + η0 )hn (1 + η0 )hn 2η0 θ0 ≥ . (1 + η0 )k

=

This contradiction shows that (∗ ∗ ∗) holds. Denote Gn = {t ∈ Gb : kn xn (t) < kx(t)}, like in Lemma 7, we can derive that µ kn xn χGb \Gn − kx −→ 0 on (G \ Gb ), i.e. for any ε > 0, µ{t ∈ (G \ Gb ) ∪ (Gb \ Gn ): |kn xn (t) − kx(t)| ≥ ε} → 0 as n → ∞ . µ

Finally we show that kn xn − kx −→ 0 on G. Indeed if suppose that for some n ⊂ Gn ε > 0, σ > 0, and all n, if necessary passing to a subsequence, there are G n = µ{t ∈ Gn : kn xn (t) < kx(t) − ε} ≥ σ . µG Taking into account that p(u) is right-continuous and nondecreasing, kn xn (t) < kx(t) (t ∈ Gn ) and (∗ ∗ ∗), we derive

On W ∗ U R point and U R point of Orlicz spaces with Orlicz norm

297

k · kn 1 ≤ lim RN p (1 + η) (xn + x) n→∞ k + kn k · kn (xn + x)χG\G ) = lim RN p((1 + η) n n→∞ k + kn k · kn (xn + x)χG ) + RN p((1 + η) n k + kn ≤ lim RN p((1 + η)kxχG\G n ) n→∞ ε + N p((1 + η)(kx(t)) − ) dµ 1+k n G

= lim RN p((1 + η)kxχG\G n ) + N p(a) µGn n→∞

≤ lim RN p((1 + η)kxχG\G ) n n→∞ n − N (p(b)) − N (p(a)) µGn + N (p(b))µG ≤ RN (p((1 + η)kx)) − N (p(b)) − N (p(a)) σ . kx )) < ∞, let η → 0, we get a contradiction: By RN (p( 1−τ

1 ≤ 1 − N (p(b)) − N (p(a)) σ . µ

µ

So kn xn −kx −→ 0 on G. From lim supn RM (kn xn χe ) = 0, we have that xn −x −→ µe→0

0.

By Theorem 1, we easily deduce the correct criteria of U R and W U R points. Theorem 2 Let x ∈ S(LM ) and k ∈ K(x). A point x is a U R (W U R) if and only if (i) M ∈ ∆2 ∩ ∇2 (ii) µ{t ∈ G: |kx(t)| ∈ R \ SM ∪ {a } ∪ {b }} = 0 (iii) if µGa > 0 then RN (p− (kx)) = 1 if µGb > 0 then RN (p(kx)) = 1 . Proof. Necessity. Since a W U R point is a W ∗ U R point, it follows that M ∈ ∇2 , (ii) and (iii) hold. On the other hand, suppose M ∈ ∆2 . 0 If x ∈ L0M \ EM , take the transversal function xn ∈ B(LM ), xn + x0 ≥ 2xn 0 → 2x0 = 2 (where xn (t) = x(t) if |x(t)| ≤ n; = 0 if |x(t)| > n). By HahnBanach theorem there is a singular functional ϕ, ϕ(x) = 0, and ϕ(xn − x) = −ϕ(x), w

so xn − x → 0.

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0 0 If x ∈ EM , take z ∈ L0M \ EM , RM (z) < ∞. Choose a singular function ϕ(z) = 0. Denote Gn = {t ∈ G: |z(t)| ≤ n}, then µ(G \ Gn ) → 0. Deﬁne

xn (t) = xχGn (t) +

1 zχG\Gn (t) . k

Then xn 0 ≤ k1 (1 + RM (kxχGn ) + RM (zχG\Gn )) → k1 (1 + RM (kx)) = x0 = 1 and xn + x0 ≥ 2xχGn 0 → 2. But ϕ(xn − x) = ϕ( k1 zχG\Gn ) = k1 ϕ(z) = 0, and w

xn − x → 0. Suﬃciency. In this case, (1) and (2) of Theorem 1 hold. By M ∈ ∆2 , it yields that RN (p((1 + τ )kx)) < ∞, i.e. (3) of Theorem 1 holds. Also by M ∈ ∆2 , we derive RM (uχe ) ≤ RM ( kx ε χe ) ≤ DRM (kxχe ) → 0 (as µe → 0) i.e., (4) of Theorem 1 holds. µ Therefore, xn + x0 → 2 implies xn − x −→ 0. By M ∈ ∆2 and [5], it yields that xn − x0 → 0, hence x is a U R point. For Orlicz sequence spaces, we obtain the same results as in function spaces, and omit the proof. Theorem 3 Let x ∈ S(lM ) and k ∈ K(x). A point x is W ∗ U R if and only if (1) M ∈ ∇2 , (2) {j: kx(j) = 0} is a single element set, or (i), (ii) and (iii) hold (i) {j: |kx(j)| ∈ R \ SM ∪ {a } ∪ {b }} = ∅ (ii) if |kx(i)| = a ∈ {a}, j =i N (p− (kx(j))) + N (p(kx(i))) > 1 if |kx(i)| = b ∈ {b}, j =i N (p(kx(j))) + N (p− (kx(i))) < 1 and kx )) < ∞ for some 0 < τ < 1 RN (p( 1−τ (iii) for any ε, ε > 0, there is n0 such that for all summable sequences {u(i)} and subsequences e of natural numbers N with min{i: i ∈ e} > n0 and for all i ∈ e ε ε ≤ ε |u(i)| ≤ k|x(i)| ≤ |u(i)|, M (u(i)) ≥ εN (p− (u(i))) M

u(i) + kx(i) 2

M (u(i)) < ε.

we have i∈e

1 M (u(i)) + M (kx(i)) > 1− n0 2

On W ∗ U R point and U R point of Orlicz spaces with Orlicz norm

299

Theorem 4 Let x ∈ S(lM ) and k ∈ K(x). A point x is U R (W U R) if and only if (1) M ∈ ∆2 ∩ ∇2 (2) {j: kx(j) = 0} is a singleton set, or (i) and (ii) hold (i) {i: |kx(i)| ∈ R \ SM ∪ {a } ∪ {b }} = ∅ (ii) if |kx(i)| = a ∈ {a}, j =i N (p− (kx(j))) + N (p(kx(i))) > 1 if |kx(i)| = b ∈ {b}, j =i N (p(kx(j))) + N (p− (kx(i))) < 1 .

References 1. T. Wang, Z. Ren, Y. Zhang, On U R (W U R) point of Orlicz spaces, J. Math. 13 (1993), 443–452. 2. Y. Li, Lee Pengyee, T. Wang, On U R (W U R) point of Orlicz sequence spaces, Ann.Math. Res. 27 (1994). 3. S. Chen, T. Wang, On U R point of Orlicz spaces, J. Harbin Norm. Univ. 3 (1992), 5–10. 4. C. Wu, T. Wang, S. Chen, Y. Wang, Geometric theory of Orlicz spaces, Print House of H.I.T., Harbin, 1986. 5. T. Wang, Y. Wu, On convergence on sphere of orlicz spaces, J. Heilongjiang Univ. 4 (1988), 1–4.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 301–306 c 1994 Universitat de Barcelona

On the weak star uniformly rotund points of Orlicz spaces

Tingfu Wang and Quandi Wang Math. Dept., Harbin Univ. Sci. Tech., Harbin, Heilongjiang, 150080, P.R. China

Abstract We obtain the necessary and sufficient condition of weak star uniformly rotund point in Orlicz spaces.

In this paper, X denotes a Banach space, B(X), S(X) denote respectively the unit ball and the unit sphere of X. x ∈ S(X) is said to be a UR (WUR, W∗ UR) point w

w∗

provided that xn ∈ B(X), xn + x → 2 ⇒ xn − x → 0 (xn − x → 0, xn − x → 0). Obviously, URP ⇒ WURP ⇒ W∗ URP. If all points of S(X) are W∗ UR points, X is locally weak star uniformly rotund. M (·), N (·) denote a pair of complemented N -functions (see [3]); “M ∈ ∆2 ” (“M ∈ ∇2 ”) means that M (·) satisﬁes the ∇2 -condition (N satisﬁes the ∇2 condition). (G, Σ, µ) stands for a non-atomic ﬁnite measure space; LM (G, Σ, µ) expresses the Orlicz space generated by M (·):

LM (G, Σ, µ) = x(t): ∃a > 0, RM with the norm

x a

=

M G

x(t) dµ < ∞ a

x x = inf a > 0: RM ≤1 . a

For an Orlicz function space LM and a sequence space lM , a criterion of UR (WUR) point was obtained in recent years [1,2]. Here we give the criterion of W∗ UR point, but the statement and method is diﬀerent to WURP. 301

302

Wang and Wang

Theorem 1 Let x ∈ S(LM ). x is a W∗ URP if and only if x )<∞ (1) ∃τ > 0, RM ( 1−τ

(2) µ {t ∈ G: |x(t)| ∈ R \ SM } = 0 ∞ ∞ {bi } = 0 or µ t ∈ G : |x(t)| ∈ {ai } = 0 and M ∈ ∇2 , (3) µ t ∈ G: |x(t)| ∈ i=1

i=1

where SM is the set of all strictly convex points of M (·), (ai , bi ) are aﬃne segments of M (·) for i = 1, 2, . . . Proof. Without loss of generality, we assume x(t) ≥ 0. x Necessity. Suppose that RM ( 1−ε ) = ∞ for all ε > 0. Take c large enough such that µGc = µ{t ∈ G: x(t) ≤ c} > 0. Deﬁne x = −xχGc + xχG\Gc ; then x = x = 1, but for all ε > 0

x(t) x + x = dµ = ∞ M RM 2(1 − ε) 1−ε G\Gc so x + x = 2. From x = x , we get a contradiction with the fact that x is a W∗ UR point, which shows that (1) is true. Since a W∗ UR point is an extreme point, from [3], we get that (2) is true. Suppose that there exist aﬃne segments [a, b], [c, d] of M (·) such that A = {t: x(t) = a}, B = {t: x(t) = d} are sets with positive measure. Take E ⊂ A, F ⊂ B satisfying (M (b) − M (a))µE = (M (d) − M (c))µF . Deﬁne x = xχG\(E∪F ) + bχE +

cχF , then x = x, RM (x) = RM (x ) = 1, and RM ( x+x 2 )= ∗ contradicts the fact that x is a W UR point.

