Abstract
We give a self-contained introduction to (quasi-)Banach modulation spaces of ultradistributions, and review results on boundedness for multiplications and convolutions for elements in such spaces. Furthermore, we use these results to study the Gabor product. As an example, we show how it appears in a phase-space formulation of the nonlinear cubic Schrödinger equation.
Access provided by Autonomous University of Puebla. Download chapter PDF
Similar content being viewed by others
Keywords
1 Introduction
Modulation spaces were introduced in Feichtinger’s seminal technical report [17], and prove themselves as useful family of Banach spaces of tempered distributions in time-frequency analysis, [4, 10, 28]. The main purpose of this survey article is to enlighten some properties of modulation spaces in a rather self-contained manner. In contrast to the most common situation, our analysis includes both quasi-Banach and Banach modulation spaces within the framework of ultradifferentiable functions and ultradistributions of Gelfand–Shilov type. For that reason we collect necessary background material in a rather detailed preliminary section.
Motivated by recent applications of modulation spaces in the context of nonlinear harmonic analysis and its applications, cf. [4,5,6, 14, 22, 38, 39, 47, 54] we focus our attention to boundedness for multiplications and convolutions for elements in such spaces. The basic results in that direction go back to the original contribution [17], and were thereafter reconsidered by many authors in different contexts. Let us give a brief, and unavoidably incomplete account on the related results.
In Sect. 3 we formulate in Theorems 3.5 and 3.7 bilinear versions of more general multiplication and convolution results in [54, Section 3]. The contents of Theorems 3.5 and 3.7 in the unweighted case for modulation spaces M p, q can be summarized as follows.
Proposition 1.1
Let p j, q j ∈ (0, ∞], j = 0, 1, 2,
Then
The general multiplication and convolution properties in Sect. 3 also overlap with results by Bastianoni, Cordero and Nicola in [2], by Bastianoni and Teofanov in [1], and by Guo et al. in [32].
The multiplication relation in Proposition 1.1 for p j, q j ≥ 1 was obtained already in [17] by Feichtinger. It is also obvious that the convolution relation was well-known since then (though a first formal proof of this relation seems to be given first in [48]). In general, these convolution and multiplication properties follow the rules
and
which goes back to [17] in the Banach space case and to [25] in the quasi-Banach case. See also [19] and [42] for extensions of these relations to more general Banach function spaces and quasi-Banach function spaces, respectively.
In Sect. 3 we basically review some results from [54]. To make this survey self-contained we give the proof of Theorem 3.7 in unweighted case. In contrast to [32], we do not deduce any sharpness for our results.
To show Proposition 1.1 in the quasi-Banach setting, apart from the usual use of Hölder’s and Young’s inequalities, additional arguments are needed. In our situation we discretize the situations in similar ways as in [2] by using Gabor analysis for modulation spaces, and then apply some further arguments, valid in non-convex analysis. This approach is slightly different compared to what is used in [32] which follows the discretization technique introduced in [55], and which has some traces of Gabor analysis.
We refer to [54] for a detailed discussion on the uniqueness of multiplications and convolutions in Proposition 1.1.
In Sect. 4 we apply the results from previous parts in the framework of the so called Gabor product. It is introduced in [14] in order to derive a phase space analogue to the usual convolution identity for the Fourier transform. The main motivation is to use such kind of products in a phase-space formulation of certain nonlinear equations. As noticed in [14], among other interesting characteristics of phase-space representations, the initial value problem in phase-space may be well-posed for more general initial distributions. This means that the phase-space formulation could contain solutions other than the standard ones. We refer to [11,12,13], where the phase-space extensions are explored in different contexts. Here we illustrate this approach by considering the nonlinear cubic Schrödinger equation, which appear for example in Bose-Einstein condensate theory [35]. We also refer to [4, Chapter 7] for an overview of results related to well-posedness of the nonlinear Schrödinger equations in the framework of modulation spaces, see also [3, 38, 39].
2 Preliminaries
In this section we give an exposition of background material related to the definition and basic properties of modulation spaces. Thus we recall some facts on the short-time Fourier transform and related projections, the (Fourier invariant) Gelfand-Shilov spaces, weight functions, and mixed-norm spaces of Lebesgue type. We also recall convolution and multiplication in weighted Lebesgue sequence spaces.
2.1 The Short-Time Fourier Transform
In what follows we let \(\mathcal F\) be the Fourier transform which takes the form
when \(f\in L^1(\mathbb R^{d})\). Here 〈 ⋅ , ⋅ 〉 denotes the usual scalar product on \(\mathbb R^{d}\). The same notation is used for the usual dual form between test functions and corresponding (ultra-)distributions. We recall that map \(\mathcal F\) extends uniquely to a homeomorphism on the space of tempered distributions \(\mathcal S^{\prime }(\mathbb R^{d})\), to a unitary operator on \(L^2(\mathbb R^{d})\) and restricts to a homeomorphism on the Schwartz space of smooth rapidly decreasing functions \(\mathcal S(\mathbb R^{d})\), cf. (29). We also observe with our choice of the Fourier transform, the usual convolution identity for the Fourier transform takes the forms
when \(f,g\in \mathcal S(\mathbb R^{d})\).
In several situations it is convenient to use a localized version of the Fourier transform, called the short-time Fourier transform, STFT for short. The short-time Fourier transform of \(f\in \mathcal S^{\prime }(\mathbb R^{d})\) with respect to the fixed window function \(\phi \in \mathcal S(\mathbb R^{d})\) is defined by
Here \(( \cdot \, ,\, \cdot \, )_{L^2}\) denotes the unique continuous extension of the inner product on \(L^2(\mathbb R^{d})\) restricted to \(\mathcal S(\mathbb R^{d})\) into a continuous map from \(\mathcal S^{\prime }(\mathbb R^{d}) \times \mathcal S(\mathbb R^{d})\) to \(\mathbb C\).
We observe that using certain properties for tensor products of distributions,
(cf. [33, 52]). If in addition \(f\in L^p(\mathbb R^{d})\) for some p ∈ [1, ∞], then
We observe that the domain of V ϕ is \(\mathcal S^{\prime }(\mathbb R^{d})\). The images are contained in \(C^\infty (\mathbb R^{2d})\), the set of smooth functions defined on the phase space \(\mathbb R^{d}\times \mathbb R^{d}\simeq \mathbb R^{2d}\).
The short-time Fourier transform appears in different contexts and under different names. In quantum mechanics it is rather common to call it the coherent state transform (see e.g. [37]). It is also closely related to the so-called Wigner distribution or radar ambiguity function (see e.g. [36]). In time-frequency analysis, it is also sometimes called the Voice transform.
The main idea with the design of short-time Fourier transform is to get the Fourier content, or the frequency resolution of localized functions and distributions. Roughly speaking, short-time Fourier transforms give a simultaneous information both on functions or distributions themselves as well as their Fourier transforms in the sense that the map
resembles on f(x), while the map
resembles on \(\widehat f(\xi )\).
As for the ordinary Fourier transform, there are several mapping properties which hold true for the short-time Fourier transform. As an elegant way to approach such properties in the framework of distributions, we may follow ideas given in [24] by Folland.
In fact, let T be the semi-conjugated tensor map
U be the linear pullback
and \(\mathcal F _2\) be the partial Fourier transform given by
Then
when \(f,\phi \in \mathcal S(\mathbb R^{d})\).
We observe that the mappings
are continuous and uniquely extendable to continuous mappings
which in turn restricts to isometric mappings
Here that T is isometric means that
It is now natural to define V ϕ f as the right-hand side of (6) when \(f,\phi \in \mathcal S^{\prime }(\mathbb R^{d})\), in which V ϕ f is well-defined as an element in \(\mathcal S^{\prime }(\mathbb R^{2d})\).
Proposition 2.1
The map
is continuous, which extends uniquely to a continuous map
which in turn restricts to an isometric map
If \(\phi \in \mathcal S(\mathbb R^{d})\) and \(f\in \mathcal S^{\prime }(\mathbb R^{d})\), then (11) shows that \(V_\phi f\in \mathcal S^{\prime }(\mathbb R^{2d})\). On the other hand, it is easy to see that the right-hand side of (2) defines a smooth function. Consequently beside (11) and (10), we also have the continuous map
For short-time Fourier transform, the Parseval identity is replaced by the so-called Moyal identity, also known as the orthogonality relation given by
when \(f,g,\phi ,\psi \in \mathcal S(\mathbb R^{d})\). The identity (14) is obtained by rewriting the short-time Fourier transforms by (2)′ and then applying the Parseval identity in suitable ways. We observe that the right-hand side makes sense also when f, g, ϕ and ψ belong to other spaces than \(\mathcal S(\mathbb R^{d})\). For example we may let
when p, p ′, q, q ′∈ [1, ∞] satisfy
By Moyal’s identity (14) it follows that if \(\phi \in \mathcal S(\mathbb R^{d})\setminus \{ 0 \}\), then the identity operator on \(\mathcal S^{\prime }(\mathbb R^{d})\) is given by
provided suitable mapping properties of the (L 2-)adjoint \(V_\phi ^*\) of V ϕ can be established. Obviously, \(V_\phi ^*\) fullfils
when \(F\in \mathcal S(\mathbb R^{2d})\) and \(g\in \mathcal S(\mathbb R^{d})\).
