Continuity properties and Bargmann mappings of quasi-Banach Orlicz modulation spaces
-
Joachim Toft
,
Rüya Üster
Abstract
We deduce continuity, compactness and invariance properties for
quasi-Banach Orlicz modulation spaces on
1 Introduction
In analysis and its applications in science and technology, it is common to
perform investigations, e.g., on continuity, in the framework on functions
and distribution spaces whose elements obey some kind of Lebesgue
norm estimates. For example, for
On the other hand, in several situations, e.g., when dealing
with non-linear functionals, it might be beneficial to replace Lebesgue
norm estimates with suitable Orlicz norm estimates. For example,
in statistics or statistical physics, the entropy applied on probability density
functions f on
When investigating
respectively (see [28, 29] and the references therein).
Orlicz spaces are important types of Banach function spaces
that are considered in mathematical analysis. They naturally
generalize
In this paper, we extend the analysis in [18, 21] concerning classical
modulation spaces
The design of Orlicz modulation spaces follows a path similar to
classical modulation spaces, and they are defined by imposing a mixed
In the restricted case when
exist, then
with continuous embedding. We remark that several properties for (Banach) Orlicz modulation spaces in [40] hold true in more general contexts where the Orlicz norms are replaced by norms from solid translation invariant Banach function spaces (see, e.g., [4, 11, 15, 16, 13]).
In our situation, allowing, more generally,
More precisely, we show that
is compact if and only if
of entire analytic functions on
Several of these properties follow from our characterizations of Orlicz modulation
spaces in terms of Gabor expansions, given in Section 5. In fact, here it is proved
that for a distribution f, some lattice
are continuous between the spaces
and that
provided that the lattice Λ is dense enough.
Here the series converges in
The paper is organized as follows. In Section 2, we recall some basic properties on Gelfand–Shilov spaces, weight functions, Pilipović spaces, Orlicz spaces, and introduce quasi-Orlicz modulation spaces. Here we also recall some properties for classical modulation spaces concerning Gabor analysis, images under pseudo-differential operators and under the Bargmann transform.
In Section 3, we deduce mapping properties of Orlicz modulation spaces under the Bargmann transform. At the same time, we prove that they are complete, and thereby quasi-Banach spaces. In Section 4, we obtain convolution estimates for quasi-Orlicz spaces.
In Section 5, we apply the convolution results in Section 4 to extend
the Gabor analysis for classical modulation spaces to quasi-Orlicz modulation
spaces. In particular, we deduce (1.3) for quasi-Young functions
2 Preliminaries
In this section, we provide a review of some basic facts. In the first part, we recall the definition and explain some well-known facts about Gelfand–Shilov and Pilipović spaces and their spaces of (ultra-)distributions. Thereafter, we consider (mixed) Orlicz and quasi-Orlicz spaces and explain some basic properties. Our family of quasi-Orlicz spaces contains the family of Orlicz spaces, but is less comprehensive than the families of quasi-Orlicz spaces given in, e.g., [25] by Harjulehto and Hästö. We refer to [36, 25, 32] for more facts about Orlicz spaces.
Then we introduce and discuss basic properties of quasi-Banach Orlicz modulation spaces, which are obtained by imposing quasi-Orlicz norm estimates on the short-time Fourier transforms of the involved functions and distributions. Finally, we recall some basic facts in Gabor frame theory, and for the Bargmann transform.
2.1 Gelfand–Shilov spaces
We start by discussing Gelfand–Shilov spaces and their properties.
Let
is finite for some
Whenever
with dense embeddings.
Here
The Gelfand–Shilov distribution spaces
From now on, we let
when
Gelfand–Shilov spaces can in convenient ways be characterized in terms of estimates of the functions and their Fourier
transforms. More precisely, in [8, 48] it is proved that
if
for some
Gelfand–Shilov spaces and their distribution spaces can also
be characterized by estimates of short-time Fourier
transforms (see, e.g., [24, 46]).
More precisely, let
If
2.2 Weight functions
A weight or weight function on
If
We let
It can be proved that if
for some
We say that
We let
Evidently,
In several situations, our weights are defined on the phase space
Remark 2.1.
The class
(see, e.g., [17]). The condition (2.9) is equivalent to
and is usually called the GRS condition, or Gelfand–Raikov–Shilov condition. A more restrictive condition on v is given by the Beurling–Domar condition
which is strongly linked to non-quasi-analytic classes that contain non-trivial compactly supported elements (see, e.g., [22]).
