Abstract
We deduce continuity properties for pseudo-differential operators with symbols in quasi-Banach Orlicz modulation spaces when rely on other quasi-Banach Orlicz modulation spaces. In particular we extend some earlier results.
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1 Introduction
In the paper we deduce continuity properties for pseudo-differential operators when acting on quasi-Banach Orlicz modulation spaces. For example, for a pseudo-differential operator \({\text {Op}}(a)\) with the symbol a we show that the following is true:
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Suppose that \(q_0\in (0,1]\), \(\Phi _{\!j}\) are quasi-Young functions which satisfy \(\Phi _{\!j}(t)\lesssim t^{q_0}\) near origin, and that a belongs to the classical modulation space \(M^{\infty ,q_0}({\textbf{R}}^{2d})\). Then \({\text {Op}}(a)\) is continuous on the quasi-Banach Orlicz modulation space \(M^{\Phi _1,\Phi _2}({\textbf{R}}^{d})\);
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Suppose that \(\Phi \) is a quasi-Young function which satisfy \(t\lesssim \Phi (t)\) near origin, and that a belongs to \(M^{\Phi }({\textbf{R}}^{2d})\). Then \({\text {Op}}(a)\) is continuous from \(M^\infty ({\textbf{R}}^{d})\) to \(M^{\Phi }({\textbf{R}}^{d})\);
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Suppose that \(\Phi _0\) is a Young function and \(\Phi _0^*\) is the complementary Young function, and that a belongs to \(M^{\Phi _0}({\textbf{R}}^{2d})\). Then \({\text {Op}}(a)\) is continuous from \(M^{\Phi _0^*} ({\textbf{R}}^{d})\) to \(M^{\Phi }({\textbf{R}}^{d})\).
(We refer to [18] and Sect. 1 for notations).
More generally, we deduce weighted versions of such continuity results. In particular we extend some continuity properties for pseudo-differential operators when acting on (ordinary) modulation spaces, e.g. in [4, 5, 14, 15, 30, 31, 33].
Essential parts of our analysis are based on [28] by Schnackers and Führ concerning Orlicz modulation spaces, and on [37] concerning quasi-Banach Orlicz modulation spaces. In these approaches, general properties and aspects on quasi-Banach Orlicz spaces given in [17] by Harjulehto and Hästö are fundamental. In this respect, we show that for mixed quasi-Banach Orlicz modulation spaces like \(M^{\Phi _1,\Phi _2}_{(\omega )}({\textbf{R}}^{2d})\) we have \(M^{\Phi ,\Phi }_{(\omega )}({\textbf{R}}^{2d}) =M^{\Phi }_{(\omega )}({\textbf{R}}^{2d})\) when \(\Phi \), \(\Phi _1\) and \(\Phi _2\) are quasi-Young functions. This leads to convenient improvement of the style of the continuity results for our pseudo-differential operators when acting on quasi-Banach Orlicz modulation spaces.
In some situations it might be beneficial to replace Lebesgue norm estimates with more refined Orlicz norm estimates. This may appear when dealing with certain non-linear functionals. For example, in statistics or statistical physics, the entropy applied on probability density functions f on \({\textbf{R}}^{d}\) is given by
When investigating \({\textsf{E}}\), it might be more efficient to replace the pair of Lebesgue spaces \((L^1,L^\infty )\) by the pair of Orlicz spaces \((L\log (L+1),L^{\cosh -1})\), where the Young functions are given by
respectively. We also observe that the Zygmund space \(L\log ^+L\) is an Orlicz space related to Hardy-Littlewood maximal functions (see [21, 22] and the references therein).
Such questions are also relevant when investigating localized Fourier transforms like short-time Fourier transforms \(V_\phi f\) because of the entropy conditions
for some constant C, when
(See [19].) We remark that such refined Fourier transforms are indispensable tools within time-frequency, signal processing and certain parts of quantum mechanics.
In time-frequency analysis and signal processing, non-stationary filters can be modelled by pseudo-differential operators \(f\mapsto {\text {Op}}(a)f\), where the symbols a are determined by time and frequency varying filters, the target functions f are the original signals and \({\text {Op}}(a)f\) are the reflected signals. In such situations it is suitable to discuss continuity properties by means of certain types of time-frequency invariant (quasi-)Banach spaces. This leads to modulation spaces.
The classical modulation spaces is a family of function and distribution spaces, introduced by Feichtinger in [6]. Here the modulation spaces are defined by imposing a weighted mixed Lebesgue norm estimate on the short-time Fourier transforms of the involved functions and distributions. The theory has thereafter been extended and generalized, especially by Feichtinger and Gröchenig in [8, 9], where the theory of (Banach) modulation spaces was put into the context of coorbit space theory. A less abstract extension of the classical modulation spaces is performed in [7], where Feichtinger replaces the mixed Lebesgue norm estimates in [6] with more general translation invariant norms of solid Banach function spaces.
Some extensions to the quasi-Banach case have thereafter been performed in e.g. [10, 26, 27, 32, 35].
In [28], Führ and Schnacker study Orlicz modulation spaces of the form \(M^{\Phi _1,\Phi _2}\), where \(\Phi _1\) and \(\Phi _2\) are Young functions. That is, they consider modulation spaces in [7], where the solid Banach function spaces are Orlicz spaces, a naturally generalization of \(L^p\) spaces which contain certain Sobolev spaces as subspaces. In particular their investigations also include the classical modulation spaces in [6], since these spaces are obtained by choosing
The analysis in [28] is extended in [37] to quasi-Banach weighted Orlicz modulation spaces, \(M^{\Phi _1,\Phi _2}_{(\omega )}({\textbf{R}} ^{d})\), where \(\Phi _1\), \(\Phi _2\) are quasi-Young functions of certain degrees and \(\omega \) is a suitable weight function on \({\textbf{R}}^{2d}\). In particular, it is here allowed to let \(\Phi _{\!j}(t)=t^p\) for every \(p>0\) (instead of \(p\ge 1\) as in [28]), which implies that any modulation space \(M^{p,q}_{(\omega )}({\textbf{R}}^{d})\) for \(p,q\in (0,\infty ] \) are included in the studies in [37].
In the paper, our deduced continuity for pseudo-differential operators, are based on the various properties of quasi-Banach Orlicz modulation spaces, obtained in [37].
2 Preliminaries
In this section we recall some facts for Gelfand-Shilov spaces, Orlicz spaces, Orlicz modulation spaces and pseudo-differential operators. First we discuss some useful properties of Gelfand-Shilov spaces. Thereafter we recall some classes of weight functions which are used later on in the definition of Orlicz modulation spaces. In Sects. 1.3 and 1.4 we define and present some properties for Orlicz spaces and Orlicz modulation spaces. We conclude the section by discussing Gabor analysis for Orlicz modulation spaces and pseudo-differential operators.
2.1 Gelfand–Shilov spaces
We start by discussing Gelfand-Shilov spaces and their properties. Let \(0<s\in {\textbf{R}}\) be fixed. Then the (Fourier invariant) Gelfand-Shilov space \({\mathcal {S}}_{s}({\textbf{R}}^{d})\) (\(\Sigma _{s}({\textbf{R}}^{d})\)) of Roumieu type (Beurling type) with parameter s consists of all \(f\in C^\infty ({\textbf{R}}^{d})\) such that
is finite for some \(h>0\) (for every \(h>0\)). Here the supremum should be taken over all \(\alpha ,\beta \in {\textbf{N}}^d\) and \(x\in {\textbf{R}}^{d}\). We equip \({\mathcal {S}}_{s}({\textbf{R}}^{d})\) (\(\Sigma _{s}({\textbf{R}}^{d})\)) by the canonical inductive limit topology (projective limit topology) with respect to \(h>0\), induced by the semi-norms in (1.1).
For any \(s,s_0>0\) such that \(\frac{1}{2} {\leqslant } s_0<s\) we have
with dense embeddings. Here \(A\hookrightarrow B\) means that the topological spaces A and B satisfy \(A\subseteq B\) with continuous embeddings. The space \(\Sigma _s({\textbf{R}}^{d})\) is a Fréchet space with seminorms \(\Vert \cdot \Vert _{{\mathcal {S}}_{s,h}}\), \(h>0\). Moreover, \(\Sigma _s({\textbf{R}}^{d})\ne \{ 0\}\), if and only if \(s>1/2\), and \({\mathcal {S}}_s({\textbf{R}}^{d})\ne \{ 0\}\), if and only if \(s\ge 1/2\).
The Gelfand-Shilov distribution spaces \({\mathcal {S}}_{s}'({\textbf{R}}^{d})\) and \(\Sigma _s'({\textbf{R}}^{d})\) are the (strong) dual spaces of \({\mathcal {S}}_{s}({\textbf{R}}^{d})\) and \(\Sigma _s({\textbf{R}}^{d})\), respectively. As for the Gelfand-Shilov spaces there is a canonical projective limit topology (inductive limit topology) for \({\mathcal {S}}_{s}'({\textbf{R}}^{d})\) (\(\Sigma _s'({\textbf{R}}^{d})\)) (cf. [11, 23, 24]).