By expressing the scalar product and the short-time Fourier transform in terms of integrals in (17), it follows by straight-forward manipulations that the adjoint in (17) is given by
when \(F\in \mathcal S(\mathbb R^{2d})\). We may now use mapping properties like (11)–(12) to extend the definition of \(V_\phi ^*F\) when F and ϕ belong to various classes of function and distribution spaces. For example, by (11), (10) and (12), it follows that the map
defines a sesqui-linear form on \(\mathcal S(\mathbb R^{2d})\times \mathcal S^{\prime }(\mathbb R^{d})\), \(\mathcal S^{\prime }(\mathbb R^{2d})\times \mathcal S(\mathbb R^{d})\) and on \(L^2(\mathbb R^{2d})\times L^2(\mathbb R^{d})\). This implies that if \(\phi \in \mathcal S(\mathbb R^{d})\), then \(V_\phi ^*\) in (17) is continuous from \(\mathcal S(\mathbb R^{2d})\) to \(\mathcal S(\mathbb R^{d})\) which is uniquely extendable to a continuous map \(\mathcal S^{\prime }(\mathbb R^{2d})\) to \(\mathcal S^{\prime }(\mathbb R^{d})\), and to \(L^2(\mathbb R^{2d})\) to \(L^2(\mathbb R^{d})\). That is, the mappings
are continuous.
2.2 STFT Projections and a Suitable Twisted Convolution
If \(\phi \in \mathcal S(\mathbb R^{d})\) satisfies \(\Vert \phi \Vert _{L^2}=1\), then (16) shows that \(V_\phi ^*\circ V_\phi \) is the identity operator on \(\mathcal S^{\prime }(\mathbb R^{d})\). If we swap the order of this composition we get certain types of projections. In fact, for any \(\phi \in \mathcal S(\mathbb R^{d})\setminus \{ 0 \}\), let P ϕ be the operator given by
We observe that P ϕ is continuous on \(\mathcal S(\mathbb R^{2d})\), \(L^2(\mathbb R^{2d})\) and \(\mathcal S^{\prime }(\mathbb R^{2d})\) due to the mapping properties for V ϕ and \(V_\phi ^*\) above.
It is clear that \(P_\phi ^*=P_\phi \), i.e. P ϕ is self-adjoint. Furthermore, P ϕ is an projection:
Hence,
which shows that P ϕ is an orthonormal projection.
The ranks of P ϕ are given by
In fact, if \(F\in \mathcal S^{\prime }(\mathbb R^{2d})\), then
where \(f=\Vert \phi \Vert _{L^2}^{-2}V_\phi ^*F\in \mathcal S^{\prime }(\mathbb R^{d})\). This shows that \(P_\phi (\mathcal S^{\prime }(\mathbb R^{2d})) \subseteq V_\phi (\mathcal S^{\prime }(\mathbb R^{d}))\). On the other hand, if \(f\in \mathcal S^{\prime }(\mathbb R^{d})\) and F = V ϕ f, then
which shows that any element in \(V_\phi (\mathcal S^{\prime }(\mathbb R^{d}))\) equals an element in \(P_\phi (\mathcal S^{\prime }(\mathbb R^{2d}))\), i.e. \(P_\phi (\mathcal S^{\prime }(\mathbb R^{2d})) = V_\phi (\mathcal S^{\prime }(\mathbb R^{d}))\). This gives the last identity in (22). In the same way, the first two identities are obtained.
Remark 2.2
Let \(F\in \mathcal S^{\prime }(\mathbb R^{2d})\). Then it follows from the last identity in (22) that F = V ϕ f for some \(f\in \mathcal S^{\prime }(\mathbb R^{d})\), if and only if
Furthermore, if (23) holds, then F = V ϕ f with
There is a twisted convolution which is linked to the projection in (20). In fact, if \(F\in \mathcal S(\mathbb R^{2d})\) and \(\phi \in \mathcal S(\mathbb R^{d})\setminus \{ 0 \}\), then it follows by expanding the integrals for V ϕ and \(V_\phi ^*\) in (20), and performing some straight-forward manipulations that
where the twisted convolution ∗V is defined by
when \(F,G\in \mathcal S(\mathbb R^{2d})\). We observe that the definition of ∗V is uniquely extendable in different ways. For example, Young’s inequality for ordinary convolution also holds for the twisted convolution. Moreover, the map (F, G)↦F ∗V G extends uniquely to continuous mappings from \(\mathcal S(\mathbb R^{2d}) \times \mathcal S^{\prime }(\mathbb R^{2d})\) or \(\mathcal S^{\prime }(\mathbb R^{2d})\times \mathcal S(\mathbb R^{2d})\) to \(\mathcal S^{\prime }(\mathbb R^{2d})\). By straight-forward computations it follows that
when \(F,H\in \mathcal S(\mathbb R^{2d})\) and \(G\in \mathcal S^{\prime }(\mathbb R^{2d})\), or \(F,H\in \mathcal S^{\prime }(\mathbb R^{2d})\) and \(G\in \mathcal S(\mathbb R^{2d})\).
Let \(f\in \mathcal S^{\prime }(\mathbb R^{d})\) and \(\phi _j\in \mathcal S(\mathbb R^{d})\), j = 1, 2, 3. By straight-forward applications of Parseval’s formula it follows that
which is some sort of reproducing kernel of short-time Fourier transforms in the background of ∗V.
2.3 Gelfand-Shilov Spaces
Before defining the Gelfand-Shilov spaces, we recall that the Schwartz space \(\mathcal S(\mathbb R^{d})\) consists of all (complex-valued) smooth functions \(f\in C^\infty (\mathbb R^{d})\) such that
for some constants C α,β > 0, which only depend on the multi-indices \(\alpha ,\beta \in \mathbb N^{d}\). The Schwartz space possess several convenient properties, and is heavily used in mathematics, science and technology. For example, the Schwartz space is invariant under Fourier transformation. By duality the same holds true for its (L 2-)dual \(\mathcal S^{\prime }(\mathbb R^{d})\), the set of tempered distributions on \(\mathbb R^{d}\).
On the other hand, we observe that there are no conditions on the growths of the constants C α,β with respect to \(\alpha ,\beta \in \mathbb N^{d}\). This implies that in the context of the spaces \(\mathcal S(\mathbb R^{d})\) and \(\mathcal S^{\prime }(\mathbb R^{d})\), it is almost impossible to investigate important properties like analyticity or related regularity properties which are stronger than pure smoothness. For investigating such stronger regularity properties, we need to modify \(\mathcal S(\mathbb R^{d})\) and the estimate (29) by imposing suitable growth conditions on the constants C α,β. This leads to the definition of Gelfand-Shilov spaces, [26, 40].
We only discuss Fourier invariant Gelfand-Shilov spaces and their properties. Let \(0<s\in \mathbb R\) be fixed. We have two different types of Gelfand-Shilov spaces. The Gelfand-Shilov space \(\mathcal S_{s}(\mathbb R^{d})\) of Roumieu type with parameter s > 0 consists of all \(f\in C^\infty (\mathbb R^{d})\) such that
for some constants C, h > 0. In the same way, the Gelfand-Shilov space \(\Sigma _{s}(\mathbb R^{d})\) of Beurling type with parameter s > 0 consists of all \(f\in C^\infty (\mathbb R^{d})\) such that for every h > 0, there is a constant C = C h > 0 such that (30) holds. Hence, in comparison with the definition of Schwartz functions, we have limited ourself to constants C α,β in (29) which are not allowed to grow faster than those of the form
when dealing with Gelfand-Shilov spaces.
It can be proved that \(\mathcal S _s(\mathbb R^{d})\) and \(\Sigma _t(\mathbb R^{d})\) are dense in \(\mathcal S(\mathbb R^{d})\) when \(s\ge \frac 12\) and \(t>\frac 12\). We call such s and t admissible. On the other hand, for the other choices of s and t we have
One has that \(\mathcal S _1(\mathbb R^{d})\) consists of real analytic functions, and that \(\Sigma _1(\mathbb R^{d})\) consists of smooth functions on \(\mathbb R^{d}\) which are extendable to entire functions on \(\mathbb C^{d}\). The topologies of \(\mathcal S_{s}(\mathbb R^{d})\) and \(\Sigma _{s}(\mathbb R^{d})\) are defined by the semi-norms
Here the supremum should be taken over all \(\alpha ,\beta \in \mathbb N^{d}\) and \(x\in \mathbb R^{d}\). We equip \(\mathcal S_{s}(\mathbb R^{d})\) and \(\Sigma _{s}(\mathbb R^{d})\) by the canonical inductive limit topology and projective limit topology, respectively, with respect to h > 0, which are induced by the semi-norms in (31).
