aa r X i v : . [ m a t h . F A ] F e b WIENER–LUXEMBURG AMALGAM SPACES
DALIMIL PEˇSA
Abstract.
In this paper we introduce the concept of Wiener–Luxemburg amalgam spaceswhich are a modification of the more classical Wiener amalgam spaces intended to address someof the shortcomings the latter face in the context of rearrangement-invariant Banach functionspaces.We introduce the Wiener–Luxemburg amalgam spaces and study their properties, including(but nor limited to) their normability, embeddings between them and their associate spaces.We also study amalgams of quasi-Banach function spaces and introduce a necessary general-isation of the concept of associate spaces. We then apply this general theory to resolve thequestion whether the Hardy–Littlewood–P´olya principle holds for all r.i. quasi-Banach functionspaces. Finally, we illustrate the asserted shortcomings of Wiener amalgam spaces by provid-ing counterexamples to certain properties of Banach function spaces as well as rearrangementinvariance. Introduction
In this paper we introduce the concept of Wiener–Luxemburg amalgam spaces which are amodification of the more classical Wiener amalgam spaces. The principal idea of both kinds ofamalgam spaces is to treat separately the local and global behaviour of a given function, in thesense that said function is required to be locally in one space and globally in a different space.The exact meaning of being locally and globally in a space varies in literature, depending onthe desired generality and personal preference.The classical Wiener amalgams approach this issue in a very general, albeit quite non-trivial,manner. They were, in their general form, first introduced by Feichtinger in [12], although theless general cases were studied earlier, see for example the paper [18] due to Holland, and somespecial cases date as far back as 1926 when the first example of such a space was introducedby Wiener in [26]. The different versions of these spaces saw many applications in the lastdecades, great surveys of which have been conducted, concerning a somewhat restricted version,by Fournier and Stewart in [16] and, concerning the more general versions, by Feichtinger in [13]and [14]. Probably the most famous example is the Tauberian theorem for the Fourier transformon the real line due to Wiener (see [27] and [28]), other examples include the theory of Fouriermultipliers (see [11]), several variation of the sampling theorem (see [15]), and the theory ofproduct convolution–operators (see [6]).One unfortunate property of Wiener amalgams is that, even in the simplest and most naturalcases, their construction does not preserve the properties of Banach function spaces, nor does itpreserve rearrangement invariance (see Appendix A). This approach is therefore unsuitable whenone wishes to work in this context. But this often is the case, since there are many situationswhen the need arises naturally to prescribe separately the conditions on local and on global
Date : February 18, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Wiener–Luxemburg amalgam spaces, Wiener amalgam spaces, embedding theo-rems, associate spaces, rearrangement-invariant Banach function spaces, quasi-Banach function spaces, Hardy–Littlewood–P´olya principle.This research was supported by the grant P201-18-00580S of the Czech Science Foundation, by the Primusresearch programme PRIMUS/21/SCI/002 of Charles University, and by the Danube Region Grant no. 8X20043TIFREFUS. behaviour of a function. One such situation is the study of optimal Sobolev type embeddingsover the entire Euclidean space in the context of rearrangement-invariant Banach function spacesas performed by Alberico, Cianchi, Pick and Slav´ıkov´a in [1]. A very natural example is theoptimal target space, in the context of rearrangement-invariant Banach function spaces, for thelimiting case of the classical Sobolev embeddings over the entire Euclidean space, which hasbeen found by Vyb´ıral in [25]. Another such situation arised during the study of generalisedLorentz–Zygmund spaces which led to the introduction of broken logarithmic functions to allowseparate treatment of local and global properties of functions in this context. For further detailsand a comprehensive study of generalised Lorentz–Zygmund spaces we refer the reader to [22].Further example of an area where this approach has been successfully employed is the theory ofinterpolation, where description of sums and intersections of spaces as de facto amalgams haveproven useful, see [2] and [3] for details.This led us to develop the theory of Wiener–Luxemburg amalgam spaces, which aims toeliminate the above mentioned limitations and to provide a general framework for separateprescription of local and global conditions in the context of rearrangement-invariant Banachfunction spaces. The starting point is provided by the non-increasing rearrangement, which isthe crucial element in the theory of said spaces and which naturally separates the local behaviourof a function from its global behaviour, at least in the sense of size. This allows us to defineWiener–Luxemburg amalgam spaces in a very easy and straightforward manner.While the main part of this paper focuses on amalgams of rearrangement-invariant Banachfunction spaces, we will in the later sections also use the recent advances in the theory ofquasi-Banach function spaces (see [21]) and extend our theory into this context. While ofindependent interest, this will also allow us to view the theory of quasi-Banach function spacesfrom a new viewpoint and provide a negative answer to an important open question whetherHardy–Littlewood–P´olya principle holds for all rearrangement-invariant quasi-Banach functionnorms.The paper is structured as follows. In Section 2 we present the basic theoretical backgroundneeded in order to build the theory in later sections.Section 3 is the main part of the paper where the theory of Wiener–Luxemburg amalgamspaces is developed in some detail for the more classical context of rearrangement-invariantBanach function spaces. We show that they too are rearrangement-invariant Banach functionspaces, then we provide a characterisation of their associate spaces, a full characterisation oftheir embeddings, and put them in relation with the concepts of sum and intersection of Banachspaces. Furthermore we refine the well known classical result that it holds for all rearrangement-invariant Banach function spaces A that L ∩ L ∞ ֒ → A ֒ → L + L ∞ by showing that L is the locally weakest and globally strongest rearrangement-invariant Banachfunction space, while L ∞ is, in the same setting, the locally strongest and globally weakest space.Needless to say, our definition of Wiener–Luxemburg amalgam spaces is general enough to coverall the spaces appearing in the applications outlined above.We then in Section 4 switch to the more abstract context of rearrangement-invariant quasi-Banach function spaces and introduce an extension of the concept of associate spaces which wecall the integrable associate spaces. This concept emerged naturally in the study of associatespaces of Wiener–Luxemburg amalgams in the context of quasi-Banach function spaces, so inthis section we lay the groundwork for Section 5. However, we believe this topic to be interestingin its own right and the treatment we provide goes beyond what is strictly necessary for ourlater work.Section 5 then contains the extension of our theory to the context of rearrangement-invariantquasi-Banach function spaces. We show that, in this context, the Wiener–Luxemburg amalgam IENER–LUXEMBURG AMALGAM SPACES 3 spaces are rearrangement-invariant quasi-Banach function spaces, we describe their integrableassociate spaces (and, in the case when it is meaningful, their associate spaces), and provide aprecise treatment of the above mentioned refinement of the result that L ∩ L ∞ ֒ → A ֒ → L + L ∞ which shows what properties of rearrangement-invariant Banach function norms are relevantfor the individual embeddings. Last but not least, we show that the validity of the Hardy–Littlewood–P´olya principle is sufficient for one of the embeddings in question. Since thereare spaces for which this embedding fails, we obtain a negative answer to the important openquestion whether the Hardy–Littlewood–P´olya principle holds for all r.i. quasi-Banach functionspaces.Finally in Appendix A we present some counterexamples which show that our claim thatWiener amalgams are in general neither Banach function spaces nor rearrangement-invariant isjustified. This serves three distinct purposes: first, it fills a gap in literature; second, it providessome insight into the thought process behind our definition of Wiener–Luxemburg amalgamspaces; and third, it is an application of our theory, since we use Wiener–Luxemburg amalgamsto show that Wiener amalgams are not rearrangement-invariant.2. Preliminaries
This section serves to establish the basic theoretical background upon which we will build ourtheory of Wiener–Luxemburg amalgam spaces. The definitions and notation is intended to beas standard as possible. The usual reference for most of this theory is [4].Throughout this paper we will denote by (
R, µ ), and occasionally by (
S, ν ), some arbitrary(totally) sigma-finite measure space. Given a µ -measurable set E ⊆ R we will denote its char-acteristic function by χ E . By M ( R, µ ) we will denote the set of all extended complex-valued µ -measurable functions defined on R . As is customary, we will identify functions that coin-cide µ -almost everywhere. We will further denote by M ( R, µ ) and M + ( R, µ ) the subsets of M ( R, µ ) containing, respectively, the functions finite µ -almost everywhere and the non-negativefunctions.For brevity, we will abbreviate µ -almost everywhere, M ( R, µ ), M ( R, µ ), and M + ( R, µ ) to µ -a.e., M , M , and M + , respectively, when there is no risk of confusing the reader.When X, Y are two topological linear spaces, we will denote by
Y ֒ → X that Y ⊆ X and thatthe identity mapping I : Y → X is continuous.As for some special cases, we will denote by λ n the classical n -dimensional Lebesgue measure,with the exception of the 1-dimensional case in which we will simply write λ . We will furtherdenote by m the counting measure over N . When p ∈ (0 , ∞ ] we will denote by L p the classicalLebesgue space (of functions in M ( R, µ )) defined by L p = (cid:26) f ∈ M ( R, µ ); Z R | f | p dµ < ∞ (cid:27) equipped with the customary (quasi-)norm k f k p = (cid:18)Z R | f | p dµ (cid:19) p , with the usual modifications when p = ∞ . In the special case when ( R, µ ) = ( N , m ) we willdenote this space by l p .Note that in this paper we consider 0 to be an element of N . IENER–LUXEMBURG AMALGAM SPACES 4
Non-increasing rearrangement.
We now present the concept of the non-increasing re-arrangement of a function and state some of its properties that will be important later in thepaper. We proceed in accordance with [4, Chapter 2].We start by introducing the distribution function.
Definition 2.1.
The distribution function µ f of a function f ∈ M is defined for s ∈ [0 , ∞ ) by µ f ( s ) = µ ( { t ∈ R ; | f ( t ) | > s } ) . The non-increasing rearrangement is then defined as the generalised inverse of the distributionfunction.
Definition 2.2.
The non-increasing rearrangement f ∗ of a function f ∈ M is defined for t ∈ [0 , ∞ ) by f ∗ ( t ) = inf { s ∈ [0 , ∞ ); µ f ( s ) ≤ t } . For the basic properties of the distribution function and the non-increasing rearrangement,with proofs, see [4, Chapter 2, Proposition 1.3] and [4, Chapter 2, Proposition 1.7], respectively.We consider those properties to be classical and well known and we will be using them withoutfurther explicit reference.An important concept used in the paper is that of equimeasurability defined below.
Definition 2.3.
We say that the functions f ∈ M ( R, µ ) and g ∈ M ( S, ν ) are equimeasurable if µ f = ν g .It is not hard to show that two functions are equimeasurable if and only if their non-increasingrearrangements coincide too.A very important classical result is the Hardy–Littlewood inequality which we will use exten-sively in the paper. For proof, see for example [4, Chapter 2, Theorem 2.2]. Theorem 2.4.
It holds for all f, g ∈ M that Z R | f g | dµ ≤ Z ∞ f ∗ g ∗ dλ. It follows directly from this result that it holds for every f, g ∈ M thatsup ˜ g ∈ M ˜ g ∗ = g ∗ Z R | f ˜ g | dµ ≤ Z ∞ f ∗ g ∗ dλ. This motivates the definition of resonant measure spaces.
Definition 2.5.
A sigma-finite measure space (
R, µ ) is said to be resonant if it holds for all f, g ∈ M ( R, µ ) that sup ˜ g ∈ M ˜ g ∗ = g ∗ Z R | f ˜ g | dµ = Z ∞ f ∗ g ∗ dλ. The property of being resonant is an important one. Luckily, there is a straightforwardcharacterisation of resonant measure spaces. For proof and further details see [4, Chapter 2,Theorem 2.7].
Theorem 2.6.
A sigma-finite measure space is resonant if and only if it is either non-atomicor completely atomic with all atoms having equal measure.
IENER–LUXEMBURG AMALGAM SPACES 5
Norms and quasinorms.