RM (x)+RM (x ) 2

= 1, which

Suppose now that there is an aﬃne segment [a, b] with positive measure set D = {t: x(t) = b} and M ∈ ∇2 , i.e. there are un ∞ such that u1 > b and n) M ( u2n ) > (1 − n1 ) M (u (n = 1, 2, . . .). Take subsets En , D ⊃ E1 ⊃ E2 ⊃ . . . 2 satisfying M (un − b)µEn + M (a)µ(D \ En ) = M (b)µD then µEn → 0. Deﬁne xn = xχG\D + aχD\En + (un − b)χEn so RM (xn ) = RM (x) = 1 (n = 1, 2, . . .) and

On the weak star uniformly rotund points of Orlicz spaces

RM

303

x + x a + b u n n = RM (xχG\D ) + M µ(D \ En ) + M µEn 2 2 2 1 M (a) + M (b) 1 µ(D \ En ) + 1− M (un )µEn ≥ RM (xχG\D ) + 2 2 n RM (xχG\D ) + M (b)µ(D \ En ) ≥ 2 RM (xχG\D ) + M (a)µ(D \ En ) + (1 − n1 )M (un − b)µEn + 2 −→ RM (x) = 1.

From x − xn , χD\E1 = (b − a)µ(D \ E1 ) > 0, we get a contradiction since x is a W∗ UR point. Suﬃciency. Suppose that xn ∈ B(LM ), xn + x → 2. We will ﬁrst show that RM (xn ) → 1 and RM ( xn2+x ) → 1. Suppose that RM (xn ) ≤ 1−δ for some δ > 0 (n = 1, 2, . . .). Take ε small enough 1+ε 1 < 1−δ . For such ε, while n is large enough we have (1+ε) xn2+x > 1, such that 1−ε xn +x and so RM ((1 + ε) 2 ) > 1. Hence applying assumption (1) we get 1 + ε xn + x 1−ε1+ε = RM xn + x 1 < RM (1 + ε) 2 2 2 1−ε 1 + ε 1 + ε 1+ε 1−ε ≤ RM (xn ) + RM x ≤ (1 − δ) 2 2 1−ε 2 1 − ε + 1 + o(ε) RM (x). 2 If ε → 0, 1 ≤ 1 − δ/2, so we get RM (xn ) → 1. Clearly xn2+x + x → 2, so we get similarly RM ( xn2+x ) → 1. w∗

µ

In the following, we show xn − x → 0, and so it is enough to show xn − x → 0. µ Denote E = {t ∈ G: x(t) ∈ SM \ ({ai } ∪ {bi })}. We ﬁrst show xn − x → 0 on E. Suppose for a contrary that there exist ε, σ > 0 with µ{t ∈ E: |xn (t)−x(t)| ≥ ε} ≥ σ. Since M xn (t) dµ ≥ M (d)µ{t: |xn (t)| > d}, 1 ≥ RM (xn ) ≥ {t:|xn (t)|≥d}

we have that for d large enough µ{t: |xn (t)| > d} < σ/3 (n = 1, 2, . . .) and µ{t: |x(t)| > d} < σ/3. Thus µEn ≥ σ/3, where

En = t ∈ E: |xn (t) − x(t)| ≥ ε, |xn (t)| ≤ d, |x(t)| ≤ d .

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Clearly there is δ > 0 such that for t ∈ En , M Hence

x (t) + x(t) M (xn (t)) + M (x(t)) n ≤ (1 − δ) . 2 2

x + x RM (xn ) + RM (x) n − RM 2 2 x (t) + x(t) M (xn (t)) + M (x(t)) n −M dµ = 2 2 G x (t) + x(t) M (xn (t)) + M (x(t)) n −M dµ ≥ 2 2 En δ δ εσ ≥ . M (xn (t) + M x(t) dµ ≥ M 2 En 2 2 3

0←

µ

This contradiction shows that xn − x → 0 on E. Denote Fa+ (n) = {t ∈ G: x(t) ∈ {ai }, xn (t) ≥ x(t)} Fa− (n) = {t ∈ G: x(t) ∈ {ai }, xn (t) < x(t)} Fb+ (n) = {t ∈ G: x(t) ∈ {bi }, xn (t) ≥ x(t) or xn (t) < 0} Fb− (n) = {t ∈ G: x(t) ∈ {bi }, xn (t) < x(t) and

xn (t) ≥ 0} .

Analogously as above we have that for any ε > 0 µ{t ∈ Fa− (n): xn (t) ≤ x(t) − ε} → 0

(n → ∞)

µ{t ∈

(n → ∞),

Fb+ (n): xn (t)

≥ x(t) + ε} → 0

whence lim sup n→∞

E∪Fa− (n)∪Fb+ (n)

M xn (t) dµ ≥ lim sup

lim sup n→∞

E∪Fa− (n)

E∪Fa− (n)∪Fb+ (n)

n→∞

Consider the case µ(Fb+ (n) ∪ Fb− (n)) = 0

(n = 1, 2, . . .). From

M xn (t) dµ ≥ lim sup

E∪Fa− (n)

n→∞

and RM (xn ) → 1 = RM (x), it follows that M xn (t) dµ ≤ lim inf lim inf n→∞

Fa+ (n)

M x(t) dµ.

n→∞

Fa+ (n)

M x(t) dµ

M x(t) dµ

On the weak star uniformly rotund points of Orlicz spaces

thus

lim inf n→∞

305

M (xn (t)) − M (x(t)) dµ = 0

Fa+ (n)

so for any ε > 0 µ{t ∈ Fa+ (n): xn (t) ≥ x(t) + ε} → 0

(n → ∞).

µ

Combining the above, we have xn − x → 0 on E ∪ Fa− (n) ∪ Fa+ (n) = G. Consider the case of µ(Fa+ (n) ∪ Fa− (n)) = 0 (n = 1, 2, . . .) and M ∈ ∇2 . We ﬁrst prove that lim sup M (xn (t))dµ = 0. Suppose that there exist ε > 0 and µe→0 n e en ⊂ G with µen 0 such that M (xn (t))dµ ≥ ε. en

Take c > 0 small enough, M (c)µG < ε/2, denote en = {t ∈ en : |xn (t)| ≥ c}. Then M xn (t) dµ = M xn (t) dµ − M xn (t) dµ en

en \en

en

≥ ε − M (c)µG ≥

ε . 2

(u) u Combining M ∈ ∇2 , for τ, δ > 0, while u ≥ c, M ( 1+τ ) ≤ (1 − δ) M 1+τ whence x + x x + x x + x n n n = RM χG\en + RM χen 1 ← RM 2 2 2 1 + τ x (t) 1 − τ x(t) RM (xn χG\en ) + RM (xχG\en ) n + + dµ M ≤ 2 2 1 +τ 2 1−τ en x (t) RM (xn χG\en ) + RM (xχG\en ) 1 + τ n + dµ M ≤ 2 2 1 +τ en x(t) 1−τ dµ M + 2 1−τ en RM (xn χG\en ) + RM (xχG\en ) 1 + τ 1 − δ + M xn (t) dµ ≤ 2 2 1 + τ en x 1−τ RM χe + 2 1−τ n RM (xn χG\en ) + RM (xχG\en ) 1 − τ RM (xn χen ) δε + o(µen ) + − ≤ 2 2 2 4 RM (xn ) + RM (x) δε ≤ − + o(µe ). 2 4 If n → ∞, it gives a contradiction: 1 ≤ 1 − δε/4. So we get lim sup M xn (t) dµ = lim sup M x(t) dµ n→∞

E∪Fb+ (n)

n→∞

E∪Fb+ (n)

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hence lim inf n→∞

From lim inf n→∞

Fb− (n)

Fb− (n)

M xn (t) dµ = lim inf n→∞

Fb− (n)

M x(t) dµ. µ

(M (x(t))−M (xn (t)))dµ = 0, it follows that xn −x → 0 on Fb− (n), µ

and consequently xn − x → 0 on G. By an analogous argumentation, we get the same result for Orlicz sequence spaces. Theorem 2 Let x ∈ S(lM ). Then x is a W∗ URP point if and only if x )<∞ (1) there is τ > 0, R( 1−τ (2) if there is “i”, |x(i)| ∈ (a, b], then M ∈ ∇2 and there is no “j”, j = i with |x(j)| ∈ [c, d), where [a, b] and [c, d] are aﬃne segments of M (·).

References 1. T. Wang, Z. Ren and Y. Zhang, On UR point and WUR point of Orlicz spaces, J. Math. 13 (1993), 443–452. 2. Y. Li, Lee Pengyee, T. Wang, On UR point and WUR point of Orlicz sequence spaces, Ann. Math. Res. 27 (1994). 3. M.A. Krasnoselskii and Ya. B. Rutiskii, Convex function and Orlicz spaces, Groningen 1961.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 307–325 c 1994 Universitat de Barcelona

P -convexity property in Musielak-Orlicz sequence spaces

Ye Yining and Huang Yafeng P.O. Box 610, Math. Dept., Harbin Univ. Sci. Tech., Harbin, Heilongjiang, 150080, P.R. China

Abstract We prove that in the Musielak-Orlicz sequence spaces equipped with the Luxemburg norm, P -convexity coincides with reflexivity.