From now on we let \({\mathscr {F}}\) be the Fourier transform which takes the form
when \(f\in L^1({\textbf{R}}^{d})\). Here \(\langle \, \cdot \, ,\, \cdot \, \rangle \) denotes the usual scalar product on \({\textbf{R}}^{d}\). The map \({\mathscr {F}}\) extends uniquely to homeomorphisms on \({\mathscr {S}}'({\textbf{R}}^{d})\), on \({\mathcal {S}}_s'({\textbf{R}}^{d})\) and on \(\Sigma _s'({\textbf{R}}^{d})\). Furthermore, \({\mathscr {F}}\) restricts to homeomorphisms on \({\mathscr {S}}({\textbf{R}}^{d})\), on \({\mathcal {S}}_s({\textbf{R}}^{d})\) and on \(\Sigma _s({\textbf{R}}^{d})\), and to a unitary operator on \(L^2({\textbf{R}}^{d})\).
Gelfand-Shilov spaces can in convenient ways be characterized in terms of estimates of the functions and their Fourier transforms. More precisely, in [3] it is proved that if \(f\in {\mathscr {S}}'({\textbf{R}}^{d})\) and \(s>0\), then \(f\in {\mathcal {S}}_s({\textbf{R}}^{d})\) (\(f\in \Sigma _s({\textbf{R}}^{d})\)), if and only if
for some \(r>0\) (for every \(r>0\)). Here \(r_1(\theta ) \lesssim r_2(\theta )\) means that \(r_1(\theta ) \le c \cdot r_2(\theta )\) holds uniformly for all \(\theta \) in the intersection of the domains of \(r_1\) and \(r_2\) for some constant \(c>0\). We write \(r_1\asymp r_2\) when \(r_1\lesssim r_2 \lesssim r_1\).
Let \(\phi \in {\mathcal {S}}_s ({\textbf{R}}^{d})\) be fixed. Then the short-time Fourier transform \(V_\phi f\) of \(f\in {\mathcal {S}}_s ' ({\textbf{R}}^{d})\) with respect to the window function \(\phi \) is the Gelfand-Shilov distribution on \({\textbf{R}}^{2d}\), defined by
If \(f ,\phi \in {\mathcal {S}}_s ({\textbf{R}}^{d})\), then it follows that
We recall that Gelfand-Shilov spaces and their distribution spaces can also be characterized by estimates of short-time Fourier transforms, (see e.g. [16, 35]).
2.2 Weight functions
A weight or weight function on \({\textbf{R}}^{d}\) is a positive function \(\omega \in L^\infty _{loc}({\textbf{R}}^{d})\) such that \(1/\omega \in L^\infty _{loc}({\textbf{R}}^{d})\). The weight \(\omega \) is called moderate, if there is a positive weight v on \({\textbf{R}}^{d}\) such that
If \(\omega \) and v are weights on \({\textbf{R}}^{d}\) such that (1.5) holds, then \(\omega \) is also called v-moderate. We note that (1.5) implies that \(\omega \) fulfills the estimates
We let \({\mathscr {P}}_E({\textbf{R}}^{d})\) be the set of all moderate weights on \({\textbf{R}}^{d}\).
It can be proved that if \(\omega \in {\mathscr {P}}_E({\textbf{R}}^{d})\), then \(\omega \) is v-moderate for some \(v(x) = e^{r|x|}\), provided the positive constant r is large enough (cf. [13]). That is, (1.5) implies
for some \(r>0\). In particular, (1.6) shows that for any \(\omega \in {\mathscr {P}}_E({\textbf{R}}^{d})\), there is a constant \(r>0\) such that
We say that v is submultiplicative if v is even and (1.5) holds with \(\omega =v\). In the sequel, v and \(v_j\) for \(j\ge 0\), always stand for submultiplicative weights if nothing else is stated.
We let \({\mathscr {P}}^{0} _E({\textbf{R}} ^{d})\) be the set of all \(\omega \in {\mathscr {P}}_E({\textbf{R}} ^{d})\) such that (1.7) holds for every \(r>0\). We also let \({\mathscr {P}}({\textbf{R}} ^{d})\) be the set of all \(\omega \in {\mathscr {P}}_E({\textbf{R}} ^{d})\) such that
for some \(r>0\). Evidently,
2.3 Orlicz spaces
In this subsection we provide an overview of some basic definitions and state some technical results that will be needed.
First we recall some facts concerning Young functions and Orlicz spaces (see [17, 25]).
Definition 1.1
A function \(\Phi :{\textbf{R}} \rightarrow {\textbf{R}} \cup \{ \infty \}\) is called convex if
when \(s_j,t_j\in {\textbf{R}}\) satisfy \(s_j \ge 0\) and \(s_1 + s_2 = 1,\ j=1,2\).
We observe that \(\Phi \) might not be continuous, because we permit \(\infty \) as function value. For example,
is convex but discontinuous at \(t=a\).
Definition 1.2
Let \(r_0\in (0,1]\), \(\Phi _0\) and \(\Phi \) be functions from \([0,\infty )\) to \([0,\infty ]\). Then \(\Phi _0\) is called a Young function if
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(1)
\(\Phi _0\) is convex,
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(2)
\(\Phi _0(0)=0\),
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(3)
\(\lim \limits _{t\rightarrow \infty } \Phi _0(t)=+\infty \).
The function \(\Phi \) is called \(r_0\)-Young function or quasi-Young function of order \(r_0\), if \(\Phi (t)=\Phi _0 (t^{r_0})\), \(t \ge 0\), for some Young function \(\Phi _0\).
It is clear that \(\Phi \) in Definition 1.2 is non-decreasing, because if \(0\le t_1\le t_2\) and \(s\in [0,1]\) is chosen such that \(t_1=st_2\) and \(\Phi _0\) is the same as in Definition 1.2, then
since \(\Phi (0) = \Phi _0(0)=0\) and \(s\in [0,1]\).
Definition 1.3
Let \((\Omega ,\Sigma ,\mu )\) be a Borel measure space, with \(\Omega \subseteq {\textbf{R}} ^{d}\), \(\Phi _0\) be a Young function and let \(\omega _0 \in {\mathscr {P}}_E({\textbf{R}}^{d})\).
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(1)
\(L^{\Phi _0}_{(\omega _0)}(\mu )\) consists of all \(\mu \)-measurable functions \(f:\Omega \rightarrow {\textbf{C}}\) such that
$$\begin{aligned} \Vert f\Vert _{L^{\Phi _0}_{(\omega _0)}(\mu )}=\inf \left\{ \, \lambda >0\, ;\, \int _\Omega \Phi _0 \left( \frac{|f(x) \cdot \omega _0 (x)|}{\lambda } \right) d\mu (x)\le 1\, \right\} \end{aligned}$$is finite. Here f and g in \(L^{\Phi _0}_{(\omega _0)}(\mu )\) are equivalent if \(f=g\) a.e.
-
(2)
Let \(\Phi \) be a quasi-Young function of order \(r_0\in (0,1]\), given by \(\Phi (t)=\Phi _0(t^{r_0})\), \(t\ge 0\), for some Young function \(\Phi _0\). Then \(L^ \Phi _{(\omega _0)}(\mu )\) consists of all \(\mu \)-measurable functions \(f:\Omega \rightarrow {\textbf{C}}\) such that
$$\begin{aligned} \Vert f\Vert _{L^{\Phi }_{(\omega _0)}(\mu )} = (\Vert |f \cdot \omega _0 |^{r_0}\Vert _{L^{\Phi _0}(\mu )})^{1/r_0} \end{aligned}$$is finite.
Remark 1.4
Let \(\Phi \), \(\Phi _0\) and \(\omega _0\) be the same as in Definition 1.2. Then it follows by straight-forward computation that
Definition 1.5
Let \((\Omega _j ,\Sigma _j ,\mu _j )\) be Borel measure spaces, with \(\Omega _j \subseteq {\textbf{R}} ^{d}\), \(r_0\in (0,1]\), \(\Phi _{j}\) be \(r_0\)-Young functions, \(j=1,2\) and let \(\omega \in {\mathscr {P}}_E ({\textbf{R}} ^{2d})\). Then the mixed quasi-norm Orlicz space \({L^{\Phi _1, \Phi _2}_{(\omega )}} = {L^{\Phi _1, \Phi _2}_{(\omega )}}(\mu _1 \otimes \mu _2)\) consists of all \(\mu _1 \otimes \mu _2\)-measurable functions \(f:\Omega _1 \times \Omega _2 \rightarrow {\textbf{C}}\) such that
is finite, where
If \(r_0=1\) in Definition 1.5, then \(L^{\Phi _1, \Phi _2}_{(\omega )} (\mu _1 \otimes \mu _2)\) is a Banach space and is called a mixed norm Orlicz space.
Remark 1.6
Suppose \(\Phi _{\!j}\) are quasi-Young functions of order \(q_j\in (0,1]\), \(j=1,2\). Then both \(\Phi _{1}\) and \(\Phi _{2}\) are quasi-Young functions of order \(r_0=\min (q_1,q_2)\).
Let \(\Lambda \subseteq {\textbf{R}}^{d}\) be a lattice, i.e., \(\Lambda \) is given by
for some basis \(e_1,\dots ,e_d\) of \({\textbf{R}}^{d}\). Then \(\ell _0'(\Lambda )\) is the set of all formal sequences
Let \(\ell _0 (\Lambda )\) be the set of all sequences \(\{ a(n)\} _{n\in \Lambda }\) such that \(a(n)\ne 0\) for at most finite numbers of n. We observe that
is a lattice in \({\textbf{R}}^{2d}\simeq {\textbf{R}}^{d}\times {\textbf{R}}^{d}\).
