Linear and bilinear Fourier multipliers on Orlicz modulation spaces

Abstract

Let $\Phi _i, \Psi _i$ be Young functions, $\omega _i$ be weights and $M^{\Phi _i,\Psi _i}_{\omega _i}(\mathbb {R} ^{d})$ be the corresponding Orlicz modulation spaces for $i=1,2,3$. We consider linear (respect. bilinear) multipliers on $\mathbb {R} ^{d}$, that is bounded measurable functions $m(\xi )$ (respect. $m(\xi ,\eta )$) on $\mathbb {R} ^{d}$ (respect. $\mathbb {R} ^{2d}$) such that

$$\begin{aligned} T_m(f)(x)=\int _{\mathbb {R} ^{d}}{\hat{f}}(\xi ) m(\xi )e^{2\pi i \langle \xi , x\rangle }d\xi \end{aligned}$$

(respect.

$$\begin{aligned} B_m(f_1,f_2)(x)=\int _{\mathbb {R} ^{d}}\int _{\mathbb {R} ^{d}} \hat{f_1}(\xi ) \hat{f_2}(\eta )m(\xi ,\eta )e^{2\pi i \langle \xi +\eta , x\rangle }d\xi d\eta \end{aligned}$$

define a bounded linear (respect. bilinear) operator from $M^{\Phi _1,\Psi _1}_{\omega _1}(\mathbb {R} ^{d})$ to $M^{\Phi _2,\Psi _2}_{\omega _2}(\mathbb {R} ^{d})$ (respect. $M^{\Phi _1,\Psi _1}_{\omega _1}(\mathbb {R} ^{d})\times M^{\Phi _2,\Psi _2}_{\omega _2}(\mathbb {R} ^{d})$ to $M^{\Phi _3,\Psi _3}_{\omega _3}(\mathbb {R} ^{d})$). In this paper we study some properties of these spaces and give methods to generate linear and bilinear multipliers between Orlicz modulation spaces.

1 Introduction

Throughout the paper $\mathcal {S}(\mathbb {R} ^{d})$ denotes the Schwartz class on $\mathbb {R} ^{d}$, i.e., $f:\mathbb {R} ^{d}\rightarrow \mathbb {C}$ such that $f\in C^\infty (\mathbb {R} ^{d})$ and $x^\alpha \partial ^\beta f(x)$ is bounded for any multi-indices $\alpha , \beta$. We write the Fourier transform by ${\mathcal {F}}(f)={\hat{f}}$ where ${{\hat{f}}}(\xi )=\int _{\mathbb {R} ^{d}} f(x)e^{-2\pi i \langle x, \xi \rangle } dx$. Here $\langle \cdot ,\cdot \rangle$ denotes the usual scalar product on $\mathbb {R} ^{d}$. We denote the translation by $\tau _yf(x)=f(x-y)$, the modulation by $M_yf(x)=f(y)e^{2\pi i \langle x,y\rangle }$ and the dilation by $D_\lambda f(x)=f(\lambda x)$ for $x,y\in \mathbb {R} ^{d}$ and $\lambda >0$. Moreover we write $f_\lambda (x)=\frac{1}{\lambda ^d}f(\frac{x}{\lambda })$. Clearly one has for each $f\in L^1(\mathbb {R} ^{d})$, $y\in \mathbb {R} ^{d}$ and $\lambda >0$

$$\begin{aligned} (\widehat{\tau _yf})(\xi )=M_{-y}{{\hat{f}}}(\xi ),\quad (\widehat{M_xf})(\xi )=\tau _x{{\hat{f}}}(\xi ),\quad (\widehat{D_\lambda f})(\xi )={\hat{f}}_\lambda (\xi ). \end{aligned}$$

One of the basic problems in Fourier Analysis is the description of linear and bilinear Fourier multipliers acting on different function spaces defined on ${\mathbb {R}}^d$. In the linear case one deals with bounded measurable functions m defined on $\mathbb {R} ^{d}$ and linear operators given by

$$\begin{aligned} T_m(f)(x)=\mathcal {F}^{-1}(m\mathcal {F}f)(x) \end{aligned}$$

defined for $f\in {\mathcal {S}}(\mathbb {R} ^{d})$, while in the bilinear formulation one deals with bounded measurable functions m defined on $\mathbb {R} ^{2d}$ and bilinear operators given by

$$\begin{aligned} B_m(f,g)(x)=\mathcal {F}_2^{-1}\Big (m({\mathcal {F}} f\otimes {\mathcal {F}} g)\Big )(x,x) \end{aligned}$$

defined for $f,g\in {\mathcal {S}}(\mathbb {R} ^{d})$ where $({\mathcal {F}} f\otimes {\mathcal {F}} g) (\xi ,\eta )={{\hat{f}}}(\xi ) {{\hat{g}}}(\eta )$ for $\xi ,\eta \in \mathbb {R} ^{d}$ and $\mathcal {F}_2$ denotes the Fourier transform in $\mathbb {R} ^{2d}$. There are two types of problems concerning these operators: either one fixes the measurable function m and study their boundedness when acting on a concrete scale of functions spaces or one starts with certain function spaces and tries to describe the symbols m for the above operators to be bounded when acting on them. The study of linear Fourier multipliers in the setting of Lebesgue spaces is rather classical (see [14, 25, 36]). The investigation of bilinear multipliers was originated in the work by Coiffman and Meyer [9] in the eighties of the last century and continued by Grafakos and Torres [15] and many others. A renewed interest appeared in the nineties after the results by Kenig and Stein on the bilinear fractional integral [17] and the celebrated result by Lacey and Thiele [21,22,23,24], solving the old standing conjecture of Calderón on the boundedness of the bilinear Hilbert transform acting on Lebesgue spaces. In the last decades the study of bilinear multipliers acting in different function spaces has been developed in several papers (see [3, 4, 12, 17, 32, 39]).

Besides many other function spaces there has been a recent interest in the case of Orlicz spaces. They naturally generalize $L^p$ spaces and contain certain Sobolev spaces as subspaces. Orlicz spaces appear in computation such as the Zygmund space $L \log ^+ L$, which is a Banach space related to Hardy–Littlewood maximal functions. Different problems on linear and bilinear multipliers on Orlicz spaces have been considered in [4,5,6, 38]. The reader is referred to [16, 30] for the general theory of Orlicz spaces and to [27,28,29] for weighted Orlicz spaces on locally compact groups.

In this paper we shall deal with spaces arising from the time-frequency analysis and Fourier (linear and bilinear) multipliers acting on them. In time–frequency analysis one is interested to measure, quantitatively, the behavior of functions and distributions in the time–frequency plane ${\mathbb {R}}^d \times {\mathbb {R}}^d$, and for this purpose a natural candidate is to deal with modulation spaces. Since their definition in early 1980’s due to Feichtinger [10] (originally based on Lebesgue spaces) and a first systematic study appeared in [13]), modulation spaces have found their way into different areas of mathematical analysis and applications, in particular in the theory of pseudo-differential operators on modulation spaces or with symbols in modulation spaces. Also the definition has been extended to cover more cases than the Lebesgue spaces, such as Orlicz spaces (see [35, 37]) or even more general function spaces (see [13, Chapter 11]). We shall concentrate in the class of Orlicz modulation spaces. Since the family of Orlicz spaces contains all Lebesgue spaces it follows that the family of Orlicz modulation spaces contain all classical modulation spaces introduced by Feichtinger in [10].

Our objective in the paper will be to deal with linear and bilinear multipliers on $\mathbb {R} ^{d}$ acting on Orlicz modulation spaces. We find examples of linear and bilinear multipliers in these classes, and get methods to produce new ones. We refer the reader to [1, 2, 11, 18] for some results on (linear) Fourier multipliers on classical modulation spaces.

The paper is organized as follows: In Sect. 2 we recall some basic definitions and notions on Orlicz and weighted Orlicz spaces, weighted mixed norm Orlicz spaces and Orlicz modulation spaces. Moreover we obtain several results to be used in sequel. In particular we present the corresponding version of Hölder’s inequality for products of functions or the version of Young’s inequality for convolution of functions in this setting (see Theorem 2.7 for weighted Orlicz spaces, Theorems 2.13 and 2.14 for weighted mixed norm Orlicz spaces and Theorem 2.27 for Orlicz modulation spaces). Also we consider in this section some results on the density of the Schwartz class in the previously mentioned classes and the norm of the dilation operator acting on Orlicz modulation spaces.

Section 3 contains the results on multipliers acting on Orlicz modulation spaces. We present there the version of Young’s inequality in the setting of Orlicz modulation spaces in Theorem 3.3. Also we give elementary examples of linear and bilinear multipliers and methods to generate them which are obtained analyzing the translation, modulation and dilation of multipliers.

Throughout the paper $\Phi , \Psi$ stand for Young functions, $\omega$ for weights and C denotes a constant that may vary from line to line.

2 Results

2.1 Weighted Orlicz spaces

First we define and recall some basic facts for Orlicz spaces and weighted Orlicz spaces (see [16, 30]).

A non-zero function $\Phi :[0,\infty ) \rightarrow [0,\infty ]$ is called a Young function if $\Phi$ is convex, $\Phi (0)=0$ and $\lim _{t\rightarrow \infty }\Phi (t)=\infty$. Given a Young function $\Phi$, its complementary function, to be denoted $\Phi ^*$, is defined by

$$\begin{aligned} \Phi ^*(s)=\sup \{ ts - \Phi (t) : t\ge 0 \} \end{aligned}$$

for $s\ge 0$. It can be seen that $\Phi ^*$ is also a Young function and the pair $(\Phi ,\Phi ^*)$ is called a complementary pair of Young functions.

By definition, a Young function can have the value $\infty$ at a point, and hence be discontinuous at such a point. However, we always consider the pair of complementary Young functions $(\Phi ,\Phi ^*)$ with being real valued and continuous on $[0,\infty )$. Note that even though $\Phi$ is continuous, it may happen that $\Phi ^*$ is not continuous.

Since $\Phi$ is a convex function and $\Phi (0)=0$, a Young function $\Phi$ is non-decreasing.