Let \(\mathcal S_{s,h}(\mathbb R^{d})\) be the Banach space which consists of all \(f\in C^\infty (\mathbb R^{d})\) such that \(\Vert f\Vert _{\mathcal S_{s,h}}\) in (31) is finite, and let \(\mathcal S_{s,h}^{\prime }(\mathbb R^{d})\) be the (L 2-)dual of \(\mathcal S_{s,h}(\mathbb R^{d})\). If \(s\ge \frac 12\), then the Gelfand-Shilov distribution space \(\mathcal S_{s}^{\prime }(\mathbb R^{d})\) of Roumieu type is the projective limit of \(\mathcal S_{s,h}^{\prime }(\mathbb R^{d})\) with respect to h > 0. If instead \(s> \frac 12\), then the Gelfand-Shilov distribution space \(\Sigma _{s}^{\prime }(\mathbb R^{d})\) of Beurling type is the inductive limit of \(\mathcal S_{s,h}^{\prime }(\mathbb R^{d})\) with respect to h > 0. Consequently, for admissible s we have
It can be proved that \(\mathcal S_{s}^{\prime }(\mathbb R^{d})\) and \(\Sigma _{s}^{\prime }(\mathbb R^{d})\) are the (strong) duals to \(\mathcal S_{s}(\mathbb R^{d})\) and \(\Sigma _{s}(\mathbb R^{d})\), respectively.
We have the following embeddings and density properties for Gelfand-Shilov and Schwartz spaces
with dense embeddings. Here A↪B means that the topological spaces A and B satisfy A ⊆ B with continuous embeddings.
The Fourier transform possess convenient mapping properties on Gelfand-Shilov spaces and their distribution spaces. In fact, the Fourier transform extends uniquely to homeomorphisms on \(\mathcal S _s^{\prime }(\mathbb R^{d})\) and on \(\Sigma _s^{\prime }(\mathbb R^{d})\) for admissible s. Furthermore, \(\mathcal F\) restricts to homeomorphisms on \(\mathcal S _s(\mathbb R^{d})\) and on \(\Sigma _s(\mathbb R^{d})\).
One of the most important characterizations of Gelfand-Shilov spaces is performed in terms of estimates of the functions and their Fourier transforms. More precisely, in [8, 15] it is proved that if \(f\in \mathcal S^{\prime }(\mathbb R^{d})\) and s > 0, then \(f\in \mathcal S _s(\mathbb R^{d})\) (\(f\in \Sigma _s(\mathbb R^{d})\)), if and only if
for some r > 0 (for every r > 0). Here \(g_1 \lesssim g_2\) means that g 1(θ) ≤ c ⋅ g 2(θ) holds uniformly for all θ in the intersection of the domains of g 1 and g 2 and for some constant c > 0, and we write g 1 ≍ g 2 when \(g_1\lesssim g_2 \lesssim g_1\).
The analysis in [8, 15] can also be applied on the Schwartz space, from which it follows that an element \(f\in \mathcal S^{\prime }(\mathbb R^{d})\) belongs to \(\mathcal S(\mathbb R^{d})\), if and only if
for every N ≥ 0. Here and in what follows we let
Remark 2.3
Several properties in Sects. 2.1–2.3 in the background of \(\mathcal S(\mathbb R^{d})\) and \(\mathcal S^{\prime }(\mathbb R^{d})\) also hold for the Gelfand-Shilov spaces and their distribution spaces. Let \(s\ge \frac 12\). By similar arguments which lead to Proposition 2.1 and (13), it follows that
is continuous, which extends uniquely to continuous mappings
and
It follows that (14) makes sense after each \(\mathcal S\) in (15) are replaced by \(\mathcal S _s\). Let \(\phi \in \mathcal S _s(\mathbb R^{d})\setminus \{ 0 \}\) be fixed. Then by similar arguments which lead to (19) give that the mappings
are continuous. For P ϕ in (20) we have that (21) still holds true and that (22) can be completed with
We also have that the twisted convolution in (26) is continuous from \(\mathcal S _s(\mathbb R^{2d})\times \mathcal S _s(\mathbb R^{2d})\) to \(\mathcal S _s(\mathbb R^{2d})\) and uniquely extendable to a continuous map \(\mathcal S _s(\mathbb R^{2d})\times \mathcal S _s^{\prime }(\mathbb R^{2d})\) or \(\mathcal S _s^{\prime }(\mathbb R^{2d})\times \mathcal S _s(\mathbb R^{2d})\) to \(\mathcal S _s^{\prime }(\mathbb R^{2d})\), and that the formulae (25)–(28) still hold true after each \(\mathcal S\) is replaced by \(\mathcal S _s\) in the attached assumptions.
If instead \(s>\frac 12\), then similar facts hold true with Σs in place of \(\mathcal S _s\) above, at each occurrence.
Remark 2.4
In similar ways as characterizing Gelfand-Shilov spaces in terms of Fourier estimates (see (33)), we may also use the short-time Fourier transform to perform similar characterizations. Moreover, the short-time Fourier transform can in addition be used to characterize spaces of Gelfand-Shilov distributions.
In fact, let \(\phi \in \mathcal S _s(\mathbb R^{d})\setminus \{ 0 \}\) (\(\phi \in \Sigma _s(\mathbb R^{d})\setminus \{ 0 \}\)) be fixed and let f be a Gelfand-Shilov distribution on \(\mathbb R^{d}\). Then the following is true:
-
1.
\(f\in \mathcal S _s(\mathbb R^{d})\) (\(f\in \Sigma _s(\mathbb R^{d})\)), if and only if
$$\displaystyle \begin{aligned} |V_\phi f(x,\xi )|\lesssim e^{-r(|x|{}^{\frac 1s} +|\xi |{}^{\frac 1s})} \end{aligned} $$(39)for some r > 0 (for every r > 0);
-
2.
\(f\in \mathcal S _s^{\prime }(\mathbb R^{d})\) (\(f\in \Sigma _s^{\prime }(\mathbb R^{d})\)), if and only if
$$\displaystyle \begin{aligned} |V_\phi f(x,\xi )|\lesssim e^{r(|x|{}^{\frac 1s} +|\xi |{}^{\frac 1s})} \end{aligned} $$(40)for every r > 0 (for some r > 0).
We refer to [31, Theorem 2.7] for the characterization 1. concerning Gelfand-Shilov functions and to [51, Proposition 2.2]) for the characterization 2. concerning Gelfand-Shilov distributions.
2.4 Weight Functions
A weight or weight function on \(\mathbb R^{d}\) is a positive function \(\omega \in L^\infty _{loc}(\mathbb R^{d})\) such that \(1/\omega \in L^\infty _{loc}(\mathbb R^{d})\). The weight ω is called moderate, if there is a positive weight v on \(\mathbb R^{d}\) and a constant C ≥ 1 such that
If ω and v are weights on \(\mathbb R^{d}\) such that (41) holds, then ω is also called v-moderate. We note that (41) implies that ω fulfills the estimates
We let \(\mathcal P _E(\mathbb R^{d})\) be the set of all moderate weights on \(\mathbb R^{d}\).
We say that v is submultiplicative if
We observe that if \(v\in \mathcal P _E(\mathbb R^{d})\) is even and satisfies
for some constant C > 0, then for v 0 = C 1∕2 v, one has that \(v_0\in \mathcal P _E(\mathbb R^{d})\) is submultiplicative and v ≍ v 0 (see e.g. [17, 19, 28]).
We also recall from [29] that if v is positive and locally bounded and satisfies (44), then \(v(x)\le C_0e^{r_0|x|}\) for some positive constants C 0 and r 0. In fact, if \(x\in \mathbb R^{d}\),
and n is an integer such that n − 1 ≤|x|≤ n, then (44) gives
which gives the statement.
Therefore, if v is a submultiplicative weight, then
for some r ≥ 0. Hence, if \(\omega \in \mathcal P _E(\mathbb R^{d})\), then (41) and (45) imply
for some r > 0. In particular, (42) shows that for any \(\omega _0\in \mathcal P_E(\mathbb R^{d})\), there is a constant r > 0 such that
If (41) holds, then there is a smallest positive even function v 0 such that (41) holds with C = 1. We remark that this v 0 is given by
and is submultiplicative (see e.g. [19, 27, 49]). Consequently, if ω is a moderate weight, then it is also moderated by a submultiplicative weight. In the sequel, v and v j for j ≥ 0, always stand for submultiplicative weights if nothing else is stated.
We also remark that in the literature it is common to define submultiplicative weights as (43) should hold, without the condition v(−x) = v(x), i.e. that v does not have to be even (cf. e.g. [17, 19, 25, 28]). However, in the sequel it is convenient for us to include this property in the definition.
There are several subclasses of \(\mathcal P _E(\mathbb R^{d})\) which are interesting for different reasons. Though our results later on are formulated in background of weights in \(\mathcal P _E(\mathbb R^{d})\), we here mention some subclasses which especially appear in time-frequency analysis. First we observe the class \(\mathcal P ^{0} _E(\mathbb R^{d})\), which consists of all \(\omega \in \mathcal P _E(\mathbb R^{d})\) such that (46) holds for every r > 0.
The class \(\mathcal P _E^0(\mathbb R^{d})\) is important when dealing with spectral invariance for matrix or convolution operators on \(\ell ^2(\mathbb Z^{d})\) (see e.g. [30]). If \(v\in \mathcal P _E(\mathbb R^{d})\) is submultiplicative, then \(v\in \mathcal P _E^0(\mathbb R^{d})\), if and only if
(see e.g. [23]). The condition (47) is equivalent to
and is usually called the GRS condition, or Gelfand-Raikov-Shilov condition.