In our discussions, we go for maximal generality of the
ingredients, both concerning Orlicz spaces and the weights.
It turns out that we will be able to deduce the searched properties
without the restrictions in (2.9), (\ref{Eq:GRSCond})’ and (2.10).
Hence, for us the most relevant weight class will be
We refer to [10, 21, 22] for more facts about weight functions.
2.3 Pilipović spaces
In the course of our work, we will also make use of
the Pilipović space
such that
for some
In the same way,
for every
We also have
when
For future references, we note that if
then the short-time Fourier transform (2.4) makes sense as a smooth function in view of [46, (2.25) and Theorem 4.1].
2.4 Quasi-Banach spaces
We recall that a quasi-norm
and
The space
From now on, we always assume that the quasi-norm of the quasi-Banach space
2.5 Orlicz spaces
Next, we define and recall some basic facts for (quasi-) Orlicz spaces (see [36, 25]). First, we give the definition of Young functions and quasi-Young functions.
Definition 2.2.
A function
for every
We observe that Φ might not be continuous, because we permit
is convex but discontinuous at
Definition 2.3.
Let
The function Φ is called
It is clear that Φ in
Definition 2.3 is
non-decreasing,
because if
since
Orlicz spaces are defined as follows.
Definition 2.4.
Let
is finite.
Let Φ be a quasi-Young function of order
is finite. We let
Remark 2.5.
Let Φ,
In several situations, we need to consider mixed quasi-normed Orlicz spaces (see, e.g., [2, 5, 6, 27, 30, 31, 40] for details in the Banach space case, i.e. on mixed norm Orlicz spaces.)
Definition 2.6.
Let
is finite, where
We let
If
Remark 2.7.
Suppose
Remark 2.8.
Let ω,
with equality in quasi-norms. Hence, the family of quasi-Orlicz spaces contains the usual Lebesgue spaces and mixed quasi-normed spaces of Lebesgue type.
Remark 2.9.
Let
for some Young functions
In fact, the completeness of
which shows that
In what follows, let
and let
Remark 2.10.
Let
when μ is the Lebesgue measure on
Evidently,
Lemma 2.11.
Let Φ,
Specifically, we observe that the constant C in Lemma 2.11 does only depend on the involved weight functions. In particular, it is independent of the choice of quasi-Young functions.
Lemma 2.11 is essentially well known (see, e.g., [12, 15, 38, 37]). In order to be self-contained, we give a proof of the second statement.
Proof.
We have
Here the inequality follows from the fact that ω is v-moderate, and the last two relations follow from the definitions. ∎
2.6 Orlicz modulation spaces
Before considering Orlicz modulation spaces, we recall the definition of classical modulation spaces (cf. [11, 12]).
Definition 2.12.
Let
We observe that
For convenience, we set
In the following proposition, we have listed some basic properties for modulation spaces. We refer to [11, 15, 18, 21, 43, 46] for the proof.
Proposition 2.13.
Let
If
If
Let
and
The quasi-norms on
and
Remark 2.14.
Let
We notice that (2.24) and (2.25) are norms when
are independent of the choices of ϕ in
Later on, we need the following proposition.
Proposition 2.15.
Let
Proof.
Let
and
The result now follows from Proposition 2.13 (i). ∎
2.7 Gabor frames
Let
We put
Definition 2.16.
Let
The analysis operator
The synthesis operator
The Gabor frame operator
We set
and
Let
is called a Gabor frame if there are constants
We observe that, by slightly
modifying the constants
The function ϕ is called the (frame) atom for the frame (2.27) (see, e.g., [20]).
If (2.27) is a Gabor frame, then there is a Gabor frame
with
where the series converges in
Remark 2.17.
Gabor frames can be obtained for general lattices
which are dense enough. Hence, the assumption above that the
configuration variable j and momentum variable ι above should belong
to separate lattices
On the other hand, in some computations later on, it is convenient to restrict ourselves to separate lattices.
The next result shows that it is possible to find suitable ϕ and ψ in the previous definition. We omit the proof since the result is a special case of [20, Theorem S].
Lemma 2.18.
Let
are dual frames to each others.
Remark 2.19.
Let
Then Lemma 2.18 guarantees that for some choice of
Lemma 2.20.