In this subsection, and also in the following one, we providethe definitions for several classes of functionals we will study in the paper. All definitions shouldbe standard or at least straightforward generalisations of standard ones.The starting point shall be the class of norms.
Definition 2.7.
Let X be a complex linear space. A functional k·k : X → [0 , ∞ ) will be calleda norm if it satisfies the following conditions:(1) it is positively homogeneous, i.e. ∀ a ∈ C ∀ x ∈ X : k ax k = | a |k x k ,(2) it satisfies k x k = 0 ⇔ x = 0 in X ,(3) it is subadditive, i.e. ∀ x, y ∈ X : k x + y k ≤ k x k + k y k .Because the definition of a norm is sometimes too restrictive we will need a class of weakerfunctionals, namely quasinorms. Definition 2.8.
Let X be a complex linear space. A functional k·k : X → [0 , ∞ ) will be calleda quasinorm if it satisfies the following conditions:(1) it is positively homogeneous, i.e. ∀ a ∈ C ∀ x ∈ X : k ax k = | a |k x k ,(2) it satisfies k x k = 0 ⇔ x = 0 in X ,(3) there is a constant C ≥
1, called the modulus of concavity of k·k , such that it is subad-ditive up to this constant, i.e. ∀ x, y ∈ X : k x + y k ≤ C ( k x k + k y k ).It is obvious that every norm is also a quasinorm with the modulus of concavity equal to 1and that every quasinorm with the modulus of concavity equal to 1 is also a norm.It is a well-known fact that every norm defines a metrizable topology on X and that it iscontinuous with respect to that topology. This is not true for quasinorms, but this can beremedied thanks to the Aoki–Rolewicz theorem which we list below. Further details can befound for example in [19] or in [5, Appendix H]. Theorem 2.9.
Let k·k X be a quasinorm over the linear space X . Then there is a quasinorm |||·||| X over X such that (1) there is a finite constant C > such that it holds for all x ∈ X that C − k x k X ≤ ||| x ||| X ≤ C k x k X , (2) there is an r ∈ (0 , such that it holds for all x, y ∈ X that ||| x + y ||| rX ≤ ||| x ||| rX + ||| y ||| rX . The direct consequence of this result is that every quasinorm defines a metrizable topologyon X and that the convergence in said topology is equivalent to the convergence with respectto the original quasinorm, in the sense that x n → x in the induced topology if and only iflim n →∞ k x n − x k = 0.Natural question to ask is when do different quasinorms define equivalent topologies. It isan easy exercise to show that the answer is the same as in the case of norms, that is that twoquasinorms are topologically equivalent if and only if they are equivalent in the following sense. Definition 2.10.
Let k·k X and |||·||| X be quasinorms over the linear space X . We say that k·k X and |||·||| X are equivalent if there is some C > x ∈ X that C − k x k X ≤ |||·||| X ≤ C k x k X . To conclude this part, we recall the concepts of sum and intersection of normed spaces.
Definition 2.11.
Let X and Y be normed linear spaces equipped with the norms k·k X and k·k Y respectively. Suppose that there is a Hausdorff topological linear space Z into which X IENER–LUXEMBURG AMALGAM SPACES 6 and Y are continuously embedded. We then define the spaces X + Y and X ∩ Y as X + Y = { z ∈ Z ; ∃ x ∈ X, ∃ y ∈ Y : z = x + y } ,X ∩ Y = { z ∈ Z ; z ∈ X, z ∈ Y } , equipped with the norms k z k X + Y = inf {k x k X + k y k Y ; x ∈ X, y ∈ Y, x + y = z } , k z k X ∩ Y = max {k z k X , k z k Y } , respectively.The concepts presented above play a crucial role in the theory of interpolation. For furtherdetails, we refer the reader to [4, Chapter 3], where one can also find the following result (as [4,Chapter 3, Theorem 1.3]). Theorem 2.12.
Let X and Y be as above. Then X + Y and X ∩ Y , when equipped with theirrespective norms, are normed linear spaces. Furthermore, if X and Y are Banach spaces, thenso are X + Y and X ∩ Y . Banach function norms and quasinorms.
We now turn our attention to the case inwhich we are interested the most, that is the case of norms and quasinorms acting on spaces offunctions. The approach taken here is the same as in [4, Chapter 1, Section 1], which meansthat it differs, at least formally, from that in the previous part.The major definitions are of course those of a Banach function norm and the correspondingBanach function space.
Definition 2.13.
Let k·k : M ( R, µ ) → [0 , ∞ ] be a mapping satisfying k | f | k = k f k for all f ∈ M . We say that k·k is a Banach function norm if its restriction to M + satisfies the followingaxioms:(P1) it is a norm, in the sense that it satisfies the following three conditions:(a) it is positively homogeneous, i.e. ∀ a ∈ C ∀ f ∈ M + : k af k = | a |k f k ,(b) it satisfies k f k = 0 ⇔ f = 0 µ -a.e.,(c) it is subadditive, i.e. ∀ f, g ∈ M + : k f + g k ≤ k f k + k g k ,(P2) it has the lattice property, i.e. if some f, g ∈ M + satisfy f ≤ g µ -a.e., then also k f k ≤ k g k ,(P3) it has the Fatou property, i.e. if some f n , f ∈ M + satisfy f n ↑ f µ -a.e., then also k f n k ↑ k f k ,(P4) k χ E k < ∞ for all E ⊆ R satisfying µ ( E ) < ∞ ,(P5) for every E ⊆ R satisfying µ ( E ) < ∞ there exists some finite constant C E , dependentonly on E , such that the inequality R E f dµ ≤ C E k f k is true for all f ∈ M + . Definition 2.14.
Let k·k X be a Banach function norm. We then define the correspondingBanach function space X as the set X = { f ∈ M ; k f k X < ∞} . It is easy to see that a Banach function norm, when restricted to the space it defines, is indeeda norm in the sense of Definition 2.7 and therefore Banach function spaces, when equipped withtheir defining norm, are normed linear spaces. Detailed study of these spaces can be found in[4].Just as with general norms, the triangle inequality is sometimes too strong a condition torequire. We therefore introduce the notions of quasi-Banach function norms and of the corre-sponding quasi-Banach function spaces.
Definition 2.15.
Let k·k : M ( R, µ ) → [0 , ∞ ] be a mapping satisfying k | f | k = k f k for all f ∈ M . We say that k·k is a quasi-Banach function norm if its restriction to M + satisfies the IENER–LUXEMBURG AMALGAM SPACES 7 axioms (P2), (P3) and (P4) of Banach function norms together with a weaker version of axiom(P1), namely(Q1) it is a quasinorm, in the sense that it satisfies the following three conditions:(a) it is positively homogeneous, i.e. ∀ a ∈ C ∀ f ∈ M + : k af k = | a |k f k ,(b) it satisfies k f k = 0 ⇔ f = 0 µ -a.e.,(c) there is a constant C ≥
1, called the modulus of concavity of k·k , such that it issubadditive up to this constant, i.e. ∀ f, g ∈ M + : k f + g k ≤ C ( k f k + k g k ) . Definition 2.16.
Let k·k X be a quasi-Banach function norm. We then define the correspondingquasi-Banach function space X as the set X = { f ∈ M ; k f k X < ∞} . As before, it is easy to see that a quasi-Banach function norm restricted to the space it definesis a quasinorm in the sense of Definition 2.8. Let us now list here some of their importantproperties we will need later.
Theorem 2.17.
Let k·k X be a quasi-Banach function norm and let X be the correspondingquasi-Banach function space. Then X is complete. Theorem 2.18.
Let k·k X and k·k Y be quasi-Banach function norms and let X and Y be thecorresponding quasi-Banach function spaces. If X ⊆ Y then also X ֒ → Y . Both of these results have been known for a long time in the context of Banach function spacesbut they have been only recently extended to quasi-Banach function spaces. Theorem 2.17 hasbeen first obtained by Caetano, Gogatishvili and Opic in [7] while Theorem 2.18 has been provedby Nekvinda and the author in [21]. The following result has also been obtained in [21]:
Theorem 2.19.
Let k·k X be a quasi-Banach function norm and let X be the correspondingquasi-Banach function space. Suppose that E ⊆ R is a set such that µ ( E ) < ∞ and that forevery constant K ∈ (0 , ∞ ) there is a non-negative function f ∈ X satisfying Z E | f | dµ > K k f k X . Then there is a non-negative function f E ∈ X such that Z E f E dµ = ∞ . The last result concerning Banach function spaces we want to list at this point concerns theproperties of the intersection of two Banach function spaces. The proof is an easy exercise.
Proposition 2.20.
Let X and Y be two Banach function spaces. Then X ∩ Y is also a Banachfunction space. Let us now define an important property that a quasi-Banach function norm can have andthat we will take a special interest in. Note that the class of quasi-Banach function normscontains that of Banach function norms so it is not necessary to provide separate definitions.
Definition 2.21.
Let k·k X be a quasi-Banach function norm. We say that k·k X is rearrangement-invariant, abbreviated r.i., if k f k X = k g k X whenever f, g ∈ M are equimeasurable (in the senseof Definition 2.3).Furthermore, if the above condition holds, the corresponding space X will be called rearrangement-invariant too. IENER–LUXEMBURG AMALGAM SPACES 8
An important property of r.i. quasi-Banach function spaces over ([0 , ∞ ) , λ ) is that the dilationoperator is bounded on those spaces, as stated in the following theorem. This is a classical resultin the context of r.i. Banach function spaces which has been recently extended to r.i. quasi-Banach function spaces by by Nekvinda and the author in [21] (for the classical version see forexample [4, Chapter 3, Proposition 5.11]). Definition 2.22.
Let t ∈ (0 , ∞ ). The dilation operator D t is defined on M ([0 , ∞ ) , λ ) by theformula D t f ( s ) = f ( ts ) , where f ∈ M ([0 , ∞ ) , λ ), s ∈ (0 , ∞ ). Theorem 2.23.
Let X be an r.i. quasi-Banach function space over ([0 , ∞ ) , λ ) and let t ∈ (0 , ∞ ) .Then D t : X → X is a bounded operator. Finally, we want to discuss here one property of some r.i. quasi-Banach function norms thatis often important in applications.
Definition 2.24.
Let k·k X be an r.i. quasi-Banach function norm. We say that the Hardy–Littlewood–P´olya principle holds for k·k X if the estimate k f k X ≤ k g k X is true for any pair offunctions f, g ∈ M satisfying Z t f ∗ dλ ≤ Z t g ∗ dλ for all t ∈ (0 , ∞ ).To put this property into the proper context we will need the following lemma: Lemma 2.25.
Let k·k X be an r.i. quasi-Banach function norm and consider the following threestatements: (1) k·k X is an r.i. Banach function norm. (2) The Hardy–Littlewood–P´olya principle holds for k·k X . (3) k·k X satisfies (P5) .Then (1) implies (2) which in turn implies (3) .Proof. That (1) implies (2), i.e. that the Hardy–Littlewood–P´olya principle holds for all r.i. Ba-nach function spaces is a well known result, see for example [4, Chapter 2, Theorem 4.6].The remaining implication will be proved by contradiction, so we assume that k·k X does notsatisfy (P5). Then it follows from Theorem 2.19 that there is a set E ⊆ R and a function f ∈ M such that µ ( E ) < ∞ , k f k X < ∞ , and Z E | f | dµ = ∞ . Hence µ ( E ) > Z µ ( E )0 f ∗ dλ = ∞ . Since f ∗ is non-increasing, we finally obtain the equality Z t f ∗ dλ = ∞ for all t ∈ (0 , ∞ ).Because we assume that the Hardy–Littlewood–P´olya principle holds for k·k X , we concludethat every function g ∈ M satisfies k g k X < ∞ . Since this includes g = ∞ χ R and since µ ( R ) ≥ µ ( E ) >
0, we obtain a contradiction with the property that quasi-Banach functionspaces contain only functions that are finite almost everywhere (see [20, Lemma 2.4] or [21,Theorem 3.4]). (cid:3)
IENER–LUXEMBURG AMALGAM SPACES 9
Remark 2.26.