Let us recall that a complementary pair of Young functions satisfies the Young’s inequality

$$\begin{aligned} ts \le \Phi (t)+\Phi ^*(s) ,\quad t,s \ge 0. \end{aligned}$$

(2.1)

The Orlicz space $L^\Phi (\mathbb {R} ^{d})$ is defined by

$$\begin{aligned} L^\Phi (\mathbb {R} ^{d})=\left\{ f:\mathbb {R} ^{d}\rightarrow \mathbb {C}: \int \limits _{\mathbb {R} ^{d}}\Phi (\alpha |f(x)|)dx<\infty \text { for some }\alpha >0\right\} \end{aligned}$$

where f indicates an equivalence class of measurable functions. The Orlicz space is a Banach space under the Luxemburg norm $N_\Phi (f)$ given by

$$\begin{aligned} N_\Phi (f)=\inf \left\{ \lambda >0:\int \limits _{\mathbb {R} ^{d}}\Phi (\frac{|f(x)|}{\lambda })dx\le 1\right\} . \end{aligned}$$

Since the Young function $\Phi$ is increasing, the Orlicz space $L^\Phi ({\mathbb {R}}^d)$ is a solid space [30, Theorem 3.1.2]. That is if any measurable function f for which there exists $g \in L^\Phi ({\mathbb {R}}^d)$ such that $|f|\le |g|$ then $f\in L^\Phi ({\mathbb {R}}^d)$ and $N_\Phi (f)\le N_\Phi (g)$.

We define $S^{\Phi }(\mathbb {R} ^{d})$ as the closure in $L^{\Phi }(\mathbb {R} ^{d})$ of the linear space of all simple functions, i.e. linear combinations of $\chi _E$ where E is a Borel set of finite measure. It is known that $(S^{\Phi }(\mathbb {R} ^{d}))^{*}$, the dual of $S^{\Phi }(\mathbb {R} ^{d})$, can be identified with $L^{\Phi ^*}(\mathbb {R} ^{d})$ in a natural way [30, Theorem 4.1.6].

Recall that a Young function $\Phi$ is said to satisfy $\Delta _2$-condition (globally) if there exists a constant $K>0$ such that

$$\begin{aligned} \Phi (2t) \le K\Phi (t), \quad t\ge 0.\end{aligned}$$

(2.2)

If $\Phi$ satisfies $\Delta _2$-condition then simple functions are dense in $L^\Phi ({\mathbb {R}}^d)$ and we have

$$\begin{aligned} L^{\Phi }(\mathbb {R} ^{d})=S^{\Phi }(\mathbb {R} ^{d}) \end{aligned}$$

(2.3)

(see [30, Corollary 3.4.5]).

Since using Lusin’s theorem one can see that $C_{c}(\mathbb {R} ^{d})$, the space of all continuous compactly supported functions on $\mathbb {R} ^{d}$ is dense in $S^\Phi (\mathbb {R} ^{d})$ (see [33, Theorem 3.14]) then we know that under the $\Delta _2$-condition of $\Phi$ we have that $C_c(\mathbb {R} ^{d})$ is dense in $L^\Phi (\mathbb {R} ^{d})$ and that $(L^{\Phi }(\mathbb {R} ^{d}))^{*}\cong L^{\Phi ^*}(\mathbb {R} ^{d})$. If in addition $\Phi ^*\in \Delta _{2}$, then the Orlicz space $L^{\Phi }(\mathbb {R} ^{d})$ is a reflexive Banach space. For more detail, see [30].

There are several inequalities to be used throughout the paper when dealing with Orlicz spaces, namely the generalizations of Hölder’s and Young’s inequalities (see [26, 30]). Let $\Phi _i$ be Young functions for $i=1,2,3$:

(1) If $f_i\in L^{\Phi _i}(\mathbb {R} ^{d})$ for $i=1,2$ and

$$\begin{aligned} \Phi _1^{-1}(t)\Phi _2^{-1}(t)\le \Phi _3^{-1}(t), \quad t\ge 0 \end{aligned}$$

(2.4)

then $f_1f_2\in L^{\Phi _3}(\mathbb {R} ^{d})$ and

$$\begin{aligned} N_{\Phi _3}(f_1f_2)\le 2N_{\Phi _1}(f_1)N_{\Phi _2}(f_2). \end{aligned}$$

(2.5)

(2) If $f_i\in L^{\Phi _i}(\mathbb {R} ^{d})$ for $i=1,2$ and

$$\begin{aligned} \Phi _1^{-1}(t)\Phi _2^{-1}(t)\le t\Phi _3^{-1}(t), \quad t\ge 0 \end{aligned}$$

(2.6)

then $f_1*f_2\in L^{\Phi _3}(\mathbb {R} ^{d})$ and

$$\begin{aligned} N_{\Phi _3}(f_1*f_2)\le 2N_{\Phi _1}(f_1)N_{\Phi _2}(f_2). \end{aligned}$$

(2.7)

Definition 2.1

Given a Young function $\Phi$ and a positive function $\omega$ such that $\omega \in L^1_{{\text {loc}}}(\mathbb {R} ^{d})$ and $1/\omega \in L^1_{{\text {loc}}}(\mathbb {R} ^{d})$ the weighted Orlicz space $L^{\Phi }_\omega (\mathbb {R} ^{d})$ is defined as the set of measurable functions f such that $f\omega \in L^{\Phi }(\mathbb {R} ^{d})$. For each $f\in L^\Phi _\omega (\mathbb {R} ^{d})$ we set the norm

$$\begin{aligned} N_{\Phi ,\omega }(f)=N_\Phi (f\omega ). \end{aligned}$$

Weighted Orlicz spaces are Banach spaces with this norm and compactly supported continuous functions belong to $L^\Phi _\omega (\mathbb {R} ^{d})$. Moreover, if $\Phi \in \Delta _2$, then $C_c(\mathbb {R} ^{d})$ is dense in $L^{\Phi }_\omega (\mathbb {R} ^{d})$ and the dual space of $(L^{\Phi }_\omega (\mathbb {R} ^{d}),\Vert \cdot \Vert _{\Phi ,\omega })$ is the Banach space $L^{\Phi ^*}_{\omega ^{-1}}(\mathbb {R} ^{d})$, under the duality

$$\begin{aligned} \langle f,g\rangle = \int \limits _{\mathbb {R} ^{d}} f(x) \overline{g(x)} dx, \quad f \in L^{\Phi }_\omega (\mathbb {R} ^{d}), \; g\in L^{\Phi ^*}_{\omega ^{-1}}(\mathbb {R} ^{d}). \end{aligned}$$

We are interested in selecting functions $\omega$ such that $L^1_\omega (\mathbb {R} ^{d})$ becomes a Banach algebra under convolution, the so-called Beurling algebras. To work in this setting we shall consider special class of functions $\omega$.

Definition 2.2

A weight on $\mathbb {R} ^{d}$ is a positive continuous function $\omega$ such that

$$\begin{aligned} \omega (x+y)\le \omega (x)\omega (y) \end{aligned}$$

(2.8)

for $x,y \in \mathbb {R} ^{d}$.

Remark 2.3

Note that (2.8) implies that $\omega (0)\ge 1$. Sometimes $\omega$ is only assumed to be a sub-multiplicative measurable function such that $\omega \ge 1$, $\frac{1}{\omega }\in L^\infty _{{\text {loc}}}(\mathbb {R} ^{d})$. In fact there is no loss of generality in assuming that the weight $\omega$ is continuous (see [31, Section 3.7]). In [27] A. Osançlıol and S. Öztop defined weighted Orlicz spaces $L^\Phi (G,\omega )$ on a locally compact group G and viewed them as Banach algebras with respect to the convolution product.

Proposition 2.4

Let $\Phi$ be Young function and $\omega$ be a weight on $\mathbb {R} ^{d}$. If $f,g\in L^1(\mathbb {R} ^{d})$ then

$$\begin{aligned}{} & {} N_{\Phi ,\omega }(\tau _x f)\le \omega (x)N_{\Phi ,\omega }(f), \quad x\in \mathbb {R} ^{d}. \end{aligned}$$

(2.9)

$$\begin{aligned}{} & {} |(f*g)\omega |\le (|f|\omega *|g|\omega ). \end{aligned}$$

(2.10)

Proof

(2.9) follows simply using that

$$\begin{aligned} |\tau _x f|\omega \le \omega (x)\tau _x( |f|\omega ), \quad x\in \mathbb {R} ^{d}\end{aligned}$$

and $L^\Phi (\mathbb {R} ^{d})$ is invariant under translations.

(2.10) is immediate from the condition $\omega (x)\le \omega (x-x')\omega (x')$. $\square$

We shall use the notation $M_\omega (\mathbb {R} ^{d})$ for the space of measures $\mu \in (C_0(\mathbb {R} ^{d}))^*$ such that $\Vert \mu \Vert _{1,\omega }=\int _{\mathbb {R} ^{d}}\omega (x)d|\mu |(x)<\infty$.

Proposition 2.5

Let $\omega$ be a weight on $\mathbb {R} ^{d}$ and let $\Phi$ be Young function satisfying $\Delta _2$. If $\mu \in M_{\omega }({\mathbb {R}}^{d})$ and $f\in L^{\Phi }_{\omega }({\mathbb {R}}^{d})$ then $f*\mu \in L^{\Phi }_{\omega }({\mathbb {R}}^{d})$. Moreover

$$\begin{aligned} N_{\Phi ,\omega }(f*\mu )\le \Vert \mu \Vert _{1,\omega }N_{\Phi ,\omega }(f).\end{aligned}$$

Proof

Standard arguments show that if $f\in C_c(\mathbb {R} ^{d})$ then $x\rightarrow \tau _x f$ is continuous from $\mathbb {R} ^{d}$ into $L_\omega ^\Phi (\mathbb {R} ^{d})$. Now density and (2.9) imply the continuity of $x\rightarrow \tau _x f$ as a mapping from $\mathbb {R} ^{d}$ into $L^\Phi _\omega (\mathbb {R} ^{d})$ for a given function $f\in L^\Phi (\mathbb {R} ^{d})$. Observe also that

$$\begin{aligned}|(f*\mu )(x) \omega (x)|\le \int _{\mathbb {R} ^{d}} \tau _{x'}(|f|\omega )(x)\omega (x')d|\mu |(x').\end{aligned}$$

Therefore from vector-valued Minkowski inequality we obtain

$$\begin{aligned} N_{\Phi ,\omega }(f*\mu )\le \int _{{\mathbb {R}}^{d}}N_{\Phi }(\tau _{x'}(|f|\omega ))\omega (x')d|\mu |(x')= N_{\Phi ,\omega }(f) \Vert \mu \Vert _{1,\omega }.\end{aligned}$$

$\square$

Corollary 2.6

Let $\omega$ be a weight and let $\Phi$ be a Young function satisfying $\Delta _2$. Then $\mathcal {S}({\mathbb {R}}^{d})$ is dense in $L^{\Phi }_{\omega }({\mathbb {R}}^{d})$.