Let
Then there is an
is a Gabor frame with canonical dual frame
where
We recall that φ in (2.31) is called the Rihaczek
function of
The next result gives some information about the roles that
Proposition 2.21.
Let
There is a constant
In Section 5, we extend Proposition 2.21 to the quasi-Banach case (see Theorem 5.10 and Proposition 5.11).
2.8 The Bargmann transform and modulation spaces
We finish this section by recalling the Bargmann transform and its mapping properties on modulation spaces.
The Bargmann kernel of dimension d is given by
Here (see [3])
It follows that
The Bargmann transform
There are several results on Bargmann images of well-known function
and distribution spaces. For example, it was already proved in [3]
that
where
If F is a measurable function on
where
We let
with topology inherited from the topology in
Proposition 2.22.
Let
Apart from Proposition 2.22, there are several characterizations of well-known
function and distribution spaces via their images under the Bargmann transform.
For example, convenient characterization of
Let
with
We also let
with topology inherited from the topology in
when
3 Continuity and Bargmann images of Orlicz modulation spaces
In this section, we extend Proposition 2.22 to more general weights and to the Orlicz case (see Theorem 3.1). At the same time, we prove that the Orlicz modulation spaces are quasi-Banach spaces, by deducing similar facts of their Bargmann images.
The extension of Proposition 2.22 is the following theorem.
Here we let
Theorem 3.1.
Let ω be a weight on
The Bargmann transform is isometric and bijective from
We need the following lemma for the proof. Here and in what follows,
we let
Lemma 3.2.
Let ω be a weight on
If
If
Proof.
For every
for some choices of
Since
we get
which in turn leads to
with continuous inclusion. ∎
Proof of Theorem 3.1.
Let
By the completeness of
Because
giving the completeness of
Due to the completeness of
By the definitions, it follows that
Since
when
Suppose
for some
giving that
Now we have the following inclusion relations between Orlicz modulation spaces, Gelfand–Shilov and Schwartz spaces and their duals. We recall Section 2.2 for notations on weight classes.
Proposition 3.3.
Let
Then
Before giving the proof of Proposition 3.3, we give the following consequence.
Corollary 3.4.
Let
and
Proof.
The result follows from the definitions, Proposition 3.3 and the fact that
where
We need the following lemma for the proof of Proposition 3.3.
Lemma 3.5.
Let
A proof of Lemma 3.5 is essentially given in [40]. In order to be self-contained, here we give the arguments.
Proof.
By the mapping
and
then
This gives
Corollary 3.6.
Let
Proof.
Since
Proof of Proposition 3.3.
If the
for some
4 Convolution estimates for quasi-Orlicz spaces
In this section, we extend the convolution estimates in [18] for Lebesgue spaces
to the case of quasi-Orlicz spaces. In the first part, we deduce discrete convolution
estimates between elements in discrete Orlicz and Lebesgue spaces. Thereafter,
we focus on the semi-continuous convolution, and prove corresponding estimates
for
Our investigations involve weighted Orlicz spaces where the weights should satisfy conditions of the form
or
for some constant
4.1 Discrete convolution estimates on discrete Orlicz spaces
We recall that the discrete convolution between
Lemma 4.1.
Let Λ be as in (2.26), let
for some constant
Lemma 4.1 follows by similar arguments to the ones in the proof of [18, Lemma 2.5] (see also [15, 38, 37]). In order to be self-contained, here we show the arguments.
Proof.
Since
Since
which gives (4.3). ∎
By choosing
Corollary 4.2.
Let Λ be as in (2.26) and let Φ,
Suppose that
Let
Next, we perform similar investigations for semi-discrete convolutions.
Definition 4.3.
Let Λ be as in (2.26).
The semi-discrete convolution of
For
The following result corresponds to Lemma 4.1 in the framework
of semi-discrete convolutions. Here and in what follows, we let
Lemma 4.4.
Let Λ be as in (2.26),
for some constant
There are several results which are similar to Lemma 4.4 (see, e.g., [15, 18, 20, 38, 37]). In order to be self-contained, we here give the arguments.
Proof.
We shall argue as in the proof of Lemma 4.1.
Since
Since
The proof is finished. ∎
By choosing
Corollary 4.5.
Let Λ be as in (2.26), let
Suppose that
Suppose that
is uniquely extendable to a continuous map from
and
In what follows, we set
Definition 4.6.