A more restrictive condition on v compared to (47)′ is given by the Beurling-Domar condition
This condition is strongly linked to non quasi-analytic classes which contain non-trivial compactly supported elements (see e.g. [29]). Any subexponential submultiplicative weight satisfies the Beurling-Domar condition. That is, suppose that θ ∈ (0, 1) and that \(v(x)=e^{r|x|{ }^\theta }\), \(x\in \mathbb R^{d}\), then (48) is fulfilled. We let \(\mathcal P _{\operatorname {BD}}(\mathbb R^{d})\) be the set of all weights which are moderated by submultiplicative weights which satisfy the Beurling-Domar condition.
Finally we let \(\mathcal P (\mathbb R^{d})\) be the set of all weights on \(\mathbb R^{d}\) which are moderated by polynomially bounded functions. That is, \(\omega \in \mathcal P (\mathbb R^{d})\), if and only if there are positive constants r and C such that
Here we observe that v(x) = (1 + |x|)r is submultiplicative.
Among these weight classes we have
In fact, it is clear that the ordering in (49) holds. On the other hand, if r > 0 and θ ∈ (0, 1), then due to
it also follows that the inclusions in (49) are strict.
We refer to [16, 28, 29, 49] for more facts about weights in time-frequency analysis.
2.5 Mixed Norm Spaces of Lebesgue Type
For every p, q ∈ (0, ∞] and weight ω on \(\mathbb R^{2d}\), we set
and
when F is (complex-valued) measurable function on \(\mathbb R^{2d}\). Then \(L^{p,q}_{(\omega )}(\mathbb R^{2d})\) (\(L^{p,q}_{*,(\omega )}(\mathbb R^{2d})\)) consists of all measurable functions F such that \(\Vert F\Vert _{L_{(\omega )}^{p,q}}<\infty \) (\(\Vert F\Vert _{L^{p,q}_{*,(\omega )}}<\infty \)).
In similar ways, let Ω1, Ω2 be discrete sets, ω be a positive function on Ω1 × Ω2 and \(\ell _0^{\prime }(\Omega _1\times \Omega _2)\) be the set of all formal (complex-valued) sequences \(c=\{ c(j,k)\} _{j\in \Omega _1,k\in \Omega _2}\). Then the discrete Lebesgue spaces, i.e. the Lebesgue sequence spaces
of mixed (quasi-)norm types consist of all \(c\in \ell _0^{\prime }(\Omega _1\times \Omega _2)\) such that \(\Vert c\Vert _{\ell _{(\omega )}^{p,q}(\Omega _1\times \Omega _2)}<\infty \) respectively \(\Vert c\Vert _{\ell _{*,(\omega )}^{p,q}(\Omega _1\times \Omega _2)}<\infty \). Here
and
when \(c\in \ell _0^{\prime }(\Omega _1\times \Omega _2)\).
2.6 Convolutions and Multiplications for Discrete Lebesgue Spaces
Next we discuss extended Hölder and Young relations for multiplications and convolutions on discrete Lebesgue spaces. The Hölder and Young conditions on Lebesgue exponent are then
respectively
Notice that, when p 1, p 2 ∈ (0, 1), then (52) becomes \( p_0 \geq \max \{ p_1, p_2 \} \), while for p 1, p 2 ≥ 1 it reduces to the common Young condition
The conditions on the weight functions are
respectively
where Λ is a lattice of the form
where e 1, …e d is a basis for \(\mathbb R^{d}\).
Proposition 2.5
Let p j, q j ∈ (0, ∞], j = 0, 1, 2, be such that (51) and (52) hold, let \(\Lambda \subseteq \mathbb R^{d}\) be a lattice and let ω j be weights on Λ, j = 0, 1, 2. Then the following is true:
-
1.
if (53) holds, then the map (a 1, a 2)↦a 1 ⋅ a 2 from ℓ 0( Λ) × ℓ 0( Λ) to ℓ 0( Λ) extends uniquely to a continuous map from \(\ell ^{q_1}_{(\omega _1)}(\Lambda ) \times \ell ^{q_2}_{(\omega _2)}(\Lambda )\) to \(\ell ^{q_0}_{(\omega _0)}(\Lambda )\) , and
$$\displaystyle \begin{aligned} \Vert a_1\cdot a_2\Vert _{\ell ^{q_0}_{(\omega _0)}} \le \Vert a_1\Vert _{\ell ^{q_1}_{(\omega _1)}} \Vert a_2\Vert _{\ell ^{q_2}_{(\omega _2)}}, \qquad a_j\in \ell ^{q_j}_{(\omega _j)}(\Lambda ),\ j=1,2 \text ; \end{aligned} $$(55) -
2.
if (54) holds, then the map (a 1, a 2)↦a 1 ∗ a 2 from ℓ 0( Λ) × ℓ 0( Λ) to ℓ 0( Λ) extends uniquely to a continuous map from \(\ell ^{p_1}_{(\omega _1)}(\Lambda ) \times \ell ^{p_2}_{(\omega _2)}(\Lambda )\) to \(\ell ^{p_0}_{(\omega _0)}(\Lambda )\) , and
$$\displaystyle \begin{aligned} \Vert a_1*a_2\Vert _{\ell ^{p_0}_{(\omega _0)}} \le \Vert a_1\Vert _{\ell ^{p_1}_{(\omega _1)}}\Vert a_2\Vert _{\ell ^{p_2}_{(\omega _2)}}, \qquad a_j\in \ell ^{p_j}_{(\omega _j)}(\Lambda ),\ j=1 ,2 \text . \end{aligned} $$(56)
The assertion 1. in Proposition 2.5 is the standard Hölder’s inequality for discrete Lebesgue spaces. The assertion 2. in that proposition is the usual Young’s inequality for Lebesgue spaces on lattices in the case when p 0, p 1, p 2 ∈ [1, ∞]. A proof of Proposition 2.5 is given in Appendix A in [54].
3 Modulation Spaces, Multiplications and Convolutions
In this section we introduce modulation spaces, and recall their basic properties, in particular in the context of Gelfand-Shilov spaces. Notice that we permit the Lebesgue exponents to belong to the full interval (0, ∞] instead of the most common choice [1, ∞], and general moderate weights which may have a (sub)exponential growth. Here we also recall some facts on Gabor expansions for modulation spaces.
Then we deduce multiplication and convolution estimates on modulation spaces. There are several approaches to multiplication and convolution in the case when the involved Lebesgue exponents belong to [1, ∞] (see [9, 17, 19, 32, 43, 48]). Here we consider the case when these exponents belong to (0, ∞) (see also [1, 2, 25, 41, 42, 50]). In addition, and in order to keep the survey style of our exposition, we focus on the bilinear case, and refer to [54] for extension of these results to multi-linear products.
3.1 Modulation Spaces
The (classical) modulation spaces, essentially introduced in [17] by Feichtinger are given in the following. (See e.g. [18] for definition of more general modulation spaces.)
Definition 3.1
Let p, q ∈ (0, ∞], \(\omega \in \mathcal P _E(\mathbb R^{2d})\) and \(\phi \in \Sigma _1 (\mathbb R^{d})\setminus \{ 0 \}\).
-
1.
The modulation space \(M^{p,q}_{(\omega )}(\mathbb R^{d})\) consists of all \(f\in \Sigma _1 ^{\prime }(\mathbb R^{d})\) such that
$$\displaystyle \begin{aligned} \Vert f\Vert _{M^{p,q}_{(\omega )}}\equiv \Vert V_\phi f\Vert _{L^{p,q}_{(\omega )}} \end{aligned}$$is finite. The topology of \(M^{p,q}_{(\omega )}(\mathbb R^{d})\) is defined by the (quasi-)norm \(\Vert \, \cdot \,\Vert _{M^{p,q}_{(\omega )}}\);
-
2.
The modulation space (of Wiener amalgam type) \(W^{p,q}_{(\omega )}(\mathbb R^{d})\) consists of all \(f\in \Sigma _1 ^{\prime }(\mathbb R^{d})\) such that
$$\displaystyle \begin{aligned} \Vert f\Vert _{W^{p,q}_{(\omega )}}\equiv \Vert V_\phi f\Vert _{L^{p,q}_{*,(\omega )}} \end{aligned}$$is finite. The topology of \(W^{p,q}_{(\omega )}(\mathbb R^{d})\) is defined by the (quasi-)norm \(\Vert \, \cdot \,\Vert _{W^{p,q}_{(\omega )}}\).