Proof

It suffices to combine (2.3) and Proposition 2.5, since for any $f\in C_c(\mathbb {R} ^{d})$ and $K\in \mathcal {S}({\mathbb {R}}^{d})$ one has that $f*K_\varepsilon \in \mathcal {S}({\mathbb {R}}^{d})$ where $K_\varepsilon (x)=\frac{1}{\varepsilon ^d}K(\frac{x}{\varepsilon })$ for $\varepsilon >0$. Now use that $N_{\Phi ,\omega }(f*K_\varepsilon - f)\rightarrow 0$ as $\varepsilon \rightarrow 0$ to conclude the result. $\square$

Let us now present a version of Hölder and Young inequalities in the weighted setting.

Theorem 2.7

Let $\omega$ be a weight on $\mathbb {R} ^{d}$, let $u_i$ for $i=1,2,3$ be measurable positive functions defined in $\mathbb {R} ^{d}$ such that

$$\begin{aligned} u_3(x)\le u_1(x)u_2(x), \quad x\in \mathbb {R} ^{d}, \end{aligned}$$

(2.11)

let $\Phi _{i}, \Psi _i$ for $i=1,2,3$ be Young functions such that

$$\begin{aligned} \Phi _{1}^{-1}(t)\Phi _{2}^{-1}(t)\le \Phi _{3}^{-1}(t), \quad t\ge 0 \end{aligned}$$

(2.12)

and

$$\begin{aligned} \Psi _{1}^{-1}(t)\Psi _{2}^{-1}(t)\le t\Psi _{3}^{-1}(t), \quad t\ge 0. \end{aligned}$$

(2.13)

(a) If $f_i\in L^{\Phi _{i}}_{u_i}({\mathbb {R}}^{d})$ for $i=1,2$ then $f_1 f_2\in L^{\Phi _{3}}_{u_3}({\mathbb {R}}^{2d})$. Moreover

$$\begin{aligned} N_{\Phi _{3},u_3}(f_1f_2)\le 2N_{\Phi _{1},u_1}(f_1)N_{\Phi _{2},u_2}(f_2). \end{aligned}$$

(2.14)

(b) If $f_i\in L^{\Psi _{i}}_\omega ({\mathbb {R}}^{d})$ for $i=1,2$ then $f_1*f_2\in L^{\Psi _{3}}_\omega ({\mathbb {R}}^{d})$. Moreover

$$\begin{aligned} N_{\Psi _{3},\omega }(f_1*f_2)\le 2N_{\Psi _{1},\omega }(f_1)N_{\Psi _{2},\omega }(f_2). \end{aligned}$$

(2.15)

Proof

(a) It follows from (2.5) applied to $f_iu_i$ for $i=1,2$ and the estimate (2.11).

(b) It follows using (2.10) and (2.7).

$\square$

2.2 Weighted mixed normed Orlicz spaces

The definition of weighted mixed normed Orlicz spaces depends on the authors. In [35] the authors use the vector-valued approach, while in [37] they use the two-variable approach. Recall that if E is a Banach ideal in $L^0(\mu )$ the space of measurable functions in a complete $\sigma$-finite measure space $(\Omega , \Sigma ,\mu )$ and X is a Banach space then E(X) stands for the set of all strongly measurable functions $f:\Omega \rightarrow X$ such that $t\rightarrow \Vert f(t)\Vert$ belongs to E and $\Vert f\Vert _{E(X)}= \Vert \Vert f(\cdot )\Vert _X\Vert _E$. On the other hand whenever $X={\tilde{E}}$ is also a Banach ideal in $L^0(\nu )$ defined on (possibly) another complete $\sigma$-finite measure space $({\tilde{\Omega }}, {\tilde{\Sigma }},\nu )$ one can define the mixed norm Banach ideal $E[{\tilde{E}}]$ as the classes of all $\mu \times \nu$-measurable functions $F:\Omega \times {\tilde{\Omega }}\rightarrow \mathbb {C}$ such that $F(t,\cdot )\in {\tilde{E}}$ for $\mu$-almost all $t\in \Omega$ and $F_{{\tilde{E}}}(t)=\Vert F(t,\cdot )\Vert _{{\tilde{E}}}$ belongs to E with the norm $\Vert F\Vert _{E[{\tilde{E}}]}=\Vert F_{{\tilde{E}}}\Vert _E.$

It is clear that $E({\tilde{E}})\subset E[{\tilde{E}}]$. It was shown by Bukhvalov [7] that $E({\tilde{E}})= E[{\tilde{E}}]$ if and only if either $\mu$ is discrete or ${\tilde{E}}$ has a continuous norm, i.e. if $\Vert x_n\Vert _{{\tilde{E}}}\rightarrow 0$ whenever $0\le x_n\in {\tilde{E}}$ and $x_n$ decreases to zero pointwise.

We shall consider here E and ${\tilde{E}}$ being Orlicz spaces. It is known that for N-functions $L^\Phi (\mu )$ has continuous norm iff $\Phi$ satisfies $\Delta _2$ condition (see [19, Theorem II.10.3].)

We choose here the vector-valued approach.

Definition 2.8

Let $\Phi _{j}$ be Young functions, $j=1,2$. Then the mixed norm Orlicz space $L^{\Phi _1, \Phi _2}(\mathbb {R} ^{2d}) = L^{\Phi _2}(\mathbb {R} ^{d}, L^{\Phi _1}(\mathbb {R} ^{d}))$ consists of functions $F:\mathbb {R} ^{2d}\rightarrow \mathbb {C}$ such that $\xi \rightarrow F(\cdot ,\xi )\in L^{\Phi _1}(\mathbb {R} ^{d})$ is strongly measurable and $\xi \rightarrow N_{\Phi _1}(F(\cdot ,\xi ))$ belongs to $L^{\Phi _2}(\mathbb {R} ^{d})$. We consider the norm

$$\begin{aligned} N_{\Phi _1,\Phi _2} (f)= N_{\Phi _2}\Big (N_{\Phi _1}(F(\cdot ,\xi ))\Big )<\infty . \end{aligned}$$

Given a positive continuous function $\omega$ defined in $\mathbb {R} ^{2d}$, $L^{\Phi _1, \Phi _2}_\omega (\mathbb {R} ^{2d})$ stands for the space of functions such that $F\omega \in L^{\Phi _1, \Phi _2}(\mathbb {R} ^{2d})$ and we set the norm $N_{\Phi _1,\Phi _2, \omega } (F)= N_{\Phi _1,\Phi _2} (F\omega ).$

We shall denote by $S_{\omega }^{\Phi _1,\Phi _2}({\mathbb {R}}^{2d})$ the closure of $C_c(\mathbb {R} ^{2d})$ in $L_{\omega }^{\Phi _1, \Phi _2}({\mathbb {R}}^{2d})$ and write $S^{\Phi _1,\Phi _2}({\mathbb {R}}^{2d})$ in case $\omega =1$.

Proposition 2.9

Let $\omega$ be a continuous positive function on ${\mathbb {R}}^{2d}$ and let $\Phi _1$ and $\Phi _2$ be Young functions satisfying $\Delta _2$. Then

$$\begin{aligned} S_{\omega }^{\Phi _1,\Phi _2}({\mathbb {R}}^{2d})=L_{\omega }^{\Phi _1,\Phi _2}({\mathbb {R}}^{2d})\end{aligned}$$

(2.16)

and

$$\begin{aligned} (L^{\Phi _1, \Phi _2}_{\omega }({\mathbb {R}}^{2d}))^*= L^{\Phi ^*_1, \Phi ^* _2}_{\omega ^{-1}}({\mathbb {R}}^{2d}).\end{aligned}$$

(2.17)

Proof

Let us first show the case $\omega =1$. If $f\in L^{\Phi _2}(\mathbb {R} ^{d}, L^{\Phi _1}(\mathbb {R} ^{d}))$ and $\varepsilon >0$ then, using that $\Phi _2$ satisfies $\Delta _2$-condition we obtain that $F=\sum _{n=1}^m g_n \chi _{A_n}$ such that $g_n\in L^{\Phi _1}(\mathbb {R} ^{d})$ and $|A_n|<\infty$ for $1\le n\le m$ with $N_{\Phi _1,\Phi _2}(f-F)<\varepsilon /2$. Now, for each $1\le n\le m$, using that $\Phi _1$ satisfies $\Delta _2$-condition we can find simple functions $s_n$ such that

$$\begin{aligned}N_{\Phi _1,\Phi _2}(g_n-s_n)<\frac{\varepsilon }{ 2M}\end{aligned}$$

for $M=\sum _{n=1}^m N_{\Phi _2}(\chi _{A_n})$. Hence $s=\sum _{n=1}^m s_n \chi _{A_n}$ is a simple function defined in $\mathbb {R} ^{2d}$ and

$$\begin{aligned} N_{\Phi _1,\Phi _2}(f-s)\le N_{\Phi _1,\Phi _2}(f-F)+ \sum _{n=1}^m N_{\Phi _1}(g_n-s_n)N_{\Phi _2}(\chi _{A_n})<\varepsilon .\end{aligned}$$

Therefore we can also conclude that $C_c({\mathbb {R}}^{2d})$ is dense in $L^{\Phi _1,\Phi _2}({\mathbb {R}}^{2d})$.

Given $F\in L_{\omega }^{\Phi _1,\Phi _2}({\mathbb {R}}^{2d})$ and $\varepsilon >0$ then there is $G\in C_c(\mathbb {R} ^{2d})$ such that $N_{\Phi _1,\Phi _2}(F\omega -G)<\varepsilon$. On the other hand since $G\omega ^{-1}\in C_c({\mathbb {R}}^{2d})$ we obtain $N_{\Phi _1,\Phi _2,\omega }(F -G\omega ^{-1})<\varepsilon .$ This gives (2.16).