In 1970, Kottman [1] introduced an important geometric property-P -convexity in order to describe a reﬂexive Banach space. We say that a Banach space (X, · ) is P -convex if X is P (nε)-convex for some positive integer n and a real number ε > 0, i.e. for any x1 , x2 , . . . , xn in the unit sphere of X, min xi −xj < 2−ε for some n and i=j

ε > 0. Moreover Kottman proved that any P -convex Banach space is reﬂexive. After P -convexity property was introduced, many people tried to give a distinct relation between P -convexity and reﬂexivity. But there are a lot of diﬀerences between them in a Banach space. In 1978 Sastry and Naidu [2] introduced a new geometric property, O-convexity intermediate between P -convexity and reﬂexivity, and proved that P -convexity implies O-convexity and O-convexity implies reﬂexivity. In 1984, D. Amir and C. Franhetti [3] gave two geometric properties, Oconvexity and H-convexity by the preceding results and proved O-convexity implies Q-convexity, Q-convexity implies reﬂexivity and H-convexity implies B-convexity and these convexities do not coincide with each other. In 1988, Yeyining, Hemiaohong and Ryszard Pluciennik [4] proved that in Orlicz spaces P -convexity coincides with reﬂexivity, and reﬂexivity coincides with P (3, ε)convexity for some ε > 0. In this paper we prove that in Musielak-Orlicz sequence spaces equipped with the Luxemburg norm P -convexity coincides with reﬂexivity. 307

308

Yining and Yafeng 0. Introduction

Let X be a Banach space equipped with the norm · and S(X) be the unit sphere of the space X, i.e. S(X) = {x ∈ X: x = 1}. Denote by N the set of positive integers and by R the set of real numbers. Let ϕ = (ϕn ) be a sequence of Young functions, i.e. for every n ∈ N, ϕn (·): R → [0, ∞] is a convex, ϕn (0) = 0, lim ϕn (u) = ∞, ϕn (·) u→∞ is continuous at 0 and not identically equal to the zero function, and there exists a real number u0 , s.t. ϕn (u0 ) < ∞. We deﬁne a modular on the family of all sequences x = (xn ) of real numbers by the following formula Iϕ (x) =

∞

ϕn (xn ).

n=1

The linear set lϕ = {x = (xn ): ∃a > 0, Iϕ (ax) < ∞} equipped with so - called Luxemburg norm x = inf{k > 0: Iϕ (k −1 x) ≤ 1} is said to be a Musielak-Orlicz sequence space. We say that ϕ = (ϕn ) satisﬁes the δ2 -condition if there are constants a, k, and a sequence (cn ) of non-negative real numbers such that ∞

cn < ∞

and ϕa (2u) ≤ kϕn (u) + cn

n=1

for all n ∈ N and u ∈ R with ϕa (u) ≤ a . The complementary function of Young function ϕ = (ϕn ) is deﬁned by ϕ∗n (v) = sup{u|v| − ϕn (u)}, u≥0

for all n ∈ N.

A Musielak-Orlicz sequence space lϕ is reﬂexive if and only if ϕ = (ϕn ) and ϕ∗ = (ϕ∗n ) satisfy the δ2 -condition. Let an = sup{u > 0: ϕn (u) ≤ 1} for all n ∈ N.

P -convexity property in Musielak-Orlicz sequence spaces

309

1. Auxiliary lemmas Lemma 1 Let ϕ = (ϕn ) satisfy the δ2 -condition, then (i) if A = inf ϕn (an ), then A > 0, n

(1)

(ii) for any l1 > 1, a1 > 0, there are k1 > 1 and a sequence (cn ) of non-negative real numbers such that ∞

(1) c(1) n < ∞ and ϕn (lu) ≤ kϕn (u) + cn

n=1

for all n ∈ N and u ∈ R with ϕn (l1 u) ≤ a1 , (2) (iii) for any k1 > 1, l2 > 1, a2 > 0, there are σ ∈ (0, l2 − 1) and a sequence (cn ) of non-negative real numbers such that ∞

c(2) n <∞

and ϕn (1 + δ)u ≤ k2 ϕn (u) + c(2) n

n=1

for all n ∈ N and u ∈ R with ϕn (l2 u) ≤ a2 . Proof. (i) Obviously A ≥ 0, so it is enough to prove A = 0. Assume that A = 0. Then for any a > 0 there is n0 ∈ N, such that ϕn0 (an0 ) < a. It is easy to see that an0 = 0 by the deﬁnition of ϕn (u). We may assume without loss of generality that a < 1. Then ϕn0 (an0 ) < 1 implies ϕn0 (2an0 ) = ∞ because ϕn (u) is a convex function and so it has the only discontinuous point u0 , such that ϕn0 (u0 − 0) < ∞ and ϕn0 (u0 +0) = ∞. By the deﬁnition of an0 and ϕn0 (an0 ) < 1 we may deduce that an0 is the discontinuous point of ϕn0 (u), so ϕn0 (2an0 ) = ∞. But this contradicts the δ2 -condition and so A > 0. (ii) Let a positive integer α satisfy 2α−1 < l1 < 2α . Since ϕ = (ϕn ) satisﬁes the δ2 -condition, there are constants k > 0, a > 0 and a sequence (cn ) of non-negative real numbers such that ∞

cn < ∞

and ϕn (2u) ≤ kϕn (u) + cn

n=1

for all n ∈ N and u ∈ R with ϕn (u) ≤ a. When ϕn (l1 u) ≤ a, ϕn (2α−1 u) ≤ ϕn (l1 u) ≤ a, then ϕn (l1 u) ≤ ϕn (2α u) ≤ kϕn (2α−1 u) + cn ≤ . . . ≤ k α ϕn (u) + (k α−1 + . . . + k + 1)cn .

310

Yining and Yafeng (1)

Let cn

∞

= (k α−1 + · · · + k + 1)cn . Obviously

(1)

cn

< ∞. Then ϕn (l1 u) ≤

n=1 (1)

k α ϕn (u) + cn with ϕn (l1 u) ≤ a. If a1 ≤ a, it is enough to put k1 = k α . Let a < ϕn (l1 u) ≤ a1 and ϕn (l1 u) = a. Then l1 < l. Hence ϕn (l1 , u) ≤ a1 = a1 a−1 a = a1 a−1 ϕn (l1 u) = a1 a−1 ϕn (l1 l2−1 l1 u) ≤ a1 a−1 [k α ϕn (l1−1 l1 u) + c(1) n ] ≤ a1 a−1 k α ϕn (u) + a1 a−1 c(1) n . Replace a1 a−1 k α by k1 , a1 a−1 cn

(1)

(1)

by cn , then

∞

(1)

cn

< ∞.

So ϕn (l1 u) ≤

n=1 (1)

k1 ϕn (u) + cn when ϕn (l1 u) ≤ a1 . (1) (iii) For l2 > 1, a2 > 0, by (ii) there are k1 > 1 and a sequence (cn ) of non-negative real numbers such that ∞ (1) c(1) n < ∞ and ϕn (l2 u) ≤ k1 ϕn (u) + cn n=1

for all n ∈ N and u ∈ R with ϕn (l2 u) ≤ a2 . Take σ satisfying σ < min l2 − 1, [(k2 − 1)/(k1 − 1)](l2 − 1) . Because ϕn (u) is convex, when ϕn (l2 u) ≤ a2 it follows that (l − 1)(l + σ) 2 u ϕn (1 + σ)u = ϕn l2 − 1 σ l2 − 1 − σ l2 u + u = ϕn l2 − 1 l2 − 1 l2 − 1 − σ σ ϕn (l2 u) + ϕn (u) ≤ l2 − 1 l2 − 1 l2 − 1 − σ σ k1 σ ϕn (u) + ϕn (u) + c(1) ≤ l2 − 1 l2 − 1 l2 − 1 n

(k − 1)σ σ 1 + 1 ϕn (u) + c(1) = l2 − 1 l2 − 1 n

(k − 1)(k − 1) 1 2 ≤ (l2 − 1) + 1 ϕn (u) (l2 − 1)(k1 − 1) (1)

cn (k2 − 1) (l2 − 1) (l2 − 1)(k1 − 1) k2 − 1 (1) c . = k2 ϕn (u) + k1 − 1 n +

(2)

(1)

Let cn = [(k2 − 1)/(k1 − 1)]cn , which completes the proof of (iii).

P -convexity property in Musielak-Orlicz sequence spaces

311

Lemma 2 If ϕ = (ϕn ) and ϕ∗ = (ϕ∗n ) satisfy the δ2 -condition, then for any l3 > 1, b > 1 (3) there are k3 > 1 and a sequence (cn ) of non-negative real numbers such that ∞

and ϕ∗n (v) <

c(3) n <∞

n=1

1 ∗ ϕ (l3 v) + c(3) n , l3 k3 n

for all n ∈ N and v ∈ R with ϕ∗n (v) ≤ b. Proof. First we prove when ϕ∗n (v) ≤ b, there is a > 0 such that ϕn (u) ≤ a for all n ∈ N where v = pn (u). Otherwise, there is a sequence {uk }∞ k=1 of real numbers such that ϕnk (uk ) → ∞ ∗ as k → ∞, while ϕnk (v) ≤ b. Notice that for some l3 > 1, there is b > 0, such that ϕ∗n (l3 v) ≤ b for all n ∈ N. It is enough to put b = 2l3 b. If ϕ∗n (l3 v) > 2l3 b, Lemma 2 obviously holds. By Lemma 1, there is σ ∈ (0, l3 − 1) such that ϕ∗nk ((1 + σ)vk ) ≤ k2 ϕ∗nk (vk ) + ck for all n ∈ N with ϕ∗nk (l3 v) ≤ b ,

where k2 > 1,

∞

ck < ∞.

n=1

Let b1 = k2 b + max ck . Then ϕ∗nk ((1 + σ)vk ) ≤ b1 for all k ∈ N. k

On the other hand, when vk = pnk (uk ), ϕ∗nk (vk ) = |uk vk | − ϕnk (uk ) > 0, and ϕnk (uk ) → ∞ as k → ∞, i.e. there is k0 ∈ N such that ϕnk (uk ) > b1 σ −1 with k > k0 . So, when k > k0 , we have ϕ∗nk (1 + σ)vk = sup (1 + σ)|vk |u − ϕnk (u) u≥0