Remark 1.7
Let \(\Lambda \subseteq {\textbf{R}}^{d}\) be a lattice, \(\Phi , \Phi _1\) and \(\Phi _2\) be \(r_0\)-Young functions, \(\omega _0, v_0 \in {\mathscr {P}}_E ({\textbf{R}} ^{d})\) and \(\omega , v \in {\mathscr {P}}_E ({\textbf{R}} ^{2d})\) be such that \(\omega _0\) and \(\omega \) are \(v_0\)- respectively v-moderate (in the sequel it is understood that all lattices contain 0). Then we set
when \(\mu \) is the Lebesgue measure on \({\textbf{R}}^{d}\). If instead \(\mu \) is the standard (Haar) measure on \(\Lambda \), i.e. \(\mu (n)=1,\ n\in \Lambda \), then we set
Evidently, \(\ell ^{\Phi _1, \Phi _2} _{(\omega )}(\Lambda \times \Lambda )\subseteq \ell _0'(\Lambda \times \Lambda )\).
Lemma 1.8
Let \(\Phi , \Phi _{\!j}\) be Young functions, \(j=1,2\), \(\omega _0, v_0 \in {\mathscr {P}}_E ({\textbf{R}} ^{d})\) and \(\omega , v \in {\mathscr {P}}_E ({\textbf{R}} ^{2d})\) be such that \(\omega _0\) is \(v_0\)-moderate and \(\omega \) is v-moderate. Then \(L^{\Phi }_{(\omega _0)}({\textbf{R}} ^{d})\) and \(L^{\Phi _{1}, \Phi _{2}}_{(\omega )}({\textbf{R}}^{2d})\) are invariant under translations, and
and
Proof
We only prove the assertion for \(L^{\Phi _{1}, \Phi _{2}}_{(\omega )}({\textbf{R}}^{2d})\). The other part follows by similar arguments and is left for the reader.
We have \(\Phi _{\!j}(t) = \Phi _{0,j}(t^{r_0}),\ t\ge 0\), for some Young functions \(\Phi _{0,j}\), \(j=1,2\). This gives
Here the inequality follows from the fact that \(\omega \) is v-moderate, and the last two relations follow from the definitions. \(\square \)
We refer to [17, 25, 28] for more facts about Orlicz spaces.
2.4 Orlicz modulation spaces
The definitions of classical modulation spaces and Orlicz modulation spaces are the following (cf. [6, 7, 28, 37]).
Definition 1.9
Let \(\phi (x) = \pi ^{-\frac{d}{4}}e^{-\frac{|x|^2}{2}},\ x\in {\textbf{R}} ^{d}\), \(p,q\in (0,\infty ]\) and \(\omega ~\in ~{\mathscr {P}}_{E}({\textbf{R}} ^{2d})\). Then the modulation spaces \(M^{p,q}_{(\omega )}({\textbf{R}} ^{d})\) is set of all \(f\in {\mathcal {S}}_{1/2}' ({\textbf{R}}^{d})\) such that \(V_\phi f\in L^{p,q}_{(\omega )}({\textbf{R}}^{2d})\). We equip these spaces with the quasi-norm
Also let \(\Phi , \Phi _1,\Phi _2\) be quasi-Young functions. Then the Orlicz modulation spaces \(M^{\Phi }_{(\omega )} ({\textbf{R}} ^{d})\) and \(M^{\Phi _{1}, \Phi _{2}}_{(\omega )}({\textbf{R}} ^{d})\) are given by
and
The quasi-norms on \(M^{\Phi }_{(\omega )}({\textbf{R}} ^{d})\) and \(M^{\Phi _{1}, \Phi _{2}}_{(\omega )}({\textbf{R}} ^{d})\) are given by
and
For conveniency we set
and \(M^p=M^{p,p}\) and \(M^p_{(\omega )}=M^{p,p}_{(\omega )}\).
We notice that (1.10) and (1.11) are norms when \(\Phi , \Phi _1\) and \(\Phi _2\) are Young functions. If \(\omega \in {\mathscr {P}}_E({\textbf{R}} ^{2d})\) as in Definition 1.9, then the conditions
are independent of the choices of \(\phi \) in \(\Sigma _1({\textbf{R}} ^{d})\setminus {0}\) and that different \(\phi \) give rise to equivalent quasi-norms (see e.g. [37, Sect. 5]).
Later on we need the following proposition.
Proposition 1.10
Let \(\Phi , \Phi _{\!j}\) be Young functions, \(j=1,2\), \(\omega _0 \in {\mathscr {P}}_E ({\textbf{R}} ^{d})\) and \(\omega \in {\mathscr {P}}_E ({\textbf{R}} ^{2d})\). Then
Proof
Let \(v_0 \in {\mathscr {P}}_E({\textbf{R}} ^{d})\) and \(v\in {\mathscr {P}}_E({\textbf{R}}^{2d})\) be chosen such that \(\omega _0\) is \(v_0\)-moderate and \(\omega \) is v-moderate. Since \(L^{\Phi }_{(\omega _0)}({\textbf{R}} ^{d})\) and \(L^{\Phi _{1}, \Phi _{2}}_{(\omega )}({\textbf{R}} ^{2d})\) are invariant under translation and modulation, we have
and
(see [12, 36, 37]). The result now follows from well-known inclusions between modulation spaces, Schwartz spaces, Gelfand-Shilov spaces, and their duals. \(\square \)
The next result gives some information about the roles that \(\Phi _1\) and \(\Phi _2\) play for \(M^{\Phi _{1},\Phi _{2}}\). We omit the proof since it can be found in [37]. See also [28] for the Banach case.
Proposition 1.11
Let \(\Phi _{\!j}\) and \(\Psi _{\!j}\), \(j=1,2\), be quasi-Young functions, \(\Lambda \) be a lattice in \({\textbf{R}}^{d}\) and \(\omega \in {\mathscr {P}}_E({\textbf{R}} ^{2d})\). Then the following conditions are equivalent:
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(1)
\(M^{\Phi _{1}, \Phi _{2}}_{(\omega )}({\textbf{R}} ^{d})\subseteq M^{\Psi _{1} ,\Psi _{2}}_{(\omega )}({\textbf{R}} ^{d})\);
-
(2)
\(\ell ^{\Phi _{1}, \Phi _{2}}_{(\omega )}(\Lambda )\subseteq \ell ^{\Psi _{1}, \Psi _{2}}_{(\omega )}(\Lambda )\);
-
(3)
\(\Psi _{\!j} (t)\lesssim \Phi _{\!j} (t)\) for every \(t\in [0, t_0]\), for some \(t_0>0\).
2.5 Gabor frames
Definition 1.12
Let \(\omega , v \in {\mathscr {P}}_E ({\textbf{R}} ^{2d})\) be such that \(\omega \) is v-moderate, \(\phi , \psi \in M^1_{(v)}({\textbf{R}} ^{d})\), \(\varepsilon >0\) and let \(\Lambda \subseteq {\textbf{R}} ^{d}\) be a lattice.
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(1)
The analysis operator \(C_{\phi }^{\varepsilon , \Lambda }\) is the operator from \(M^\infty _{(\omega )}({\textbf{R}} ^{d})\) to \(\ell ^\infty _{(\omega )} (\varepsilon \Lambda ^2)\), given by
$$\begin{aligned} C_{\phi }^{\varepsilon , \Lambda } f \equiv \{ V_\phi f( j,\iota )\} _{j,\iota \in \varepsilon \Lambda }. \end{aligned}$$ -
(2)
The synthesis operator \(D_ \psi ^{\varepsilon , \Lambda }\) is the operator from \(\ell ^\infty _{(\omega )}(\varepsilon \Lambda ^2)\) to \(M^\infty _{(\omega )}({\textbf{R}} ^{d})\), given by
$$\begin{aligned} D_{\psi }^{\varepsilon , \Lambda } c \equiv \sum _{j,\iota \in \varepsilon \Lambda '} c(j,\iota )e^{i \langle \, \cdot \, ,\iota \rangle }\psi (\, \cdot \, - j). \end{aligned}$$ -
(3)
The Gabor frame operator \(S_{\phi ,\psi }^{\varepsilon , \Lambda }\) is the operator on \(M^\infty _{(\omega )}({\textbf{R}} ^{d})\), given by \(D_{\psi }^{\varepsilon , \Lambda } \circ C_{\phi }^{\varepsilon , \Lambda },\) i.e.
$$\begin{aligned} S_{\phi ,\psi }^{\varepsilon , \Lambda } f \equiv \sum _{j,\iota \in \varepsilon \Lambda '} V_\phi f(j,\iota ) e^{i\langle \, \cdot \, ,\iota \rangle }\psi (\, \cdot \, - j). \end{aligned}$$
The next result shows that it is possible to find suitable \(\phi \) and \(\psi \) in the previous definition.
Lemma 1.13
Let \(\Lambda \subseteq {\textbf{R}} ^{d}\) be a lattice, \(v\in {\mathscr {P}}_E({\textbf{R}}^{2d})\) be submultiplicative and \(\phi \in M^1_{(v)}({\textbf{R}}^{d}) \setminus {0}\). Then there is an \(\varepsilon >0\) and \(\psi \in M^1_{(v)}({\textbf{R}}^{d}) \setminus {0}\) such that
are dual frames to each others.