Let
belongs to
For convenience, we set
for every measurable function F on
Remark 4.7.
Let
It holds
where
It holds
where
The next lemma corresponds to [18, Lemma 3.8] and gives suitable Orlicz estimates of samples in terms of
Wiener norm estimates. Here and in what follows, we let
Lemma 4.8.
Let
A more general result of Lemma 4.8 can be found in [37], and in the Banach space case in [15]. In order to be self-contained, here we give a proof.
Proof.
The map
Since there are at most
By the definition of the
for
Since there are at most
By the definition of the
Lemma 4.9.
Let
and
for some constant
Lemma 4.9 is similar to [38, Theorem 5.1] when Y in [38] is an Orlicz space. Lemma 4.9 also essentially generalizes parts of [18, Lemma 2.9].
Proof.
We have
and we have that
Let
where
Here the suprema in (4.8) are taken with respect to
By (4.8), Corollary 4.2 and the fact that
for some constant
where all hidden constants only depend on d. This gives (4.7), and the result follows. ∎
By similar arguments, we get the following semi-discrete convolution relation. The result essentially generalizes [18, Lemma 2.10], while more general results are (implicitly) available in [37].
Lemma 4.10.
Let
extends uniquely to a continuous mapping from
and
Proof.
Let
then, for
In particular, since
The result now follows by similar arguments to the ones in the proof of Lemma 4.9, and the fact that
is dense in
5 Gabor analysis of Orlicz modulation spaces
In this section, we extend the Gabor analysis in [18] to Orlicz modulation spaces. We show that the quasi-norms
are equivalent when
and that the corresponding synthesis operator is continuous from
In the end, we are able to prove that an element belongs to
We also remark that our investigations are related to those general results in [38, 37] by Rauhut on quasi-Banach co-orbit space theory, but note that Rauhut’s results do not cover our situation. For example, our weight functions are allowed to grow and decay exponentially, which is not the case in [38, 37].
5.1 Comparisons between
∥
V
ϕ
1
f
∥
L
(
ω
)
Φ
1
,
Φ
2
and
∥
V
ϕ
2
f
∥
W
(
L
(
ω
)
Φ
1
,
Φ
2
)
The next extension of [18, Theorem 3.1] shows that the condition
is independent of the choice of window function ϕ, and that different ϕ give rise to equivalent norms.
Theorem 5.1.
Let
and
where the constant
Proof.
Assume that
is a Gabor frame for
with unconditional convergence in
where
in view of [44, Theorem 3.7], we obtain
for some constants
Remark 5.2.
Let
By [20, Theorem S],
provided that the lattices of Gabor frames are dense enough. This implies that
Proposition 5.3.
Let
for some constant C which only depends on ω and d.
Proof.
Let
For each
such that
We have
For any
and
let
and let
for some constants
In order to estimate the left-hand side of (5.3), we apply the latter estimates on
with
By (5.5), (5.6) and the definition of the
for some constant C which only depends on ω and d.
Let
Then, by Minkowski’s inequality and again using Jensen’s inequality,
for some suitable
By the definition of the
where C only depends on ω and d. Hence, (5.4) and (5.7) give
for a suitable constant C. ∎
Now we have the following extension of Proposition 5.3, [18, Theorem 3.3] and [44, Proposition 3.4].
Theorem 5.4.
Let
for some constant
Proof.
Let
By Lemma 4.9, Theorem 5.1 and Proposition 5.3, we obtain
for some constants
Hence, if
for some constants
Now we may deduce suitable continuity properties for analysis and synthesis operators, related to [37, Theorem 5.6] and to some results in [15], and which extend [18, Theorem 3.5].
Theorem 5.5.
Let
The analysis operator
(5.8)for some constant
The synthesis operator
(5.9)for some constant
Proof.
Since
which proves (i).
It remains to prove (ii). Let
when
we get
Lemma 4.10 implies that
and (ii) follows from Theorem 5.4. ∎
The next theorem is the main result of this section, and it shows that the Gabor analysis in [18] for modulation spaces also holds for quasi-Orlicz modulation spaces.
Theorem 5.6.
Let
are dual frames to each others. Then the following is true:
The Gabor frame operator
If
with unconditional convergence in
Furthermore, if
for some constant
Again, we observe that the constant C in Theorem 5.6 does not depend on
the quasi-Young functions
Proof.