For convenience we set \(M^{p,q}=M^{p,q}_{(\omega )}\) and \(W^{p,q}=W^{p,q}_{(\omega )}\) when the weight ω is trivial, i.e. when ω(x, ξ) = 1 for every \(x,\xi \in \mathbb R^{d}\). We also set
Remark 3.2
Modulation spaces possess several convenient properties. Let p, q ∈ (0, ∞], \(\omega \in \mathcal P _E(\mathbb R^{2d})\) and \(\phi \in \Sigma _1(\mathbb R^{d})\setminus \{ 0 \}\). Then the following is true (see [17,18,19,20, 25, 28] and their analyses for verifications):
-
the definitions of \(M_{(\omega )}^{p,q}(\mathbb R^{d})\) and \(W_{(\omega )}^{p,q}(\mathbb R^{d})\) are independent of the choices of \(\phi \in \Sigma _1 (\mathbb R^{d})\setminus \{ 0 \}\), and different choices give rise to equivalent quasi-norms;
-
the spaces \(M^{p,q}_{(\omega )}(\mathbb R^{d})\) and \(W^{p,q}_{(\omega )}(\mathbb R^{d})\) are quasi-Banach spaces which increase with p and q, and decrease with ω. If in addition p, q ≥ 1, then they are Banach spaces;
-
If p, q ≥ 1, then the \(L^2(\mathbb R^{d})\) scalar product, \(( \cdot \, ,\, \cdot \, )_{L^2(\mathbb R^{d})}\), on \(\Sigma _1 (\mathbb R^{d})\times \Sigma _1 (\mathbb R^{d})\) is uniquely extendable to dualities between \(M^{p,q}_{(\omega )}(\mathbb R^{d})\) and \(M^{p^{\prime },q^{\prime }}_{(1/\omega )}(\mathbb R^{d})\), and between \(W^{p,q}_{(\omega )}(\mathbb R^{d})\) and \(W^{p^{\prime },q^{\prime }}_{(1/\omega )}(\mathbb R^{d})\). If in addition p, q < ∞, then the dual spaces of \(M^{p,q}_{(\omega )}(\mathbb R^{d})\) and \(W^{p,q}_{(\omega )}(\mathbb R^{d})\) can be identified with \(M^{p^{\prime },q^{\prime }}_{(1/\omega )}(\mathbb R^{d})\) respectively \(W^{p^{\prime },q^{\prime }}_{(1/\omega )}(\mathbb R^{d})\), through the form \(( \cdot \, ,\, \cdot \, )_{L^2(\mathbb R^{d})}\);
-
if ω 0(x, ξ) = ω(−ξ, x), then \(\mathcal F\) on \(\Sigma _1 ^{\prime }(\mathbb R^{d})\) restricts to a homeomorphism from \(M^{p,q}_{(\omega )}(\mathbb R^{d})\) to \(W^{q,p}_{(\omega _0)}(\mathbb R^{d})\).
-
The inclusions
$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} \Sigma _1 (\mathbb R^{d}) &\subseteq & M^{p,q}_{(\omega )}(\mathbb R^{d}),W^{p,q}_{(\omega )}(\mathbb R^{d}) &\subseteq &\Sigma _1 ^{\prime}(\mathbb R^{d}) \quad &\text{when} & \quad \omega \in \mathcal P _E(\mathbb R^{2d}), {} \end{array}\end{aligned} $$(57)$$\displaystyle \begin{aligned}\begin{array}{r*{20}l}[1ex] \mathcal S _1 (\mathbb R^{d}) &\subseteq & M^{p,q}_{(\omega )}(\mathbb R^{d}),W^{p,q}_{(\omega )}(\mathbb R^{d}) &\subseteq &\mathcal S _1 ^{\prime}(\mathbb R^{d}) \quad &\text{when} & \quad \omega \in \mathcal P _E^0(\mathbb R^{2d}) {} \end{array}\end{aligned} $$(58)and
$$\displaystyle \begin{aligned}\begin{array}{r*{20}l} \mathcal S(\mathbb R^{d}) &\subseteq & M^{p,q}_{(\omega )}(\mathbb R^{d}),W^{p,q}_{(\omega )}(\mathbb R^{d}) &\subseteq & \, \mathcal S^{\prime}(\mathbb R^{d}) \quad &\text{when} & \quad \omega \in \mathcal P (\mathbb R^{2d}) {} \end{array}\end{aligned} $$(59)are continuous. If in addition p, q < ∞, then these inclusions are dense.
We recall from [49] that the embeddings (57)–(59), are essentially special cases of certain characterizations of the Schwartz space, Gelfand-Shilov spaces and their distribution spaces in terms of suitable unions and intersections of modulation spaces. In fact, let p, q ∈ (0, ∞] and s ≥ 1 be fixed and set
where r > 0. Then
and
The topologies of the spaces on the left-hand sides of (61)–(66) are obtained by replacing each intersection by projective limit with respect to r > 0 and each union with inductive limit with respect to r > 0.
The relations (61)–(66) are essentially special cases of [49, Theorem 3.9], see also [31, 45, 46]. In order to be self-contained we here give a proof of (62).
Proof of (62)
Since
it suffices to prove the result for p = q = ∞. Let \(\phi \in \Sigma _1(\mathbb R^{d})\setminus \{ 0 \}\) be fixed. First suppose that
Then it follows from the definition of modulation space norm that (39) holds for some r > 0. By Remark 2.4 it follows that \(f\in \mathcal S _s(\mathbb R^{d})\), and we have proved
Suppose instead that \(f\in \mathcal S _s(\mathbb R^{d})\). Then (39) holds for some r > 0, giving that \(f\in M^\infty _{(v_{r,s})}(\mathbb R^{d})\). Hence (67) holds with reversed inclusion, and the result follows. □
Example 3.3
Let p = q = 1 and ω = 1. Then \(M^{1,1} _{(\omega )} (\mathbb R^{d}) = M^{1} (\mathbb R^{d})\) is the Feichtinger algebra, probably the most prominent example of a modulation space. We refer to a recent survey [34] for a detailed account on \(M^{1} (\mathbb R^{d})\), and to [14, Lemma 11] for a list of its basic properties.
Familiar examples arise when p = q = 2 and ω = 1. Then \(M_{(\omega )} ^{2,2} (\mathbb R^{d}) =M^{2} (\mathbb R^{d}) = L^2 (\mathbb R^{d})\), and
where ω s(ξ) = 〈ξ〉s, and \(H^s (\mathbb R^{d})\) is the Sobolev space (also known as the Bessel potential space) of distributions \(f \in \mathcal S^{\prime }(\mathbb R^{d}) \) such that
cf. [28, Proposition 11.3.1]. Furthermore, if v s(x, ξ) = 〈(x, ξ)〉s, then \( M_{(v_{s})} ^{2,2} (\mathbb R^{d}) = Q_s (\mathbb R^{d})\), \(s \in \mathbb {R},\) [7, Lemma 2.3]. Here Q s denotes the Shubin-Sobolev space, [44].
Finally we remark that modulation spaces can be conveniently discretized in terms of Gabor expansions. In order for explaining some basic issues on this, in a similar way as in Subsection 1.5 in [54], we limit ourself to the case when the involved weights are moderated by subexponential functions. That is, we suppose that ω in \(M^{p,q}_{(\omega )}(\mathbb R^{d})\) satisfies
for some s > 1 and r > 0. We observe that this implies that
in vew of (42), (61) and (66). For more general approaches we refer to [19, 27, 28, 42, 50].
Since s > 1, it follows from Sections 1.3 and 1.4 in [33] that there are \(\phi ,\psi \in \Sigma _s (\mathbb R^{d})\) with values in [0, 1] such that
and
Let \(f\in \Sigma _s^{\prime }(\mathbb R^{d})\). Then x↦f(x)ϕ(x − j) belongs to \(\Sigma _s^{\prime }(\mathbb R^{d})\) and is supported in \(j+[-\frac 34,\frac 34]^d\). Hence, by periodization it follows from Fourier analysis that
where
Since ψ = 1 on the support of ϕ, (73) gives
By (72) it now follows that
where
which is the Gabor expansion of f with respect to the Gabor pair (ϕ, ψ) and lattice Λ, i.e. with respect to the Gabor atom ϕ and the dual Gabor atom ψ. Here the series converges in \(\Sigma _s^{\prime }(\mathbb R^{d})\). By duality and the fact that compactly supported elements in \(\Sigma _s(\mathbb R^{d})\) are dense in \(\Sigma _s^{\prime }(\mathbb R^{d})\) we also have
with convergence in \(\Sigma _s^{\prime }(\mathbb R^{d})\).
Let T be a linear continuous operator from \(\Sigma _s(\mathbb R^{d})\) to \(\Sigma _s^{\prime }(\mathbb R^{d})\) and let \(f\in \Sigma _s(\mathbb R^{d})\). Then it follows from (74) that
and
A combination of these expansions show that
where A = (a(j, k))j,k ∈ Λ is the Λ × Λ-matrix, given by
By the Gabor analysis for modulation spaces we get the following restatement of [54, Proposition 1.8]. We refer to [17, 19,20,21, 25, 27, 28, 50] for details.
Proposition 3.4
Let s > 1, p, q ∈ (0, ∞], \(\omega \in \mathcal P _E (\mathbb R^{2d})\) be such that (68) holds for some r > 0, \(\phi ,\psi \in \Sigma _s (\mathbb R^{d}) \) with values in [0, 1] be such that (70), (71) and (72) hold true, and let \(f\in \Sigma _s^{\prime }(\mathbb R^{d})\) . Then the following is true:
-
1.
\(f\in M^{p,q}_{(\omega )}(\mathbb R^{d})\) , if and only if \(\Vert V_\phi f\Vert _{\ell _{(\omega )}^{p,q}(\mathbb Z^{d}\times \pi \mathbb Z^{d})}<\infty \);
-
2.