≥ (1 + σ)|vk uk | − ϕnk (uk ) ≥ (1 + σ)ϕnk (uk ) − ϕnk (uk ) = σϕnk (uk ) > b1 . This contradicts the inequality ϕ∗nk ((1 + σ)vk ) ≤ b1 . Therefore, there is a > 0 such that ϕn (u) ≤ a for all n ∈ N with ϕ∗n (v) ≤ b. Hence by ϕ∗n (l3 v) ≤ b there is a > 0 such that ϕn (l3 u) ≤ a for all n ∈ N. By Lemma 1 (iii) for k2 = l3 , l2 = l3 , a2 = a , there are ε ∈ (0, l3 − 1) and a (2) sequence (cn ) of non-negative real numbers such that ∞ n=1

c(2) n <∞

and ϕn (1 + ε)u ≤ l3 ϕn (u) + c(2) n

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Yining and Yafeng

for all n ∈ N and u ∈ R with ϕn (l3 u) ≤ a . Then ϕ∗n (v) = sup u|v| − ϕn (u): u ≥ 0, ϕ∗n (l3 v) ≤ b ≤ sup u|v| − ϕn (u): ϕn (l3 u) ≤ a (2) ϕn (u + ε)u) − cn ≤ sup u|v| − l3 u≥0 c(2) l3 |v| 1 n (1 + ε)u − ϕn (1 + ε)u + = sup l3 u≥0 1 + ε l3 (2) 1 l3 v cn + = ϕ∗n l3 1+ε l3 (2)

< (3)

1 cn ϕ∗n (l3 v) + . l3 (1 + ε) l3 (2)

Let k3 = 1 + ε, cn = cn /l3 , which completes the proof of Lemma 2. Lemma 3 If ϕ = (ϕn ) and ϕ∗ = (ϕ∗n ) satisfy the δ2 -condition, then there is a sequence ∞ (cn ) of non-negative real numbers such that ϕn (cn ) < ∞, and if n=1

dn = sup α(u, n): ϕn

u 1 ≥ ϕn (u), cn ≤ |u| ≤ an , n = 1, 2, . . . α(u, n) 2

d1 = lim sup dn , m→∞ n>m

then d1 < 2. (3)

Proof. Let l3 = 2, b = 1 in Lemma 2. Then there are k3 > 1 and a sequence (cn ) of non-negative real numbers such that ∞ n=1

c(3) n <∞

and ϕn (u) ≤

1 ϕn (2u) + c(3) n 2k3

(1)

for all n and u with ϕn (u) ≤ 1. In Lemma 1 (iii) let k2 = (k3 + 1)/2, l2 = 2, a2 = 1. There are ε ∈ (0, 1) and a ∞ sequence (βn ) of positive numbers such that βn < ∞, and when ϕn (2u) ≤ 1, n=1

1 ϕn (1 + ε)u < (k3 + 1)ϕn (u) + βn . 2

(2)

P -convexity property in Musielak-Orlicz sequence spaces Let cn = Obviously

∞ n=1

313

2k3 (k3 + 1) (3) 4k3 cn + βn . k3 − 1 k3 − 1

cn < ∞.

Since A = inf ϕn (an ) > 0 is true by Lemma 1 (i), so there is n0 ∈ N such that n

cn < A for n > n0 . We deﬁne a sequence (cn ) by 0 when n ≤ n0 cn = ϕ−1 n (cn ) when n > n0 . ∞ ∞ Then ϕn (cn ) ≤ cn < ∞. n=1

n=1

We will show the sequence (cn ) satisﬁes Lemma 3. Obviously d1 ≤ 2. If d1 = 2, for n > n0 there are subsequence {un }n>n0 and {α(un , n)}n>n0 (let the subsequence be {un } and {α(un , n)}) such that 1 u 0 (3) ≥ ϕn (un ), cn ≤ |un | < an ϕn α(un , n) 2 and α(un , n) → 2 as n → ∞. So there is n1 ∈ N, such that 2/α(un , n) < 1 + ε for n > n1 . Let αn = α(un , n). By formula (2) it follows that u u n k3 + 1 un n < ϕn + βn . ≤ ϕn (1 + ε) ϕn αn 2 2 2 By (1), we get u k + 1 1 k3 + 1 k3 + 1 (3) n 3 cn + βn . < + βn = ϕn (un ) + c(3) ϕn (un ) + ϕn n αn 2 2k3 4k3 2 By (3), we have 1 k3 + 1 k3 + 1 (3) ϕn (un ) + ϕn (un ) < cn + βn , 2 4k3 2 i.e. 2k3 (k3 + 1) (3) 3k3 ϕn (un ) < cn + βn . (4) k3 − 1 k3 − 1 But when n > max(n0 , n1 ), we have 2k3 (k3 + 1) (3) 4k3 cn + βn . k3 − 1 k3 − 1 This contradicts (4), so Lemma 3 is true. ϕn (un ) ≥ ϕn (cn ) = cn =

314

Yining and Yafeng 2. Result

Theorem A Musielak-Orlicz sequence space lϕ is P -convex if and only if lϕ is reﬂexive. Proof. We may obtain necessity according to paper [1], so it is enough to prove suﬃciency. Assume suﬃciency is false. Let lϕ be reﬂexive i.e. ϕ = (ϕn ) and ϕ∗ = (ϕ∗n ) satisfy the δ2 -condition but lϕ is not P -convex. Then for any ε > 0 and positive integer N1 , there is a set X = {xi } having N1 elements in S(lϕ ) such that xi − xj ≥ 2(1 − ε); i = j, i, j = 1, 2, . . . , N1 . We will complete the proof of theorem in two steps. Step 1. There is ε0 > 0 such that xn < (1 − ε0 )an for any x = (xn ) ∈ X and all n ∈ N. (1a) We deﬁne some constants. By Lemma 3, there are a sequence (cn ) of non-negative real numbers, N ∈ N, d > 0 ∞ such that cn < ∞, d1 < d < 2 and dn < d with n > N . Let β = ε0 /4, then n=1

β < 1. By Lemma 1 (ii), for l1 = 1/β and a1 = 1, there are k1 > 1 and a sequence (2) (cn ) of non-negative real numbers such that ∞

c(2) n <∞

and ϕn (u/β) ≤ k1 ϕn (u) + c(2) n

(1)

n=1

for all n ∈ N and u ∈ R with ϕn (u/β) ≤ 1. Let λ1 = (2−d)/(24k1 ), λ2 = (2−d)/2d. By Lemma 1 (iii), for k2 = 1 + min(λ1 , λ2 ), l2 > 1 and a = 1, there are a ∈ (0, l − 1) (3) and a sequence (cn ) of non-negative real numbers such that ∞

c(3) n <∞

and ϕn (1 + δ)u ≤ k2 ϕn (u) + c(3) n

n=1

for all n ∈ N and u ∈ R with ϕn (l2 u) ≤ 1.

(2)

P -convexity property in Musielak-Orlicz sequence spaces

315 (1)

By Lemma 1 (ii), for l1 = 2, and a1 = 1, there are k > 1 and a sequence (cn ) of non-negative real numbers such that ∞

c(1) n <∞

and ϕn (2u) ≤ kϕn (u) + c(1) n

(3)

n=1

for all n ∈ N and u ∈ R with ϕn (2u) ≤ 1. Let h1 be, such that 0 < h1 < 1. Let 2 − d 2 − d h2 = min , 8k 4 1−h h2 (1 − h1 ) 1 , r1 = min 4(1 + k1 ) 12kk1 h2 (1 − h1 ) r2 = . 12(3k + 1) By

∞

ϕn (cn ) < ∞ and (1), (2), (3), there is N0 > N , such that

n=1 ∞ n=1

ϕn (cn ) < r1 ,

∞

c(i) n < r, i = 1, 2, 3.

(4)

n=N0

(1b) Now we will prove that for any h1 , 0 < h1 < 1, there do not exist three elements x1 , x2 and x3 in X, such that ∞

ϕn (xin ) ≥ Iϕ (xi ) − h1 = 1 − h1 , i = 1, 2, 3.

(5)

n=1

Assume (1b) is false: (i) If 0 < ε < ε0 /4, then ϕn ((xin −xjn )/2(1−ε)) < ∞ for all n ∈ N, i = j, i, j = 1, 2, 3. Let un = max{|x1n |, |x2n |, |x3n |}, wn = min{|x1n |, |x2n |, |x3n |}, vn be the arithmetic mean of un and wn . Since un vn ≥ 0, or un wn ≥ 0, or vn wn ≥ 0 is true, we ﬁrst consider vn , wn ≥ 0. Divide positive integers n ≥ N0 into the following sets:

v n I1 = n: ≥ β and |vn | ≥ cn u

vn n I2 = n: ≥ β and |vn | < cn u

vn n I3 = n: < β and |un | ≥ cn u vn

n I4 = n: < β and |un | < cn . un

316

Yining and Yafeng

When n ∈ I1 , by formula (2) for l2 = (1 − ε0 /2)/[(1 − ε0 )(1 − ε)], if σ = 1/(1 − ε) − 1, then σ < l2 − 1. Since u −v 1 − ε /2 n n 0 = ϕn ϕn l2 2 1 − ε0 1 − ε /2 0 ≤ ϕn 1 − ε0 /4

un − vn 2(1 − ε) 2un · ≤ ϕn (an ) ≤ 1 2(1 − ε0 ) ·

by (2) and k2 = 1 + min(λ1 , λ2 ), it follows that ϕn

u − v u − v un − vn n n n n = ϕn (1 + σ) ≤ k2 ϕn + c(3) n 2(1 − ε) 2 2 u − v n n + c(3) ≤ (1 + λ1 )ϕn n 2 ϕn (un ) + ϕn (vn ) ≤ (1 + λ1 ) + c(3) n 2 1 1 < ϕn (un ) + ϕn (vn ) + λ1 ϕn (un ) + c(3) n . 2 2

(6)

By the same argumentation, we get ϕn

u − w 1 1 n n ≤ ϕn (un ) + ϕn (wn ) + λ1 ϕn (un ) + c(3) n . 2(1 − ε) 2 2

(7)

By vn , wn ≥ 0 and |vn | ≥ |wn |, it follows that v − w v v n n n n ≤ ϕn ≤ (1 + λ1 )ϕn + c(3) n . 2(1 − ε) 2(1 − ε) 2