None of the implications in Lemma 2.25 can be reversed. In the first case, we canshow this by considering the functional k·k X = k·k L p,q , where L p,q is the Lorentz space, since forthe choice of parameters p ∈ (1 , ∞ ), q ∈ (0 , Associate space.
An important concept in the theory of Banach function spaces and theirgeneralisations is that of an associate space. The detailed study of associate spaces of Banachfunction spaces can be found in [4, Chapter 1, Sections 2, 3, and 4].We will approach the issue in a slightly more general way. The very definition of an associatespace requires no assumptions on the functional defining the original space.
Definition 2.27.
Let k·k X : M → [0 , ∞ ] be some non-negative functional and put X = { f ∈ M ; k f k X < ∞} . Then the functional k·k X ′ defined for f ∈ M by(2.1) k f k X ′ = sup g ∈ X k g k X Z R | f g | dµ, where we interpret = 0 and a = ∞ for any a >
0, will be called the associate functional of k·k X while the set X ′ = { f ∈ M ; k f k X ′ < ∞} will be called the associate space of X .As suggested by the notation, we will be interested mainly in the case when k·k X is at leasta quasinorm, but we wanted to emphasize that such an assumption is not necessary for thedefinition. In fact, it is not even required for the following result, which is the H¨older inequalityfor associate spaces. Theorem 2.28.
Let k·k X : M → [0 , ∞ ] be some non-negative functional and denote by k·k X ′ its associate functional. Then it holds for all f ∈ M that Z R | f g | dµ ≤ k g k X k f k X ′ provided that we interpret · ∞ = −∞ · ∞ = ∞ on the right-hand side. The convention at the end of the preceding theorem is necessary because the lack of assump-tions on k·k X means that we allow some pathological cases that need to be taken care of. Tobe more specific, 0 · ∞ = ∞ is necessary because we allow k g k X = 0 even for non-zero g while −∞ · ∞ = ∞ is needed because the set X can be empty, in which case k f k X ′ = sup ∅ = −∞ .The last result we will present in this generality is the following proposition concerning em-beddings. Although the proof is an easy modification of that in [4, Chapter 2, Proposition 2.10]we provide it to show that it truly does not require any assumptions on the original functional. Proposition 2.29.
Let k·k X : M → [0 , ∞ ] and k·k Y : M → [0 , ∞ ] be two non-negative func-tionals satisfying that there is a constant C > such that it holds for all f ∈ M that k f k X ≤ C k f k Y . Then the associate functionals k·k X ′ and k·k Y ′ satisfy, with the same constant C , k f k Y ′ ≤ C k f k X ′ for all f ∈ M . IENER–LUXEMBURG AMALGAM SPACES 10
Proof.
Our assumptions guarantee that Y ⊆ X and therefore k f k Y ′ = sup g ∈ Y k g k Y Z R | f g | dµ ≤ sup g ∈ Y C k g k X Z R | f g | dµ ≤ sup g ∈ X C k g k X Z R | f g | dµ = C k f k X ′ . (cid:3) Let us now turn our attention to the case when k·k X is a quasi-Banach function norm. Notethat in this case the definition of the associate functional does not change when one replaces thesupremum in (2.1) by one taken only over the unit sphere in X .The following result, due to Gogatishvili and Soudsk´y in [17], shows that the conditions theoriginal functional needs to satisfy in order for the associate functional to be a Banach functionnorm are quite mild. Specially, they are satisfied by any quasi-Banach function norm thatsatisfies the axiom (P5). This special case was observed earlier in [10, Remark 2.3.(iii)]. Theorem 2.30.
Let k·k X : M → [0 , ∞ ] be a functional that satisfies the axioms (P4) and (P5) from the definition of Banach function norms and which also satisfies for all f ∈ M that k f k X = k | f | k X . Then the functional k·k X ′ is a Banach function norm. In addition, k·k X isequivalent to a Banach function norm if and only if there is some constant C ≥ such that itholds for all f ∈ M that (2.2) k f k X ′′ ≤ k f k X ≤ C k f k X ′′ , where k·k X ′′ denotes the associate functional of k·k X ′ . Additionally, if k·k X is a Banach function norm then (2.2) holds with C = 1. This is a classicalresult of Lorenz and Luxemburg, proof of which can be found for example in [4, Chapter 1,Theorem 2.7]. Theorem 2.31.
Let k·k X be a Banach function norm, then k·k X = k·k X ′′ where k·k X ′′ is theassociate functional of k·k X ′ . Let us point out that even in the case when k·k X , satisfying the assumptions of Theorem 2.30,is not equivalent to any Banach function norm we still have one interesting embedding, asformalised in the following statement. The proof is an easy exercise. Proposition 2.32.
Let k·k X satisfy the assumptions of Theorem 2.30. Then it holds for all f ∈ M that k f k X ′′ ≤ k f k X , where k·k X ′′ denotes the associate functional of k·k X ′ . We will also use in the paper the following version of Landau’s resonance theorem. This resultwas first obtained in full generality in [21].
Theorem 2.33.
Let k·k X be a quasi-Banach function norm, let X be the corresponding quasi-Banach function space and let k·k X ′ and X ′ , respectively, be the associate norm of k·k X and thecorresponding associate space. Then a function f ∈ M belongs to X ′ if and only if it satisfies Z R | f g | dµ < ∞ for all g ∈ X . IENER–LUXEMBURG AMALGAM SPACES 11
To conclude this section, we observe that, provided the underlying measure space is resonant,the associate functional of an r.i. quasi-Banach function norm can be expressed in terms ofnon-increasing rearrangement. The proof is the same as in [4, Chapter 2, Proposition 4.2].
Proposition 2.34.
Let k·k X be an r.i. quasi-Banach function norm over a resonant measurespace. Then its associate functional k·k X ′ satisfies k f k X ′ = sup g ∈ X k g k X Z ∞ f ∗ g ∗ dλ. An obvious consequence of Proposition 2.34 is that an associate space of an r.i. quasi-Banachfunction space (over a resonant measure space) is also rearrangement-invariant.3.
Wiener–Luxemburg amalgam spaces
Throughout this section we restrict ourselves to the case when (
R, µ ) = ([0 , ∞ ) , λ ). This al-lows us to make the proofs more elegant and less technical as well as ensures that the underlyingmeasure space is resonant. Note that this comes at no loss of generality, since any r.i. Banachfunction space over an arbitrary resonant measure space can be represented by some r.i. Banachfunction space over ([0 , ∞ ) , λ ), as follows from the classical Luxemburg representation theo-rem (see for example [4, Chapter 2, Theorem 4.10]), and any r.i. Banach function space over([0 , ∞ ) , λ ) represents an r.i. Banach function space over any resonant measure space (see forexample [4, Chapter 2, Theorem 4.9]).3.1. Wiener–Luxemburg quasinorms.Definition 3.1.
Let k·k A and k·k B be r.i. Banach function norms. We then define the Wiener–Luxemburg quasinorm k·k W L ( A,B ) , for f ∈ M , by(3.1) k f k W L ( A,B ) = k f ∗ χ [0 , k A + k f ∗ χ (1 , ∞ ) k B and the corresponding Wiener–Luxemburg amalgam space W L ( A, B ) as
W L ( A, B ) = { f ∈ M ; k f k W L ( A,B ) < ∞} . Furthermore, we will call the first summand in (3.1) the local component of k·k
W L ( A,B ) whilethe second summand will be called the global component of k·k W L ( A,B ) .For the sake of brevity we will sometimes write just Wiener–Luxemburg amalgams instead ofWiener–Luxemburg amalgam spaces.Let us at first note that this concept somewhat generalises the concept of the r.i. Banachfunction spaces in the sense that every r.i. Banach function space is, up to equivalence of thedefining functionals, a Wiener–Luxemburg amalgam of itself. Remark 3.2.
Let k·k A be an r.i. Banach function norm. Then k f k A ≤ k f k W L ( A,A ) ≤ k f k A for every f ∈ M .Consequently, it makes a good sense to talk about local and global components of arbitraryr.i. Banach function norms.The local component of an arbitrary r.i. Banach function norm is worth separate attention.As shown in the following proposition, this component is itself an r.i. Banach function normand therefore any unpleasant behaviour of the Wiener–Luxemburg quasinorm must be causedby its global element. Second part of the proposition then illustrates the interesting fact thatthe global element of L ∞ puts no additional condition on the size of a given function, in thesense that the spaces W L ( A, L ∞ ) consists of exactly those functions that are locally in A . IENER–LUXEMBURG AMALGAM SPACES 12
Proposition 3.3.
Let k·k A be an r.i. Banach function norm. Then the functional f
7→ k f ∗ χ [0 , k A is also an r.i. Banach function norm.Furthermore, there is a constant C > such that it holds for all f ∈ M that (3.2) k f ∗ χ [0 , k A ≤ k f k W L ( A,L ∞ ) ≤ C k f ∗ χ [0 , k A . Proof.
That the functional in question satisfies the axioms (P2), (P3) and (P4) as well as parts(a) and (b) of the axiom (P1) is an easy consequence of the respective properties of k·k A andthe properties of non-increasing rearrangement. Furthermore, the rearrangement invariance isobvious.As for (P5), fix some set E ⊆ [0 , ∞ ) of finite measure. We may, without loss of generality,assume that λ ( E ) >
1, because otherwise the proof is similar but simpler. Then, by Hardy–Littlewood inequality (Theorem 2.4), it holds for every f ∈ M that Z E f dλ ≤ Z λ ( E )0 f ∗ dλ = Z f ∗ dλ + Z λ ( E )1 f ∗ dλ ≤ Z f ∗ dλ + ( λ ( E ) − f ∗ (1) ≤ λ ( E ) Z f ∗ dλ ≤ λ ( E ) C [0 , k f ∗ χ [0 , k A , where C [0 , is the constant from the property (P5) of k·k A for the set [0 , k·k A (see Theorem 2.31 and Proposition 2.34) and the fact that [0 , ∞ ) is resonant to get for anarbitrary pair of functions f, g ∈ M that k ( f + g ) ∗ χ [0 , k A = sup k h k A ′ ≤ Z ∞ ( f + g ) ∗ χ [0 , h ∗ dλ = sup k h k A ′ ≤ sup ˜ h ∗ = h ∗ χ [0 , Z ∞ ( f + g )˜ h dλ ≤ sup k h k A ′ ≤ sup ˜ h ∗ = h ∗ χ [0 , Z ∞ f ˜ h dλ + sup k h k A ′ ≤ sup ˜ h ∗ = h ∗ χ [0 , Z ∞ g ˜ h dλ = sup k h k A ′ ≤ Z ∞ f ∗ χ [0 , h ∗ dλ + sup k h k A ′ ≤ Z ∞ g ∗ χ [0 , h ∗ dλ = k f ∗ χ [0 , k A + k g ∗ χ [0 , k A . Thus we have shown that the functional in question is an r.i. Banach function norm. Itremains to show (3.2).The first inequality in (3.2) is trivial. For the second estimate, it suffices to observe that k f ∗ χ (1 , ∞ ) k L ∞ = f ∗ (1) ≤ Z f ∗ dλ ≤ C [0 , k f ∗ χ [0 , k A , where C [0 , is the constant from the property (P5) of k·k A for the set [0 , (cid:3) While the local component is an r.i. Banach function norm, the global component is much lesswell behaved. Indeed, it is fairly easy to see that it cannot have the properties (P1) and (P5) (in(P1) only part (a) can possibly hold), because it cannot distinguish from zero any function thatis supported on a set of measure less than one. Thus it makes no sense to consider it separately.The following theorem shows that although Wiener–Luxemburg quasinorm needs not to be anorm, it satisfies all the remaining axioms of r.i. Banach function norms. Note that this resultis not redundant because in order to show that Wiener–Luxemburg amalgams are normable(see Corollary 3.6) we will use Theorem 2.18 and Theorem 2.30 and it is thus necessary to
IENER–LUXEMBURG AMALGAM SPACES 13 establish first that Wiener–Luxemburg quasinorms are quasi-Banach function norms that satisfythe axiom (P5).