On the other hand using the duality

$$\begin{aligned} \langle F,G\rangle =\int _{{\mathbb {R}}^{2d}}F(x,\xi )\overline{G(x,\xi )}dxd\xi \end{aligned}$$

we know that (see [30, Theorem 5, VII.7.5]),

$$\begin{aligned}(L^{\Phi _1, \Phi _2}({\mathbb {R}}^{2d}))^*=L^{\Phi ^*_1, \Phi ^*_2}({\mathbb {R}}^{2d}).\end{aligned}$$

From this easily follows (2.17) using that $F\in L^{\Phi _1, \Phi _2}_{w}({\mathbb {R}}^{2d})$ iff $F\omega \in L^{\Phi _1, \Phi _2}({\mathbb {R}}^{2d})$ and $G\in L^{\Phi ^*_1, \Phi ^*_2}({\mathbb {R}}^{2d})$ iff $G\omega ^{-1}\in L_{\omega }^{\Phi ^*_1, \Phi ^* _2}({\mathbb {R}}^{2d})$. $\square$

We now present the following results which can be shown as in the case of weighted Orlicz spaces:

Proposition 2.10

Let $\Phi _1,\Phi _2$ be Young functions and $\omega$ be a weight on $\mathbb {R} ^{2d}$. If $F\in L^{\Phi _1,\Phi _2}_\omega (\mathbb {R} ^{2d})$ and $z\in \mathbb {R} ^{2d}$ then

$$\begin{aligned} N_{\Phi _1,\Phi _2,\omega }(\tau _z F)\le \omega (z)N_{\Phi _1,\Phi _2,\omega }(F), \quad z\in \mathbb {R} ^{2d}. \end{aligned}$$

(2.18)

Proposition 2.11

Let $\omega$ be a weight and let $\Phi _1,\Phi _2$ be Young functions satisfying $\Delta _2$. If $\mu \in M_{\omega }({\mathbb {R}}^{2d})$ and $F\in L^{\Phi _1,\Phi _2}_{\omega }({\mathbb {R}}^{2d})$ then $F*\mu \in L^{\Phi _1,\Phi _2}_{\omega }({\mathbb {R}}^{2d})$. Furthermore

$$\begin{aligned} N_{\Phi _1,\Phi _2,\omega }(F*\mu )\le \Vert \mu \Vert _{1,\omega }N_{\Phi _1,\Phi _2,\omega }(F).\end{aligned}$$

Corollary 2.12

Let $\omega$ be a weight and let $\Phi _1, \Phi _2$ be Young functions satisfying $\Delta _2$. Then $\mathcal {S}({\mathbb {R}}^{2d})$ is dense in $L^{\Phi _1,\Phi _2}_{\omega }({\mathbb {R}}^{2d})$.

Theorem 2.13

Let $u_i$ be measurable positive functions defined in $\mathbb {R} ^{2d}$ such that

$$\begin{aligned} u_3(z)\le u_1(z)u_2(z), \quad z\in \mathbb {R} ^{2d}\end{aligned}$$

(2.19)

and let $\Phi _{i}, \Psi _i$ be Young functions for $i=1,2,3$ such that

$$\begin{aligned} \Phi _{1}^{-1}(t)\Phi _{2}^{-1}(t)\le \Phi _{3}^{-1}(t), \quad t\ge 0 \end{aligned}$$

(2.20)

and

$$\begin{aligned} \Psi _{1}^{-1}(t)\Psi _{2}^{-1}(t)\le \Psi _{3}^{-1}(t), \quad t\ge 0. \end{aligned}$$

(2.21)

If $F_1\in L^{\Phi _{1}, \Psi _{1}}_{u_1}({\mathbb {R}}^{2d})$ and $F_2\in L^{\Phi _{2}, \Psi _{2}}_{u_2}({\mathbb {R}}^{2d})$ then $F_1 F_2\in L^{\Phi _{3}, \Psi _{3}}_{u_3}({\mathbb {R}}^{2d})$. Moreover

$$\begin{aligned} N_{\Phi _{3},\Psi _{3},u_3}(F_1F_2)\le 4N_{\Phi _{1},\Psi _{1},u_1}(F_1)N_{\Phi _{2},\Psi _{2},u_2}(F_2). \end{aligned}$$

(2.22)

We include for the sake of completeness the Young’s inequalities in this setting.

Theorem 2.14

Let $\Phi _{i}, \Psi _i$ be Young functions for $i=1,2,3$ satisfying $\Delta _2$ such that

$$\begin{aligned} \Phi _{1}^{-1}(t)\Phi _{2}^{-1}(t)\le t\Phi _{3}^{-1}(t), \quad t\ge 0 \end{aligned}$$

(2.23)

and

$$\begin{aligned} \Psi _{1}^{-1}(t)\Psi _{2}^{-1}(t)\le t\Psi _{3}^{-1}(t), \quad t\ge 0 \end{aligned}$$

(2.24)

and let $\omega$ be a weight defined in ${\mathbb {R}}^{2d}$.

If $F_1\in L^{\Phi _{1}, \Psi _{1}}_\omega ({\mathbb {R}}^{2d})$ and $F_2\in L^{\Phi _{2}, \Psi _{2}}_\omega ({\mathbb {R}}^{2d})$ then $F_1*F_2\in L^{\Phi _{3}, \Psi _{3}}_\omega ({\mathbb {R}}^{2d})$. Moreover

$$\begin{aligned} N_{\Phi _{3},\Psi _{3},\omega }(F_1*F_2)\le 4N_{\Phi _{1},\Psi _{1},\omega }(F_1)N_{\Phi _{2},\Psi _{2},\omega }(F_2). \end{aligned}$$

(2.25)

Proof

Let $F_1$ and $F_2$ be Schwartz functions. Hence, using (2.10) we have

$$\begin{aligned} |(F_1*F_2)(x,\xi )\omega (x,\xi )|\le & {} (|F_1|\omega *|F_2|\omega )(x,\xi )\\= & {} \int _{\mathbb {R} ^{d}}\int _{\mathbb {R} ^{d}} |F_1|\omega (x-x',\xi -\xi ')|F_2|\omega (x',\xi ')dx'd\xi '\\= & {} \int _{\mathbb {R} ^{d}} |F_1|\omega (\cdot ,\xi -\xi ')*|F_2|\omega (\cdot ,\xi ')(x)d\xi '.\end{aligned}$$

Then using solidity and Minkowski’s inequality combined with (2.7), which follows from condition (2.23), give for each $\xi \in \mathbb {R} ^{d}$,

$$\begin{aligned} N_{\Phi _{3}}\Big ((F_1*F_2)(\cdot ,\xi )\omega (\cdot ,\xi )\Big )\le & {} \int _{\mathbb {R} ^{d}} 2 N_{\Phi _{1}}(F_1\omega )(\cdot ,\xi -\xi ')N_{\Phi _{2}}(F_2\omega )(\cdot ,\xi ') d\xi '\\= & {} 2 \gamma _1* \gamma _2(\xi ) \end{aligned}$$

where $\gamma _i(\xi )=N_{\Phi _{i}}(F_i\omega )(\cdot ,\xi )$.

Applying the same argument again we obtain

$$\begin{aligned} N_{\Phi _{3},\Psi _{3},\omega }(F_1*F_2)\le 4 N_{\Phi _{1},\Psi _{1},\omega }(F_1)N_{\Phi _{2},\Psi _{2},\omega }(F_2).\end{aligned}$$

The result now follows by density. $\square$

2.3 Orlicz modulation spaces

We start this section by recalling some facts on the short time Fourier transform $V_\phi (f)$. For a function $\phi \in {\mathcal {S}}(\mathbb {R} ^{d})$ and a tempered distribution $f\in {\mathcal {S}}'(\mathbb {R} ^{d})$ we write

$$\begin{aligned}V_\phi (f)(x,\xi )=\int _{\mathbb {R} ^{d}}f(u)\overline{\phi (u-x)}e^{-2\pi i \langle u, \xi \rangle }du=\langle f, M_\xi \tau _x \phi \rangle \end{aligned}$$

which is called the short time Fourier transform of f with window $\phi$.

Let us mention some properties of the STFT to be used in the sequel;

Using the notation $\phi ^*(t)=\overline{\phi (-t)}$, we have

$$\begin{aligned}{} & {} V_\phi (f)(x,\xi )= ({{\hat{f}}}* M_{-x}{\hat{\phi }}^*)(\xi )=e^{-2\pi i x\xi }(f*M_\xi \phi ^*)(x), \end{aligned}$$

(2.26)

$$\begin{aligned}{} & {} |V_{\phi _1*\phi _2}(f_1*f_2)(x,\xi )|= |(f_1*M_\xi \phi _1^*)*(f_2*M_\xi \phi _2^*)(x)|, \end{aligned}$$

(2.27)

$$\begin{aligned}{} & {} V_{\phi _1\phi _2}(f_1f_2)(x,\xi )= ({{\hat{f}}}_1*M_{-x}{\hat{\phi }}_1^*)*({{\hat{f}}}_2*M_{-x}{\hat{\phi }}_2^*)(\xi ) \end{aligned}$$

(2.28)

for $f, f_1,f_2\in \mathcal {S}'(\mathbb {R} ^{d})$ and $\phi , \phi _1,\phi _2\in \mathcal {S}(\mathbb {R} ^{d})$.

Let us also recall the inversion formula for the STFT [13, Corollary 11.2.7].

Assume that $\phi ,\psi \in \mathcal {S}(\mathbb {R} ^{d})$ such that $\langle \phi ,\psi \rangle \ne 0$. If $|F(x,\xi )|\le C(1+|x|+|\xi |)^N$ for some constants $C,N\ge 0$, then the integral $\int _{{\mathbb {R}}^{2d}}F(x,\xi )M_\xi \tau _x\psi \,dx d\xi$ defines a tempered distribution $f\in \mathcal {S}'(\mathbb {R} ^{d})$ in the sense that for all $\varphi \in \mathcal {S}(\mathbb {R} ^{d})$

$$\begin{aligned} \langle f, \varphi \rangle =\int _{{\mathbb {R}}^{2d}} F(x,\xi )\langle M_\xi \tau _x\psi ,\varphi \rangle dx d\xi . \end{aligned}$$

In particular, if $F=V_\phi f$ for some $f\in \mathcal {S}'(\mathbb {R} ^{d})$, then

$$\begin{aligned} \langle \phi ,\psi \rangle f=\int _{{\mathbb {R}}^{2d}} V_\phi (f)(x,\xi )M_\xi \tau _x\psi dxd\xi . \end{aligned}$$

(2.29)

Note that if $f\in \mathcal {S}(\mathbb {R} ^{d})$ then for all $n\ge 0$ there is $C_n>0$ such that $|(V_\phi f)(x,\xi )|\le C_n (1+|x|+|\xi |)^{-n}$ [13, Theorem 11.2.5]. Hence using above arguments the integral in (2.29) converges absolutely (for more detail see [13, Corollary 11.2.7]).