ϕ

By |vn | ≥ cn and the deﬁnition of d, we get ϕn

v n

2

so ϕn Let

= ϕn

d v d v d n n · ≤ ϕn ≤ ϕ4 (vn ), 2 d 2 d 4

v − w d n n ≤ (1 + λ1 )ϕn (vn ) + c(3) n . 2(1 − ε) 4 u − w u − v v − w n n n n n n + ϕn +ϕ 2(1 − ε) 2(1 − ε) 2(1 − ε) − ϕn (un ) − ϕn (vn ) − ϕn (wn ).

f (n) = ϕn

(8)

P -convexity property in Musielak-Orlicz sequence spaces

317

(2)

By (1) we get ϕn (un ) ≤ k1 ϕn (βun ) + cn . By (6), (7) and (8) it follows

d 1 f (n) ≤ − (v ) ϕ 2λ1 ϕn (un ) + (1 + λ1 )ϕn (vn ) + 3c(3) n n n 4 2 n∈I1 n∈I1

2−d ϕn (vn ) + 3 3λ1 ϕn (un ) − ≤ c(3) n 4 n∈I1 n∈I1

2−d ϕ(βun ) + 3 3λ1 ϕn (un ) − ≤ c(3) n 4 n∈I1 n∈I1 2−d

2−d ≤ ϕn (un ) + c(3) 3λ1 ϕn (un ) − n 4k1 4k1 n∈I1 n∈I1 +3 c(3) n

(9)

n∈I1

=

2−d 2 − d (3) ϕn (un ) + cn + 3 c(3) n . 8k1 4k1 n∈I1

When n ∈

n∈I1

n∈I1

I2 , | uvnn |

≥ β, |vn | < cn . Since 2u u n n ϕn ≤ ϕn ≤ ϕn (an ) ≤ 1, 2(1 − ε) 1 − ε0

by (3) we get ϕn so ϕn

u 2un n (1) ≤ kϕn + c(1) n ≤ kϕn (un ) + cn , 2(1 + ε) 2(1 − ε)

u − v 2u n n n ≤ ϕn ≤ kϕn (un ) + c(1) n 2(1 − ε) 2(1 − ε) (1) (2) (1) ≤ kk1 ϕn (βun ) + kc(2) n + cn ≤ kk1 ϕn (cn ) + kcn + cn .

We have also

u − w n n (2) ≤ kk1 ϕn (cn ) + c(1) n + kcn 2(1 − ε) v − w n n (2) ≤ kk1 ϕn (cn ) + c(1) ϕn n + kcn , 2(1 − ε)

ϕn

so we get n∈I2

u − v ) v − w u − w n n n n n n + ϕn + ϕn 2(1 − ε) 2(1 − ε) 2(1 − ε) n∈I2 ϕn (cn ) + 3 c(1) + 3k c(3) ≤ 3kk1 n n .

f (n) ≤

ϕn

n∈I2

n∈I2

n∈I3

(10)

318

Yining and Yafeng When n ∈ I3 , | uvnn | < β, |un | ≥ cn , by ϕn

u − v (1 + ε /4)u n n 0 0 ≤ ϕn , 2(1 − ε) 2(1 − ε)

denoting (1 + ε0 /4)/(1 − ε) = 1/(1 − ε ), σ = 1/(1 − ε ) − 1, we get as in (6), ϕn

u − v u un n n n ≤ ϕn (1 + σ ) ≤ (1 + λ2 )ϕn + c(3) n 2(1 − ε) 2 2 d ≤ (1 + λ2 )ϕn (un ) + c(3) n 4

and ϕn

u − w d n n ≤ (1 + λ2 )ϕn (un ) + c(3) n . 2(1 − ε) 4

vn n −wn By ϕn ( u2(1−ε) ) ≤ ϕn ( 2(1−ε) ) ≤ ϕn (vn ) we get

d d ϕn (un ) + λ2 ϕn (un ) + 2c(3) − ϕ (u ) n 4 n 2 2 n∈I3 2−d 2−d ϕn (un ) + ϕn (un ) + 2 − ≤ c(3) n 2 4

f (n) ≤

n∈I1

n∈I2

=−

2−d ϕn (un ) + 2 c(3) n . 4 n∈I3

(11)

n∈I3

n∈I3

When n ∈ I4 , |un | < cn , as in the case of n ∈ I2 , we get u − v n n (1) ≤ kϕn (un ) + c(1) n ≤ kϕn (cn ) + cn 2(1 − ε) u − v n n ≤ kϕn (cn ) + c(1) ϕn n 2(1 − ε) u − w n n ϕn ≤ kϕn (cn ) + c(1) m . 2(1 − ε) ϕn

Then

n∈I4

f (n) ≤ 3k

n∈I4

ϕn (cn ) + 3

n∈I4

c(1) n .

(12)

P -convexity property in Musielak-Orlicz sequence spaces

319

By (9), (10), (11) and (12), we get ∞

∞

f (n) ≤ −h2

n=N0

ϕn (un ) + h2

ϕn (un )

n∈I2 ∪I4

n=N0 ∞

(3) (c(1) n + cn )

+3

(13)

n=N0

+ 3kk1

n=N0

∞ 2 − d (2) ϕn (cn ) + 3k + cn . 4k1 n=N0

(2)

When n ∈ I2 , since (1) implies ϕn (un ) ≤ k1 ϕn (cn ) + cn , then ϕn (un ) = h2 ϕn (un ) + h2 ϕn (un ) h2 n∈I2 ∪I4

n∈I2

≤ h2

n∈I4

[k1 ϕn (cn ) + c(2) n ] + h2

n∈I2

≤ h2 (k1 + 1)

ϕn (cn )

n∈I4 ∞

ϕn (cn ) + h2

n=N0

(14)

c(2) n .

n∈I2

It we put (14) into (13), by (4) and (5), we get ∞

∞

f (n) ≤ −h2

n=N0

n=N0

ϕn (cn )

n=N0 ∞

+ 3kk1 +3

∞

ϕn (un ) + h2 (k1 + 1) ϕn (cn )

n=N0 ∞

(2) (c(1) n + cn ) + (3k + 1)

n=N0

∞

(15) c(2) n

n=N0

< −h2 (1 − h1 ) + h2 (k1 + 1)r1 + 3kk1 r1 + 3(3k + 1)r2 h2 (1 − h1 ) . <− 4 (ii) Formula (5) implies

N 0 −1 n=1

ϕn (xin ) < h, i = 1, 2, 3. We deduce that |2xin | < an

for all n < N, and i = 1, 2, 3. Let α = min ϕ−1 n n
Then k = max

max ϕn (u)/ϕn ( u2 ) < ∞.

n
h 2 . 48N0

320

Yining and Yafeng

So when |2un | ∈ [α , an ], ϕn (2un ) ≤ k ϕn (un ); when |2un | < α , ϕn (2un ) ≤ ϕn (α ). Hence N 0 −1

f (n) <

N 0 −1

n=1

ϕn

n=1

≤3

N 0 −1

u − v v − w u − w n n n n n n + ϕn + ϕn 2(1 − ε) 2(1 − ε) 2(1 − ε)

ϕn (2un ) ≤ 3k

n=1

and when h1 < N 0 −1

1 3k1

·

h2 16

N 0 −1

ϕn (un ) + 3

n=1

N 0 −1

ϕn (α )

n=1

· h1 < 12 , then

f (n) < 3k h1 + 3N0

n=1

By (15) and (16), we get

∞

h2 h2 h2 h2 (1 − h1 ) h2 + = < . ≤ 48N0 16 16 8 4

(16)

f (n) < 0, i.e.

n=1

Iϕ

x1 − x2 2(1 − ε)

+ Iϕ

x2 − x3 x1 − x3 + Iϕ − Iϕ (x1 ) − Iϕ (x2 ) − Iϕ (x3 ) < 0. 2(1 − ϕ) 2(1 − ε)

x −x x −x x −x ) < 1, or Iϕ ( 2(1−ε) ) < 1, or Iϕ ( 2(1−ε) ) < 1, Since Iϕ (xi ) = 1, i = 1, 2, 3, so Iϕ ( 2(1−ε) 1 2 2 3 1 3 and this implies x − x < 2(1 − ε) or x − x < 2(1 − ε), or x − x < 2(1 − ε). This contradicts the assumption in the theorem, so result (1b) is true. Repeating the same argumentation, we may prove result (1b) in case of uw > 0 and uv > 0. 1

2

2

3

1

3

(1c) Let N1 = 2N0 + 1, N1 is the number of elements of X. Result (1b) implies that there are at least 2N0 − 1 elements in X such that N 0 −1

ϕn (xn ) > h1 .

(17)

n=1

Let α1 =

α1 1 h1 , u0 = min ϕ−1 . n n
The fact that a continuous function is uniformly continuous in a closed interval implies that there is δn > 0 such that u α1 ϕ ≤ ϕn (u) + , n =, 1, 2, . . . , N0 − 1 (18) 1−δ 4(N0 − 1) for all δ < δn and u ∈ [u0 , an ].

P -convexity property in Musielak-Orlicz sequence spaces

321

Let δ = min δn . Take ε < ε0 /4 and 0 < ε < δ . Among the elements satisfying n
(17), there are three ones x1 , x2 , x3 and n0 < N0 such that ϕn0 (xin0 ) >

h1 , i = 1, 2, 3 N0 − 1

this is because 2N0 − 1 elements satisfy (17) in the former N0 − 1 components, then there are three elements satisfying the above formula in the same component. Since there are at least two elements having same sign among x1n0 , x2n0 , x3n0 and without loss of generality we have x1n0 x2n0 ≥ 0

and |x1n0 | ≥ |x2n0 |.