Remark 1.14
There are several ways to achieve dual frames (1.12). In fact, let \(v, v_0\in {\mathscr {P}}_E({\textbf{R}} ^{2d})\) be submultiplicative such that \(\omega \) is v-moderate and \(L^1_{(v_0)}({\textbf{R}} ^{2d})\subseteq L^r({\textbf{R}} ^{2d}),\ r\in (0,1]\). Then Lemma 1.13 guarantees that for some choice of \(\phi , \psi \in M^1_{(v_0 v)}({\textbf{R}} ^{d})\subseteq M^r_{(v)}({\textbf{R}} ^{d})\) and lattice \(\Lambda \) , the set in (1.12) are dual frames to each others, and that \(\psi = (S^\Lambda _{\phi ,\phi })^{-1}\phi \). (Cf. [33, Proposition 1.5 and Remark 1.6].)
Lemma 1.15
Let \(\Lambda \subseteq {\textbf{R}} ^{d}\) be a lattice, \(v\in {\mathscr {P}}_E({\textbf{R}}^{4d})\) be submultiplicative, \(\phi _1,\phi _2 \in \Sigma _1({\textbf{R}} ^{d})\setminus {0}\) and
Then there is an \(\varepsilon >0\) such that
is a Gabor frame with canonical dual frame
where \(\psi = (S_{\varphi ,\varphi }^{\Lambda ^2\times \Lambda ^2})^{-1} \varphi \) belongs to \(M_{(v)}^r({\textbf{R}}^{2d})\) for every \(r>0\).
The next result shows that Gabor theory is suitable when dealing with Orlicz modulation spaces. We omit the proof since the result follows from [37, Theorem 4.7]. See also [28] for the Banach case.
Proposition 1.16
Let \(\Lambda \subseteq {\textbf{R}} ^{d}\) be a lattice, \(v\in {\mathscr {P}}_E({\textbf{R}}^{4d})\) be submultiplicative, \(\Phi _1,\Phi _2\) be quasi-Young functions of order \(r_0 \in (0,1]\), \(\omega , v \in {\mathscr {P}}_E ({\textbf{R}} ^{2d})\) be such that \(\omega \) is v-moderate and let \(\phi , \psi \in M^{r_0}_{(v)}({\textbf{R}} ^{d})\) and \(\varepsilon >0\) be chosen such that
are dual frames to each others. If \(f\in M_{(\omega )}^{\Phi _1, \Phi _2}({\textbf{R}} ^{d})\), then
with unconditionally convergence in \(M^{\Phi _1,\Phi _2}_{(\omega )}({\textbf{R}}^{d})\) when \({\mathscr {S}}({\textbf{R}} ^{2d})\) is dense in \(L^{\Phi _1,\Phi _2}({\textbf{R}} ^{2d})\), and with convergence in \(M^\infty _{(\omega )}({\textbf{R}} ^{d})\) with respect to the weak\(^*\) topology otherwise. It holds
We also recall that the previous result was heavily based on the following consequence of Theorems 4.5 and 4.6 in [37]. The proof is therefore omitted.
Proposition 1.17
Let \(\Lambda \subseteq {\textbf{R}} ^{d}\) be a lattice, \(\varepsilon >0\), \(\phi ,\psi \in \Sigma _1({\textbf{R}}^{d})\), \(\Phi _1,\Phi _2\) be quasi-Young functions of order \(r_0 \in (0,1]\), and let \(\omega , v \in {\mathscr {P}}_E ({\textbf{R}} ^{2d})\) be such that \(\omega \) is v-moderate. Then the the following is true:
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(1)
The analysis operator \(C_{\phi }^{\varepsilon ,\Lambda }\) is continuous from \(M^{\Phi _1,\Phi _2}_{(v)}({\textbf{R}} ^{d})\) into \(\ell ^{\Phi _1,\Phi _2}_{(\omega )}(\varepsilon \Lambda ^2)\), and
$$\begin{aligned} \Vert C_{\phi }^{\varepsilon ,\Lambda }f\Vert _{\ell ^{\Phi _1,\Phi _2}_{(\omega )}} \lesssim \Vert f\Vert _{ M_{(\omega )}^{\Phi _1,\Phi _2}}, \quad f\in M_{(\omega )}^{\Phi _1,\Phi _2}({\textbf{R}} ^{d})\text{; } \end{aligned}$$ -
(2)
The synthesis operator \(D_{\psi }^{\varepsilon ,\Lambda }\) is continuous from \(\ell ^{\Phi _1,\Phi _2}_{(\omega )}(\varepsilon \Lambda ^2)\) into \(M^{\Phi _1,\Phi _2}_{(\omega )}({\textbf{R}} ^{d})\), and
$$\begin{aligned} \Vert D_{\psi }^{\varepsilon ,\Lambda } c\Vert _{M^{\Phi _1,\Phi _2}_{(\omega )}} \lesssim \Vert c\Vert _{\ell ^{\Phi _1,\Phi _2}_{(\omega )}}, \quad c\in \ell ^{\Phi _1,\Phi _2}_{(\omega )}(\varepsilon \Lambda ^2). \end{aligned}$$
2.6 Pseudo-differential operators
Let \({\textbf{M}}(d,\Omega )\) be the set of all \(d\times d\)-matrices with entries in the set \(\Omega \), and let \(s\ge 1/2\), \(a\in {\mathcal {S}}_s ({\textbf{R}}^{2d})\) and \(A\in {\textbf{M}}(d,{\textbf{R}})\) be fixed. Then the pseudo-differential operator \({\text {Op}}_A(a)\) is the linear and continuous operator on \({\mathcal {S}}_s ({\textbf{R}}^{d})\), given by
when \(f\in {\mathcal {S}}_s({\textbf{R}}^{d})\). For general \(a\in {\mathcal {S}}_s'({\textbf{R}}^{2d})\), the pseudo-differential operator \({\text {Op}}_A(a)\) is defined as the linear and continuous operator from \({\mathcal {S}}_s({\textbf{R}}^{d})\) to \({\mathcal {S}}_s'({\textbf{R}}^{d})\) with distribution kernel given by
Here \({\mathscr {F}}_2F\) is the partial Fourier transform of \(F(x,y)\in {\mathcal {S}}_s'({\textbf{R}}^{2d})\) with respect to the y variable. This definition makes sense, since the mappings
are homeomorphisms on \({\mathcal {S}}_s'({\textbf{R}}^{2d})\). In particular, the map \(a\mapsto K_{a,A}\) is a homeomorphism on \({\mathcal {S}}_s'({\textbf{R}}^{2d})\).
An important special case appears when \(A=t\cdot I\), with \(t\in {\textbf{R}}\). Here and in what follows, \(I\in {\textbf{M}}(d,{\textbf{R}})\) denotes the \(d\times d\) identity matrix. In this case we set
The normal or Kohn-Nirenberg representation, a(x, D), is obtained when \(t=0\), and the Weyl quantization, \({\text {Op}}^w(a)\), is obtained when \(t=\frac{1}{2}\). That is,
For any \(K\in {\mathcal {S}}'_s({\textbf{R}}^{d_1+d_2})\), we let \(T_K\) be the linear and continuous mapping from \({\mathcal {S}}_s({\textbf{R}}^{d_1})\) to \({\mathcal {S}}_s'({\textbf{R}}^{d_2})\), defined by the formula
It is well-known that if \(A\in {\textbf{M}}(d,{\textbf{R}})\), then it follows from Schwartz kernel theorem that \(K\mapsto T_K\) and \(a\mapsto {\text {Op}}_A(a)\) are bijective mappings from \({\mathscr {S}}'({\textbf{R}}^{2d})\) to the set of linear and continuous mappings from \({\mathscr {S}}({\textbf{R}}^{d})\) to \({\mathscr {S}}'({\textbf{R}}^{d})\) (cf. e.g. [18]). Furthermore, by e.g. [20, Theorem 2.2] it follows that the same holds true if each \({\mathscr {S}}\) and \({\mathscr {S}}'\) are replaced by \({\mathcal {S}}_s\) and \({\mathcal {S}}_s'\), respectively, or by \(\Sigma _s\) and \(\Sigma _s'\), respectively.
In particular, for every \(a_1\in {\mathcal {S}}_s '({\textbf{R}}^{2d})\) and \(A_1,A_2\in {\textbf{M}}(d,{\textbf{R}})\), there is a unique \(a_2\in {\mathcal {S}}_s '({\textbf{R}}^{2d})\) such that \({\text {Op}}_{A_1}(a_1) = {\text {Op}}_{A_2} (a_2)\). The following result explains the relations between \(a_1\) and \(a_2\).
Proposition 1.18
Let \(a_1,a_2\in {\mathcal {S}}_{1/2}'({\textbf{R}}^{2d})\) and \(A_1,A_2\in {\textbf{M}}(d,{\textbf{R}})\). Then
In [32], a proof of the previous proposition is given, which is similar to the proof of the case \(A=t\cdot I\) in [18, 29, 38].
Let \(a\in {\mathcal {S}}_s '({\textbf{R}}^{2d})\) be fixed. Then a is called a rank-one element with respect to \(A\in {\textbf{M}}(d,{\textbf{R}})\), if \({\text {Op}}_A(a)\) is an operator of rank-one, i.e.
for some \(f_1,f_2\in {\mathcal {S}}_s '({\textbf{R}}^{d})\). By straight-forward computations it follows that (1.20) is fulfilled if and only if \(a=(2\pi )^{\frac{d}{2}}W_{f_1,f_2}^A\), where \(W_{f_1,f_2}^A\) is the A-Wigner distribution, defined by the formula
which takes the form
when \(f_1,f_2\in {\mathcal {S}}_s ({\textbf{R}}^{d})\). By combining these facts with (1.19), it follows that
for every \(f_1,f_2\in {\mathcal {S}}_s '({\textbf{R}}^{d})\) and \(A_1,A_2\in {\textbf{M}}(d,{\textbf{R}})\). Since the Weyl case is particularly important, we set \(W_{f_1,f_2}^{A}=W_{f_1,f_2}\) when \(A=\frac{1}{2}I\), i.e. \(W_{f_1,f_2}\) is the usual (cross-) Wigner distribution of \(f_1\) and \(f_2\).