As usual we denote

$$\begin{aligned}V_\psi ^*(F)=\int _{{\mathbb {R}}^{2d}} F(x,\xi )M_\xi \tau _x\psi dxd\xi \end{aligned}$$

for each $F\in \mathcal {S}({\mathbb {R}}^{2d})$ and $\psi \in \mathcal {S}(\mathbb {R} ^{d})$. It is known that for each $F\in \mathcal {S}({\mathbb {R}}^{2d})$ one has that $V^*_\psi (F)\in \mathcal {S}({\mathbb {R}}^{d})$ (see [13, Theorem 11.2.4]) and, from (2.29)

$$\begin{aligned} \langle \phi ,\psi \rangle f= V_\psi ^*(V_\phi (f))\end{aligned}$$

whenever $\langle \phi ,\psi \rangle \ne 0$ and $f\in {\mathcal {S}}(\mathbb {R} ^{d})$.

Although modulation spaces $M^{\Phi _1,\Phi _2}_m(\mathbb {R} ^{2d})$ (see [13] for the case $\Phi _1(t)=t^p$ and $\Phi _2(t)=t^q$) can be defined for more general weights m, to keep in the setting of tempered distributions we shall deal in this paper as in [13] and we deal with certain weights to develop the theory.

Definition 2.15

We denote by ${\mathcal {W}}_d$ the class of weights $\omega$ in ${\mathbb {R}}^{d}$ which are of polynomial growth, that is to say $\omega$ is a continuous positive function defined in $\mathbb {R} ^{d}$ such that

$$\begin{aligned}\omega (x+y)\le \omega (x)\omega (y), \quad x,y\in \mathbb {R} ^{d}\end{aligned}$$

and there exist $C>0$ and $r>0$ such that

$$\begin{aligned} \omega (x) \le C (1+|x|)^r,\qquad x\in \mathbb {R} ^{d}. \end{aligned}$$

Definition 2.16

Let $\omega \in {\mathcal {W}}_{2d}$ and $u,v\in {\mathcal {W}}_d$. We denote

$$\begin{aligned} \omega _1(x)=\omega (x,0) \quad \hbox {and} \quad \omega _2(\xi )=\omega (0, \xi )\end{aligned}$$

and

$$\begin{aligned}u\otimes v(x,\xi )=u(x)v(\xi ).\end{aligned}$$

Clearly $\omega _1, \omega _2\in {\mathcal {W}}_d$, $u\otimes v\in {\mathcal {W}}_{2d}$, $(u\otimes v)_1=v(0)u$ and $(u\otimes v)_2=u(0)v$.

Definition 2.17

Fix a non-zero window $\varphi \in \mathcal {S}(\mathbb {R} ^{d})$ and let $\Phi ,\Phi _1,\Phi _2$ be Young functions and $\omega \in {\mathcal {W}}_{2d}$. The Orlicz modulation spaces, to be denoted $M^\Phi _\omega (\mathbb {R} ^{d})$ and $M^{\Phi _1,\Phi _2}_\omega (\mathbb {R} ^{d})$ respectively, are defined by

$$\begin{aligned} M^{\Phi }_\omega (\mathbb {R} ^{d})=\{f\in \mathcal {S}'(\mathbb {R} ^{d}):V_\varphi (f)\in L^{\Phi }_\omega (\mathbb {R} ^{2d})\} \end{aligned}$$

and

$$\begin{aligned} M^{\Phi _1,\Phi _2}_\omega (\mathbb {R} ^{d})=\{f\in \mathcal {S}'(\mathbb {R} ^{d}):V_\varphi (f)\in L^{\Phi _1,\Phi _2}_\omega (\mathbb {R} ^{2d})\}. \end{aligned}$$

The norm on $M^\Phi _\omega (\mathbb {R} ^{d})$ and $M^{\Phi _1,\Phi _2}_\omega (\mathbb {R} ^{d})$ is given by $\Vert f\Vert _{M^{\Phi }_\omega }=N_{\Phi ,\omega }(V_\varphi f)$ and $\Vert f\Vert _{M^{\Phi _1,\Phi _2}_\omega }=N_{\Phi _1,\Phi _2,\omega }(V_\varphi f)$.

Remark 2.18

We would like to point out that for the definition of modulation spaces in Definition 2.17 we might have chosen $f\in \mathcal {S}'(\mathbb {R} ^{d})$ such that $\omega V_\varphi (f)\in L^{\Phi _2}[L^{\Phi _1}]$ instead of $\omega V_\varphi (f)\in L^{\Phi _2}(L^{\Phi _1}).$ In the case $\Phi _1$ is N-function satisfying $\Delta _2$-condition we have $L^{\Phi _2}[L^{\Phi _1}]=L^{\Phi _2}(L^{\Phi _1})$. But it still remains an open question in the case $\Phi _1$ is a general Young function.

We observe that using the similar techniques in [13, Theorem 11.3.5], one can have the modulation spaces $M^\Phi _\omega (\mathbb {R} ^{d})$ and $M^{\Phi _1,\Phi _2}_\omega (\mathbb {R} ^{d})$ are Banach space.

The reader is referred to [35, 37] for the case of Orlicz modulation spaces with $\omega (z)=1$ and for a more general class of v-moderate weight functions respectively.

It is known that, if $\Phi _1,\Phi _2$ satisfy $\Delta _2$ then $(M^{\Phi _1,\Phi _2})^*= M^{\Phi ^*_1,\Phi ^*_2}$ using the duality

$$\begin{aligned}\langle f,g\rangle =\int _{\mathbb {R} ^{2d}} V_\varphi (f)(x,\xi )\overline{V_\varphi (g)(x,\xi )} dxd\xi .\end{aligned}$$

The above result for $\omega =1$ was mentioned in [35, Theorem 9] under slightly stronger assumptions, but using (2.17) one can easily see that

$$\begin{aligned}(M^{\Phi _1,\Phi _2}_\omega )^*= M^{\Phi ^*_1,\Phi ^*_2}_{1/\omega }.\end{aligned}$$

Next result is known for weighted Lebesgue modulation spaces [13, Proposition 11.3.4] and for weighted Lorentz spaces [34]. We present here a proof for weighted Orlicz modulation spaces.

Theorem 2.19

Let $\Phi , \Phi _1,\Phi _2$ be Young functions satisfying $\Delta _2$-condition, $\phi \in \mathcal {S}(\mathbb {R} ^{d})\setminus \{0\}$ and $\omega \in {\mathcal {W}}_{2d}$. Then $\mathcal {S}(\mathbb {R} ^{d})$ is dense in $M^{\Phi }_\omega (\mathbb {R} ^{d})$ and $M^{\Phi _1,\Phi _2}_\omega (\mathbb {R} ^{d})$.

Proof

Both cases follow similarly. We do only the second one. It is known that $f\in \mathcal {S}(\mathbb {R} ^{d})$ implies that $V_\phi (f)\in \mathcal {S}({\mathbb {R}}^{2d})$ [13, Theorem 11.2.5]. This shows that $\mathcal {S}(\mathbb {R} ^{d})\subset M^{\Phi _1,\Phi _2}_\omega (\mathbb {R} ^{d})$. To show the density, given $f\in M^{\Phi _1,\Phi _2}_\omega (\mathbb {R} ^{d})$ and $\varepsilon >0$, we can first use Corollary 2.12 to obtain $G\in \mathcal {S}({\mathbb {R}}^{2d})$ such that $N_{\Phi _1,\Phi _2,\omega }(V_\phi (f)- G)<\varepsilon .$ Now select $\psi \in \mathcal {S}(\mathbb {R} ^{d})$ such that $\langle \phi ,\psi \rangle =1$ and define $g=V^*_\psi (G)\in \mathcal {S}(\mathbb {R} ^{d})$. We have that $V_\phi (g)=G$ and

$$\begin{aligned} \Vert f-g\Vert _{M^{\Phi _1,\Phi _2}_\omega }=N_{\Phi _1,\Phi _2,\omega }(V_\phi (f)- V_\phi (g))<\varepsilon .\end{aligned}$$

$\square$

Let us now study the action of modulation and translation for these spaces.

Proposition 2.20

Let $\Phi ,\Phi _1,\Phi _2$ be Young functions, $\omega \in {\mathcal {W}}_{2d}$, $f\in \mathcal {S}(\mathbb {R} ^{d})$ and $y,\eta \in \mathbb {R} ^{d}$. Then

$$\begin{aligned}{} & {} \Vert \tau _y f\Vert _{M^{\Phi }_\omega }\le \omega _1(y)\Vert f\Vert _{M^{\Phi }_\omega }, \quad \Vert \tau _y f\Vert _{M^{\Phi _1,\Phi _2}_\omega }\le \omega _1(y)\Vert f\Vert _{M^{\Phi _1,\Phi _2}_\omega }. \end{aligned}$$

(2.30)

$$\begin{aligned}{} & {} \Vert M_\eta f\Vert _{M^{\Phi }_\omega }\le \omega _2(\eta )\Vert f\Vert _{M^{\Phi }_\omega }, \quad \Vert M_\eta f\Vert _{M^{\Phi _1,\Phi _2}_\omega }\le \omega _2(\eta )\Vert f\Vert _{M^{\Phi _1,\Phi _2}_\omega }. \end{aligned}$$

(2.31)

Proof

It is well known (see [13]) that

$$\begin{aligned} V_\varphi (\tau _y f)=M_{(0,-y)}\tau _{(y,0)}V_\varphi (f), \quad y\in \mathbb {R} ^{d}. \end{aligned}$$

(2.32)

$$\begin{aligned} V_\varphi (M_{\eta } f)=\tau _{(0,\eta )}V_\varphi (f), \quad \eta \in \mathbb {R} ^{d}. \end{aligned}$$

(2.33)

Using (2.18) we have

$$\begin{aligned}{} & {} \Vert \tau _y f\Vert _{M^{\Phi _1,\Phi _2}_\omega }= N_{\Phi _1,\Phi _2,\omega }(M_{(0,-y)}\tau _{(y,0)}V_\varphi (f))\le \omega _1(y)\Vert f\Vert _{M^{\Phi _1,\Phi _2}_\omega }. \\{} & {} \Vert M_\eta f\Vert _{M^{\Phi _1,\Phi _2}_\omega }= N_{\Phi _1,\Phi _2,\omega }(\tau _{(0,\eta )}V_\varphi (f))\le \omega _2(\eta )\Vert f\Vert _{M^{\Phi _1,\Phi _2}_\omega }.\end{aligned}$$

Similar argument can be used for the case $L^\Phi _\omega (\mathbb {R} ^{2d})$. $\square$

It was pointed out without proof in the unweighted case in [35, Theorem 7] that the norm in the Orlicz modulation spaces does not depend on the window. Let us get a precise statement which work even in the weighted case.