\(f\in M^{p,q}_{(\omega )}(\mathbb R^{d})\) , if and only if \(\Vert V_\psi f\Vert _{\ell _{(\omega )}^{p,q}(\mathbb Z^{d}\times \pi \mathbb Z^{d})}<\infty \);
-
3.
the quasi-norms
$$\displaystyle \begin{aligned} f\mapsto \Vert V_\phi f\Vert _{\ell _{(\omega )}^{p,q}(\mathbb Z^{d}\times \pi \mathbb Z^{d})} \quad \mathit{\text{and}}\quad f\mapsto \Vert V_\psi f\Vert _{\ell _{(\omega )}^{p,q}(\mathbb Z^{d}\times \pi \mathbb Z^{d})} \end{aligned}$$are equivalent to \(\Vert \, \cdot \,\Vert _{M^{p,q}_{(\omega )}}\).
The same holds true with \(W^{p,q}_{(\omega )}\) and \(\ell _{*,(\omega )}^{p,q}\) in place of \(M^{p,q}_{(\omega )}\) respectively \(\ell _{(\omega )}^{p,q}\) at each occurrence.
3.2 Multiplications and Convolutions in Modulation Spaces
As a first step for approaching multiplications and convolutions for elements in modulation spaces, we reformulate such products in terms of short-time Fourier transforms. Let \(\phi _0,\phi _1 ,\phi _2\in \Sigma _1 (\mathbb R^{d})\) be fixed such that
and let \(f_1,f_2\in \Sigma _1 (\mathbb R^{d})\). Then the multiplication f 0 = f 1 f 2 can be expressed by
where
In fact, by Fourier’s inversion formula we get
We also observe that we may extract f 0 = f 1 f 2 by the formula
provided ϕ 0 is not trivially equal to 0.
In the same way, let \(\phi _0,\phi _1,\phi _2\in \Sigma _1 (\mathbb R^{d})\) be fixed such that
and let \(f_1,f_2,g\in \Sigma _1 (\mathbb R^{d})\). Then the convolution f 0 = f 1 ∗ f 2 can be expressed by
where F j are given by (81), and that we may extract f 0 = f 1 ∗ f 2 from (82).
Next we discuss convolutions and multiplications for modulation spaces, and start with the following convolution result for modulation spaces. For multiplications of elements in modulation spaces we need to swap the conditions for the involved Lebesgue exponents compared to (51) and (52). That is, these conditions become
or
The conditions on the weight functions are
respectively
Theorem 3.5
Let p j, q j ∈ (0, ∞) and \(\omega _j\in \mathcal P _E(\mathbb R^{2d})\) , j = 0, 1, 2, be such that (85) and (87) hold. Then (f 1, f 2)↦f 1 f 2 from \(\Sigma _1 (\mathbb R^{d})\times \Sigma _1(\mathbb R^{d})\) to \(\Sigma _1 (\mathbb R^{d})\) is uniquely extendable to a continuous map from \(M^{p_1,q_1}_{(\omega _1)}(\mathbb R^{d}) \times M^{p_2,q_2}_{(\omega _2)}(\mathbb R^{d})\) to \(M_{(\omega _0)}^{p_0,q_0}(\mathbb R^{d})\) , and
Theorem 3.6
Let p j, q j ∈ (0, ∞) and \(\omega _j\in \mathcal P _E(\mathbb R^{2d})\) , j = 0, 1, 2, be such that (86) and (87) hold. Then (f 1, f 2)↦f 1 f 2 from \(\Sigma _1 (\mathbb R^{d})\times \Sigma _1(\mathbb R^{d})\) to \(\Sigma _1 (\mathbb R^{d})\) is uniquely extendable to a continuous map from \(W^{p_1,q_1}_{(\omega _1)}(\mathbb R^{d}) \times W_{(\omega _2)}^{p_2,q_2}(\mathbb R^{d})\) to \(W_{(\omega _0)}^{p_0,q_0}(\mathbb R^{d})\) , and
The corresponding results for convolutions are the following. Here the conditions on the involved Lebesgue exponents are swapped as
or
Theorem 3.7
Let p j, q j ∈ (0, ∞) and \(\omega _j\in \mathcal P _E(\mathbb R^{2d})\) , j = 0, 1, 2, be such that (88) and (92) hold. Then (f 1, f 2)↦f 1 ∗ f 2 from \(\Sigma _1 (\mathbb R^{d})\times \Sigma _1(\mathbb R^{d})\) to \(\Sigma _1 (\mathbb R^{d})\) is uniquely extendable to a continuous map from \(M^{p_1,q_1}_{(\omega _1)}(\mathbb R^{d}) \times M^{p_2,q_2}_{(\omega _2)}(\mathbb R^{d})\) to \(M_{(\omega _0)}^{p_0,q_0}(\mathbb R^{d})\) , and
Theorem 3.8
Let p j, q j ∈ (0, ∞) and \(\omega _j\in \mathcal P _E(\mathbb R^{2d})\) , j = 0, 1, 2, be such that (88) and (91) hold. Then (f 1, f 2)↦f 1 ∗ f 2 from \(\Sigma _1 (\mathbb R^{d})\times \Sigma _1(\mathbb R^{d})\) to \(\Sigma _1 (\mathbb R^{d})\) is uniquely extendable to a continuous map from \(W^{p_1,q_1}_{(\omega _1)}(\mathbb R^{d}) \times W^{p_2,q_2}_{(\omega _2)}(\mathbb R^{d})\) to \(W_{(\omega _0)}^{p_0,q_0}(\mathbb R^{d})\) , and
We observe that Theorems 3.2–3.5 in [54] are multi-linear versions of the previous results. In particular, Theorems 3.5 and 3.6 are Fourier transformations of Theorems 3.7 and 3.8. Hence it suffices to prove the last two theorems, cf. [54]. To shed some ideas of the arguments, we give a proof in the unweighted case of Theorem 3.7. We will use Proposition A.1 from Appendix A, which is a special case of [54, Proposition 3.6].
Proof of Theorem 3.7
Suppose \(f_j\in \mathcal S(\mathbb R^{d})\), \(\phi _j\in \mathcal S(\mathbb R^{d})\), j = 0, 1, 2 be such that
and let F j be the same as in (81). Then
in view of (84).
We have
where Q d,r is the cube
and χ E is the characteristic function with respect to the set E.
Set
and
Then
and
in view of (A.5) and Proposition A.1 in Appendix A (see also [25, Theorem 3.3])).
By (84) we have
We observe that
where
and
Hence, if x = l in (97), we get
where \(e_0=(1,\dots ,1)\in \mathbb Z^{d}\) and I = {0, 1, 2, 3}d.
If we apply the \(\ell ^{p_0}\) quasi-norm on (98) with respect to the l variable, then Proposition 2.5 (2) and the fact that I is finite set give
By applying the \(\ell ^{q_0}\) quasi-norm and using Proposition 2.5 (1) we now get
This is the same as
A combination of this estimate with (95) and (96) gives that f 1 ∗ f 2 is well-defined and that (93) holds.
The uniqueness now follows from that (93) holds for \(f_1,f_2\in \mathcal S(\mathbb R^{d})\), and that \(\mathcal S(\mathbb R^{d})\) is dense in \(M^{p,q}(\mathbb R^{d})\) when p, q < ∞. □
4 Gabor Products and Modulation Spaces
In this section we give an illustration how the multiplication properties for modulation spaces can be used when treating certain nonlinear problems. We consider the Gabor product which is connected to such multiplication properties. It is introduced in [14] in order to derive a phase space analogue to the usual convolution identity for the Fourier transform. We will prove a formula related to (80), and then use results from previous section to extend the Gabor product initially defined on \( M^1 (\mathbb R^{2d}) \times M^1 (\mathbb R^{2d}) \) to some other spaces. Finally, we show how the Gabor product gives rise to a phase-space formulation of the qubic Schrödinger equation.
Definition 4.1
Let \(\phi \in M^1 (\mathbb R^{d}) \setminus \{ 0 \}\), and let \(F_1 , F_2 \in M^1 (\mathbb R^{2d})\). Then the Gabor product ♮ ϕ is given by
In the proof of [14, Lemma 13] it is justified that the Gabor product in (99) is well-defined, and that
is a continuous map.
The Gabor product is particularly well-suited in the context of the STFT.
Theorem 4.2
Let \(\phi , \phi _1, \phi _2 \in M^1 (\mathbb R^{d}) \backslash \left \{0 \right \}\) . Then
Moreover, \( V_{\phi } (f_1 \cdot f_2) \in M^1 (\mathbb R^{2d})\).
Proof
We have
where
By Parseval’s formula we get
By inserting this into (102) and using Fubini’s theorem we get
where
Hence, by evaluating the integral with respect to ζ, and using Fourier’s inversion formula, we get
which gives (100), and the result follows. □
The formula (100) is closely related to (80). In fact, the windows \( \phi _j \in \Sigma _1 (\mathbb R^{d})\), j = 0, 1, 2, in (80) should satisfy the condition (79), while (100) is valid for arbitrary non-zero elements from \( M^1 (\mathbb R^{d})\). For example, when ϕ = ϕ 1 = ϕ 2 and \(\Vert \phi \Vert _{L^2 (\mathbb R^{d})}=1\), then (100) reduces to
while (80) does not allow such choice of windows.
One of the main goals of [14] are extensions of the Gabor product to some function spaces \(\mathcal {F}_j (\mathbb R^{2d})\), j = 0, 1, 2, so that ♮ ϕ maps \(\mathcal {F}_1 \times \mathcal {F}_2\) into \(\mathcal {F}_0\), with:
This can be considered as a phase space form of the Young convolution inequality.