For future references we note the link
between pseudo-differential operators and Wigner distributions, which follows by straight-forward computations (see e.g. [34] and the references therein).
For any \(A\in {\textbf{M}}(d,{\textbf{R}})\), the A-product, \(a{\#}_Ab\) between \(a\in {\mathcal {S}}_s' ({\textbf{R}}^{2d})\) and \(b\in {\mathcal {S}}_s'({\textbf{R}}^{2d})\) is defined by the formula
provided the right-hand side makes sense as a continuous operator from \({\mathcal {S}}_s ({\textbf{R}}^{d})\) to \({\mathcal {S}}_s '({\textbf{R}}^{d})\).
3 More general Orlicz modulation spaces
In this section we analyse more general Orlicz modulation spaces, parameterized with more quasi-Young functions, compared to what is introduced in Sect. 1. We prove that if two consecutive quasi-Young functions are the same, then the Orlicz modulation space remains the same if one of these parameterizing quasi-Young functions are removed. In particular it follows \(M^{\Phi ,\Phi }_{(\omega )} =M^{\Phi }_{(\omega )}\) for the Orlicz modulation spaces considered in Sect. 1.
Definition 2.1
Let \(\mu _j\) be (Borel) measures on \({\textbf{R}}^{d_j}\), \(\mu = \mu _1 \otimes \cdots \otimes \mu _N\), \(\Phi _{\!j}\) be quasi-Young functions, \(j=1,\dots ,N\), \(\omega \) be a weight function and f be measurable on \({\textbf{R}} ^{d_1+ \cdots + d_N}\). Then \(\Vert f\Vert _{L^{\Phi _1, \dots , \Phi _N}_{(\omega )}(\mu )} = \Vert f_{{N-1},\omega }\Vert _{L^{\Phi _N}(\mu )}\) where \(f_{k,\omega }\), \(k=1,\ldots ,N-1\) are inductively defined by
The space \(L^{\Phi _1, \dots , \Phi _N}_{(\omega )}(\mu )\) consists of all measurable functions f on \({\textbf{R}}^{d_1+ \cdots + d_N}\) such that \(\Vert f\Vert _{L^{\Phi _1, \dots , \Phi _N}_{(\omega )}(\mu )}\) is finite, and the topology of \(L^{\Phi _1, \dots , \Phi _N}_{(\omega )}(\mu )\) is induced by the quasi-norm \(\Vert \, \cdot \, \Vert _{L^{\Phi _1, \dots , \Phi _N}_{(\omega )}(\mu )}\).
Let
For 
, let
with \(\mu =dx _1 \otimes \cdots \otimes dx_N\) with \(x_j\in {\textbf{R}}^{d_j}\).
If \(\Lambda _j\subseteq {\textbf{R}}^{d_j}\) are lattices and \(\mu _j\) is the standard discrete or Haar measure on \(\Lambda _j\), then we set
as usual.
When discussing modulation spaces, it is suitable that 
should belong to \(I_{2d,N}^0\), which consists of all \((d_1,\dots ,d_N) \in I_{2d,N}\) such that
for some \(k\in \{ 1,\dots ,N-1\}\), when \(N\ge 2\). We observe that (2.1) implies
We observe that \(I_{2d,1}=\{ 2d\}\), and for convenience, we put \(I_{2d,1}^0=\{ 2d\}\).
Now suppose that 
, 
, k is chosen such that (2.1) holds, and let
Then we write \(\Lambda ^2=\Lambda \times \Lambda \) and
Let \(\Phi _{\!j}\) be quasi-Young functions, \(j=1,\dots ,N\), \(\omega \in {\mathscr {P}}_E({\textbf{R}}^{2d})\), 
and \(\phi \in \Sigma _1({\textbf{R}}^{d})\setminus 0\). Then the Orlicz modulation space
consists of all \(f\in \Sigma _1'({\textbf{R}}^{d})\) such that
is finite. By similar arguments as in [37] it follows that 
is a quasi-Banach space with quasi-norm 
, which is a Banach space and norm, respectively, when \(\Phi _j\) is a Young function for every \(j\in \{1,\dots ,N\}\).
A common situation is when 
for some integer \(d_0\ge 1\), and then we put
Remark 2.2
For future references we observe that Proposition 1.16 carry over to Orlicz modulation spaces of the form 
when \(\Phi _{\!j}\) are quasi-Young functions, \(j=1,\dots ,N\), \(\omega \in {\mathscr {P}}_E({\textbf{R}}^{2d})\) and 
. In particular it follows that (1.14) takes the form
Proposition 2.3
Let 
be lattices in 
, 
and let
Also let \(\omega \) be a weight on 
and 
be quasi-Young functions such that
If 
then
and
For the proof we recall that for the sequence a on \({\textbf{Z}}^{d_1+d_2}\) it holds
for some \(c >0\). This implies that
for some \(c_1>0\) and a positive sequence \(c_2\) on \({\textbf{Z}}^{d_2}\).
Proof
We only prove the result in the case \(N=2\) and for \(\Lambda _j={\textbf{Z}}^{d_j}\). The general case follows by these arguments and induction, and is left for the reader.
Let \(r_0 \in (0,1]\) be chosen such that \(\Phi _{0,j}(t)=\Phi _{\!j}(t^{\frac{1}{r_0}})\) are Young functions. Then
Furthermore, \(\Vert |a|\Vert _{\ell ^{\Phi _1 \Phi _2}} = \Vert a\Vert _{\ell ^{\Phi _1 \Phi _2}}\). This reduce the result to the case when \(\Phi \) is a Young function, \(\omega =1\) and \(a \ge 0\).
The result is obviously true when \(\Phi = 0\) near origin. In fact for such \(\Phi \),
in view of Proposition 1.11. In the same way, If \(\lim _{t \rightarrow 0+}\left( \frac{\Phi (t)}{t}\right) >0\), then
and the result follows in this case as well (see, e.g. [28, 37]). It remains to consider the case when 
, and when \(\lim _{t \rightarrow 0+} \left( \frac{\Phi (t)}{t}\right) =0\), Since \(\ell ^\Phi \) and \(\ell ^{\Phi ,\Phi }\) do not change when \(\Phi (t)\) is replaced by an increasing convex function which is equal to \(c \cdot \Phi (t)\) near \(t=0\), where \(c>0\) is a constant, it follows from Proposition 1.11 that we may assume that \(\Phi (t) \le t\) and that \(\Phi \) is increasing.
This gives
when \(c_1,c_2>0\) are constants.
Hence if
for some constant \(c>0\), then
for some constants \(c_1,c_2>0\). By (2.3) and (2.4) we get
We need to deduce the reversed inclusion in (2.5).
First we assume that a has finite support, i.e. \(a(j_1,j_2)\ne 0\) for at most finite numbers of \((j_1,j_2)\). Since \(\Phi (t) >0\) when \(t>0~\text {and}~\lim _{t \rightarrow 0+} \left( \frac{\Phi (t)}{t}\right) =0,\) it follows that the complementary Young function \(\Phi ^*\) to \(\Phi \) fulfills the same properties.
By Propositions 3 and 4 in Section 3.3 in [25], we have
and
By a combination of these relations and (2.5) we get
and the searched estimate follows for sequences with finite support.
For general \(a\ge 0\), let \(a_j\), \(j\ge 1\) be sequences such that
Then Beppo-Levi’s theorem gives
For general a, we may split up a into positive and negative real and imaginary parts and use (2.6) to get
This implies \(\ell ^{\Phi ,\Phi } ({\textbf{Z}}^{d_1+d_2}) \hookrightarrow \ell ^\Phi ({\textbf{Z}}^{d_1+d_2})\) and the result follows. \(\square \)
By combining Propositions 1.16, 2.3 and Remark 2.2 we get the following. The details are left for the reader.
Theorem 2.4
Let N and \(j_0\) be positive integers such that \(1\le j_0\le N-1\), \(\Phi _{\!j}\) and \(\Psi _k\), \(j=1,\dots ,N\), \(k=1,\dots ,N-1\), be quasi-Young functions such that (2.2) holds and let \(\omega \in {\mathscr {P}}_E({\textbf{R}}^{2d})\). Also let
and
Then
and
Corollary 2.5
Let N be a positive integer, 
, \(\Phi \) be a quasi-Young function and \(\omega \in {\mathscr {P}}_E({\textbf{R}}^{2d})\). Then
and
4 Continuity of pseudo-differential operators on Orlicz modulation spaces
In this section we deduce continuity properties of pseudo-differential operators when acting on Orlicz modulation spaces. The main results are Theorems 3.7 and 3.10 which deal with such operators with symbols in \(M^{\infty ,r_0}_{(\omega )}({\textbf{R}}^{2d})\) and \(M^{\Phi ,\Phi }_{(\omega )}({\textbf{R}}^{2d})\), respectively, where \(r_0\in (0,1]\) and \(\Phi \) is a quasi-Young functions.