Proposition 2.21

Let $\Phi , \Phi _1,\Phi _2$ be Young functions satisfying $\Delta _2$-condition, $\omega \in {\mathcal {W}}_{2d}$, $f\in \mathcal {S}'(\mathbb {R} ^{d})$ and $\varphi ,\phi , \psi \in \mathcal {S}(\mathbb {R} ^{d})$ such that $\langle \varphi ,\psi \rangle \langle \phi ,\psi \rangle \ne 0$. Denote

$$\begin{aligned}C_{\varphi ,\phi ,\psi }= \frac{\Vert (V_\psi (\phi ))^*\Vert _{L^1_\omega }}{|\langle \varphi ,\psi \rangle |}.\end{aligned}$$

Then

$$\begin{aligned} C_{\phi ,\varphi ,\psi }^{-1}N_{\Phi ,\omega }(V_\varphi (f))\le N_{\Phi ,\omega }(V_\phi (f))\le C_{\varphi ,\phi ,\psi }N_{\Phi ,\omega }(V_\varphi (f))\end{aligned}$$

and

$$\begin{aligned} C_{\phi ,\varphi ,\psi }^{-1}N_{\Phi _1,\Phi _2,\omega }(V_\varphi (f))\le N_{\Phi _1,\Phi _2,\omega }(V_\phi (f))\le C_{\varphi ,\phi ,\psi }N_{\Phi _1,\Phi _2,\omega }(V_\varphi (f)).\end{aligned}$$

Proof

Let first observe that for any couple $\phi ,\psi$ of Schwartz functions in ${\mathcal {S}}(\mathbb {R} ^{d})$ and $F\in {\mathcal {S}}'(\mathbb {R} ^{2d})$ we have

$$\begin{aligned} |V_\phi (V^*_\psi (F))|\le |F|*|(V_\psi (\phi ))^*|. \end{aligned}$$

(2.34)

Indeed, using (2.32) and (2.33) we have

$$\begin{aligned} V_\phi (V^*_\psi (F))(x,\xi )= & {} \langle V_\psi ^*(F), M_\xi \tau _x\phi \rangle \\= & {} \langle F, V_\psi (M_\xi \tau _x\phi )\rangle \\= & {} \langle F, \tau _{(0,\xi )}M_{(0,-x)}\tau _{(x,0)}V_\psi (\phi )\rangle \\= & {} \int _{\mathbb {R} ^{2d}} F(y,\eta ) e^{2\pi i(\eta -\xi )x}\overline{V_\psi (\phi )(y-x,\eta -\xi )} dyd\eta . \end{aligned}$$

Therefore we obtain (2.34).

Now (2.34) and Propositions 2.5 and 2.11 imply

$$\begin{aligned} N_{\Phi ,\omega }(V_\phi (V^*_\psi (F))\le \Vert (V_\psi (\phi ))^*\Vert _{L^1_\omega } N_{\Phi ,\omega }(F)\end{aligned}$$

(2.35)

and

$$\begin{aligned} N_{\Phi _1,\Phi _2,\omega }(V_\phi (V^*_\psi (F))\le \Vert (V_\psi (\phi ))^*\Vert _{L^1_\omega } N_{\Phi _1,\Phi _2,\omega }(F).\end{aligned}$$

(2.36)

Select now $\psi$ such that $\langle \varphi ,\psi \rangle \ne 0$ and $\langle \phi ,\psi \rangle \ne 0$. Therefore, applying (2.35) and the inversion formula we obtain

$$\begin{aligned} N_{\Phi ,\omega }(V_\phi (f))= & {} \frac{1}{|\langle \varphi ,\psi \rangle |}N_{\Phi ,\omega }(V_\phi (V^*_\psi (V_\varphi f)))\\\le & {} \frac{\Vert (V_\psi (\phi ))^*\Vert _{L^1_\omega }}{|\langle \varphi ,\psi \rangle |} N_{\Phi ,\omega }(V_\varphi f). \end{aligned}$$

Similar estimate can be applied using (2.36).

The converse inequalities follow changing the rules of $\phi$ and $\varphi$. $\square$

Let us now study the behavior of the dilation acting on modulation spaces. To such a purpose, let us introduce the following constants associated the following dilations: Given measurable functions f and F defined in $\mathbb {R} ^{d}$ and $\mathbb {R} ^{2d}$ respectively and $\lambda ,\lambda _1,\lambda _2>0$ we shall write

$$\begin{aligned} D_\lambda f(x)=f(\lambda x), \quad D_{(\lambda _1,\lambda _2)}F(x,\xi )=F(\lambda _1 x,\lambda _2 \xi ).\end{aligned}$$

(2.37)

Definition 2.22

Given Young functions $\Phi , \Phi _1, \Phi _2$, $\omega \in {\mathcal {W}}_{d}$ and $\lambda , \lambda _1,\lambda _2>0$ we shall write

$$\begin{aligned}C_{\Phi ,\omega ,d}(\lambda )=\Vert D_\lambda \Vert _{L^\Phi _{w}(\mathbb {R} ^{d})\rightarrow L^\Phi _{w}(\mathbb {R} ^{d})}=\sup \{ N_{\Phi ,\omega }(D_\lambda (f)): N_{\Phi ,\omega }(f)\le 1\},\end{aligned}$$

(to be denoted $C_{\Phi ,d}$ for $\omega =1$)

$$\begin{aligned}C_{\Phi }(\lambda _1,\lambda _2)=\Vert D_{(\lambda _1,\lambda _2)}\Vert _{L^{\Phi }(\mathbb {R} ^{2d})\rightarrow L^{\Phi }(\mathbb {R} ^{2d})}\end{aligned}$$

and

$$\begin{aligned}C_{\Phi _1,\Phi _2}(\lambda _1,\lambda _2)=\Vert D_{(\lambda _1,\lambda _2)}\Vert _{L^{\Phi _1,\Phi _2}(\mathbb {R} ^{2d})\rightarrow L^{\Phi _1,\Phi _2}(\mathbb {R} ^{2d})}.\end{aligned}$$

Remark 2.23

Notice that $C_{\Phi ,\omega ,d}(\lambda )$ is non-increasing, submultiplicative and $C_{\Phi ,\omega ,d}(1)=1$. Notice that the definition of $C_{\Phi ,\omega ,d}(\lambda )$ depends on the norms used in $L^\Phi _{\omega }(\mathbb {R} ^{d})$.

Hence, since $\langle D_\lambda f,g\rangle = \frac{1}{\lambda ^d}\langle f,D_{1/\lambda } g \rangle$, it follows that

$$\begin{aligned} C_{\Phi ,\omega ,d}(\lambda )\approx \frac{1}{\lambda ^d} C_{\Phi ^*, 1/\omega ,d}(1/\lambda )\end{aligned}$$

(2.38)

but this becomes an equivalent constant when we replace $N_{\Phi ,\omega }(\cdot )$ by an equivalent norm.

Remark 2.24

It follows easily that

$$\begin{aligned} C_{\Phi _1,\Phi _2}(\lambda _1,\lambda _2)\le C_{\Phi _1,d}(\lambda _1)C_{\Phi _2,d}(\lambda _2),\quad \lambda _1,\lambda _2>0 \end{aligned}$$

(2.39)

and

$$\begin{aligned} C_{\Phi }(\lambda _1,\lambda _2)= C_{\Phi ,2d}(\sqrt{\lambda _1\lambda _2}).\end{aligned}$$

(2.40)

In particular $C_\Phi (\lambda ,1/\lambda )=1$ and in the cases $\Phi (t)=t^{p}$, $\Phi _1(t)=t^{p_1}$ and $\Phi _2(t)=t^{p_2}$ with $1\le p,p_1,p_2<\infty$ we obtain

$$\begin{aligned}{} & {} C_{\Phi }(\lambda _1,\lambda _2)=(\lambda _1\lambda _2)^{-d/p}. \end{aligned}$$

(2.41)

$$\begin{aligned}{} & {} C_{\Phi _1,\Phi _2}(\lambda _1,\lambda _2)=\lambda _1^{-d/p_1}\lambda _2^{-d/p_2}. \end{aligned}$$

(2.42)

Proposition 2.25

Let $\Phi$ be a Young function. Then there exists $K>0$ such that

$$\begin{aligned}\Vert D_\lambda f\Vert _{M^{\Phi }}\le K \Big (\frac{\lambda ^2+1}{\lambda ^2}\Big )^{d/2} \Vert f\Vert _{M^{\Phi }}, \quad \lambda >0\end{aligned}$$

for all $f\in \mathcal {S}(\mathbb {R} ^{d})$.

Proof

It is straightforward to see that for any $\phi \in {\mathcal {S}}(\mathbb {R} ^{d})$

$$\begin{aligned} V_\phi (D_\lambda f)=D_{(\lambda ,\frac{1}{\lambda })}V_{\phi _\lambda }(f), \quad \lambda >0. \end{aligned}$$

(2.43)

Now from Proposition 2.21 we know that

$$\begin{aligned}N_{\Phi }(V_{\varphi _\lambda }(f))\le \frac{\Vert V_\psi (\varphi _\lambda )\Vert _{L^1}}{|\langle \varphi ,\psi \rangle |}N_{\Phi }(V_\varphi (f))\end{aligned}$$

for any $\psi$ such that $\langle \varphi ,\psi \rangle \ne 0$.

Select $\psi =\varphi$ and $\varphi (x)=e^{-\pi |x|^2}$. It was shown in [8, Proposition 3.3] that

$$\begin{aligned} |V_\varphi (D_{\sqrt{\lambda }}\varphi )(x,\xi )|=(\lambda +1)^{-d/2}e^{-\frac{\pi (\lambda |x|^2+|\xi |^2)}{\lambda +1}} \end{aligned}$$

(2.44)

and

$$\begin{aligned} \Vert V_\varphi (D_{\sqrt{\lambda }}\varphi )\Vert _{L^1}\approx \Big (\frac{\lambda +1}{\lambda }\Big )^{d/2}. \end{aligned}$$

(2.45)

Therefore, since $\varphi _\lambda =\lambda ^{-d}D_{1/\lambda }\varphi$, we obtain

$$\begin{aligned} \Vert V_\varphi (\varphi _\lambda )\Vert _{L^1}\approx \Big (\frac{\lambda ^2+1}{\lambda ^2}\Big )^{d/2}. \end{aligned}$$

(2.46)

Hence

$$\begin{aligned}\Vert D_\lambda f\Vert _{M^\Phi (\mathbb {R} ^{d})}=N_\Phi (V_\phi (D_\lambda f))\le K\Big (\frac{\lambda ^2+1}{\lambda ^2}\Big )^{d/2}C_\Phi (1/\lambda ,\lambda ) \Vert f\Vert _{M^\Phi (\mathbb {R} ^{d})}.\end{aligned}$$

Finally use that $C_\Phi (1/\lambda ,\lambda )=1$ to obtain the result.