Next we discuss continuity of the Gabor product on certain spaces involving superpositions of short-time Fourier transforms. In the end we deduce properties similar to [14, Theorem 29]. Instead of modulation spaces of the form \(M^{p,q}_{(\omega )}(\mathbb R^{d})\), p, q ∈ [1, ∞), \(\omega \in \mathcal P _E(\mathbb R^{2d})\), here we consider modulation spaces of Wiener amalgam types \(W^{p,q}_{(\omega )}(\mathbb R^{d})\), and allow the “quasi-Banach” choice for Lebesgue parameters, i.e. p and q are allowed to be smaller than one.
Thus, in what follows we assume that p, q ∈ (0, ∞), \(\omega \in \mathcal P _E(\mathbb R^{2d})\) is v-moderate, and consider \(L^{p,q}_{*,(\omega )}(\mathbb R^{2d})\) spaces rather than \(L^{p,q}_{(\omega )}(\mathbb R^{2d})\) which are treated in [14].
We need some additional notation. Let s > 1, \(N\in \mathbb {N}\) be given, and let
be an orthonormal basis of \( L^2 (\mathbb R^{d})\). Then let \(\mathcal {V}_{\mathcal {G},\omega }^{(N),p,q}(\mathbb R^{2d})\) be the closure of
with respect to the \(L^{p,q}_{*,(\omega )}(\mathbb R^{2d})\) norm. In particular, if N = 1, ϕ = ϕ 1 and p, q ≥ 1, then this reduces to the closure
of
with respect to the \(L^{p,q}_{*,(\omega )}(\mathbb R^{2d})\) norm.
By [14, Theorem 26], it follows that for every \( F \in \mathcal {V}_{\mathcal {G},\omega }^{(N),p,q}(\mathbb R^{2d})\) there exist \(f_n \in W_{(\omega )}^{p,q}(\mathbb R^{d})\), n = 1, 2, …, N, and such that
Theorem 4.3
Let p j, q j ∈ (0, ∞) and \(\omega _j\in \mathcal P _E(\mathbb R^{2d})\) be v j –moderate, j = 0, 1, 2, and such that (86) and (87) hold, and let \(\phi \in \Sigma _s (\mathbb R^{d}) \) , s > 1. Then the Gabor product ♮ ϕ from \( {\mathcal V}_{\mathcal {G}}^{(N)} \left (\mathbb R^{2d}\right ) \times {\mathcal V}_{\mathcal {G}}^{(N)} \left (\mathbb R^{2d}\right ) \) to \(W_{(v)}^{1,1}(\mathbb R^{2d})\) , extends uniquely to a continuous map from \(\mathcal {V}_{\mathcal {G},\omega _1}^{(N),p_1,q_1} (\mathbb R^{2d}) \times \mathcal {V}_{\mathcal {G},\omega _2}^{(N),p_2,q_2}(\mathbb R^{2d})\) to the closure of \({P_\phi }(L^{p_0,q_0}_{*,(\omega _0 )}(\mathbb R^{2d}) )\) , and
for all \(F_j \in \mathcal {V}_{\mathcal {G}, \omega _j}^{(N),p_j,q_j}(\mathbb R^{2d}) \) , j = 1, 2.
In particular, if F j = V ϕ f j , j = 1, 2, and \(\| \phi \|{ }_{L^2}= 1\) , then (107) reduces to
We omit the proof which is a slight modification of the proof of Theorem 29 in [14].
We end the paper by formally demonstrating how the Gabor product arises in a phase space version of the cubic Schrödinger equation. Consider the elliptic nonlinear Schrödinger equation (NLSE) given by
subject to the initial condition:
Here λ = ±1 stands for an attracting (λ = +1) or repulsive (λ = −1) power-law nonlinearity, and the Laplacian is given by
Thus we consider ψ = ϕ(x, t) with \( x \in \mathbb R^{d},\) and t in an open interval \(I \subseteq \mathbb {R}\).
Using the following intertwining relations
j = 1, ⋯ , d, and assuming that ϕ is a real-valued window, we obtain upon application of the STFT V ϕ to (109) that
Here, \(D_{x_j}=-i \frac {\partial }{\partial x_j}\),
and \(\widetilde F \) is given by
By considering (110) the phase-space formulation of the initial value problem may be well-posed for more general initial distributions. This means that the phase-space formulation “contains” the solutions of the standard NLSE, but it is richer, as it admits other solutions. We refer to [11,12,13], where phase-space extensions are explored in several different contexts.
Let us conclude by noticing that (110) contains the triple product. Thus, its qualitative analysis calls for a multilinear extension of Theorems 3.6 and 4.3. Then the conditions (86) and (87) become more involved, see [54]. Such analysis demands a more technical tools and arguments and goes beyond the scope of this survey article.
References
F. Bastianoni, N. Teofanov, Subexponential decay and regularity estimates for eigenfunctions of localization operators. J. Pseudo-Differ. Oper. Appl. 12,, Paper no. 19, 28 (2021)
F. Bastianoni, E. Cordero, F. Nicola Decay and smoothness for eigenfunctions of localization operators. J. Math. Anal. Appl. 492, 124480 (2020)
Á. Bényi, K. Okoudjou, Local well-posedness of nonlinear dispersive equations on modulation spaces. Bull. Lond. Math. Soc. 41, 549–558 (2009)
Á. Bényi, K. Okoudjou, Modulation Spaces. With Applications to Pseudodifferential Operators and Nonlinear Schrödinger Equations. Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, New York, 2020)
Á. Bényi, L. Grafakos, K.H. Gröchenig, K. Okoudjou A class of Fourier multipliers for modulation spaces. Appl. Comput. Harmon. Anal. 19, 131–139 (2005)
Á. Bényi, K. H. Gröchenig, K. Okoudjou, L. Rogers, Unimodular Fourier multipliers for modulation spaces. J. Func. Anal. 246, 366–384 (2007)
P. Boggiatto, E. Cordero, K. Gröchenig, Generalized anti-Wick operators with symbols in distributional Sobolev spaces. Integr. Eq. Oper. Theory 48, 427–442 (2004)
J. Chung, S.-Y. Chung, D. Kim, Characterizations of the Gelfand-Shilov spaces via Fourier transforms. Proc. Am. Math. Soc. 124, 2101–2108 (1996)
E. Cordero, K.H. Gröchenig, Time-frequency analysis of localization operators. J. Funct. Anal. 205, 107–131 (2003)
E. Cordero, L. Rodino Time-Frequency Analysis of Operators. Studies in Mathematics, vol. 75 (De Gruyter, Berlin, Boston, 2020)
N.C. Dias, M. de Gosson, F. Luef, J.N. Prata. A Pseudo-differential calculus on non-standard symplectic space; spectral and regularity results in modulation spaces. J. Math. Pur. Appl. 96, 423–445 (2011)
N.C. Dias, M. de Gosson, F. Luef, J.N. Prata, Quantum mechanics in phase space: the Schrödinger and the Moyal representations. J. Pseudo-Differ. Oper. Appl. 3, 367–398 (2012)
N.C. Dias, M. de Gosson, J.N. Prata, Dimensional extension of pseudo-differential operators: properties and spectral results. J. Func. Anal. 266, 3772–3796 (2014)
N.C. Dias, J.N. Prata, N. Teofanov, Short-time Fourier transform of the pointwise product of two functions with application to the nonlinear Schrödinger equation (2022). Preprint (arXiv:2108.04985)
S.J.L. Eijndhoven, Functional analytic characterizations of the Gelfand-Shilov spaces \(S^\beta _\alpha \). Nederl. Akad. Wetensch. Indag. Math. 49, 133–144 (1987)
H.G. Feichtinger, Gewichtsfunktionen auf lokalkompakten Gruppen. Sitzber. d. österr. Akad. Wiss. 188, 451–471 (1979)
H.G. Feichtinger, Modulation spaces on locally compact abelian groups. Technical report, University of Vienna, Vienna, 1983; also in ed. by M. Krishna, R. Radha, S. Thangavelu. Wavelets and Their Applications (Allied Publishers Private Limited, NewDehli, 2003), pp. 99–140
H.G. Feichtinger, Modulation spaces: looking back and ahead. Sampl. Theory Signal Image Process. 5, 109–140 (2006)
H.G. Feichtinger, K.H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, I. J. Funct. Anal. 86, 307–340 (1989)
H.G. Feichtinger, K.H. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions, II. Monatsh. Math. 108, 129–148 (1989)
H.G. Feichtinger, K.H. Gröchenig, Gabor frames and time-frequency analysis of distributions. J. Funct. Anal. 146, 464–495 (1997)
H.G. Feichtinger, G. Narimani, Fourier multipliers of classical modulation spaces. Appl. Comput. Harmon. Anal. 21, 349–359 (2006)
C. Fernandez, A. Galbis, J. Toft, Characterizations of GRS-weights, and consequences in time-frequency analysis. J. Pseudo-Differ. Oper. Appl. 6, 383–390 (2015)
G.B. Folland, Harmonic Analysis in Phase Space (Princeton University Press, Princeton, 1989)
Y.V. Galperin, S. Samarah, Time-frequency analysis on modulation spaces \(M^{p,q}_m\), 0 < p, q ≤∞. Appl. Comput. Harmon. Anal. 16, 1–18 (2004)