In the first part we deduce related continuity results for suitable matrix operators. In the second part we combine these results and Gabor analysis results from the previous section to establish the continuity results for the pseudo-differential operators.
In the following definition we recall some matrix classes, considered in [33]. Here we observe that we may identify \(\Lambda \times \Lambda \) matrices with sequences on \(\Lambda \times \Lambda \), when \(\Lambda \) is a lattice in \({\textbf{R}}^{d}\).
Definition 3.1
Let \(p,q \in (0,\infty ]\), \(\Phi _1,\Phi _2\) be quasi-Young functions, \(\omega \in {\mathscr {P}}_E({\textbf{R}}^{2d})\), \(\Lambda \) be a lattice in \({\textbf{R}}^{d}\) and let T be the map on \(\ell _0'(\Lambda \times \Lambda )\), given by
-
(1)
The set \({\mathbb {U}}_0' (\Lambda \times \Lambda )\) consists of all (formal) matrices
$$\begin{aligned} A=(a(j,k))_{j,k\in \Lambda } \end{aligned}$$(3.1)with entries a(j, k) in \({\textbf{C}}\), and \({\mathbb {U}}_0 (\Lambda \times \Lambda )\) consists of all A in (3.1) such that at most finite numbers of a(j, k) are nonzero.
-
(2)
The set \({\mathbb {U}}^{p,q} _{(\omega )}(\Lambda \times \Lambda )\) consists of all matrices \(A=(a(j,k))_{j,k\in \Lambda }\) such that
$$\begin{aligned} \Vert A\Vert _{{\mathbb {U}}^{p,q}_{(\omega )}} \equiv \Vert T(a\cdot \omega )\Vert _{\ell ^{p,q}}, \end{aligned}$$is finite.
-
(3)
The set \({\mathbb {U}}^{\Phi _1,\Phi _2} _{(\omega )}(\Lambda \times \Lambda )\) consists of all matrices \(A=(a(j,k))_{j,k\in \Lambda }\) such that
$$\begin{aligned} \Vert A\Vert _{{\mathbb {U}}^{\Phi _1,\Phi _2}_{(\omega )}} \equiv \Vert T(a\cdot \omega )\Vert _{\ell ^{\Phi _1,\Phi _2}}, \end{aligned}$$is finite.
Remark 3.2
Let \(p\in (0,\infty ]\), \(\Phi \) be a quasi-Young function and \(\omega \in {\mathscr {P}}_E({\textbf{R}}^{2d})\). Then it follows from Proposition 2.3 and straight-forward changes of variables that the following is true. The details are left for the reader.
-
(1)
If \(A_0=(a(j,k))_{j,k\in {\textbf{Z}}^{d}}\) is a matrix, then \(A_0\in {\mathbb {U}} ^{p,p}_{(\omega )}({\textbf{Z}}^{2d})\), if and only if \(a\in \ell ^{p,p}_{(\omega )}({\textbf{Z}}^{2d}) = \ell ^p_{(\omega )}({\textbf{Z}}^{2d})\), and
$$\begin{aligned} \Vert A_0\Vert _{{\mathbb {U}} ^{p,p}_{(\omega )}} = \Vert a\Vert _{\ell ^{p,p}_{(\omega )}} = \Vert a\Vert _{\ell ^{p}_{(\omega )}}. \end{aligned}$$ -
(2)
If \(A_0=(a(j,k))_{j,k\in {\textbf{Z}}^{d}}\) is a matrix, then \(A_0\in {\mathbb {U}} ^{\Phi ,\Phi }_{(\omega )}({\textbf{Z}}^{2d})\), if and only if \(a\in \ell ^{\Phi ,\Phi }_{(\omega )}({\textbf{Z}}^{2d}) = \ell ^\Phi _{(\omega )}({\textbf{Z}}^{2d})\), and
$$\begin{aligned} \Vert A_0\Vert _{{\mathbb {U}} ^{\Phi ,\Phi }_{(\omega )}} = \Vert a\Vert _{\ell ^{\Phi ,\Phi }_{(\omega )}} = \Vert a\Vert _{\ell ^{\Phi }_{(\omega )}}. \end{aligned}$$
Next we discuss continuity for certain matrix operators when acting on discrete Orlicz spaces. We recall that if \(\Lambda \subseteq {\textbf{R}}^{d}\) is a lattice, \(\omega _1, \omega _2 \in {\mathscr {P}}_E ({\textbf{R}} ^{2d})\) and \(\omega \in {\mathscr {P}}_E ({\textbf{R}}^{4d})\) are such that
\(r_0\in (0,1]\) and \(p,q\in [r_0,\infty ]\), then [33, Theorem 2.3] shows that \(A_0\) from \(\ell _0(\Lambda ^2)\) to \(\ell _0'(\Lambda ^2)\) is uniquely extendable to a continuous map from \(\ell _{(\omega _1)}^{p,q}(\Lambda ^2)\) to \(\ell _{(\omega _2)}^{p,q}(\Lambda ^2)\). The following result extends this result to discrete Orlicz spaces.
Theorem 3.3
Let \(\varepsilon >0\), \(N\ge 1\) be an integer, 
, \(\Phi _1,\dots ,\Phi _N\) be quasi Young functions of order \(r_0 \in (0,1]\), \(\omega _1, \omega _2 \in {\mathscr {P}}_E ({\textbf{R}} ^{2d})\), \(\omega \in {\mathscr {P}}_E ({\textbf{R}}^{4d})\) be such that (3.2) holds. If \(A\in {\mathbb {U}}^{\infty ,r_0} _{(\omega )}(\varepsilon {\textbf{Z}}^{4d})\), then A from \(\ell _{(\omega _1)}^\infty (\varepsilon {\textbf{Z}}^{2d})\) to \(\ell _{(\omega _2)}^\infty (\varepsilon {\textbf{Z}}^{2d})\) restricts to a continuous map from 
to 
and
We need the following lemma for the proof of Theorem 3.3. We omit the proof since the result is a consequence of [37, Lemma 3.1].
Lemma 3.4
Let \(\Lambda \subseteq {\textbf{R}}^{d}\) be a lattice, \({\mathscr {B}}\subseteq \ell _0'(\Lambda )\) be a quasi-Banach space of order \(r_0\in (0,1]\), with quasi-norm \(\Vert \, \cdot \, \Vert _{{\mathscr {B}}}\). If
then the discrete convolution map \((f,g)\mapsto f*_\Lambda g\) from \(\ell ^{r_0}(\Lambda )\times \ell ^{r_0}(\Lambda )\) to \(\ell ^{r_0}(\Lambda )\) extends uniquely to a continuous map from \({\mathscr {B}}\times \ell ^{r_0}(\Lambda )\) to \({\mathscr {B}}\), and
Proof of Theorem 3.3
We only prove the result in the case \(N=2\). For general N, the result follows by similar arguments, and is left for the reader. Let \(f\in \ell ^{\Phi _1,\Phi _2}_{(\omega _1)} (\varepsilon {\textbf{Z}}^{2d})\) and set \(g=Af\).
First we consider the case when \(A\in {\mathbb {U}}_0 (\varepsilon {\textbf{Z}}^{4d})\) and let
and
We get
where \(h_{\omega }(k)=\sup \limits _{j\in \varepsilon {\textbf{Z}}^{2d}} a_{\omega }(j,j-k)\).
By Lemma 3.4 we get
and the result follows in this case.
For general \(A\in {\mathbb {U}}^{\infty ,r_0} _{(\omega )}(\varepsilon {\textbf{Z}}^{4d})\) we decompose A and f into
where \(A_j\) and \(f_k\) only have non-negative entries, chosen as small as possible. By Beppo-Levi’s theorem and the estimates above it follows that \(A_j f_k\) is uniquely defined as an element in \(\ell ^{\Phi _1\Phi _2}_{(\omega _2)}(\varepsilon {\textbf{Z}}^{2d})\). It also follows from these estimates that (3.3) holds. \(\square \)
Remark 3.5
Let
Then \(A_n\) in (3.4) are given by
where
and \(f_n = \{ f_n(j)\} _{j\in \varepsilon {\textbf{Z}}^{2d}}\), are obtained in the same way after each \(a_n(j,k)\) and a(j, k) are replaced by \(f_n(j)\) and f(j), respectively.
Before we discuss continuity properties of pseudo-differential operators on Orlicz modulation spaces, we have the following result concerning operator classes
of continuous operators from \(\Sigma _1({\textbf{R}}^{d})\) to \(\Sigma _1'({\textbf{R}}^{d})\). Here recall [36, Proposition 1.9] for analogous relations for pseudo-differential operators with symbols in (ordinary) modulation spaces. Here and in what follow, \(A^*\) denotes the transpose of the matrix A.
Proposition 3.6
Let \(N\ge 1\) be an integer, 
, \(A\in {\textbf{M}}(d,{\textbf{R}})\), \(\Phi _1,\dots ,\Phi _N\) be quasi-Young functions, \(\omega \in {\mathscr {P}}_E({\textbf{R}}^{4d})\) and let
Then the following is true:
-
(1)
The map \(e^{i\langle AD_\xi ,D_x\rangle }\) from \(M^\infty _{(\omega )}({\textbf{R}}^{2d})\) to \(M^\infty _{(\omega _A)}({\textbf{R}}^{2d})\) restricts to a homeomorphism from
to 
; -
(2)
The set
of operators from \(\Sigma _1({\textbf{R}}^{d})\) to \(\Sigma _1'({\textbf{R}}^{d})\) is independent of \(A\in {\textbf{M}}(d,{\textbf{R}})\).
to 
;
Proof
We only prove the result in the case \(N=2\). For general N, the result follows by similar arguments and is left for the reader.