$\square$

Repeating the proof in Proposition 2.25 and using (2.39) we obtain the following result.

Proposition 2.26

Let $\Phi _1,\Phi _2$ be Young functions. Then there exists $K>0$ such that

$$\begin{aligned} \Vert D_\lambda f\Vert _{M^{\Phi _1,\Phi _2}}\le K (\frac{1+\lambda ^2}{\lambda ^2})^{d/ 2} C_{\Phi _1,d}(\lambda )C_{\Phi _2,d}(\frac{1}{\lambda }) \Vert f\Vert _{M^{\Phi _1,\Phi _2}}, \quad \lambda >0\end{aligned}$$

for all $f\in \mathcal {S}(\mathbb {R} ^{d})$.

In particular for $\Phi _1(t)=t^{p_1}$ and $\Phi _2(t)=t^{p_2}$ with $1\le p_1,p_2<\infty$ we obtain

$$\begin{aligned} \Vert D_\lambda \Vert _{M^{p_1,p_2}(\mathbb {R} ^{d})\rightarrow M^{p_1,p_2}(\mathbb {R} ^{d})}\le K (1+\lambda ^2)^{d/ 2} \lambda ^{-d(1/p_1+1/p'_2)}, \quad \lambda >0\end{aligned}$$

where $1/p_2+1/p_2'=1$.

From now on we concentrate on the weight $\omega \in {\mathcal {W}}_{2d}$ in Definition 2.16.

We shall present now a result on multiplication of functions in Orlicz modulation spaces.

Theorem 2.27

Let $\Phi _i,\Psi _i$, $i=1,2,3$, be Young functions such that satisfy $\Delta _2$-condition for $i=1,2$, let $\omega \in {\mathcal {W}}_{2d}$ and $0<\alpha ,\beta <1$ with $\alpha +\beta =1$. Assume that

$$\begin{aligned} \Phi _1^{-1}(t)\Phi _2^{-1}(t)\le \Phi _3^{-1}(t), \quad t\ge 0\end{aligned}$$

(2.47)

and

$$\begin{aligned} \Psi _1^{-1}(t)\Psi _2^{-1}(t)\le t\Psi _3^{-1}(t),\quad t\ge 0.\end{aligned}$$

(2.48)

If $f_1\in M^{\Phi _1,\Psi _1}_{(\omega _1)^\alpha \otimes \omega _2}(\mathbb {R} ^{d})$, $f_2\in M^{\Phi _2,\Psi _2}_{(\omega _1)^\beta \otimes \omega _2}(\mathbb {R} ^{d})$ then $f_1f_2\in M^{\Phi _3,\Psi _3}_{\omega }(\mathbb {R} ^{d})$ and

$$\begin{aligned} \Vert f_1f_2\Vert _{ M^{\Phi _3,\Psi _3}_{\omega }}\le C\Vert f_1\Vert _{ M^{\Phi _1,\Psi _1}_{(\omega _1)^\alpha \otimes \omega _2}}\Vert f_2\Vert _{ M^{\Phi _2,\Psi _2}_{(\omega _1)^{\beta }\otimes \omega _2}} \end{aligned}$$

where $\omega _1(x)=\omega (x,0)$ and $\omega _2(\xi )=\omega (0,\xi )$.

Proof

We assume first that $f_1,f_2\in \mathcal {S}(\mathbb {R} ^{d})$. Taking into account that different windows give equivalent norms as shown in Proposition 2.21 we shall see, writing $u_\alpha =(\omega _1)^\alpha \otimes \omega _2$ and $u_\beta =(\omega _1)^\beta \otimes \omega _2$, that if $\phi \in \mathcal {S}(\mathbb {R} ^{d})$ then

$$\begin{aligned} N_{\Phi _3,\Psi _3,\omega }(V_{\phi ^2}(f_1f_2))\le 4N_{\Phi _1,\Psi _1,u_\alpha }(V_{\phi }(f_1))N_{\Phi _2,\Psi _2,u_\beta }(V_{\phi }(f_2)). \end{aligned}$$

(2.49)

From (2.28) we have that

$$\begin{aligned} |V_{\phi ^2}(f_1f_2)(x,\xi )|\omega (x,\xi )= & {} |({{\hat{f}}}_1*M_{-x}{\hat{\phi }}^*)*({{\hat{f}}}_2*M_{-x}{\hat{\phi }}^*)(\xi )|\omega (x,\xi )\\\le & {} |\int _{\mathbb {R} ^{d}} V_\phi (f_1)(x,\xi -\xi ')V_\phi (f_2)(x,\xi ')d\xi '|\omega _1(x)\omega _2(\xi )\\\le & {} \int _{\mathbb {R} ^{d}} F_1(x,\xi -\xi ') F_2(x,\xi ')d\xi '\end{aligned}$$

where

$$\begin{aligned}F_1(x,\xi )=|V_\phi (f_1)(x,\xi )|(\omega _1(x))^\alpha \omega _2(\xi ))\end{aligned}$$

and

$$\begin{aligned}F_2(x,\xi )=|V_\phi (f_2)(x,\xi )|(\omega _1(x))^\beta \omega _2(\xi ).\end{aligned}$$

Now using (2.47) we have, denoting for $i=1,2$

$$\begin{aligned}{} & {} \gamma _i(\xi )=N_{\Phi _i}\Big (F_i(\cdot , \xi )\Big )\\{} & {} N_{\Phi _3}\Big (|V_{\phi ^2}(f_1f_2)(\cdot ,\xi )|\omega (\cdot ,\xi )\Big )\le 2\int _{\mathbb {R} ^{d}} \gamma _1(\xi -\xi ')\gamma _2(\xi ')d\xi '=2\gamma _1*\gamma _2(\xi ).\end{aligned}$$

Finally, we invoke (2.48) to obtain

$$\begin{aligned}N_{\Phi _3,\Psi _3,\omega }(V_{\phi ^2}(f_1 f_2))\le 4N_{\Psi _1}(\gamma _1)N_{\Psi _2}(\gamma _2)\end{aligned}$$

which corresponds to (2.49). The general case now follows by density. $\square$

3 Multipliers on Orlicz modulation spaces

3.1 Linear multipliers

We fix a notation for this section. We shall write ${\varvec{\Phi }}=(\Phi _1,\Phi _2)$ and ${\varvec{\Psi }}=(\Psi _1,\Psi _2)$ for couples of Young functions where we assume that $\Phi _i,\Psi _i$ satisfy $\Delta _2$ and $\textbf{w}=(\omega ,{\tilde{\omega }} )\in {\mathcal {W}}_{2d}\times {\mathcal {W}}_{2d}$.

Definition 3.1

Let $m:\mathbb {R} ^{d}\rightarrow \mathbb {C}$ be a bounded measurable function and define

$$\begin{aligned} T_m(f)(x)=\int \limits _{\mathbb {R} ^{d}} {\widehat{f}}(\xi )m(\xi )e^{2\pi i \langle \xi ,x\rangle }d\xi \end{aligned}$$

for any $f\in \mathcal {S}(\mathbb {R} ^{d})$.

We denote by $\mathcal {M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$ the class of functions m such that $T_m$ extends to a linear bounded map from $M^{\Phi _1,\Psi _1}_{\omega }(\mathbb {R} ^{d})$ to $M^{\Phi _2,\Psi _2}_{{\tilde{\omega }}}(\mathbb {R} ^{d})$ and we write $\Vert m\Vert _{\mathcal {M}}=\Vert T_m\Vert$. We denote by $\mathcal {M}({\varvec{\Phi }},{\varvec{\Psi }})$ the case where $\omega ={\tilde{\omega }}=1$.

We start pointing out a basic example of multiplier in $\mathcal {M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$.

Proposition 3.2

Let $\Phi ,\Psi$ be a couple of Young functions satisfying $\Delta _2$ and $\omega \in {\mathcal {W}}_{2d}$. Let $\mu \in M_{\omega _1}(\mathbb {R} ^{d})$, that is $\mu \in M(\mathbb {R} ^{d})$ and $\int _{\mathbb {R} ^{d}} \omega _1(y)d|\mu |(y)<\infty$.

If $m(\xi )={\hat{\mu }}(\xi )$ then $m\in \mathcal {M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$ for ${\varvec{\Phi }}=(\Phi ,\Phi )$, ${\varvec{\Psi }}=({\varvec{\Psi }},{\varvec{\Psi }})$ and $\textbf{w}=(\omega ,\omega )$ and $\Vert m\Vert _{\mathcal {M}}\le \int _{\mathbb {R} ^{d}}\omega _1(y)d|\mu |(y).$

In particular

$$\begin{aligned}\{{{\hat{g}}}: g\in L^1(\mathbb {R} ^{d})\}\subset \mathcal {M}({\varvec{\Phi }},{\varvec{\Psi }}).\end{aligned}$$

Proof

We simply need to write

$$\begin{aligned}T_m(f)=\int _{\mathbb {R} ^{d}} \tau _y f d\mu (y)\end{aligned}$$

for $f\in {\mathcal {S}}(\mathbb {R} ^{d})$ and use Minkowski’s inequality and (2.30) to obtain

$$\begin{aligned}\Vert T_m(f)\Vert _{M^{\Phi ,\Psi }_{\omega }}\le \int _{\mathbb {R} ^{d}} \Vert \tau _y f\Vert _{M^{\Phi ,\Psi }_{\omega }}d|\mu |(y)\le \Big (\int _{\mathbb {R} ^{d}} \omega _1(y)d|\mu |(y)\Big ) \Vert f\Vert _{M^{\Phi ,\Psi }_{\omega }}. \end{aligned}$$

In particular if $g\in L^1(\mathbb {R} ^{d})$ and $m(\xi )={{\hat{g}}}(\xi )$ then $T_m(f)=f*g$ for any $f\in {\mathcal {S}}(\mathbb {R} ^{d})$ and we obtain $\Vert m\Vert _{{\mathcal {M}}}\le \Vert g\Vert _{L^1}.$ $\square$

Let us now study convolution on Orlicz modulation spaces and get the analogue to Young’s inequality in this context.