Theorem 3.4.
The Wiener–Luxemburg quasinorms, as defined in Definition 3.1, are rearrangement-invariant quasi-Banach function norms and they also satisfy the axiom (P5) from the definitionof Banach function norms. Consequently, the corresponding Wiener–Luxemburg amalgam spacesare rearrangement-invariant quasi-Banach function spaces.Proof.
The properties (P2), (P3) and (P4) as well as those from parts (a) and (b) of the axiom(Q1) are easy consequences of the respective properties of k·k A and k·k B and the properties ofnon-increasing rearrangement. Furthermore, the rearrangement invariance is obvious.To show (P5), fix some set E ⊆ [0 , ∞ ) of finite measure. We may, without loss of generality,assume that λ ( E ) >
1, since otherwise the proof is similar but simpler. Then, by the Hardy–Littlewood inequality (Theorem 2.4), it holds for every f ∈ M + that Z E f dλ ≤ Z λ ( E )0 f ∗ dλ = Z f ∗ dλ + Z λ ( E )1 f ∗ dλ ≤ C [0 , k f ∗ χ [0 , k A + C (1 ,λ ( E )) k f ∗ χ (1 , ∞ ) k B , where C [0 , is the constant from the property (P5) of k·k A for the set [0 ,
1] and C (1 ,λ ( E )) is theconstant from the same property of k·k B for the set (1 , λ ( E )).Finally, for the triangle inequality up to a multiplicative constant (part (c) of the axiom(Q1)), consider the dilation operator D , as defined in Definition 2.22, and use at first only theappropriate properties of non-increasing rearrangement and those of k·k A and k·k B to calculate k f + g k W L ( A,B ) = k ( f + g ) ∗ χ [0 , k A + k ( f + g ) ∗ χ (1 , ∞ ) k B ≤ k ( D f ∗ + D g ∗ ) χ [0 , k A + k ( D f ∗ + D g ∗ ) χ (1 , ∞ ) k B ≤ k D f ∗ χ [0 , k A + k D g ∗ χ [0 , k A + k D f ∗ χ (1 , ∞ ) k B + k D g ∗ χ (1 , ∞ ) k B , which shows that it is sufficient to prove that there is some C ∈ (0 , ∞ ) such that k D f ∗ χ [0 , k A + k D f ∗ χ (1 , ∞ ) k B ≤ C k f k W L ( A,B ) for all f ∈ M + . Actually, it suffices to show(3.3) k D f ∗ χ (1 , ∞ ) k B ≤ C k f k W L ( A,B ) , because D is bounded on A (by Theorem 2.23), and thus k D f ∗ χ [0 , k A = k D ( f ∗ χ [0 , ] ) k A ≤ k D kk f ∗ χ [0 , ] k A ≤ k D kk f ∗ χ [0 , k A . To show (3.3), fix some f ∈ M + and calculate k D f ∗ χ (1 , ∞ ) k B = k D ( f ∗ χ ( , ∞ ) ) k B ≤ k D kk f ∗ χ ( , ∞ ) k B ≤ k D k ( k f ∗ χ (1 , ∞ ) k B + k f ∗ χ ( , k B ) ≤ k D k ( k f ∗ χ (1 , ∞ ) k B + f ∗ ( ) k χ ( , k B ) ≤ k D k ( k f ∗ χ (1 , ∞ ) k B + k χ ( , k B k χ (0 , ) k − A k f ∗ ( ) χ (0 , ) k A ≤ k D k ( k f ∗ χ (1 , ∞ ) k B + k χ ( , k B k χ (0 , ) k − A k f ∗ χ [0 , k A ) ≤ k D k max { , k χ ( , k B k χ (0 , ) k − A }k f k W L ( A,B ) . IENER–LUXEMBURG AMALGAM SPACES 14
This concludes the proof. (cid:3)
Associate spaces of Wiener–Luxemburg amalgams.
Let us now turn our attentionto the associate spaces of Wiener–Luxemburg amalgams. The natural intuition here is that anassociate space of an amalgam should be an amalgam of the respective associate spaces. Thisintuition turns out to be correct, as can be observed from the theorem presented below.Note that while the conclusion of this theorem is natural, its proof is in fact quite involved.The difficulty stems from the fact that for a general function f ∈ M the restriction of itsnon-increasing rearrangement f ∗ χ (1 , ∞ ) is not necessarily non-increasing and thus the quasinorm k f ∗ χ (1 , ∞ ) k W L ( A,B ) actually depends not only on k·k B but also on k·k A . This complicates thingsgreatly and an entirely new method had to be developed to resolve the ensuing problems. Theorem 3.5.
Let k·k A and k·k B be r.i. Banach function norms and let k·k A ′ and k·k B ′ betheir respective associate norms. Then there is a constant C > such that the associate norm k·k ( W L ( A,B )) ′ of k·k W L ( A,B ) satisfies (3.4) k f k ( W L ( A,B )) ′ ≤ k f k W L ( A ′ ,B ′ ) ≤ C k f k ( W L ( A,B )) ′ for every f ∈ M .Consequently, the corresponding associate space satisfies ( W L ( A, B )) ′ = W L ( A ′ , B ′ ) , up to equivalence of defining functionals.Proof. We begin by showing the first inequality in (3.4). To this end, fix some f ∈ M andarbitrary g ∈ M satisfying k g k W L ( A,B ) < ∞ . Then it follows from the H¨older inequality forassociate spaces (Theorem 2.28) that Z ∞ f ∗ g ∗ dλ = Z ∞ f ∗ χ [0 , g ∗ dλ + Z ∞ f ∗ χ (1 , ∞ ) g ∗ dλ ≤ k f ∗ χ [0 , k A ′ k g ∗ χ [0 , k A + k f ∗ χ (1 , ∞ ) k B ′ k g ∗ χ (1 , ∞ ) k B ≤ max {k f ∗ χ [0 , k A ′ , k f ∗ χ (1 , ∞ ) k B ′ } · k g k W L ( A,B ) ≤ k f k W L ( A ′ ,B ′ ) k g k W L ( A,B ) . The desired inequality now follows by dividing both sides of the inequality by k g k W L ( A,B ) , takingthe supremum over W L ( A, B ) and using Proposition 2.34.The second inequality in (3.4) is more involved. We obtain it indirectly, showing first that(
W L ( A, B )) ′ ⊆ W L ( A ′ , B ′ ) and then using Theorem 2.18.Suppose that f / ∈ W L ( A ′ , B ′ ). Then at least one of the following holds: f ∗ χ [0 , / ∈ A ′ or f ∗ χ (1 , ∞ ) / ∈ B ′ . We treat these two cases separately and show that either is sufficient for f / ∈ ( W L ( A, B )) ′ .If f ∗ χ [0 , / ∈ A ′ then we get by Theorem 2.33 that there is a non-negative function g ∈ A suchthat Z ∞ f ∗ χ [0 , g dλ = ∞ . Now, g ∗ χ [0 , ∈ W L ( A, B ) because k g ∗ χ [0 , k W L ( A,B ) = k g ∗ χ [0 , k A ≤ k g ∗ k A = k g k A < ∞ and we have by the Hardy–Littlewood inequality (Theorem 2.4) the following estimate: ∞ = Z ∞ f ∗ χ [0 , g dλ ≤ Z ∞ f ∗ g ∗ χ [0 , dλ. We have thus shown that f / ∈ ( W L ( A, B )) ′ . IENER–LUXEMBURG AMALGAM SPACES 15
Suppose now that f ∗ χ (1 , ∞ ) / ∈ B ′ . We may assume that f ∗ (1) < ∞ , because otherwise f ∗ χ [0 , = ∞ χ [0 , / ∈ A ′ and thus f / ∈ ( W L ( A, B )) ′ by the argument above. As in the previouscase, we get by Theorem 2.33 that there is some non-negative function g ∈ B such that Z ∞ f ∗ χ (1 , ∞ ) g dλ = ∞ . Now, it holds for all t ∈ (0 , ∞ ) that( f ∗ χ (1 , ∞ ) ) ∗ ( t ) = f ∗ ( t + 1) , which, when combined with the Hardy–Littlewood inequality (Theorem 2.4), yields ∞ = Z ∞ f ∗ χ (1 , ∞ ) g dλ ≤ Z ∞ f ∗ ( t + 1) g ∗ ( t ) dt = Z ∞ f ∗ ( t ) g ∗ ( t − dt. If we now put ˜ g ( t ) = ( t ∈ [0 , ,g ∗ ( t −
1) for t ∈ (1 , ∞ ) , we immediately see that ˜ g ∗ = g ∗ and thus we have found a function ˜ g ∈ B that is zero on [0 , , ∞ ) and that satisfies Z ∞ f ∗ χ (1 , ∞ ) ˜ g dλ = ∞ . Furthermore, we may estimate Z f ∗ ˜ g dλ ≤ f ∗ (1) Z ˜ g dλ ≤ f ∗ (1) C [1 , k ˜ g k B < ∞ , where C [1 , is the constant from property (P5) of k·k B . It follows that Z ∞ f ∗ ˜ g dλ = ∞ , because ∞ = Z ∞ f ∗ χ (1 , ∞ ) ˜ g dλ = Z f ∗ ˜ g dλ + Z ∞ f ∗ ˜ g dλ. Finally, put h = ˜ g (2) χ [0 , + min { ˜ g, ˜ g (2) } . Note that ˜ g (2) < ∞ as follows from ˜ g ∈ B andthat h is therefore a finite non-increasing function. Hence, we get that k h k W L ( A,B ) = ˜ g (2) k χ (0 , k A + k min { ˜ g, ˜ g (2) }k B ≤ ˜ g (2) k χ (0 , k A + k ˜ g k B < ∞ , while by the arguments above we have Z ∞ f ∗ h ∗ dλ ≥ Z ∞ f ∗ h ∗ dλ = Z ∞ f ∗ ˜ g dλ = ∞ . We therefore get that f / ∈ ( W L ( A, B )) ′ . This covers the last case and establishes the desiredinclusion ( W L ( A, B )) ′ ⊆ W L ( A ′ , B ′ ).Because we already know from Theorem 3.4 that W L ( A ′ , B ′ ) is a quasi-Banach function spaceand from Theorem 2.30 that ( W L ( A, B )) ′ is a Banach function space, we may use Theorem 2.18to obtain ( W L ( A, B )) ′ ֒ → W L ( A ′ , B ′ ), i.e. that there is a constant C > f ∈ M that k f k W L ( A ′ ,B ′ ) ≤ C k f k ( W L ( A,B )) ′ , which concludes the proof. (cid:3) As a corollary, we obtain normability of Wiener–Luxemburg amalgam spaces.
IENER–LUXEMBURG AMALGAM SPACES 16
Corollary 3.6.
Let k·k A and k·k B be r.i. Banach function norms. Then the Wiener–Luxemburgquasinorm k·k W L ( A,B ) is equivalent to an r.i. Banach function norm. Consequently, the Wiener–Luxemburg amalgam space W L ( A, B ) is an r.i. Banach function space.Proof. It follows from Theorem 3.5 that
W L ( A, B ) =
W L ( A ′′ , B ′′ ) = ( W L ( A ′ , B ′ )) ′ , where the space on the right-hand side is a Banach function space thanks to Theorem 2.30 andTheorem 3.4. (cid:3) Embeddings.
We now examine the embeddings of Wiener–Luxemburg amalgams. Firstwe characterise the embeddings between two Wiener–Luxemburg amalgams, then between aWiener–Luxemburg amalgam of two spaces and the sum or intersection of these spaces andfinally we examine the case when the local or the global component is L or L ∞ .The first theorem provides the characterisation of embeddings among Wiener–Luxemburgamalgams. Theorem 3.7.