Since (5.10) are dual frames, it follows that
shows that
By interchanging the roles for ϕ and ψ, we deduce the other relations in (5.11). ∎
Remark 5.7.
Let
Here we note that the hidden constants in inequalities (5.8), (5.9)
and (5.11) heavily depend on the weights
ω, v, the parameter ε and the Young functions
We also note in the spirit of [20, Theorem S], that, for ω,
v fixed and ϕ as above, there is an
Remark 5.8.
Let
Since
By straightforward computations, it follows that
A sufficient condition for
5.2 Some consequences
Next we present some consequences of the previous results, and begin with
showing the independence of
Theorem 5.9.
Let
are quasi-norms on
Recall that
when
We observe that ϕ in Theorem 5.9 exists in view of Remark 5.7.
Theorem 5.10.
Suppose that
exist and are finite,
Proof.
By Theorem 5.6, it suffices to prove the first embedding in (5.13).
Since
Let
and
We claim
also in the topological sense.
In fact, let
Then
This gives
and (5.14) follows.
By (5.15) and the fact that
we get
and the result follows. ∎
By Theorem 5.10 and its proof, we may now extend Proposition 2.21 to the quasi-Banach case as follows. The details are left to the reader.
Proposition 5.11.
Let
There is a constant
We observe that in the Banach space case the equivalence between (i) and (ii) in Proposition 5.11 is a special case of [16, Theorem 8.4] (which, among others, hosts Orlicz modulation spaces with general moderate weights which are Banach spaces).
Next, we discuss compact embeddings of Orlicz modulation spaces.
The next result follows by similar arguments to the ones in the proof of
[33, Theorem 3.9], using the fact that
Theorem 5.12.
Let
is compact if and only if
Funding source: Vetenskapsrådet
Award Identifier / Grant number: 2019-04890
Funding statement: The first author was supported by Vetenskapsrådet (Swedish Science Council) within the project 2019-04890. The second author thanks Tinçel Kültür Vakfı and İstanbul University for the support during her stay at Linnæus University during Spring 2019.
Acknowledgements
We are very grateful to H. Feichtinger from Vienna university, for valuable advice which led to several improvements of the content and the style of the original manuscript.
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Articles in the same Issue
- Frontmatter
- Recognizing pro-𝖱 closures of regular languages
- Reverse Stein–Weiss, Hardy–Littlewood–Sobolev, Hardy, Sobolev and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups
- Non-perturbative graph languages, halting problem and complexity
- Spectral synthesis via moment functions on hypergroups
- A function field analogue of the Stark conjecture
- Continuity properties and Bargmann mappings of quasi-Banach Orlicz modulation spaces
- Krull dimension for C*-algebras
- The sharp bound of the third Hankel determinant for starlike functions
- The role of gentle algebras in higher homological algebra
- Classifying spaces for the family of virtually abelian subgroups of orientable 3-manifold groups
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- Common reducing subspaces and decompositions of contractions
- A restriction estimate on Morrey spaces
- On ideals in group algebras: An uncertainty principle and the Schur product
- Spectrality of self-similar measures with product-form digits
- Borderline Lipschitz regularity for bounded minimizers of functionals with (p,q)-growth
- On the L q spectra of in-homogeneous self-similar measures
Articles in the same Issue
- Frontmatter
- Recognizing pro-𝖱 closures of regular languages
- Reverse Stein–Weiss, Hardy–Littlewood–Sobolev, Hardy, Sobolev and Caffarelli–Kohn–Nirenberg inequalities on homogeneous groups
- Non-perturbative graph languages, halting problem and complexity
- Spectral synthesis via moment functions on hypergroups
- A function field analogue of the Stark conjecture
- Continuity properties and Bargmann mappings of quasi-Banach Orlicz modulation spaces
- Krull dimension for C*-algebras
- The sharp bound of the third Hankel determinant for starlike functions
- The role of gentle algebras in higher homological algebra
- Classifying spaces for the family of virtually abelian subgroups of orientable 3-manifold groups
-
- Common reducing subspaces and decompositions of contractions
- A restriction estimate on Morrey spaces
- On ideals in group algebras: An uncertainty principle and the Schur product
- Spectrality of self-similar measures with product-form digits
- Borderline Lipschitz regularity for bounded minimizers of functionals with (p,q)-growth
- On the L q spectra of in-homogeneous self-similar measures