By analogy of the former proof we get ∞

ϕn

n=N0

∞ ∞ x1 − x2 1 1 α1 n n < . ϕn (x1n ) + ϕn (x2n ) + 2(1 − ε) 2 2 4 n=N0

(19)

n=N0

Divide the positive integers of n < N0 (n = n0 ) into three sets: I5 = n: max(|x1n |, |x2n |) ≥ 2u0 and x1n x2n < 0 I6 = n: max(x1n |, |x2n |) ≥ 2u0 and x1n x2n ≥ 0 I7 = n: max(|x1n |, |x2n |) < 2u0 . When n ∈ I5 , |

x1n −x2n | 2

≥

1 2

max(|x1n |, |x2n |) ≥ u0 , we get by ε ≤ δn and (18)

x1 − x2 x1 − x2 α1 n n n ≤ ϕn n + 2(1 − ε) 2 4(N0 − 1) 1 α1 1 . ≤ ϕn (x1n ) + ϕn (x2n ) + 2 2 4(N0 − 1)

(20)

x1 x1 − x2 x2 n n n n ϕn ≤ max , ϕn 2(1 − ε) 2(1 − ε) 2(1 − ε) 1 α1 1 . ≤ ϕn (x1n ) + ϕn (x2n ) + 2 2 4(N0 − 1)

(21)

ϕn

When n ∈ I6 ,

When n ∈ I7 , ϕn

x1 − x2 4u α1 0 n n ≤ ϕn ≤ ϕn (4u0 ) ≤ 2(1 − ε) 2(1 − ε) 4(N0 − 1)

(22)

322

Yining and Yafeng

since ϕ n0

x1 − x2 x1 x1 α1 n0 n n < ϕ n0 ≤ ϕn0 n0 + 2(1 − ε) 2(1 − ε) 2 4(N0 − 1) α 1 1 ≤ ϕn0 (x1n0 ) + 2 4(N0 − 1)

notice ϕn0 (x2n0 ) >

Iϕ

h1 N0 −1

(23)

= α1 , by (19) and (23)

0 −1 x1 − x2 x1 − x2 N x1 − x2 n0 n n n ϕn n = ϕn 0 n 0 + 2(1 − ε) 2(1 − ε) 2(1 − ε) n=1

n=n0

+

∞ n=N0

<

x1 − x2 n ϕn n 2(1 − ε)

α1 1 ϕn0 (x1n0 ) + 2 4(N0 − 1) 1 1 α1 ϕn (x1n ) + ϕn (x2n ) + + 2 2 4(N0 − 1) n
+

1

n=N0

1 = Iϕ (x1 ) + 2 1 < Iϕ (x1 ) + 2

α 1 1 ϕn (x1n ) + ϕn (x2n ) + 2 2 4 1 1 α1 α1 Iϕ (x2 ) − ϕn0 (x2n0 ) + + 2 2 4 4 1 2 Iϕ (x ) = 1 2

so x1 − x2 < 2(1 − ε), and we get a contradiction again. Steps (1b) and (1c) complete the proof of theorem. Step 2. We discuss the general case without the restriction of step 1. For any ε ≤ 1/4, let A = inf ϕn ((1 − ε)an ). By the proof of Lemma 1 (i) we get A > 0. Let n

N2 = [1/A], i.e. N2 be the integer part of 1/A. If lϕ is reﬂexive but not P -convex, then for any ε : 0 < ε < ε/4, there is a set X consisted of any ﬁnite elements in S(Iϕ ) such that xi − xj ≥ 2(1 − ε ), i = j. Let the number of X be (2N0 + 1)2(N2 +1)N2 /2 where N0 is the positive integer satisfying (4).

P -convexity property in Musielak-Orlicz sequence spaces

323

Take any element x0 in X. The deﬁnition of A implies that x0 has at most N2 numbers of components, such that |x0n | ≥ (1 − ε)an ; hence Iϕ (x0 ) =

∞

ϕn (x0n ) ≥ (N2 + 1)A >

n=1

1 · A = 1, A

this leads to contradiction. Without loss of generality we have |x0n | ≥ (1 − ε)an x for n ≤ N2 . For any x ∈ X, we deﬁne a map: x → (r1x , r2x , . . . , rN ), i.e. for 2 n = 1, 2, . . . , N2 rnx

1,

when x0n xn < 0 and |xn | ≥ (1 − ε)an

0,

otherwise.

=

This makes us classify the elements of X into 2N1 categories, we say that the category mapping the vector (0, 0, . . . , 0) is 0-category. First we assume: apart from 0-category, the number of elements in other category is less than (2N0 + 1)2(N +1+1)N1 /2 /2N2 = (2N0 + 1)2N2 (N2 −1)/2 . Take another element from 0-category and let it be x0 , then classify X again by the former program. After we classify each time, if the number of the elements in category, except 0-category, is less than (2N0 + 1)2N1 (N1 −1)/2 , when we classify (2N0 + 1)−times we get a set X0 having (2N0 + 1) elements such that xin xjn > 0

or

|xin | ≥ (1 − ε)an

and |xjn | ≥ (1 − ε)an

(24)

for any xi , xj ∈ X0 (i = j) and n ∈ N, then x1 − x2 a + (1 − ε)a 2−ε n n n n an < an , < = 2(1 − ε) 2(1 − ε/4) 2 − ε/2 i.e. |xin | < (1 − ε )an for all n ≤ N2 , and this is the case of section 1. But in section 1, we proved that there is no set X having (2N0 + 1) elements such that xi − xj ≥ 2(1 − ε), i = j, xi , xj ∈ X, so we deduce that apart from 0-category there is a category X1 such that the number of elements in X is (2N0 + 1)2N1 (N2 −1)/2 and the element x of x1 satisﬁes rnx1 = 1 for some n1 ≤ N2 .

324

Yining and Yafeng

Apart from n1 -th component, any x = (xn ) in X1 has at most (N2 −1) numbers of components such that |xn | ≥ (1 − ε)an . Let |xn | ≥ (1 − ε)an for n = N2 + 1, N2 + 2, . . . , 2N2 − 1. x For any x ∈ X1 , deﬁne a map: x → (r1x , r2x , . . . , rN ), i.e. for n = N2 +1, N2 + 1 −1 2, . . . , 2N2 − 1 rnx

1,

when x0n xn < 0 and |xn | ≥ (1 − ε)an

0,

otherwise

=

then we may classify X1 into 2N2 −1 categories. If the number of elements in category except 0-category is less than (2N0 + 1)2(N1 −1)(N2 −2)/2 , we take one element from those mapping 0-category and let it be x0 , and then classify X1 by the former program. When we classify (2N0 + 1) times, the number of elements in the category except 0-category is less than (2N0 + 1)2(N2 −1)(N2 −2)/2 , then we get a set having (2N0 + 1) elements such that (24), which leads a contradiction again. We assume there a category X2 having (2N0 +1)2(N1 −1)(N2 −2)/2 elements except 0-category. Repeating the same discussion, when we classify N2 -times we get a category XN2 having (2N0 + 1) elements such that xin xjn > 0

and |xin | ≥ (1 − ε)an , |xjn | ≥ (1 − ε)an

for any xi , xj ∈ XN2 , i = j.n = n1 , n2 , . . . , nN2 . Then for any x ∈ XN2 I = Iϕ (x) =

j≤N2

≥

ϕnj (xnj ) +

ϕnj (1 − ε)anj + ϕn (xn ) ≥ N2 A + ϕn (xn )

j≤N0

i.e.

n=nj

ϕn (xn )

n=nj

ϕn (xn ) ≤ 1 − N1 A =

n=nj

n=nj

A I − A < A = inf ϕn (1 − ε)an n A A

so |xn | < (1 − ε)an with n = nj , but when n = nj xin xjn > 0(i = j). This shows that (24) is true for any x ∈ XN2 and all n ∈ N, which leads to a contradiction again. Section 1 and section 2 complete the proof of theorem.

P -convexity property in Musielak-Orlicz sequence spaces

325

Now we give an example of a Musielak-Orlicz sequence space which is P -convex but not P (3, ε)-convex. Let a Young function ϕ = (ϕn ) and ϕ∗ = (ϕ∗n ) satisfy the δ2 -condition, and such that there are two positive integers n1 and n2 (n1 < n2 ) ϕn1 (an1 ) + ϕn2 (an2 ) ≤ 1

and ϕn1 (an1 ) > 0, ϕn2 (an2 ) > 0.

By Theorem we know that the Iϕ generated by ϕ is P -convex but not P (3, ε)convex. Let x1 = (0, . . . , 0, an1 , 0, . . . , 0, an2 , 0, . . .) x2 = (0, . . . , 0, an1 , 0, . . . , 0, −an2 , 0, . . .) x3 = (0, . . . , 0, −an1 , 0, . . . , 0, an1 , 0, . . .). Then x1 , x2 , x3 ∈ S(Iϕ ). But for any ε > 0 x −x 2a 1 2 n2 = ϕn2 >1 2(1 − ε) 2(1 − ε) x −x 2a 1 i n1 = ϕn 1 >1 Iϕ 2(l − ε) 2(1 − ε) x ix 2a 2a 2 j n1 n1 = ϕn1 + ϕ n2 >1 Iϕ 2(1 − ε) 2(1 − ε) 2(1 − ε) Iϕ

so x1 − x2 ≥ 2(1 − ε), x2 − x3 ≥ 2(1 − ε), x1 − x4 ≥ 2(1 − ε), hence lϕ is not P (3, ε)-convex.

References 1. C.A. Kottman, Packing and reflexivity in Banach spaces, Trans. Amer. Math. Soc. 150 (1970), 565–576. 2. K.P.R. Sastry and S,V. R. Naidu, Convexity conditions in normed linear spaces, J. Reine Ange Math. 297 (1978), 35–53. 3. D. Amir and C. Franchetti, The radius ratio and convexity properties in normed linear spaces, Trans. Amer. Math. Soc. 282 (1984), 275–291. 4. Ye Yining, He Miaohong and R. Pluciennik, P -convexity and reflexivity of Orlicz spaces, Comm. Math. XXX 1 (1991), 203–216. 5. H. Hudzik and A. Kaminska, On uniformly convexifiable and B -convex Musielak-Orlicz spaces, Comment. Math. (Prace Mat.) 24 (1985), 59–75.

Collectanea Mathematica (electronic version): http://www.imub.ub.es/collect Collect. Math. 44 (1993), 327–337 c 1994 Universitat de Barcelona

Bounded variation functions of order k on sequence spaces

Zhao Linsheng Dept. of Basic Sciences, Heilongjiang Commercial College, Dao Li District, 150076 Harbin, China

Abstract In this paper, we generalize some results concerning bounded variation functions on sequence spaces.