It suffices to prove (1) in view of Proposition 1.18.
Let \(a\in M^{\Phi _1,\Phi _2} _{(\omega )}({\textbf{R}}^{2d})\), \(\phi \in \Sigma _1({\textbf{R}}^{d})\), \(\psi = e^{i\langle AD_\xi ,D_x\rangle }\phi \) and \(b=e^{i\langle AD_\xi ,D_x\rangle }a\). Then it follows from Theorem 3.1 and (3.1) in [1] that \(\psi \in \Sigma _1({\textbf{R}}^{d})\) and
By applying the \(L^{\Phi _1}\) quasi-norm with respect to the \((x,\xi )\) variables we obtain
and applying the \(L^{\Phi _2}\) quasi-norm with respect to the \((y,\eta )\) variable on the last equality gives
This gives
and the result follows. \(\square \)
We have now the following continuity result for pseudo-differential operators acting on Orlicz modulation spaces. Here the involved weight functions should satisfy
Theorem 3.7
Let \(A\in M(d,{\textbf{R}})\), \(\Phi _1,\Phi _2\) be quasi Young functions of order \(r_0 \in (0,1]\), \(\omega \in {\mathscr {P}}_E ({\textbf{R}}^{4d})\) and \(\omega _1,\omega _2 \in {\mathscr {P}}_E({\textbf{R}}^{2d})\) be such that (3.5) holds, and let \(a\in M^{\infty ,r_0}_{(\omega )}({\textbf{R}}^{2d})\). Then \({\text {Op}}_A(a)\) from \(\Sigma _1({\textbf{R}}^{d})\) to \(\Sigma _1'({\textbf{R}}^{d})\) is uniquely extendable to a continuous map from \(M^{\Phi _1,\Phi _2}_{(\omega _1)}({\textbf{R}} ^{d})\) to \(M^{\Phi _1,\Phi _2}_{(\omega _2)}({\textbf{R}} ^{d})\), and
The previous result can be generalized into the following.
Theorem 3.8
Let \(A\in M(d,{\textbf{R}})\), N be a positive integer, 
, \(\Phi _{\!j}\), \(j=1,\dots ,N\), be quasi-Young functions of order \(r_0 \in (0,1]\), \(\omega \in {\mathscr {P}}_E ({\textbf{R}}^{4d})\) and \(\omega _1,\omega _2 \in {\mathscr {P}}_E({\textbf{R}}^{2d})\) be such that (3.5) holds, and let \(a\in M^{\infty ,r_0}_{(\omega )}({\textbf{R}}^{2d})\). Then \({\text {Op}}_A(a)\) from \(\Sigma _1({\textbf{R}}^{d})\) to \(\Sigma _1'({\textbf{R}}^{d})\) is uniquely extendable to a continuous map from 
to 
, and
We only prove Theorem 3.7. Theorem 3.8 follows by similar arguments and is left for the reader.
We need some preparations for the proof of Theorem 3.7. First we have the following extension of [33, Lemma 3.3] to the case of Orlicz modulation spaces.
Lemma 3.9
Let \(\Lambda \), \(\phi _1\), \(\phi _2\), \(\varphi \), \(\psi \) and \(\varepsilon >0\) be as in Lemma 1.15. Also let \(v\in {\mathscr {P}}_E ({\textbf{R}}^{4d})\), \(a\in M^\infty _{(1/v)}({\textbf{R}}^{2d})\),
and let \(A_a\) be the matrix \(A_a=(c_0({{\varvec{j}}},{{\varvec{k}}})) _{{{\varvec{j}}},{{\varvec{k}}}\in \varepsilon \Lambda ^2}\). Then the following is true:
-
(1)
If \(\Phi _1\), \(\Phi _2\) are quasi-Young functions and \(\omega ,\omega _0\in {\mathscr {P}}_E({\textbf{R}}^{4d})\) satisfy
$$\begin{aligned} \omega (x,\xi ,y,\eta )\asymp \omega _0(x,\eta ,\xi -\eta ,y-x), \end{aligned}$$(3.8)then \(a\in M^{\Phi _1,\Phi _2}_{(\omega _0)}({\textbf{R}}^{2d})\), if and only if \(A_a\in {\mathbb {U}}^{\Phi _1,\Phi _2}_{(\omega )} (\varepsilon (\Lambda ^2\times \Lambda ^2) )\), and then
$$\begin{aligned} \Vert a\Vert _{M^{\Phi _1,\Phi _2}_{(\omega _0)}}\asymp \Vert A_a\Vert _{{\mathbb {U}}^{\Phi _1,\Phi _2}_{(\omega )}(\varepsilon (\Lambda ^2\times \Lambda ^2))} \text{; } \end{aligned}$$ -
(2)
\({\text {Op}}(a)\) as map from \(\Sigma _1({\textbf{R}}^{d})\) to \(\Sigma _1'({\textbf{R}}^{d})\) is given by
$$\begin{aligned} {\text {Op}}(a) = D_{\phi _1}^{\varepsilon ,\Lambda } \circ A_a \circ C_{\phi _2}^{\varepsilon ,\Lambda }. \end{aligned}$$(3.9)
Proof
We have
Hence, Proposition 1.16 gives
and (1) follows.
The assertion (2) is the same as assertion (2) in [33, Lemma 3.3]. The proof is therefore omitted. \(\square \)
Proof of Theorem 3.7
By Proposition 3.6 we may assume that \(A=0\).
Let a, \(A_a\), \(\phi _1\) and \(\phi _2\) be the same as in Proposition 1.17 and Lemma 3.9. Then by Proposition 1.17, Theorem 3.3 and Lemma 3.9 we get
where
and
This gives the asserted continuity. The uniqueness follows from the facts that
in view of Proposition 1.11 and that \({\text {Op}}(a)\) is uniquely defined as a continuous operator from \(M^{\infty }_{(\omega _1)}({\textbf{R}} ^{d})\) to \(M^{\infty }_{(\omega _2)}({\textbf{R}} ^{d})\), in view of [33, Theorem 3.1]. \(\square \)
We have also the following.
Theorem 3.10
Let \(A\in M(d,{\textbf{R}})\), \(\Phi _0\) be a Young function, \(\Phi _0^*\) the complementary Young function of \(\Phi _0\), \(\Phi \) be a quasi-Young function such that
is finite and let \(\omega \in {\mathscr {P}}_E ({\textbf{R}}^{4d})\) and \(\omega _1,\omega _2 \in {\mathscr {P}}_E({\textbf{R}}^{2d})\) be such that (3.5) holds. Then the following is true:
-
(1)
if \(a \in M^{\Phi _0} _{(\omega )}({\textbf{R}}^{2d})\), then \({\text {Op}}_A(a)\) from \(M^1_{(\omega _1)}({\textbf{R}}^{d})\) to \(M^\infty _{(\omega _2)}({\textbf{R}}^{d})\) is extendable to a continuous map from \(M^{\Phi _0^* }_{(\omega _1)}({\textbf{R}} ^{d})\) to \(M^{\Phi _0}_{(\omega _2)}({\textbf{R}} ^{d})\) and
$$\begin{aligned} \Vert {\text {Op}}_A(a)\Vert _{M^{\Phi _0^* }_{(\omega _1)}({\textbf{R}} ^{d})\rightarrow M^{\Phi _0}_{(\omega _2)}({\textbf{R}} ^{d})} \lesssim \Vert a\Vert _{M^{\Phi _0}_{(\omega )}({\textbf{R}}^{2d})}\text{; } \end{aligned}$$ -
(2)
if \(a \in M^{\Phi } _{(\omega )}({\textbf{R}}^{2d})\), then \({\text {Op}}_A(a)\) from \(M^1_{(\omega _1)}({\textbf{R}}^{d})\) to \(M^\infty _{(\omega _2)}({\textbf{R}}^{d})\) is uniquely extendable to a continuous map from \(M^{\infty }_{(\omega _1)}({\textbf{R}} ^{d})\) to \(M^{\Phi }_{(\omega _2)}({\textbf{R}} ^{d})\), and
$$\begin{aligned} \Vert {\text {Op}}_A(a)\Vert _{M^{\infty }_{(\omega _1)}({\textbf{R}} ^{d})\rightarrow M^{\Phi }_{(\omega _2)}({\textbf{R}} ^{d})} \lesssim \Vert a\Vert _{M^{\Phi }_{(\omega )}({\textbf{R}}^{2d})}. \end{aligned}$$
Proof
By Proposition 3.6 we may assume that \(A=0\).
Let \(\Lambda \subseteq {\textbf{R}}^{d}\) be a lattice, \(A_0=(a(j,k))_{j,k\in \Lambda } \in {\mathbb {U}} ^{\Phi _0,\Phi _0} _{(\omega )}(\Lambda \times \Lambda )\) and \(f\in \ell ^{\Phi _0^*}_{(\omega _1)}(\Lambda )\) be such that \(a(j,k)\ge 0\) and \(f(j)\ge 0\) for every \(j,k\in \Lambda \). We have
By applying the \(\ell ^{\Phi _0}\) norm and using Remark 3.2 we get
which implies that \(A_0f\) makes sense as an element in \(\ell ^{\Phi _0}_{(\omega _2)}(\Lambda )\).
