Pre-dual of Fofana's spaces
Hans Georg Feichtinger, Justin Feuto

TL;DR
This paper characterizes the pre-dual of Fofana's Wiener amalgam-based spaces, focusing on their dilation properties and minimal invariant spaces, advancing the understanding of their duality structure.
Contribution
It provides a novel characterization of the pre-dual of Fofana's spaces using minimal invariant spaces related to dilation operators.
Findings
Pre-dual characterized via minimal invariant spaces
Spaces exhibit dilation behavior similar to $L^eta(\
Abstract
It is the purpose of this paper to give a characterization of the pre-dual of the spaces introduced by I.~Fofana on the basis of Wiener amalgam spaces. Those spaces have a specific dilation behaviour similar to the spaces . The characterization of the pre-dual will be based on the idea of minimal invariant spaces (with respect to such a group of dilation operators).
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Taxonomy
TopicsMathematical Analysis and Transform Methods · Advanced Harmonic Analysis Research · Holomorphic and Operator Theory
Pre-dual of Fofana’s spaces
Hans G. Feichtinger
Faculty of Mathematics, University of Vienna (Autriche), Oskar-Morgenstern-Patz 1, 1090 Wien.
and
Justin Feuto
Laboratoire de Mathématiques Fondamentales, UFR Mathématiques et Informatique, Université Félix Houphouët-Boigny-Cocody, 22 B.P 1194 Abidjan 22. Côte d’Ivoire
Abstract.
It is the purpose of this paper to give a characterization of the pre-dual of the spaces introduced by I. Fofana on the basis of Wiener amalgam spaces. Those spaces have a specific dilation behaviour similar to the spaces . The characterization of the pre-dual will be based on the idea of minimal invariant spaces (with respect to such a group of dilation operators).
Key words and phrases:
Amalgam spaces, Fofana spaces, predual.
1991 Mathematics Subject Classification:
43A15; 46B10
Hans G. Feichtinger is grateful to J. Feuto and I. Fofana for the opportunity to visit Abidjan and give a course on Banach Gelfand Triples at the Conference Harmonic Analysis and Applications
1. Introduction
Let be a fix positive integer and the -dimensional Euclidean space, equipped with its Lebesgue measure . For the amalgam of and is the space of function which are locally in and such that the sequence belongs to , where . The map denotes the usual norm on the Lebesgue space on while stands for the characteristic function of the subset of .
Amalgam spaces have being introduced by N. Wiener in 1926 (see [31]), but their systematic study began with the work of F. Holland [23] in 1975. Since then, they have been widely studied (see [2, 21, 11, 12, 30, 17] and the references therein). It is easy to see that the usual Lebesgue space coincides with the amalgam space . Also, the Lebesgue space is known to be invariant under dilations. In fact, for , the dilation operator is isometric. Proper amalgam spaces don’t have this property. Even worse, for we can’t found such that , although for all , and (see e.g. [3] or [24]). In order to compensate this shortfall, Fofana introduced (see [18]) in the years 1980’s the functions spaces , which consist of satisfying (see Section 2 for more precision).
These spaces can be viewed as some generalized Morrey spaces, and we will always refer to them as Fofana spaces.
Many classical results for Lebesgue and the classical Morrey spaces have been extended to the setting of the spaces (see [18, 19, 20, 14, 15, 16, 27, 5, 29, 4]). Although the dual space of Lebesgue spaces ( and amalgam spaces () are well known ( and respectively with ), the one of is still unknown. But in the case and , there are already four characterizations of the pre-dual of (see [1, 22, 26, 32]) which are equal.
The purpose of this paper is to give a beginning of answer by the determination of a pre-dual of for .
In doing so we will make use of the idea of minimal invariant Banach spaces of functions which has already quite some tradition and has shown to be useful in a variety of situations, see [6, 7, 8, 9, 10] or [25].
The paper is organized as follows : In Section 2, we give some basic facts about amalgams and minimal Banach spaces. The third section is devoted to pre-dual of Fofana spaces as well as certain properties of these spaces.
2. Some basic facts about Amalgam and Minimal Banach spaces
For any normed space , we denote by its topological dual space. Given , the amalgam space is equipped with the norm . For any , we put
[TABLE]
with It is clear that .
We have the following well known properties (see for example [21]).