Some properties of bounded variation functions on sequence spaces were investigated by Wu Congxin [1], [2], [3]. Later on, he and Zhao Linsheng in [4], [5], [6] introduced and discussed bounded variation functions of order 2 on sequence spaces. In this paper, we generalize these results to bounded variation functions of order k on sequence spaces. Let λ be a real linear sequence space. The K¨ othe dual λ∗ of λ is the real linear sequence space consisting of all real sequences U = (u1 , u2 , . . .) satisfying ∞ |uk xk | < ∞ for all X = (x1 , x2 , . . .) ∈ λ. When λ = λ∗∗ , we say that λ is a k=1

perfect space. For a real function x(t) deﬁned on [a, b] and k + 1 diﬀerent points t0 , t1 , . . . , tk ∈ [a, b], we denote k x(ti ) Qk (x; t0 , t1 , . . . , tk ) = k i=0 (ti − tj ) j=0 i=j

Definition 1 [7, p. 87]. The variation of order k of a function x(t) deﬁned on [a, b] is n−k b ∆ = (x) |Qk−1 (x; ti , . . . , ti+k−1 ) − Qk−1 (x; ti+1 , . . . , ti+k )| . sup V a k

π

i=0

327

328

Linsheng Where the “sup” is taken over all partitions π: a = t0 < t1 < · · · < tn = b of b

[a, b]. When V (x) < ∞, we say that x(t) is a bounded variation function of order a k

k and denote by x(t) ∈ Vk [a, b]. Lemma 1 [7, p. 88]. For any x ∈ Vk [a, b] and k diﬀerent points a0 , a1 , . . . , ak−1 in [a, b], we have b

|Qk−1 (x; t0 , 11 , . . . , tk−1 )| ≤ |Qk−1 (x; a0 , . . . , ak−1 )| + 2 V (x). a k

Lemma 2 [7, p. 79]. Qr−1 (x; t1 , t2 , . . . , tr ) − Qr−1 (x; t0 , t1 , . . . , tr−1 ) = (tr − t0 )Qr (x; t0 , . . . , tr ). Lemma 3 b

b

a k

a 2

For any k ≥ 3, we have (k − 1)!V (x) = V (x(k−2) ). Lemma 4 [11, p. 179]. Let x: [a, b] → R, y: [a, b] → R, then Qk (xy; t0 , t1 , . . . , tk ) = y(x0 )Qk (x; t0 , t1 , . . . , tk ) + Q1 (y; t0 , t1 )Qk−1 (x; t1 , . . . , tk ) + Q2 (y; t0 , t1 , t2 )Qk−2 (x; t2 , t3 , . . . , tk ) + . . . + Qk (y; t0 , t1 , . . . , tk )x(tk ). Lemma 5 Suppose x: [a, b] → R and a0 , a1 , . . . , ar ∈ [a, b](ai = aj when i = j), then |Qr (x; a0 , a1 , . . . , ar )| ≤

1

r

min |ai − aj |r

i=0

j=i

|x(ai )| r = 1, 2, . . .

Definition 2 Let X(t) = (x1 (t), x2 (t), . . .) be an abstract function from [a, b] to a sequence space λ. If for each U = (u1 , u2 , . . .) ∈ λ∗ , we have b

∆

V (X, U ) = sup a k

π

n−k ∞

um [Qk−1 (x; ti , . . . , ti+k−1 )

i=0 m=1

− Qk−1 (x; ti+1 , . . . , ti+k )] < ∞

then X(t) is called a bounded variation function of order k and denoted by X(t) ∈ Vk ([a, b], λ).

Bounded variation functions of order k on sequence spaces

329

Theorem 1 X(t) ∈ Vk ([a, b], λ) iﬀ 1 xm (t) ∈ Vk [a, b], m = 1, 2, . . . and ∞ b {V (xm )} ∈ λ∗∗ . 20 0

m=1 a k

i−1

Proof. Necessity. 1 Pick U = (0, 0, . . . , 0, 1, 0, 0, . . .) ∈ λ∗ , then from 0

sup

n−1 ∞

π

|um ||ti+k − ti | |Qk (xm ; ti , . . . , ti+k )| =

i=0 m=1

= sup

n−k

π

|ti+k − ti | |Qk (xm ; ti , . . . , ti+k )| < ∞

i=0

we see xs ∈ Vk [a, b], s = 1, 2, . . . (0)

(0)

Next we turn to 20 . If 20 is not true, then there exist U (0) = (u1 , u2 , . . .) ∈ (0) λ∗ , um = 0, m = 1, 2, . . . and Nn ≥ 1, εn > 0 such that Nn

b

|u(0) m | V (xm ) = n + εn . a k

m=1

(m)

Since xm ∈ Vk [a, b], m = 1, 2, . . . , Nn , there exists a partition πm : a = t0 (m) < . . . < tnm = b such that b

V (xm ) ≤

n m −k

a k

(m)

|ti+k − ti | |Qk (xm ; ti

i=0

εn

(m)

, . . . , ti+k )| +

(0) 2m+1 |um |

(m)

(m)

< t1

.

(Nn )

Let π be the partition consisting of all points {ti i ≤ nm , m ≤ Nn }: π: a = s0 (N ) (N ) s1 n < · · · < sl(Nnn ) = b then by Theorem 3 in [8], we have b

V (xm ) ≤ a k

l(Nn )−k

i=1

(N )

(Nn )

|si+kn − si

(Nn )

| |Qk (xm ; si

(N )

, . . . , si+kn | +

εn (0) 2m+1 |um |

.

<

330

Linsheng

Hence l(Nn )−k ∞

(N )

(Nn )

n |u(0) m | |si+k − si

(Nn )

| |Qk (xm ; si

(N )

, . . . , si+kn )|

m=1

i=0

≥

l(Nn )−k Nn

≥

(Nn )

(Nn )

| |Qk (xm ; si

(N )

, . . . , si+kn )|

m=1

i=0

Nn l(N n )−k m=1

(N )

n |u(0) m | |si+k − si

(N )

(Nn )

n |u(0) m | |si+k − si

(Nn )

| |Qk (xm ; si

(N )

, . . . , si+kn )|

i=0

Nn

|u(0) m |

m=1

b

V (xm ) − a

Nn m=1

εn m+1 2

≥n

contradicting that X(t) ∈ Vk ([a, b], λ). Suﬃciency. Notice that sup π

n−k ∞

|um | |ti+k − ti | |Qk (xm ; ti , . . . , ti+k )|

i=0 m=1

≤ =

∞ m=1 ∞ m=1

sup π

n−k

|um | |ti+k − ti | |Qk (xm ; ti , . . . , ti+k )|

i=0 b

|un | V (xm ) < ∞ a k

we ﬁnd that X(t) is bounded variation of order k. Theorem 2 Vk ([a, b], λ) ⊂ Vr ([a, b], λ) for all 1 ≤ r < k. Proof. It is suﬃcient to consider the case r = k − 1. By Theorem 1, X(t) = {xm (t)}∞ m=1 ∈ Vk ([a, b], λ) implies xm (t) ∈ Vk [a, b]

b

and {V (xm )} ∈ λ∗∗ . a k

For k diﬀerent points a0 < a1 < · · · < ak−1 in [a, b], by Lemma 1, we have b

|Qk−1 (xm ; ti , . . . , ti+k−1 )| ≤ |Qk−1 (xm ; a0 , . . . , ak−1 )| + 2 V (xm ) . a k

Bounded variation functions of order k on sequence spaces

331

Hence sup

n−k+1 ∞

π

|um | |ti+k−1 − ti | |Qk−1 (xm ; ti , . . . , ti+k−1 )|

m=1

i=0

≤ sup π

n−k+1 ∞

|um | |ti+k−1 − ti | (|Qk−1 (xm ; a0 , a1 , . . . , ak−1 )|

m=1

i=0 b

+ 2 V (xm )) ≤ sup π

a k n−k+1

|ti+k−1 − ti |

∞

|um | (|Qk−1 (xm ; a0 , a1 , . . . , ak−1 )|

m=1

i=0 b

+ 2 V (xm )) a k

≤ k(b − a)

∞

1 min

i=j,i,j−0,1,r,...,k−1

+2

∞ m=1

|ai − aj |k−1

|um |

m=1

k−1

|xm (ai )|

i=0

b |um V (xm )| < ∞ . a k

It follows that X(t) ∈ Vk−1 ([a, b], λ). Corollary 1 b

b

a k

a r

If {V (xm )} ∈ λ∗∗ , then {V (xm )} ∈ λ∗∗ for all 1 ≤ r < k. Theorem 3 Suppose k ≥ 3, then X(t) ∈ Vk ([a, b], λ) iﬀ X (t) ≡ {xm (t)} ∈ Vk−1 ([a, b], λ). Proof. Necessity. By Theorem 1, xm (t) ∈ Vk−1 [a, b]. Moreover, by Lemma 3, when b

b

a 3

a 2

k = 3, from {V (xm )} ∈ λ∗∗ , we have {V (xm )} ∈ λ∗∗ , and when k > 3, we have b

(k − 2)| V

a k−1

b

Hence, {V

a k−1

b

(xm ) =V (x(k−2) ). m a 2

(xm )} ∈ λ∗∗ and the conclusion follows from Theorem 1.

332

Linsheng Suﬃciency. By Theorem 1 in [9], we have xm (t) ∈ Vk [a, b]. Observing that b

b

a 3

a 2

k = 3 implies 2! V (xm ) = V (xm ) and k > 3 implies b

(k − 2)! V

a k−1

b

b

a 2

a k

(xm ) =V (x(k−2) ) = (k − 1)! V (xm ) m

b

we ﬁnd that {V (xm )} ∈ λ∗ (k ≥ 3) and that Theorem 1 implies X(t) ∈ Vk [a, b], λ). a k

Theorem 1 and Theorem 3 imply Corollary 2 Let k ≥ 3 then the following are equivalent 10 X(t) ∈ Vk ([a, b], λ); 2 0 ∀2 ≤ r < k, X (k−r) (t) − X (k−r) (a) ∈ Vr ([a, b], λ); 3 0 ∃2 ≤ r < k, such that X (k−r) (t) − X (k−r) (a) ∈ Vr ([a, b], λ); 4 0 ∀2 ≤ r < k, we have (k−r)

(i) xm b

(t) ∈ Vr [a, b], m = 1, 2, . . . (k−r)

(ii) {V (xm a r

)} ∈ λ∗∗

5 0 ∃2 ≤ r < k such that (k−r)

(i) xm b

(t) ∈ Vr [a, b], m = 1, 2, . . . (k−r)

(ii) {V (xm a r

} ∈ λ∗∗ .