For general \(A_0=(a(j,k))_{j,k\in \Lambda } \in {\mathbb {U}} ^{\Phi _0,\Phi _0} _{(\omega )}(\Lambda \times \Lambda )\) and \(f\in \ell ^{\Phi _0^*}_{(\omega _1)} (\Lambda )\), we define \(A_0f\) in similar ways as in the proof of Theorem 3.3, by splitting up \(A_0\) and f into positive and negative parts of their real and imaginary parts. By (3.14) we obtain
Now let \(a\in M^{\Phi _0}_{(\omega )}({\textbf{R}}^{2d})\) and \(f\in M^{\Phi _0^*}_{(\omega _1)}({\textbf{R}}^{d})\). Then we define \({\text {Op}}(a)f\) by (3.9). The asserted continuity in (1) now follows from Proposition 2.3, (3.10), (3.12) and (3.15).
Next let \(\Phi \) be as in (2) and let \(A_0=(a(j,k))_{j,k\in \Lambda } \in {\mathbb {U}} ^{\Phi ,\Phi }_{(\omega )} (\Lambda \times \Lambda )\) and \(f\in \ell ^{\infty }_{(\omega _1)}(\Lambda )\) be such that \(a(j,k)\ge 0\) and \(f(j)\ge 0\) for every \(j,k\in \Lambda \). Then
where the last inequality follows from Proposition 1.11 and (3.13). By applying the \(\ell ^\Phi \) quasi-norm and splitting up general \(A_0=(a(j,k))_{j,k\in \Lambda } \in {\mathbb {U}} ^{\Phi ,\Phi }_{(\omega )} (\Lambda \times \Lambda )\) and \(f\in \ell ^{\infty }_{(\omega _1)}(\Lambda )\) into positive and negative real and imaginary parts, we obtain
The asserted continuity in (2) now follows by combining Proposition 2.3, (3.10), (3.12) and (3.16).
The asserted uniqueness follows from the fact that if \(a\in M^\Phi _{(\omega )}({\textbf{R}}^{2d})\), then \(a\in M^1_{(\omega )}({\textbf{R}}^{2d})\) in view of Proposition 1.11 and (3.13). Hence, if \(f\in M^\infty _{(\omega _1)}({\textbf{R}}^{d})\), then \({\text {Op}}(a)f\) is uniquely defined as an element in \(M^1_{(\omega _2)}({\textbf{R}}^{d})\) (see e.g. [33, Theorem 3.1]). This in turn implies that \({\text {Op}}(a)f\) is uniquely defined as an element in \(M^\Phi _{(\omega _2)}({\textbf{R}}^{d})\), and the result follows. \(\square \)
5 Symbol product estimates on Orlicz modulation spaces
In this section we show that if \(\omega _j\) are suitable weights, \(j=0,1,2\), \(\Phi _1,\Phi _2\) are quasi-Young functions of order \(r_0\in (0,1]\), \(a_1\in M^{\Phi _1,\Phi _2}_{(\omega _1)}\) and \(a_2\in M^{\infty ,r_0}_{(\omega _2)}\), then \({\text {Op}}_A(a_1)\circ {\text {Op}}_A(a_2)\) equals \({\text {Op}}_A(b)\) for some \(a_1\in M^{\Phi _1,\Phi _2}_{(\omega _0)}\).
An essential condition on the weight functions is
where
Theorem 4.1
Let \(A\in M(d,{\textbf{R}})\) and suppose that \(\omega _k \in {\mathscr {P}}_E({\textbf{R}}^{4d})\), \(k=0,1,2\), satisfy (4.1) and (4.2). Let \(\Phi _1,\Phi _2\) be quasi Young functions of order \(r_0 \in (0,1]\). Then the map \((a_1 ,a_2)\mapsto a_1{\#}_Aa_2\) from \(\Sigma _1({\textbf{R}}^{2d})\times \Sigma _1({\textbf{R}}^{2d})\) to \(\Sigma _1({\textbf{R}}^{2d})\) is uniquely extendable to a continuous map from \(M^{\Phi _1,\Phi _2}_{(\omega _1)}({\textbf{R}}^{2d}) \times M^{\infty ,r_0}_{(\omega _2)}({\textbf{R}}^{2d})\) to \(M^{\Phi _1 ,\Phi _2}_{(\omega )}({\textbf{R}}^{2d})\), and
We need some preparations for the proof. By [2, Proposition 3.2] it follows that the map \((A_1,A_2)\mapsto A_1\circ A_2\) is uniquely defined and continuous from \({\mathbb {U}}^{\infty ,\infty }_{(\omega _1)} (\Lambda \times \Lambda )\times {\mathbb {U}}^{\infty ,r_0}_{(\omega _2)}(\Lambda \times \Lambda )\) to \({\mathbb {U}}^{\infty ,\infty }_{(\omega )}(\Lambda \times \Lambda )\) when \(\Lambda \subseteq {\textbf{R}}^{d}\) is a lattice, \(r_0 \in (0,1]\) and \(\omega , \omega _1, \omega _2 \in {\mathscr {P}}_E ({\textbf{R}} ^{2d})\) satisfy
The following lemma extends certain parts of this continuity to matrix classes satisfying Orlicz estimates.
Lemma 4.2
Let \(\Lambda \subseteq {\textbf{R}}^{d}\) be a lattice, \(\Phi _1,\Phi _2\) be quasi Young functions of order \(r_0 \in (0,1]\), and let \(\omega , \omega _1, \omega _2 \in {\mathscr {P}}_E ({\textbf{R}} ^{2d})\) satisfy (4.4). Then \((A_1,A_2)\mapsto A_1\circ A_2\) from \({\mathbb {U}}^{\infty ,\infty }_{(\omega _1)}(\Lambda \times \Lambda )\times {\mathbb {U}}^{\infty ,r_0}_{(\omega _2)}(\Lambda \times \Lambda )\) to \({\mathbb {U}}^{\infty ,\infty }_{(\omega )}(\Lambda \times \Lambda )\) restricts to a continuous map from \({\mathbb {U}}^{\Phi _1,\Phi _2}_{(\omega _1)}(\Lambda \times \Lambda )\times {\mathbb {U}}^{\infty ,r_0}_{(\omega _2)}(\Lambda \times \Lambda )\) to \({\mathbb {U}}^{\Phi _1,\Phi _2}_{(\omega )}(\Lambda \times \Lambda )\) and
Proof
Let \(A_1=({{\varvec{a}}}_1(j,k))_{j,k\in \Lambda }\), \(A_2=({{\varvec{a}}}_2(j,k))_{j,k\in \Lambda }\) be matrices, let the matrix elements of \(B=A_1\circ A_2\) be denoted by \({{\varvec{b}}}(j,k)\), and set
and
Then
and
By a similar application of Beppo-Levi’s theorem, and splitting up \(A_\textit{j}\) as in Remark 3.5, the result follows if we prove
when \(a_1,a_2\in \mathbb U_0(\Lambda \times \Lambda )\) have non-negative entries.
Let \(\Phi _{0,j}\) be Young functions such that \(\Phi _j(t)=\Phi _{0,j}(t^{r_0})\), \(t\ge 0\), \(j=1,2\), and let
By (4.6) and the fact that \(\Phi _{0,1}\) is convex we get
This gives
in view of the definition of \(\ell ^{\Phi _1,\Phi _2}\) norm.
By (4.7) we get
\(\square \)
Proof of Theorem 4.1
By Proposition 3.6 we may assume that \(A=0\).
Let \(\varepsilon >0\), \(\phi _1\), \(\phi _2\) and \(\Lambda \) be the same as in the proofs of Theorem 3.7 and Lemma 3.9, \(a_1\in M^{\Phi _1,\Phi _2}_{(\omega _1)}({\textbf{R}} ^{2d})\) and \(a_2\in M^{\infty ,r_0}_{(\omega _2)}({\textbf{R}} ^{2d})\). By Theorem 2.17 we have
where
and
The condition (4.1) means for the weights \(\vartheta _m\), \(m=0,1,2\),
Pick \(v_1 \in {\mathscr {P}}_E({\textbf{R}} ^{d})\) such that \(\omega _2\) is \(v_2\)-moderate, where
Also let \(v =v^2_{1}\otimes v^2_{1} \in {\mathscr {P}}_E({\textbf{R}} ^{2d})\) and
Then
where
and \(C=C_{\phi _2}^{\varepsilon ,\Lambda }\circ D_{\phi _1}^{\varepsilon ,\Lambda }\) is a matrix of the form \((\pmb {c}({{\varvec{j}}},{{\varvec{k}}}))_{{{\varvec{j}}},{{\varvec{k}}}\in \varepsilon \Lambda ^2}\) with matrix elements \(\pmb {c}({{\varvec{j}}},{{\varvec{k}}})\), \({{\varvec{j}}},{{\varvec{k}}}\in \varepsilon \Lambda ^2\).
By [2, Lemma 3.3] we get
Thus
Then we obtain from Lemmas 3.9 and 4.2
\(\square \)
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Toft, J., Üster, R. Pseudo-differential operators on Orlicz modulation spaces. J. Pseudo-Differ. Oper. Appl. 14, 6 (2023). https://doi.org/10.1007/s11868-022-00492-5
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DOI: https://doi.org/10.1007/s11868-022-00492-5