- (1)
For , is a norm on equivalent to . With respect to these norms the amalgam spaces are Banach spaces. 2. (2)
The spaces are (strictly) increasing with the global exponent and (strictly) decreasing with a growing local exponent ; more precisely
[TABLE] 3. (3)
For , Hölder’s inequality is fulfilled :
[TABLE]
where and are conjugate exponent of and respectively: . When , is isometrically isomorphic to the dual of in the sense that for any element of , there is an unique element of such that
[TABLE]
and furthermore
[TABLE]
We recall that \left\|T\right\|:=\sup\left\{\left|\left\langle T,f\right\rangle\right|{\,\big{|}\,}f\in L^{q}_{loc}(\mathbb{R}^{d})\text{ and }\left\|f\right\|_{q,p}\leq 1\right\}.
Next we summarize a couple of properties of dilation operators. We assume that .
- (1)
For any real number , maps into itself. 2. (2)
. 3. (3)
.
In other words, is a commutative group of operators on , isomorphic to the multiplicative group . As mentioned in the introduction, we have for
[TABLE]
In other words, each of those normalizations is isometric on exactly one of the family of -spaces.
For amalgam spaces, direct computations (see for example (2.1) and Proposition 2.2 in [16]) give the following results:
[TABLE]
and therefore
[TABLE]
It follows that the space can be defined by
[TABLE]
We recall that for , the space is exactly the classical Morrey space introduced by Morrey in 1938, see [28], and defined for by
[TABLE]
Fofana spaces have the following properties (cf. [18], [19]):
- (1)
is a Banach space which is non trivial if and only if , 2. (2)
if then with equivalent norms, 3. (3)
if then , where is the weak Lebesgue space on defined by
[TABLE]
with . We denote by , the Lebesgue measure of a measurable subset of .
For any , the dilation map isometrically the space to itself. More precisely,
[TABLE]
Remark 2.1**.**
It is easy to see that, if and is an element of such that belongs to , then belongs to and
[TABLE]
Searching for a dilation invariant version of the Segal algebra (introduced in [8]) Feichtinger and Zimmermann introduced a certain exotic minimal space in [13]. We will use later on the following result of that paper:
Theorem 2.2** ([13], Theorem 2.1).**
Let be a Banach space, and a (not necessarily countable) bounded family in . Define
[TABLE]
and let
[TABLE]
Then is a Banach space continuously embedded into .
3. A pre-dual of Fofana spaces
Definition 3.1**.**
Let . The space is defined as the set of all elements of for which there exist a sequence of elements of such that
[TABLE]
We will always refer to any sequence of elements of satisfying (3.1) as -decomposition of .
For any element of , we set
[TABLE]
where the infimum is taken over all -decomposition of .
Proposition 3.2**.**
Let . endowed with is a Banach space continuously embedded into .
Proof.
Since , it comes from (2.2) and (2.3) that for any element ,
[TABLE]
Therefore, by (2.6)
[TABLE]
The result follows from Theorem 2.2, using the Banach space and its bounded subset
[TABLE]
For and . Zorko proved in [32] that the Morrey space is the dual space of the space , which consists of all the functions on which can be written in the form , where is a sequence in , and is a sequence of functions on satisfying for each ,
- •
,
- •
Notice that for and , the space is continuously embedded into . In fact, let with the sequences and as in the Introduction. For any , we put
[TABLE]
where is the radius of the ball associated to . There exists a constant (one can take ), such that
[TABLE]
We prove next that for , the space is a certain minimal Banach space.
Proposition 3.3**.**
For , the space is a minimal Banach space, isometrically invariant under the family and such that
[TABLE]
with continuous inclusions.
Proof.
- (1)
Let us first prove that is isometrically invariant by for all .
Let . For and any -decomposition of , we have
[TABLE]
so that
[TABLE]
Thus, is an isometric automorphism of . 2. (2)
First we verify that is continuously embedded into . For any we have
[TABLE]
and therefore belongs to and satisfies
[TABLE]
Thus our claim is verified. 3. (3)
It remains to prove that this space is minimal. Let be another Banach space continuously embedded into such that
- (a)
is isometrically invariant by , i.e., if then , 2. (b)
is continuously embedded into , i.e., there exists such that for , and .