Let k·k A , k·k B , k·k C and k·k D be r.i. Banach function norms. Then the followingassertions are true: (1) The embedding
W L ( A, C ) ֒ → W L ( B, C ) holds if and only if the local component of k·k A is stronger that that of k·k B , in the sense that for every f ∈ M the implication k f ∗ χ [0 , k A < ∞ ⇒ k f ∗ χ [0 , k B < ∞ holds. (2) The embedding
W L ( A, B ) ֒ → W L ( A, C ) holds if and only if the global component of k·k B is stronger that that of k·k C , in the sense that for every f ∈ M such that f ∗ (1) < ∞ theimplication k f ∗ χ (1 , ∞ ) k B < ∞ ⇒ k f ∗ χ (1 , ∞ ) k C < ∞ holds. (3) The embedding
W L ( A, B ) ֒ → W L ( C, D ) holds if and only if the local component of k·k A is stronger than that of k·k C and the global component of k·k B is stronger than that of k·k D .Proof. In the first two cases the sufficiency follows directly from Theorem 2.18 and Definition 3.1,only in the second case one has to realise that all f ∈ W L ( A, B ) satisfy f ∗ (1) < ∞ . The thirdcase then follows, because we can use what we already proved to get W L ( A, B ) ֒ → W L ( A, D ) ֒ → W L ( C, D ) . The necessity in the case (1) can be shown in a following way. Fix some f ∈ M such that k f ∗ χ [0 , k A < ∞ but k f ∗ χ [0 , k B = ∞ . Then f = f ∗ χ [0 , belongs to W L ( A, C ), since k f ∗ χ [0 , k A = k f ∗ χ [0 , k A < ∞ , k f ∗ χ (1 , ∞ ) k C = k k C = 0 , but not to W L ( B, C ), since k f ∗ χ [0 , k B = k f ∗ χ [0 , k B = ∞ . As for the case (2), fix some f ∈ M such that f ∗ (1) < ∞ and k f ∗ χ (1 , ∞ ) k B < ∞ while k f ∗ χ (1 , ∞ ) k C = ∞ . Then f = f ∗ (1) χ [0 , + f ∗ χ (1 , ∞ ) belongs to W L ( A, B ), since k f ∗ χ [0 , k A = k f ∗ (1) χ [0 , k A = f ∗ (1) k χ [0 , k A < ∞ , k f ∗ χ (1 , ∞ ) k B = k f ∗ χ (1 , ∞ ) k B < ∞ , but not to W L ( A, C ), since k f ∗ χ (1 , ∞ ) k C = k f ∗ χ (1 , ∞ ) k C = ∞ . IENER–LUXEMBURG AMALGAM SPACES 17
Finally for the case (3) one needs to combine the steps presented above. To be precise, if thecondition on local components is violated one finds the counterexample as in the case (1) whilein the case when the condition on the global components gets violated one finds it as in thecase (2). (cid:3)
To provide an example we turn to the classical Lebesgue spaces. It is well known that Lebesguespaces over [0 , ∞ ) are not ordered, but it is easy to show that their local and global componentare, as is formalised in the following remark. Remark 3.8.
Let p, q ∈ (0 , ∞ ]. Then it holds that(1) the local component of k·k L p is stronger than that of k·k L q if and only if p ≥ q ,(2) the global component of k·k L p is stronger than that of k·k L q if and only if p ≤ q .As a second example we present a similar statement about Lorentz spaces. The proof is easyand uses only standard techniques. Remark 3.9.
Let p , p , q , q ∈ (0 , ∞ ] such that p = p and let k·k L p ,q and k·k L p ,q be thecorresponding Lorentz functionals. Then it holds that(1) the local component of k·k L p ,q is stronger than that of k·k L p ,q if and only if p > p ,(2) the global component of k·k L p ,q is stronger than that of k·k L p ,q if and only if p < p .A third example might be found among Orlicz spaces. The following remark contains sufficientconditions for the ordering of their respective local and global components. It is again easy toprove, using only the already well-known methods which have been originally developed forcharacterising the embeddings between Orlicz spaces and which can be found for example in[23, Theorem 4.17.1]. We also refer the reader to [23, Chapter 4] for an extensive treatment ofOrlicz spaces. Remark 3.10.
Let Φ and Φ be two Young functions and let k·k Φ and k·k Φ be the corre-sponding Orlicz norms. Then(1) if there are some constants c, T ∈ (0 , ∞ ) such thatΦ ( t ) ≤ Φ ( ct )for all t ∈ [ T, ∞ ) then the local component of k·k Φ is stronger than that of k·k Φ ,(2) if there are some constants c, T ∈ (0 , ∞ ) such thatΦ ( t ) ≤ Φ ( ct )for all t ∈ [0 , T ] then the global component of k·k Φ is stronger than that of k·k Φ .We now put W L ( A, B ) in relation with the sum and intersection of A and B . We first showthat W L ( A, B ) is always sandwiched between them.
Theorem 3.11.
Let k·k A and k·k B be r.i. Banach function norms. Then A ∩ B ֒ → W L ( A, B ) ֒ → A + B. Proof.
Fix some f ∈ M . Then k f k W L ( A,B ) = k f ∗ χ [0 , k A + k f ∗ χ (1 , ∞ ) k B ≤ k f k A + k f k B ≤ k f k A ∩ B which establishes the first embedding.As for the second embedding, note that we may consider f to be non-negative, since it is easyto show that it holds for every f ∈ M that k f k A + B = k | f | k A + B .Consider now functions g and h defined by g = max { f − f ∗ (1) , } ,h = min { f, f ∗ (1) } . IENER–LUXEMBURG AMALGAM SPACES 18
Then f = g + h and thus k f k A + B ≤ k g k A + k h k B = k g ∗ k A + k h ∗ k B thanks to rearrangement invariance of both k·k A and k·k B . Furthermore, thanks to f beeingnon-negative, it is an exercise to verify that g ∗ = ( f ∗ − f ∗ (1)) χ [0 , ,h ∗ = f ∗ (1) χ [0 , + f ∗ χ (1 , ∞ ) , and therefore k f k A + B ≤ k f ∗ χ [0 , k A + k f ∗ (1) χ [0 , k B + k f ∗ χ (1 , ∞ ) k B ≤ k f k W L ( A,B ) + k χ [0 , k B Z f ∗ dλ ≤ (1 + C [0 , k χ [0 , k B ) k f k W L ( A,B ) , where C [0 , is the constant from the property (P5) of k·k A for the set [0 , (cid:3) Moreover, in the case when we have proper relations between the respective components of A and B we can describe their sum and intersection in terms of Wiener–Luxemburg amalgams,at least in the set theoretical sense. Corollary 3.12.
Let k·k A and k·k B be r.i. Banach function norms. Suppose that the localcomponent of k·k A is stronger than that of k·k B while the global component of k·k B is strongerthan that of k·k A . Then A ∩ B = W L ( A, B ) up to equivalence of quasinorms, while (3.5) A + B = W L ( B, A ) as sets.Proof. Thanks to Proposition 2.20, Theorem 2.18 and Theorem 3.11 it suffices to prove that
W L ( A, B ) ⊆ A ∩ B and A + B ⊆ W L ( B, A ). But this is provided by Theorem 3.7 andRemark 3.2, which, thanks to our assumptions, yield
W L ( A, B ) ֒ → A,W L ( A, B ) ֒ → B,A ֒ → W L ( B, A ) ,B ֒ → W L ( B, A ) . When combined with the fact that
W L ( B, A ) is a linear set, this is sufficient for the inclusions. (cid:3)
The reason for the equality (3.5) holding only in the set theoretical sense is of course thefact that A + B is not necessarily a Banach function space. This also motivates the followingobservation. Corollary 3.13.
Let k·k A and k·k B be r.i. Banach function norms. Suppose that the localcomponent of k·k A is stronger than that of k·k B while the global component of k·k B is strongerthan that of k·k A . Then there is an r.i. Banach function norm k·k X such that there is a constant C > satisfying k f k A + B ≤ C k f k X for all f ∈ A + B and such that the corresponding r.i. Banach function space X satisfies X = A + B IENER–LUXEMBURG AMALGAM SPACES 19 as a set.
In the next theorem, we show that the classical Lebesgue space L has the weakest localcomponent, as well as the strongest global component, among all r.i. Banach function spaces,while L ∞ has, in the same context, the strongest local component as well as the weakest globalcomponent. Theorem 3.14.
Let k·k A and k·k B be r.i. Banach function norms and let A and B be thecorresponding r.i. Banach function spaces. Then (1) W L ( L ∞ , B ) ֒ → W L ( A, B ) , (2) W L ( A, L ) ֒ → W L ( A, B ) , (3) W L ( A, B ) ֒ → W L ( L , B ) , (4) W L ( A, B ) ֒ → W L ( A, L ∞ ) .Proof. Fix f ∈ M . The first embedding follows from the estimate k f ∗ χ [0 , k A ≤ k f ∗ χ [0 , k L ∞ k χ [0 , k A and part (1) of Theorem 3.7.The third embedding also uses part (1) of Theorem 3.7 but this time paired with the estimate k f ∗ χ [0 , k L = Z f ∗ χ [0 , dλ ≤ C [0 , k f ∗ χ [0 , k A , where C [0 , is the constant from property (P5) of k·k A .The fourth embedding is a trivial consequence of Proposition 3.3, specifically of (3.2).The second embedding is most involved. Denote by k·k B ′ the associate norm of k·k B and by B ′ the associate space of B . Then we know from part (4), which has already been proved, andRemark 3.2 that B ′ ֒ → W L ( B ′ , L ∞ ). Thus it follows from Theorem 3.5 and Proposition 2.29that W L ( B, L ) = ( W L ( B ′ , L ∞ )) ′ ֒ → B ′′ = B. Hence, it follows from part (2) of Theorem 3.7 that the global component of k·k L is strongerthan that of k·k B which, by the same theorem, implies the desired embedding. (cid:3) As a corollary, we obtain the following well-known classical result, for which we thus providean alternative proof.
Corollary 3.15.
Let A be an r.i. Banach function space. Then L ∩ L ∞ ֒ → A ֒ → L + L ∞ . Proof.
The assertion is a direct consequence of Remark 3.8, Corollary 3.12, Theorem 3.14,Theorem 2.18 and the fact that L + L ∞ is an r.i. Banach function space. (cid:3) The final result of this section is a more precise version of Proposition 2.29.
Proposition 3.16.
Let k·k A and k·k B be r.i. Banach function norms and denote by k·k A ′ and k·k B ′ the respective associate norms. Suppose that the local component of k·k A is stronger thanthat of k·k B . Then the local component of k·k B ′ is stronger than that of k·k A ′ .Similarly, if the global component of k·k A is stronger than that of k·k B , then the global com-ponent of k·k B ′ is stronger than that of k·k A ′ Proof.
By our assumption and part (1) of Theorem 3.7 we get that
W L ( A, L ∞ ) ֒ → W L ( B, L ∞ ) . Consequently, it follows from Theorem 3.5 and Proposition 2.29 that
W L ( B ′ , L ) = ( W L ( B, L ∞ )) ′ ֒ → ( W L ( A, L ∞ )) ′ = W L ( A ′ , L ) , IENER–LUXEMBURG AMALGAM SPACES 20 that is, the local component of k·k B ′ is stronger than that of k·k A ′ .The second claim can be proved in similar manner, only using W L ( L , A ) and W L ( L , B )instead of W L ( A, L ∞ ) and W L ( B, L ∞ ). (cid:3) Integrable associate spaces
In this section we introduce a generalisation of the concept of associate spaces (see Defini-tion 2.27) the need of which arose naturally during the study of associate spaces of Wiener–Luxemburg amalgams of r.i. quasi-Banach function spaces. We then study some of its properties,mainly those directly needed for our purposes.Unlike in the other parts of the paper, we will now work in a more abstract setting and assumeonly that (
R, µ ) is a resonant measure space.Our terminology and notation in this section is inspired by the down associate norm which isderived from the concept of down norms. To those interested in this topic we suggest the papers[10] and [24].4.1.