I.M. Gelfand, G.E. Shilov, Generalized Functions, II–III (Academic Press, NewYork, 1968). Reprinted by AMS (2016)
K.H. Gröchenig, Describing functions: atomic decompositions versus frames. Monatsh. Math. 112, 1–42 (1991)
K. Gröchenig, Foundations of Time-Frequency Analysis (Birkhäuser, Boston, 2001)
K. Gröchenig, Composition and spectral invariance of pseudodifferential operators on modulation spaces. J. Anal. Math. 98, 65–82 (2006)
K. Gröchenig, Weight functions in time-frequency analysis, in ed. by L. Rodino, M.W. Wong. Pseudodifferential Operators: Partial Differential Equations and Time-Frequency Analysis. Fields Institute Communications, vol. 52 (American Mathematical Society, Providence, 2007), pp. 343–366
K. Gröchenig, G. Zimmermann, Spaces of test functions via the STFT. J. Funct. Spaces Appl. 2, 25–53 (2004)
W. Guo, J. Chen, D. Fan, G. Zhao, Characterizations of some properties on weighted modulation and wiener amalgam spaces. Michigan Math. J. 68, 451–482 (2019)
L. Hörmander, The Analysis of Linear Partial Differential Operators, vol I–III. (Springer, Berlin, 1983, 1985)
M.S. Jakobsen, On a (no longer) new segal algebra: a review of the feichtinger algebra. J. Fourier Anal. Appl. 24, 1579–1660 (2018)
P.G. Kevrekidis, D.J. Frantzeskakis, R. Carretero-Gonzalez (eds.), Emergent Nonlinear Phenomena in Bose-Einstein Condensation (Springer, Berlin, 2008)
E.H. Lieb, Integral bounds for radar ambiguity functions and Wigner distributions. J. Math. Phys. 31, 594–599 (1990)
E.H. Lieb, J.P. Solovej, Quantum coherent operators: a generalization of coherent states. Lett. Math. Phys. 22, 145–154 (1991)
T. Oh, Y. Wang, Global well-posedness of the one-dimensional cubic nonlinear Schrödinger equation in almost critical spaces. J. Diff. Eq. 269, 612–640 (2020)
T. Oh, Y. Wang, On global well-posedness of the modified KdV equation in modulation spaces. Discrete Continuous Dyn. Syst. 41, 2971–2992 (2021)
S. Pilipović, Tempered ultradistributions. Boll. U.M.I. 7, 235–251 (1988)
H. Rauhut, Wiener amalgam spaces with respect to quasi-Banach spaces. Colloq. Math. 109, 345–362 (2007)
H. Rauhut, Coorbit space theory for quasi-Banach spaces. Stud. Math. 180, 237–253 (2007)
M. Ruzhansky, M. Sugimoto, J. Toft, N. Tomita, Changes of variables in modulation and Wiener amalgam spaces. Math. Nachr. 284, 2078–2092 (2011)
M.A. Shubin, Pseudodifferential Operators and Spectral Theory, 2nd edn. (Springer, Berlin, 2001)
N. Teofanov, Ultradistributions and time-frequency analysis, in ed. by P. Boggiatto et al. Pseudo-Differential Operators and Related Topics. Operator Theory Advances and Applications, vol. 164 (Birkhäuser Verlag, Basel, 2006), pp. 173–191
N. Teofanov, Modulation spaces, Gelfand-Shilov spaces and pseudodifferential operators. Sampl. Theory Signal Image Process. 5, 225–242 (2006)
N. Teofanov, Bilinear localization operators on modulation spaces. J. Funct. Spaces 2018, Art. ID 7560870, 10 (2018)
J. Toft, Convolutions and embeddings for weighted modulation spaces in ed. by R. Ashino, P. Boggiatto, M.W. Wong. Advances in Pseudo-Differential Operators. Operator Theory Advances and Applications, vol. 155 (Birkhäuser Verlag, Basel, 2004), pp. 165–186
J. Toft, The Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators. J. Pseudo-Differ. Oper. Appl. 3, 145–227 (2012)
J. Toft, Gabor analysis for a broad class of quasi-Banach modulation spaces, in ed. by S. Pilipović, J. Toft. Pseudo-Differential Operators, Generalized Functions. Operator Theory Advances and Applications, vol. 245 (Birkhäuser, Basel, 2015), pp. 249–278
J. Toft, Images of function and distribution spaces under the Bargmann transform. J. Pseudo-Differ. Oper. Appl. 8, 83–139 (2017)
J. Toft, Tensor products for Gelfand-Shilov and Pilipović distribution spaces. J. Anal. 28, 591–613 (2020)
J. Toft, The Zak transform on Gelfand-Shilov and modulation spaces with applications to operator theory. Complex Anal. Oper. Theory 15, Paper no. 2, 42 (2021)
J. Toft, Step multipliers, Fourier step multipliers and multiplications on quasi-Banach modulation spaces. J. Func. Anal. 282, Paper no. 109343, 46 (2022)
B. Wang, C. Huang, Frequency-uniform decomposition method for the generalized BO, KdV and NLS equations. J. Diff. Equ. 239, 213–250 (2007)
Acknowledgements
The work of N. Teofanov is partially supported by TIFREFUS Project DS 15, and MPNTR of Serbia Grant No. 451–03–68/2022–14/200125. Joachim Toft was supported by Vetenskapsrådet (Swedish Science Council) within the project 2019–04890.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendix A: Some Properties of Wiener Amalgam Spaces
Appendix A: Some Properties of Wiener Amalgam Spaces
There are convenient characterizations of modulation spaces in the framework of Gabor analysis.
Let \(\omega _0\in \mathcal P _E(\mathbb R^{d})\), \(\omega \in \mathcal P _E(\mathbb R^{2d})\), p, q, r ∈ (0, ∞], Q d = [0, 1]d be the unit cube, and set for measurable f on \(\mathbb R^{d}\),
when
and for measurable F on \(\mathbb R^{2d}\),
when
The Wiener amalgam space
consists of all measurable \(f\in L^r _{loc}(\mathbb R^{d})\) such that \(\Vert F\Vert _{\mathsf {W} ^r(\omega _0,\ell ^{p})}\) is finite, and the Wiener amalgam spaces
consist of all measurable \(F\in L^r _{loc}(\mathbb R^{2d})\) such that \(\Vert F\Vert _{\mathsf {W} ^r(\omega ,\ell ^{p,q})}\) respectively \(\Vert F\Vert _{\mathsf {W} ^r(\omega ,\ell ^{p,q}_*)}\) are finite. We observe that W r(ω 0, ℓ p) is often denoted by \(W(L^r,\ell _{(\omega )}^p)\) in the literature (see e. e. [17, 19, 25, 41]).
The topologies are defined through their corresponding quasi-norms in (A.1) and (A.2). For conveniency we set
and if in addition ω = 1, we set
Obviously, W r(ω 0, ℓ p) and W r(ω, ℓ p, q) increase with p, q, decrease with r, and
and
On the other hand, for modulation spaces we have
with
The same holds true with \(W^{p,q}_{(\omega )}\), \(L^{p,q}_{*,(\omega )}\) and \(\mathsf {W} (\omega ,\ell ^{p,q}_*)\) in place of \(M^{p,q}_{(\omega )}\), \(L^{p,q}_{(\omega )}\) and W(ω, ℓ p, q), respectively, at each occurrence. (For r = ∞ , see [28] when p, q ∈ [1, ∞], [25, 50] when p, q ∈ (0, ∞], and for r ∈ (0, ∞], see [53].)
We have now the following result on the projection operator P ϕ in (20) when acting on Wiener amalgam spaces.
Proposition A.1
Let p, q ∈ (0, ∞] and \(\phi \in \mathcal S(\mathbb R^{d})\setminus \{ 0 \}\) . Then P ϕ from \(\mathcal S^{\prime }(\mathbb R^{2d})\) to \(\mathcal S^{\prime }(\mathbb R^{2d})\) , and \(V_\phi ^*\) from \(\mathcal S^{\prime }(\mathbb R^{2d})\) to \(\mathcal S^{\prime }(\mathbb R^{d})\) restrict to continuous mappings
and
We refer to [54, Proposition 3.6] for the proof of Proposition A.1 and to [19, 21, 28, 41, 42, 54] for some facts about the operators P ϕ and \(V_\phi ^*\),
For p, q ≥ 1, i.e. the case when all spaces are Banach spaces, proofs of Proposition A.1 can be found in e.g. [28] as well as in abstract forms in [19]. In the general case when p, q > 0, we refer to [25, 42], since proofs of Proposition A.1 are essentially given there.
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Teofanov, N., Toft, J. (2022). An Excursion to Multiplications and Convolutions on Modulation Spaces. In: Aron, R.M., Moslehian, M.S., Spitkovsky, I.M., Woerdeman, H.J. (eds) Operator and Norm Inequalities and Related Topics. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-02104-6_18
Download citation
DOI: https://doi.org/10.1007/978-3-031-02104-6_18
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-031-02103-9
Online ISBN: 978-3-031-02104-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)