For and any -decomposition
[TABLE]
Thus
[TABLE]
and therefore, as is a Banach space,
[TABLE]
From this and (3.2) it follows that
[TABLE]
Thus is continuously included in .
Our main result can be stated as follows.
Theorem 3.4**.**
Let . The operator defined by (3.6) is an isometric isomorphism of into .
For the proof, we need some intermediate results.
Remark 3.5**.**
It is clear that if is a -decomposition of , then is a sequence of elements of which converges to in . Hence is a dense subspace of .
Proposition 3.6**.**
Let , and . Then belongs to and
[TABLE]
Proof.
Let be a -decomposition of .
By using successively (2.11), (2.4) and (2.7), we obtain for any ,
[TABLE]
Therefore we have
[TABLE]
This implies that belongs to and
[TABLE]
Taking the infimum with respect to all -decompositions of , we get
[TABLE]
Remark 3.7**.**
Let and set :
[TABLE]
By Prop. 3.6 and the fact that belongs to , that :
- (1)
for any element of , belongs to , 2. (2)
* is linear and bounded mapping from into satisfying , that is :*
[TABLE]
Now we can prove our main result.
Proof of Theorem 3.4.
We know (see Remark 3.7) that is a bounded linear application of into such that
[TABLE]
Let be an element of . From (3.4) it follows that the restriction of to belongs to . Furthermore, we have . So, by (2.5), there is an element of such that
[TABLE]
Hence , for and we have
[TABLE]
[TABLE]
by (3.3) and (3.4). From this and (2.5) it follows that
[TABLE]
and therefore, by (2.7),
[TABLE]
From (3.8), the density of in (see Remark 3.5) and Prop. 3.6, we get
[TABLE]
This completes the proof.
We end this paper by stating some interesting properties of the spaces .
Proposition 3.8**.**
Let .
- (1)
*The space is a Banach -module: *
for
[TABLE] 2. (2)
If and with and then
[TABLE]
Consequently is an essential Banach -module. 3. (3)
For the Schwartz space of rapidly decreasing functions or the space of compactly supported -functions are dense subspaces of . 4. (4)
For , the Banach space is separable.
Proof.
Let and .
- (1)
Let be a -decomposition of . A direct computation shows that
[TABLE]
Hence, combining the fact that is a Banach module and Relation (2.6), we obtain
[TABLE]
This implies that
[TABLE]
Moreover
[TABLE]
Therefore
[TABLE]
in the sense of , with
[TABLE]
That is belongs to and satisfies
[TABLE] 2. (2)
Let us assume that and . For any real number , we set . Let us consider , a -decomposition of . b We know that the sequence defined by
[TABLE]
converges to in . Let us fix any real number . There is a positive integer satisfying
[TABLE]
Moreover, for any real number , we have
[TABLE]
The last term is not greater than
[TABLE]
where
[TABLE]
It comes that
[TABLE]
By hypothesis, we have and therefore
[TABLE]
So we have
[TABLE]
and therefore
[TABLE]
This inequality being true for any real number , we actually have . 3. (3)
Approximating a given function in first by some function with compact support and then convolving it by some compactly supported, infinitely differentiable test function provides an approximation by a test function, which also belongs to the Schwartz space. 4. (4)
For , we have . But it is well known that is separable. Thus the result follows from the density of in .
For , we have defined a pre-dual of the Fofana space using some atomic decomposition method developed by Feichtinger, and proved that for and , the pre-dual of classical Morrey space is embedded in our space. But our further goal is to find the dual space of and their interpolation spaces. This appears not as an easy task and has to be left for future work.
4. Conclusion
In summary we have used techniques concerning minimal invariant Banach spaces of functions in order to characterize the pre-dual of certain Fofana spaces which have not been known so far. Starting from a characterization of a Fofana space as a (dense) subspace of a Wiener amalgam space under a certain group of (suitably normalized) dilation operators one can generate the predual space by starting from the predual of the mentioned Wiener amalgam spaces and then describing the predual via atomic decompositions, using the (adjoint) group of dilation operators.
Acknowledgement: During the period of the preparation of the material for this manuscript (spring and summer of 2018) the first author held a guest position at TU Muenich, Dept. of Theoretical Information Sciences (H. Boche). The second author is thankful to Fofana Ibrahim for drawing his attention to the separability of the predual, and many helpful discussions.
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