Basic properties.
Our definition of integrable associate spaces is rather indirect. Weproceed by first introducing a certain subspace of an arbitrary r.i. quasi-Banach function spaceand then defining the integrable associate space as an associate space of this subspace.
Definition 4.1.
Let k·k X be an r.i. quasi-Banach function norm and let X be the correspondingr.i. quasi-Banach function space. Then the functional k·k X i , defined for f ∈ M by k f k X i = max {k f k X , k f k W L ( L ,L ∞ ) } , will be called the integrable norm of k·k X , while the space X i = { f ∈ M ; k f k X i < ∞} will be called integrable subspace of X . Theorem 4.2.
Let k·k X be an r.i. quasi-Banach function norm and let X be the correspondingr.i. quasi-Banach function space. Denote by C the modulus of concavity of k·k X . Then thefunctional k·k X i is an r.i. quasi-Banach function norm that has the property (P5) . Furthermore,if C = 1 , then k·k X i is an r.i. Banach function norm.Moreover, the space X i satisfies (4.1) X i = (cid:26) f ∈ X ; Z f ∗ dλ < ∞ (cid:27) and thus X i ֒ → X . On the other hand, if k·k X has the property (P5) then X i = X up toequivalence of quasinorms.Proof. Given that we know from Proposition 3.3 that k·k
W L ( L ,L ∞ ) is an r.i. Banach functionnorm, it is an easy exercise to show that the first part of the theorem holds, i.e. that k·k X i hasthe asserted properties. The characterisation of X i also follows from Proposition 3.3 while theembedding X i ֒ → X can then be obtained via Theorem 2.18 (or directly from the definition of k·k X i ).The last part is slightly more involved. Note that we cannot simply use the Luxemburgrepresentation theorem to show that every f ∈ X belongs to X i , since k·k X needs not to be anorm. We thus proceed as follows.We will assume that µ ( R ) ≥
1; the remaining case is easier. If k·k X has the property (P5),then its associate norm k·k X ′ is an r.i. Banach function norm and we can use it, as well as ourassumption that the underlying measure space is resonant, to obtain an estimate on Z f ∗ dλ. IENER–LUXEMBURG AMALGAM SPACES 21
To this end, we use the σ -finiteness of ( R, µ ) to find some a ∈ [1 , ∞ ) such that there is a set E with µ ( E ) = a . Then it follows from the H¨older inequality for associate spaces (Theorem 2.28)that(4.2) Z f ∗ dλ ≤ Z a f ∗ dλ = sup E ⊆ Rµ ( E )= a Z E | f | dµ ≤ sup E ⊆ Rµ ( E )= a k χ E k X ′ k f k X = C a k f k X where C a is the norm k χ E k X ′ for any set E ⊆ R such that µ ( E ) = a . Hence, it follows from (4.1)that the sets X and X i coincide. The equivalence of quasinorms then follows from Theorem 2.18(or directly from (4.2)). (cid:3) Definition 4.3.
Let k·k X be an r.i. quasi-Banach function norm, let X be the correspondingr.i. quasi-Banach function space and let k·k X i and X i , respectively, be the corresponding inte-grable norm and integrable subspace. Then the associate norm k·k X ′ i of k·k X i will also be calledthe integrable associate norm of k·k X , while the associate space X ′ i of X i will also be called theintegrable associate space of X .The next two results describe the properties of the integrable associate spaces and theirrelation to the associate spaces. Corollary 4.4.
Let k·k X be an r.i. quasi-Banach function norm, let X be the correspondingr.i. quasi-Banach function space and let k·k X ′ i and X ′ i , respectively, be the corresponding inte-grable associate norm and integrable associate space. Then k·k X ′ i is an r.i. Banach functionnorm and X ′ i is an r.i. Banach function space.Proof. This result follows from Theorem 4.2 and Theorem 2.30. (cid:3)
Corollary 4.5.
Let k·k X be an r.i. quasi-Banach function norm and let X be the correspondingr.i. quasi-Banach function space. If k·k X has the property (P5) then X ′ i = X ′ up to equivalenceof norms.Proof. This result follows directly from Theorem 4.2. (cid:3)
Next we obtain an analogue to Proposition 2.29.
Corollary 4.6.
Let k·k X and k·k Y be a pair of r.i. quasi-Banach function norms and let X and Y , respectively, be the corresponding r.i. quasi-Banach function spaces. Suppose that X ֒ → Y .Then the respective integrable associate spaces X ′ i and Y ′ i satisfy X ′ i ← ֓ Y ′ i .Proof. Thanks to our assumptions the respective integrable subspaces X i and Y i of X and Y satisfy X i ֒ → Y i and thus the result follows from Proposition 2.29. (cid:3) Finally, we now formulate the appropriate versions of H¨older inequality and Landau’s res-onance theorem. Note that while the latter can be formulated only in terms of the originalquasinorm, the H¨older inequality requires us to work with the integrable norm which is ratherunfortunate.
Corollary 4.7.
Let k·k X be an r.i. quasi-Banach function norm and let k·k X i and k·k X ′ i , re-spectively, be the corresponding integrable norm and integrable associate norm. Then it holdsfor every pair of functions f, g ∈ M that Z R | f g | dµ ≤ k f k X ′ i k g k X i . Corollary 4.8.
Let k·k X be an r.i. quasi-Banach function norm, let X be the correspondingr.i. quasi-Banach function space and let k·k X ′ i and X ′ i , respectively, be the corresponding inte-grable associate norm and integrable associate space. Then arbitrary function f ∈ M belongs to IENER–LUXEMBURG AMALGAM SPACES 22 X ′ i if and only if it satisfies (4.3) Z R | f g | dµ < ∞ for all g ∈ X such that (4.4) Z g ∗ dλ < ∞ . Proof.
This result is a consequence of Theorem 2.33 and (4.1). (cid:3)
The second integrable associate space.
We now answer the natural question how doesthe second integrable associate space relate to the original r.i. quasi-Banach function space. Itturns out that the answer is much more interesting than in the case of associate spaces.
Definition 4.9.
Let k·k X be an r.i. quasi-Banach function norm, let X be the correspondingr.i. quasi-Banach function space and let k·k X ′ i and X ′ i , respectively, be the corresponding in-tegrable associate norm and integrable associate space. We then define the second integrableassociate norm k·k X ′′ i as the integrable associate norm of k·k X ′ i and the second integrable asso-ciate space X ′′ i as the integrable associate space of X ′ i .It follows directly from Corollary 4.4 and Corollary 4.5 that the second integrable associatenorm is equivalent to the associate norm of k·k X ′ i . An analogous claim holds for the secondintegrable associate space. Theorem 4.10.
Let k·k X be an r.i. quasi-Banach function norm and let X be the correspondingr.i. quasi-Banach function space. Then: (1) if k·k X is a Banach function norm, then X = X ′′ i with equivalent norms; (2) if k·k X has the property (P5) but it is not equivalent to a Banach function norm, then X ֒ → X ′′ i and X ′′ i ֒ → X ; (3) if k·k X has the property (P1) but not the property (P5) , then X ′′ i ֒ → X and X ֒ → X ′′ i ; (4) if k·k X has neither the property (P1) nor the property (P5) , then the spaces X and X ′′ i cannot, in general, be compared.Proof. The assertion (1) follows immediately from Corollary 4.5 and Theorem 2.31.The positive part of assertion (2) follows from Corollary 4.5 and Proposition 2.32 while thenegative part follows from the fact that X ′′ i is a Banach function space by Corollary 4.4.In the case (3) we know from Theorem 4.2 that the integrable subspace X i is a Banachfunction space, hence it follows from Corollary 4.5, and Theorem 2.31 that X ′′ i = ( X i ) ′′ = X i ֒ → X. On the other hand, since X ′′ i is a Banach function space by Corollary 4.4, we get from Theo-rem 2.19 that there is a function in X that does not belong to X ′′ i .Finally, to observe that there needs not to be any embedding in the case (4) one only has toconsider the Lebesgue space L p with p <
1. Indeed, in this case the space ( L p ) ′′ i is a Banachfunction space and therefore we have from Theorem 3.14 that W L ( L ∞ , L ) ֒ → ( L p ) ′′ i ֒ → W L ( L , L ∞ ) , while there are some functions in W L ( L ∞ , L ) that do not belong to L p and at the same timethere are some functions in L p that do not belong to W L ( L , L ∞ ). (cid:3) IENER–LUXEMBURG AMALGAM SPACES 23 Wiener–Luxemburg amalgams of quasi-Banach function spaces
In this section we extend the theory of Wiener–Luxemburg amalgams to the context ofr.i. quasi-Banach function spaces. This is possible thanks to the recent advances of the gen-eral theory of quasi-Banach function spaces developed in [21]. We focus on those areas wherethe generalisation leads to new results and insights.Throughout this section we restrict ourselves to the case when (
R, µ ) = ([0 , ∞ ) , λ ), whichensures that the underlying measure space is resonant. This restriction is necessary becausewe need to work with the non-increasing rearrangement and, in contrast to the situation inSection 3, the representation theory is not available for r.i. quasi Banach function spaces.5.1. Wiener–Luxemburg quasinorms for quasi-Banach function spaces.
The definitionand the basic properties of Wiener–Luxemburg quasinorms are subject only to the most naturaland expected changes.
Definition 5.1.
Let k·k A and k·k B be r.i. quasi-Banach function norms. We then define theWiener–Luxemburg quasinorm k·k W L ( A,B ) , for f ∈ M , by(5.1) k f k W L ( A,B ) = k f ∗ χ [0 , k A + k f ∗ χ (1 , ∞ ) k B and the corresponding Wiener–Luxemburg amalgam space W L ( A, B ) as
W L ( A, B ) = { f ∈ M ; k f k W L ( A,B ) < ∞} . Furthermore, we will call the first summand in (5.1) the local component of k·k
W L ( A,B ) whilethe second summand will be called the global component of k·k W L ( A,B ) . Remark 5.2.
Let k·k A be an r.i. quasi-Banach function norm and denote by C its modulus ofconcavity. Then k f k A ≤ k f k W L ( A,A ) ≤ C k f k A for every f ∈ M .Consequently, it makes a good sense to talk about local and global components of arbitraryr.i. quasi-Banach function norms. Theorem 5.3.
Let k·k A and k·k B be r.i. quasi-Banach function norms. Then the Wiener–Luxemburg quasinorm k·k W L ( A,B ) is an r.i. quasi-Banach function norm. Consequently, the cor-responding Wiener–Luxemburg amalgam W L ( A, B ) is a rearrangement-invariant quasi-Banachfunction space.Moreover, k·k W L ( A,B ) has the property (P5) if and only if k·k A does.Proof. The proof of the first part, that is that k·k
W L ( A,B ) has all the properties of an r.i. quasi-Banach function norm, is analogous to that of Theorem 3.4.The proof that if k·k A has the property (P5) then the same is true for k·k W L ( A,B ) is analogousto the appropriate part of the proof of Proposition 3.3.Consider now the case when k·k A does not have the property (P5) and fix some E ⊆ [0 , ∞ )that serves as an appropriate counterexample. It follows from Theorem 2.19 that there is anon-negative function f E ∈ A such that ∞ = Z E f E dλ ≤ Z λ ( E )0 f ∗ E dλ, where the last estimate follows from the Hardy–Littlewood inequality (Theorem 2.4). Thefunction f = f ∗ E χ [0 , then satisfies k f k W L ( A,B ) < ∞ while Z f dλ = ∞ . (cid:3) IENER–LUXEMBURG AMALGAM SPACES 24
Integrable associate spaces of Wiener–Luxemburg amalgams.