Theorem 4 Assume that λ is a perfect space, then X(t) ∈ Vk ([a, b], λ) iﬀ there exist convex functions X (i) (t) ∈ Vk ([a, b], λ) (i = 1, 2) of order k such that X(t) = X (1) (t) − X (2) (t)

(t ∈ [a, b])

(X(t) is called a convex function of order k, if for each natural number m, xm (t) is a usual convex function of order k and x(t) is called a usual convex function of order k, if for any partition π: a = t0 < t1 < · · · < tk = b of [a, b], we have Qk (x; t0 , t1 , . . . , tk ) ≥ 0).

Bounded variation functions of order k on sequence spaces

333

Proof. The suﬃciency is obvious. Now we prove the necessity. The necessity is already known for k = 1 and k = 2. Suppose that the condition is necessary for k = m − 1, we investigate the case k = m. Since by Theorem 3, X (t) ∈ Vm−1 ([a, b], λ) by the assumption, there exist convex functions Y (i) (t) ∈ Vm−1 ([a, b], λ) of order m − 1 (i = 1, 2) such that X (t) = Y (1) (t) − Y (2) (t). For any c ∈ (a, b), we have

t

X (s)ds − X(c) =

X(t) = 0

Set X

Y

(1)

(s)ds −

0

(1)

t

(t) =

Y

(s)ds,

X

Y (2) (s)ds − X(c). 0

t (1)

t

(2)

0

t

Y (2) (s)ds + X(c)

(t) = 0

then by Theorem 13 in [8], X (i) (t) is convex of order m, i = 1, 2. But (X (i) (t)) ∈ Vm−1 ([a, b], λ), by Theorem 3, X (i) (t) ∈ Vm ([a, b], λ) i = 1, 2. Clearly, X(t) = X (1) (t) − X (2) (t), and X (1) (t) ∈ λ (and thus X (2) (t) ∈ λ) can be deduced as follows

b

|x(1) m (t)| ≤

(1) |ym (s)|ds ≤ a

b

b

(1) (1) |ym (a)|+ V (ym )ds a

a

(1) b (1) (a)|+ V (ym ) + (b − a) |ym a

therefore, {Xm (t)} ∈ λ∗∗ = λ. (1)

Theorem 5 Let λ be perfect, then X(t), Y (t) ∈ Vk ([a, b], λ) implies X(t)Y (t) ∈ Vk ([a, b], λ) iﬀ for any Z(t) ∈ Vk ([a, b]), λ), U ∈ λ∗ and c ∈ [a, b], we have

b b b |um | |Zm (c)|+ V (Zm )+ V (Zm ) + · · · + V (Zm ) ∈ λ∗ .

a

a 2

a k

Proof. Suﬃciency. Let X(t) ∈ Vk ([a, b], λ), Y (t) ∈ Vk ([a, b], λ), by Lemma 4,

334

Linsheng

sup π

n−k ∞

|um | |ti+k − ti | |Qk (xm ym ; ti , ti+1 , . . . , ti+k )|

i=1 m=1

= sup π

n−k ∞

|um | |ti+k − ti | |ym (ti )Qk (xm ; ti , . . . , ti+k )

i=1 m=1

+ Q1 (ym ; ti , ti+1 )Qi−1 (xm ; ti+1 , . . . , ti+k ) + · · · + Qk (ym ; ti , . . . , ti+k )xm (ti+k )| ≤ sup π

n−k ∞

|um | |ti+k − ti | ym (ti )Qk (xm ; ti , . . . , ti+k )

i=1 m=1

+ Q1 (ym ; ti , ti+1 )Qi−1 (xm ; ti+1 , . . . , ti+k ) + · · · + Qk (ym ; ti , . . . , ti+k )xm (ti+k ) ≤ sup π

n−k ∞

|um | |ti+k − ti | |ym (a0 )|

i=1 m=1

b + 2 V (yn ) |Qk (xm ; ti , . . . , ti+k )| a

+ sup π

n−k ∞

|um | |ti+k − ti |(|Q1 (ym ; a0 , a1 )|

i=1 m=1

b

b

+ V (ym )(|Qk−1 (xm ; a0 , . . . , at−1 )|+ V (xm ) a 2

a k

+ · · · + sup π

n−k ∞

b

|um | |ti+k − ti | |Qk (ym ; ti , . . . , ti+k )|(|xm (a0 )| + 2 V (xm )), a

i=1 m=1

where {ai }k−1 i=0 are diﬀerent points in (a, b). For the ﬁrst term, we have sup π

n−k ∞

b |um | |ti+k − ti | |ym (a0 )| + 2 V (ym ) |Qk (xm ; ti , . . . , ti+k )| a

i=1 m=1

≤ =

∞ m=1 ∞ m=1

b

|um |c|ym (a0 )| + 2 V (ym )) sup a

π

n−k

|ti+k − ti | |Qk (xm ; t1 , . . . , ti+k )|

i=0

b b |um | |ym (a0 )| + 2 V (ym ) V (xm ) < ∞ a

a k

(note that {|um |(|ym (a0 )| + 2 Vab (ym ))} ∈ λ∗ , {Vab k (xm )} ∈ λ∗∗ ) .

Bounded variation functions of order k on sequence spaces

335

Similarly, for the last term, we have sup π

n−k ∞

b |um | |ti+k − ti | |Qk (ym ; ti , . . . , ti+k )| |xm (a0 )| + 2 V (xm ) < ∞. a

i=1 m=1

Now, we show that the other terms are also bounded. Without loss of generality, we only consider the term sup π

n−k ∞

|um | |ti+k − ti |(|Q1 ((ym ; a0 , a1 )|

i=1 m=1 b

b

a 2

a k

+ 2 V (ym ))(|Qk−1 (xm ; a0 , . . . , ai−1 )| + 2 V (xm )) . By Lemma 5, sup π

n−k ∞

|um | |ti+k − ti |(|Q1 ((ym ; a0 , a1 )|

i=1 m=1 b

b

+ 2 V (ym ))(|Qk−1 (xm ; a0 , . . . , ai−1 )| + 2 V (xm )) a 2

≤ k(b − a)

a k

∞

|un |(|Q1 (ym ; a0 , a1 )|

m=1 b

b

a 2

a k

+ 2 V (ym ))(|Qk−1 (xm ; a0 , . . . , ak−1 )| + 2 V (xm )) < ∞. Thus, X(t)Y (t) ∈ Vk ([a, b], λ). Necessity. By Theorem 2.6 in [7], the condition is necessary for k = 1. Now, suppose k ≥ 2. Deﬁne

b b xm (t) = |Zm (c)|+ V (Zm ) + . . . + V (Zm ) tk−1 a

ym (t) =

a k

(a ≤ t ≤ b)

|x(0) m |t

then from

b b x(k−1) (t) = (k − 1)! |Zm (c)|+ V (Zm ) + · · · + V (Zm ) m a

(k−1) ym (t)

=

a k

(0)

|xm |, k = 2 0,

k>2

336

Linsheng

we have x(k−2) ∈ V2 [a, b], m

(k−2) ym ∈ V2 [a, b].

Hence, Theorem 1 in [9] implies xm , ym ∈ Vk [a, b], and Proposition 3.4 in [7] claims b

b

b

b

a k

a k

a k

a k

V (xm ) = V (ym ) = 0. Thus, {V (xm )} ∈ λ∗∗ and {V (ym )} ∈ λ∗∗ , and so, by Theorem 1,

X(t) = {xm (t)} ∈ Vk ([a, b], λ) y(t) = {ym (t)} ∈ Vk ([a, b], λ). Therefore X(t)y(t) ∈ Vk ([a, b]), λ). For any t0 = t1 = · · · = tk in (a, b), by Proposition 3.5 in [7] p. 82,

b b Qk (xm ym ; t0 , . . . , tk ) = |x(0) m | |Zm (c)|+ V (Zm ) + · · · + V (Zm )| a

a k

and from b

∞

a k

m=1

∞ >V (XY ; U ) ≥ = |tk − t0 |

∞

b b |um | |Zm (c)+ V (Zm ) + · · · + V (Zk )| |x(0) m | a

m=1

we ﬁnd

|um | |tk − t0 | |Qk (xm ym ; t0 , . . . , tk )|

a k

b b |um | |Zm (c)|+ V (Zm ) + · · · + V (Zm ) ∈ λ∗ . a

a k

References 1. W. Congxin, Bounded variation functions on sequence spaces (I), J. Harbin Inst. of Tech. no. 2 (1959), 93–100. 2. W. Congxin, Bounded variation functions on sequence spaces, Acta Math. Sinica 13 (1963), 548–557. 3. W. Congxin, Bounded variation functions on sequence spaces (II), Scientia Sinica 13 (1964), 1359–1380 (in Russian). 4. W. Congxin, Z. Linsheng, Variation functions of order 2 on sequence spaces (I), Science Exploration, no. 2 (1982), 71–73. 5. W. Congxin, Z. Linsheng, Variation functions of order 2 on sequence spaces, Math. Research and Exploration, no. 4 (1982), 143–150.

Bounded variation functions of order k on sequence spaces

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6. W. Congxin, Z. Linsheng, Variation functions of order 2 on sequence spaces (II), Ibid., no. 1 (1984), 98–106. 7. W. Congxin, Z. Linsheng, L. Tiefu, Bounded Variation Functions and Their Generalizations and Applications, Heilongjiang Science and Technology Press (1988). 8. A.M. Russell, Functions of bounded k th variation, Proc. London Math. Soc. 26 (1973), 547–563. 9. W. Congxin, L. Tiefu, Some Properties of bounded Variation functions of order k , J. Liaoning Univ. (1986), 1–10.