An interesting ques-tion is how to describe the associate space of a Wiener–Luxemburg amalgam
W L ( A, B ) in thecase when A has the property (P5) but B does not. It follows from Theorems 2.30 and 5.3that it will be a Banach function space, but it cannot be described as W L ( A ′ , B ′ ) since in thiscase B ′ = { } . Trying to answer this question is what motivates the introduction of integrableassociate spaces in the previous section.The answer to this question will follow as a corollary to the following general theorem, inwhich we describe the integrable associate spaces of Wiener–Luxemburg amalgams. Theorem 5.4.
Let k·k A and k·k B be r.i. quasi-Banach function norms and let k·k A ′ i and k·k B ′ i be their respective integrable associate norms. Then there is a constant C > such that theintegrable associate norm k·k ( W L ( A,B )) ′ i of k·k W L ( A,B ) satisfies (5.2) k f k ( W L ( A,B )) ′ i ≤ k f k W L ( A ′ i ,B ′ i ) ≤ C k f k ( W L ( A,B )) ′ i for every f ∈ M .Consequently, the corresponding integrable associate space satisfies ( W L ( A, B )) ′ i = W L ( A ′ i , B ′ i ) , up to equivalence of defining functionals.Proof. We begin by showing the first inequality in (5.2). To this end, fix some f ∈ M andarbitrary g ∈ M satisfying k g k W L ( A,B ) i < ∞ . It then follows from Corollary 4.7 that Z ∞ f ∗ g ∗ dλ = Z ∞ f ∗ χ [0 , g ∗ dλ + Z ∞ f ∗ χ (1 , ∞ ) g ∗ dλ ≤ k f ∗ χ [0 , k A ′ i k g ∗ χ [0 , k A i + k f ∗ χ (1 , ∞ ) k B ′ i k g ∗ χ (1 , ∞ ) k B i ≤ k f k W L ( A ′ i ,B ′ i ) · max {k g ∗ χ [0 , k A i , k g ∗ χ (1 , ∞ ) k B i }≤ k f k W L ( A ′ i ,B ′ i ) k g k W L ( A,B ) i . The desired inequality now follows by dividing both sides by k g k W L ( A,B ) i , taking the appropriatesupremum and using Proposition 2.34.The second inequality in (5.2) is more involved. We obtain it indirectly, showing first that( W L ( A, B )) ′ ⊆ W L ( A ′ , B ′ ) and then using Theorem 2.18.Suppose that f / ∈ W L ( A ′ i , B ′ i ). Then f ∗ χ [0 , / ∈ A ′ i or f ∗ χ (1 , ∞ ) / ∈ B ′ i . We treat these two casesseparately.If f ∗ χ [0 , / ∈ A ′ i then we get by Corollary 4.8 that there is a non-negative function g ∈ A suchthat Z g ∗ dλ < ∞ , Z ∞ f ∗ χ [0 , g dλ = ∞ . Now, g ∗ χ [0 , ∈ W L ( A, B ) because k g ∗ χ [0 , k W L ( A,B ) = k g ∗ χ [0 , k A ≤ k g ∗ k A = k g k A < ∞ . Moreover, it obviously holds that Z ( g ∗ χ [0 , ) ∗ dλ = Z g ∗ dλ < ∞ and we get, by the Hardy–Littlewood inequality (Theorem 2.4), the following estimate: ∞ = Z ∞ f ∗ χ [0 , g dλ ≤ Z ∞ f ∗ g ∗ χ [0 , dλ. IENER–LUXEMBURG AMALGAM SPACES 25
It thus follows from Corollary 4.8 that f / ∈ ( W L ( A, B )) ′ i .Suppose now that f ∗ χ (1 , ∞ ) / ∈ B ′ i . We may assume that f ∗ (1) < ∞ , because otherwise f ∗ χ [0 , = ∞ χ [0 , / ∈ A ′ i (see [20, Lemma 2.4] or [21, Theorem 3.4]) and thus f / ∈ ( W L ( A, B )) ′ i by the argument above. As in the previous case, we get by Corollary 4.8 that there is somenon-negative function g ∈ B such that Z g ∗ dλ < ∞ , (5.3) Z ∞ f ∗ χ (1 , ∞ ) g dλ = ∞ . Now, it holds for all t ∈ (0 , ∞ ) that( f ∗ χ (1 , ∞ ) ) ∗ ( t ) = f ∗ ( t + 1) , which, when combined with the Hardy–Littlewood inequality (Theorem 2.4), yields ∞ = Z ∞ f ∗ χ (1 , ∞ ) g dλ ≤ Z ∞ f ∗ ( t + 1) g ∗ ( t ) dt = Z ∞ f ∗ ( t ) g ∗ ( t − dt. If we now put ˜ g ( t ) = ( t ∈ [0 , ,g ∗ ( t −
1) for t ∈ (1 , ∞ ) , we immediately see that ˜ g ∗ = g ∗ and thus we have found a function ˜ g ∈ B that is zero on [0 , , ∞ ) and that satisfies Z ∞ f ∗ χ (1 , ∞ ) ˜ g dλ = ∞ . Furthermore, we may estimate by (5.3) Z f ∗ ˜ g dλ ≤ f ∗ (1) Z ˜ g dλ = f ∗ (1) Z g ∗ dλ < ∞ . It follows that Z ∞ f ∗ ˜ g dλ = ∞ , because ∞ = Z ∞ f ∗ χ (1 , ∞ ) ˜ g dλ = Z f ∗ ˜ g dλ + Z ∞ f ∗ ˜ g dλ. Finally, put h = ˜ g (2) χ [0 , +min { ˜ g, ˜ g (2) } . Note that ˜ g (2) < ∞ as follows from ˜ g ∈ B (see again[20, Lemma 2.4] or [21, Theorem 3.4]) and that h is therefore a finite non-increasing function.Hence, we get that k h k W L ( A,B ) = ˜ g (2) k χ (0 , k A + k min { ˜ g, ˜ g (2) }k B ≤ ˜ g (2) k χ (0 , k A + k ˜ g k B < ∞ , Z h ∗ dλ = ˜ g (2) < ∞ , while by the arguments above we have Z ∞ f ∗ h ∗ dλ ≥ Z ∞ f ∗ h ∗ dλ = Z ∞ f ∗ ˜ g dλ = ∞ . We therefore get from Corollary 4.8 that f / ∈ ( W L ( A, B )) ′ i . This covers the last case andestablishes the desired inclusion ( W L ( A, B )) ′ i ⊆ W L ( A ′ i , B ′ i ).Because we already know from Theorem 5.3 that W L ( A ′ i , B ′ i ) is a quasi-Banach function spaceand from Theorem 4.4 that ( W L ( A, B )) ′ i is a Banach function space, we may use Theorem 2.18 IENER–LUXEMBURG AMALGAM SPACES 26 to obtain (
W L ( A, B )) ′ i ֒ → W L ( A ′ i , B ′ i ), i.e. that there is a constant C > f ∈ M that k f k W L ( A ′ i ,B ′ i ) ≤ C k f k ( W L ( A,B )) ′ i , which concludes the proof. (cid:3) Corollary 5.5.
Let k·k A be an r.i. quasi-Banach function norm that has the property (P5) , let k·k B be an r.i. quasi-Banach function norm, let k·k A ′ be the associate norm of k·k A and let k·k B ′ i be the integrable associate norm of k·k B . Then there is a constant C > such that the associatenorm k·k ( W L ( A,B )) ′ of k·k W L ( A,B ) satisfies k f k ( W L ( A,B )) ′ ≤ k f k W L ( A ′ ,B ′ i ) ≤ C k f k ( W L ( A,B )) ′ for every f ∈ M .Consequently, the corresponding associate space satisfies ( W L ( A, B )) ′ = W L ( A ′ , B ′ i ) , up to equivalence of defining functionals.Proof. The result follows by combining Theorems 5.3 and 5.4 and Corollary 4.5. (cid:3)
Embeddings.
The characterisation of embeddings remains the same and this is also truefor its proof (which we thus omit). We state it here only because we will use it later.
Theorem 5.6.
Let k·k A , k·k B , k·k C and k·k D be r.i. quasi-Banach function norms. Then thefollowing assertions are true: (1) The embedding
W L ( A, C ) ֒ → W L ( B, C ) holds if and only if the local component of k·k A is stronger than that of k·k B , in the sense that for every f ∈ M the implication k f ∗ χ [0 , k A < ∞ ⇒ k f ∗ χ [0 , k B < ∞ holds. (2) The embedding
W L ( A, B ) ֒ → W L ( A, C ) holds if and only if the global component of k·k B is stronger than that of k·k C , in the sense that for every f ∈ M such that f ∗ (1) < ∞ theimplication k f ∗ χ (1 , ∞ ) k B < ∞ ⇒ k f ∗ χ (1 , ∞ ) k C < ∞ holds. (3) The embedding
W L ( A, B ) ֒ → W L ( C, D ) holds if and only if the local component of k·k A is stronger than that of k·k C and the global component of k·k B is stronger than that of k·k D . The following theorem generalises Theorem 3.14 and also provides better insight into therelationships between the individual embeddings and the specific properties a quasi-Banachfunction norm can possess. We formulate it in a simpler way than Theorem 3.14, but this comesat no loss of generality thanks to Theorem 5.6.
Theorem 5.7.
Let k·k A be an r.i. quasi-Banach function norm. Then (1) W L ( L ∞ , A ) ֒ → A , (2) if k·k A has the property (P1) then W L ( A, L ) ֒ → A , (3) A ֒ → W L ( L , A ) if and only if k·k A has the property (P5) , (4) A ֒ → W L ( A, L ∞ ) .Proof. The first and last embeddings are proved in the same way as in Theorem 3.14.The sufficiency in part (3) follows exactly as in the part (3) of Theorem 3.14. For the necessity,assume that
A ֒ → W L ( L , A ). It then holds for any E ⊆ [0 , ∞ ) of finite measure and any f ∈ A that Z E | f | dλ ≤ Z λ ( E )0 f ∗ dλ ≤ max { , λ ( E ) } Z f ∗ dλ < ∞ , IENER–LUXEMBURG AMALGAM SPACES 27 where the first estimate is due to the Hardy–Littlewood inequality (Theorem 2.4), the secondestimate is due to f ∗ being non-increasing, and the last estimate is due to part (1) of Theorem 5.6.We now obtain from Theorem 2.19 that k·k A must have the property (P5).As for the part (2), we know from Theorem 4.2 that if k·k A has the property (P1) then itsintegrable subspace A i is an r.i. Banach function space. Hence, we obtain from parts (1) and(2) of Theorem 3.14 that W L ( L ∞ , L ) ֒ → A i ֒ → A. The desired conclusion now follows by combining parts (3) and (2) of Theorem 5.6. (cid:3)
Remark 5.8.
Unlike in part (3) of the preceding theorem, there is no equivalence in part (2).This can be observed by considering A = L p,q , where L p,q is a Lorenz space, because if we choose p ∈ (1 , ∞ ), q ∈ (0 ,
1) then L p,q satisfies the embedding (see Remark 3.9) but is not normable(see [9, Theorem 2.5.8] and the references therein).An alternative sufficient condition for the embedding W L ( A, L ) ֒ → A is provided in thefollowing theorem. The relevant term is defined in Definition 2.24 and put into context inLemma 2.25 and Remark 2.26. Theorem 5.9.
Let k·k A be an r.i. quasi-Banach function norm and assume that the Hardy–Littlewood–P´olya principle holds for k·k A . Then the global component of k·k L is stronger thanthat of k·k A .Proof. Fix some f ∈ M such that f ∗ (1) < ∞ and k f ∗ χ (1 , ∞ ) k L < ∞ . We want to show that k f ∗ χ (1 , ∞ ) k A < ∞ . To this end, consider the non-increasing function f = f ∗ (1) χ [0 , + f ∗ χ (1 , ∞ ) which belongs to L and the norm of which satisfies f ∗ (1) ≤ k f k L < ∞ . Hence, we may define a function h = k f k L χ [0 , and observe that h ∈ A , h is non-increasing,and Z t f ∗ dλ ≤ Z t h ∗ dλ for all t ∈ (0 , ∞ ). It thus follows from our assumption on k·k A that f ∈ A and consequently k f ∗ χ (1 , ∞ ) k A < ∞ , as desired. (cid:3) As a corollary to this theorem we obtain a negative answer to the previously open questionwhether the Hardy–Littlewood–P´olya principle holds for every r.i. quasi-Banach function norm.This also serves as an example of application of Wiener–Luxemburg amalgams, since they appearonly in the proof of the corollary, not in its statement.
Corollary 5.10.
There is an r.i. quasi-Banach function norm over M ((0 , ∞ ) , λ ) which has theproperty (P5) and for which the Hardy–Littlewood–P´olya principle does not hold.Proof. Let p ∈ (0 , W L ( L , L p ) have the desired properties. (cid:3) Finally, we present a result that generalises Proposition 3.16 and which is useful when onewants to find an integrable associate space to a given r.i. quasi-Banach function space.
Proposition 5.11.
Let k·k A and k·k B be r.i. quasi-Banach function norms and denote by k·k A ′ i and k·k B ′ i the respective integrable associate norms. Suppose that the local component of k·k A isstronger than that of k·k B . Then the local component of k·k B ′ i is stronger than that of k·k A ′ i .Similarly, if the global component of k·k A is stronger than that of k·k B , then the global com-ponent of k·k B ′ i is stronger than that of k·k A ′ i IENER–LUXEMBURG AMALGAM SPACES 28
Proof.
By our assumption and part (1) of Theorem 5.6 we get that
W L ( A, L ∞ ) ֒ → W L ( B, L ∞ ) . Consequently, it follows from Theorem 5.4, Corollary 4.5 and Corollary 4.6 that
W L ( B ′ i , L ) = ( W L ( B, L ∞ )) ′ i ֒ → ( W L ( A, L ∞ )) ′ i = W L ( A ′ i , L ) , that is, the local component of k·k B ′ i is stronger than that of k·k A ′ i .The second claim can be proved in similar manner, only using W L ( L , A ) and W L ( L , B )instead of W L ( A, L ∞ ) and W L ( B, L ∞ ). (cid:3) Appendix A. Some remarks on Wiener amalgams
In the appendix we again restrict ourselves to the case when (
R, µ ) = ([0 , ∞ ) , λ ) and considerWiener amalgams of the classical Lebesgue spaces. We show that even this very simple settingis sufficient to obtain an example of a Wiener amalgam which is constructed from a pair ofr.i. Banach function spaces and which is neither rearrangement invariant nor a Banach functionspace. The reason for this behaviour is that Wiener amalgams treat locality and globality inthe topological sense, while Banach function spaces do so in the measure-theoretic sense.We start by providing the standard simple definition of a Wiener amalgam of Lebesgue spaces(see for example [16] or [18]). Definition A.1.
Let p, q ∈ [1 , ∞ ]. Consider the classical Lebesgue spaces L p of functionsbelonging to M ([0 , ∞ ) , λ ) and l q of sequences belonging to M ( N , m ). We then define, for all f ∈ M ([0 , ∞ ) , λ ), the Wiener norm k·k W ( L p ,l q ) by k f k W ( L p ,l q ) = ∞ X n =0 k f χ [ n,n +1) k qL p ! /q for q ∈ [1 , ∞ ) , k f k W ( L p ,l q ) = sup n ∈ N k f χ [ n,n +1) k L p for q = ∞ , and the corresponding Wiener amalgam space (or just Wiener amalgam) by W ( L p , l q ) = { f ∈ M ([0 , ∞ ) , λ ); k f k W ( L p ,l q )) < ∞} . The next proposition shows what properties the Wiener amalgams of Lebesgue spaces dohave. Note that the difference in behaviour is precisely that the measure-theoretic conditionson E in (P4) and (P5) are replaced with topological ones. Proposition A.2.
Let p, q ∈ [1 , ∞ ] . Then the Wiener norm k·k W ( L p ,l q ) is indeed a norm and italso satisfies the axioms (P2) and (P3) of Banach function norms together with weaker versionsof axioms (P4) and (P5) , namely (P4’) it holds for every bounded E ⊆ [0 , ∞ ) that k χ E k W ( L p ,l q ) < ∞ , (P5’) it holds for every bounded E ⊆ [0 , ∞ ) that there is a constant C E < ∞ satisfying Z E | f | dλ ≤ C E k f k W ( L p ,l q ) for every f ∈ M ([0 , ∞ ) , λ ) .Proof. Only the properties (P4’) and (P5’) in the second assertion require a proof. We will coveronly the case q ∈ [1 , ∞ ) since the remaining case is easier.Fix a bounded set E ⊆ [0 , ∞ ). Then there is n ∈ N such that E ∩ [ n, n + 1) = ∅ for every n ≥ n . Thus the assertion (P4’) follows from the properties (P1), (P2) and (P4) of k·k L p , sincethey imply that all the summands in the definition of k χ E k W ( L p ,l q ) are finite and only finitelymany of them are greater than zero. IENER–LUXEMBURG AMALGAM SPACES 29
Similarly, the property (P5’) follows from the property (P5) of k·k L p , since it allows us toestimate Z E | f | dλ = ∞ X n =0 Z E | f | χ [ n,n +1) dλ ≤ n X n =0 C E k f χ [ n,n +1) k L p ≤ C C E n X n =0 k f χ [ n,n +1) k qL p ! q ≤ C k f k W ( L p ,l q ) , where C E is the constant from the property (P5) of k·k L p for the set E and C is the constantfrom the equivalence of k·k l and k·k l q norms on R n . (cid:3) We now show that the stronger requirement on E in (P4’) and (P5’) was necessary, i.e. thatthe Wiener amalgams of Lebesgue spaces do not in general have the properties (P4) and (P5). Proposition A.3. (1)
Let ≤ q < p ≤ ∞ . Then the norm k·k W ( L p ,l q ) does not satisfy (P4) . (2) Let ≤ p < q ≤ ∞ . Then the norm k·k W ( L p ,l q ) does not satisfy (P5) .Consequently, the norm k·k W ( L p ,l q ) is a Banach function norm if and only if ≤ p = q ≤ ∞ ,in which case it coincides with the classical Lebesgue norm k·k p .Proof. We will show part (1) only for p < ∞ since the remaining case is easier. Fix arbitrary a ∈ (cid:16) , pq (cid:17) and define E = [ n ∈ N (cid:20) n, n + 1 n a (cid:21) . Then, by our assumptions on a , λ ( E ) = ∞ X n =0 n − a < ∞ but k χ E k qW ( L p ,l q ) = ∞ X n =0 n − a qp = ∞ . As for part (2), we will show it only for q < ∞ since the remaining case is easier. Fix arbitrary a ∈ (cid:16) pq , (cid:17) , b ∈ (cid:16) , p − ap − (cid:17) (if p = 1 then any b ∈ (1 , ∞ ) will suffice) and define E = [ n ∈ N (cid:20) n, n + 1 n b (cid:21) ,f = ∞ X n =0 n b − ap χ [ n,n + n − b ] . Then, by our assumptions on a and b , λ ( E ) = ∞ X n =0 n − b < ∞ , k f k qW ( L p ,l q ) = ∞ X n =0 (cid:18)Z n +1 n | f | p dλ (cid:19) qp = ∞ X n =0 n − a qp < ∞ but Z E | f | dλ = ∞ X n =0 n b − ap − b = ∞ . IENER–LUXEMBURG AMALGAM SPACES 30 (cid:3)
As for the rearrangement invariance, we consider the functional f
7→ k f ∗ k W ( L p ,l q ) . It is obviousthat k·k W ( L p ,l q ) can be equivalent to a rearrangement-invariant norm only if it is equivalent to thisfunctional. But as we show in the following theorem, this functional is in fact equivalent to theWiener–Luxemburg amalgam quasinorm k·k W L ( L p ,L q ) , which is always equivalent to a Banachfunction norm by Corollary 3.6, and thus by the previous remark the required equivalence canhold only if p = q . Theorem A.4.
Let p, q ∈ [1 , ∞ ] . Then the functional (A.1) f
7→ k f ∗ k W ( L p ,l q ) is an r.i. quasi-Banach function norm. Furthermore, there is a constant C ∈ (0 , ∞ ) such that (A.2) C − k f ∗ k W ( L p ,l q ) ≤ k f k W L ( L p ,L q ) ≤ C k f ∗ k W ( L p ,l q ) , for every f ∈ M ([0 , ∞ ) , λ ) .Proof. It is obvious that the functional defined by (A.1) has the properties (P2) and (P3) andthat it is rearrangement-invariant. Furthermore, if we have E ⊆ [0 , ∞ ) such that λ ( E ) < ∞ then χ ∗ E = χ [0 ,λ ( E )) and we have by the Hardy–Littlewood inequality (Theorem 2.4) that Z E | f | dλ ≤ Z [0 ,λ ( E )) f ∗ dλ. The properties (P4) and (P5) therefore follow from Proposition A.2.As for the property (Q1), the parts (a) and (b) are obvious while the part (c) deserves furthercomment. Let us fix some f, g ∈ M . We may use a similar argument as in Proposition 3.4 toobtain for any n ∈ N that k ( f + g ) ∗ χ [ n,n +1) k L p ≤ k ( D f ∗ + D g ∗ ) χ [ n,n +1) k L p ≤ k D f ∗ χ [ n,n +1) k L p + k D g ∗ χ [ n,n +1) k L p ≤ p (cid:16) k f ∗ χ [ n , n +12 ) k L p + k g ∗ χ [ n , n +12 ) k L p (cid:17) . From this we obtain the desired estimate k ( f + g ) ∗ k W ( L p ,l q ) ≤ p + q (cid:0) k f ∗ k W ( L p ,l q ) + k g ∗ k W ( L p ,l q ) (cid:1) . It remains to verify (A.2), which will follow from Theorem 2.18 once we show that it holdsfor all f ∈ M that k f ∗ k W ( L p ,l q ) < ∞ if and only if k f k W L ( L p ,L q ) < ∞ .Suppose that k f ∗ k W ( L p ,l q ) < ∞ . Then we immediately observe that k f ∗ χ [0 , k L p < ∞ . Fur-thermore, k f ∗ χ (1 , ∞ ) k L q = ∞ X n =1 Z n +1 n ( f ∗ ) q dλ ! q ≤ ∞ X n =1 ( f ∗ ( n )) q ! q ≤ ∞ X n =1 k f ∗ χ [ n − ,n ) k qL p ! /q < ∞ , and thus k f k W L ( L p ,L q ) < ∞ . IENER–LUXEMBURG AMALGAM SPACES 31
Conversely, suppose that k f k W L ( L p ,L q ) < ∞ . Then k f ∗ k W ( L p ,l q ) = ∞ X n =0 k f ∗ χ [ n,n +1) k qL p ! /q ≤ k f χ [0 , k qL p + ∞ X n =1 f ∗ ( n ) q ! /q ≤ k f χ [0 , k qL p + ∞ X n =2 Z nn − ( f ∗ ) q dλ ! /q = (cid:18) k f χ [0 , k qL p + Z ∞ ( f ∗ ) q dλ (cid:19) /q < ∞ and thus k f ∗ k W ( L p ,l q ) < ∞ . (cid:3) Corollary A.5.
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Dalimil Peˇsa, Department of Mathematical Analysis, Faculty of Mathematics and Physics,Charles University, Sokolovsk´a 83, 186 75 Praha 8, Czech Republic
Email address : [email protected] ORCiD ::