Theorem 3.3

Let $\Phi _i,\Psi _i$, $i=1,2,3$, be Young functions such that satisfy $\Delta _2$-condition for $i=1,2$, let $\omega \in {\mathcal {W}}_{2d}$ and $0<\alpha ,\beta <1$ with $\alpha +\beta =1$. Assume

$$\begin{aligned} \Phi _1^{-1}(t)\Phi _2^{-1}(t)\le t\Phi _3^{-1}(t), \quad t\ge 0\end{aligned}$$

(3.1)

and

$$\begin{aligned} \Psi _1^{-1}(t)\Psi _2^{-1}(t)\le \Psi _3^{-1}(t), \quad t\ge 0.\end{aligned}$$

(3.2)

If $f_1\in M^{\Phi _1,\Psi _1}_{\omega _1\otimes (\omega _2)^\alpha }(\mathbb {R} ^{d})$, $f_2\in M^{\Phi _2,\Psi _2}_{\omega _1\otimes (\omega _2)^\beta }(\mathbb {R} ^{d})$ then $f_1*f_2\in M^{\Phi _3,\Psi _3}_{\omega }(\mathbb {R} ^{d})$ and there exists $C>0$ such that

$$\begin{aligned} \Vert f_1*f_2\Vert _{ M^{\Phi _3,\Psi _3}_{\omega }}\le C\Vert f_1\Vert _{ M^{\Phi _1,\Psi _1}_{\omega _1\otimes (\omega _2)^\alpha }}\Vert f_2\Vert _{ M^{\Phi _2,\Psi _2}_{\omega _1\otimes (\omega _2)^\beta }}. \end{aligned}$$

(3.3)

Proof

We assume first that $f_1,f_2\in \mathcal {S}(\mathbb {R} ^{d})$. As above it suffices to compute the norm with respect to different windows. We shall use that if $\phi \in \mathcal {S}(\mathbb {R} ^{d})$ then

$$\begin{aligned} N_{\Phi _3,\Psi _3,\omega }(V_{\phi *\phi }(f_1*f_2))\le 4N_{\Phi _1,\Psi _1,\omega _1\otimes (\omega _2)^\alpha }(V_{\phi }(f_1))N_{\Phi _2,\Psi _2,\omega _1\otimes (\omega _2)^\beta }(V_{\phi }(f_2)). \end{aligned}$$

Using (2.27) we have

$$\begin{aligned}|V_{\phi *\phi }(f_1*f_2)(x,\xi )|\omega (x,\xi )\le |(f_1*M_\xi \phi ^*)*(f_2*M_\xi \phi ^*)(x)|\omega _1(x)\omega _2(\xi )\end{aligned}$$

and, using that $\omega _1$ is a weight on $\mathbb {R} ^{d}$ we have, for $0<\alpha ,\beta <1$ such that $\alpha +\beta =1$,

$$\begin{aligned}{} & {} |V_{\phi *\phi }(f_1*f_2)(x,\xi )|\omega (x,\xi )\le \\\le & {} \Big |(\omega _2(\xi ))^\alpha (f_1*M_\xi \phi ^*)\omega _1\Big |*\Big |(\omega _2(\xi ))^\beta (f_2*M_\xi \phi ^*)\omega _1\Big |(x)\\= & {} \Big |V_\phi (f_1)(\cdot ,\xi )\omega _1(\cdot )(\omega _2(\xi ))^\alpha \Big |*\Big |V_\phi (f_2)(\cdot ,\xi )\omega _1(\cdot )(\omega _2(\xi ))^\beta \Big |(x).\end{aligned}$$

Therefore, using the assumption (3.1) we have

$$\begin{aligned} \gamma _3(\xi )\le 2\gamma _1(\xi )\gamma _2(\xi ) \end{aligned}$$

(3.4)

where

$$\begin{aligned}{} & {} \gamma _1(\xi )=N_{\Phi _1}\Big (V_\phi (f_1)(\cdot ,\xi )\omega _1(\cdot )\omega _2(\xi )^\alpha \Big ),\\{} & {} \gamma _2(\xi )=N_{\Phi _2}\Big (V_\phi (f_2)(\cdot ,\xi )\omega _1(\cdot )\omega _2(\xi )^\beta \Big )\end{aligned}$$

and

$$\begin{aligned}\gamma _3(\xi )=N_{\Phi _3}\Big (V_{\phi *\phi }(f_1*f_2)(\cdot ,\xi )\omega (\cdot ,\xi )\Big ).\end{aligned}$$

Using now (3.4) and assumption (3.2) we conclude

$$\begin{aligned}N_{\Psi _3}(\gamma _3)\le 4N_{\Psi _1}(\gamma _1)N_{\Psi _2}(\gamma _2)\end{aligned}$$

and then we obtain (3.3) and the proof is complete. Now the result follows for general functions using the density result in Theorem 2.19.

$\square$

Corollary 3.4

Let $\Phi _i,\Psi _i$, $i=1,2,3$ be Young functions such that satisfy $\Delta _2$-condition. Assume

$$\begin{aligned} \Phi _1^{-1}(t)\Phi _3^{-1}(t)\le t\Phi _2^{-1}(t), \quad t\ge 0\end{aligned}$$

(3.5)

and

$$\begin{aligned} \Psi _1^{-1}(t)\Psi _3^{-1}(t)\le \Psi _2^{-1}(t), \quad t\ge 0.\end{aligned}$$

(3.6)

If $g\in L^1(\mathbb {R} ^{d})\cap M^{\Phi _3,\Psi _3}(\mathbb {R} ^{d})$ then $m={{\hat{g}}} \in \mathcal {M}({\varvec{\Phi }},{\varvec{\Psi }})$ and there exists $C>0$ such that

$$\begin{aligned} \Vert m\Vert _\mathcal {M}\le C\Vert g\Vert _{ M^{\Phi _3,\Psi _3}}. \end{aligned}$$

Corollary 3.5

Let $1\le p_i,q_i<\infty$ for $i=1,2,3$ and let $\omega \in {\mathcal {W}}_{2d}$ and assume

$$\begin{aligned} \frac{1}{p_1}+\frac{1}{p_2}=1+\frac{1}{p_3} \end{aligned}$$

(3.7)

and

$$\begin{aligned} \frac{1}{q_1}+\frac{1}{q_2}=\frac{1}{q_3} .\end{aligned}$$

(3.8)

If $f_1\in M^{p_1,q_1}_{\omega _1\otimes \sqrt{\omega _2}}(\mathbb {R} ^{d})$ and $f_2\in M^{p_2,q_2}_{\omega _1\otimes \sqrt{\omega _2}}(\mathbb {R} ^{d})$ then $f_1*f_2\in M^{p_3,q_3}_{\omega }(\mathbb {R} ^{d})$.

3.2 Bilinear multipliers

In this section we shall write ${\varvec{\Phi }}=(\Phi _1,\Phi _2,\Phi _3)$, ${\varvec{\Psi }}=(\Psi _1,\Psi _2,\Psi _3)$ and $\textbf{w}=(\omega ,{\tilde{\omega }}, v)$ for triples of Young functions and weights where we assume that $\Phi _i,\Psi _i$ satisfy $\Delta _2$ and that $\omega ,{\tilde{\omega }}, v\in {\mathcal {W}}_{2d}$, that is weights of polynomial growth. As above we shall use the notation $\omega _1(x)=\omega (x,0)$ and $\omega _2(\xi )=\omega (0,\xi )$ if $\omega$ is a weight in $\mathbb {R} ^{2d}$.

Definition 3.6

Let $m:\mathbb {R} ^{2d}\rightarrow \mathbb {C}$ be a bounded measurable function and define

$$\begin{aligned} B_m(f_1,f_2)(x)=\int \limits _{\mathbb {R} ^{d}}\int \limits _{\mathbb {R} ^{d}} \widehat{f_1}(\xi )\widehat{f_2}(\eta )m(\xi ,\eta )e^{2\pi i \langle \xi +\eta ,x\rangle }d\xi d\eta \end{aligned}$$

for any $f_1,f_2\in \mathcal {S}(\mathbb {R} ^{d})$.

We denote by $\mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$ the class of functions m such that $B_m$ extends to a bilinear bounded map from $M^{\Phi _1,\Psi _1}_{\omega }(\mathbb {R} ^{d})\times M^{\Phi _2,\Psi _2}_{{\tilde{\omega }}}(\mathbb {R} ^{d})$ to $M^{\Phi _3,\Psi _3}_{v}(\mathbb {R} ^{d})$. We write $\Vert m\Vert _{\mathcal{B}\mathcal{M}}=\Vert B_m\Vert$.

Let us provide a first and simple example using measures defined on $\mathbb {R} ^{d}$.

Theorem 3.7

Let ${\varvec{\Phi }},{\varvec{\Psi }}$ triples of Young functions as above and assume

$$\begin{aligned} \Phi _1^{-1}(t)\Phi _2^{-1}(t)\le \Phi _3^{-1}(t), \quad t\ge 0\end{aligned}$$

(3.9)

and

$$\begin{aligned} \Psi _1^{-1}(t)\Psi _2^{-1}(t)\le t\Psi _3^{-1}(t), \quad t\ge 0.\end{aligned}$$

(3.10)

Let $\omega \in {\mathcal {W}}_{2d}$ and $\textbf{w}=(\sqrt{\omega _1}\otimes \omega _2 , \sqrt{\omega _1}\otimes \omega _2, \omega )$, $\alpha ,\beta \in {\mathbb {R}}$ and $\mu \in M(\mathbb {R} ^{d})$. If $m(\xi ,\eta )={\hat{\mu }} (\alpha \xi +\beta \eta )$ and

$$\begin{aligned}\int _{\mathbb {R} ^{d}} \Big (\max \{\omega _1(\alpha t),\omega _1(\beta t)\}\Big ) d|\mu |(t)<\infty ,\end{aligned}$$

then $m \in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$ and there exists $C>0$ such that

$$\begin{aligned} \Vert m\Vert _{\mathcal{B}\mathcal{M}}\le C\int _{\mathbb {R} ^{d}} \Big (\max \{\omega _1(\alpha t),\omega _1(\beta t)\}\Big ) d|\mu |(t). \end{aligned}$$

Proof

Let $f_1,f_2\in \mathcal {S}(\mathbb {R} ^{d})$. It is elementary to see that

$$\begin{aligned}B_m(f_1,f_2)(x)=\int \limits _{\mathbb {R} ^{d}}\tau _{\alpha t}f_1(x)\tau _{\beta t}f_2(x)d\mu (t). \end{aligned}$$

Now, using Theorem 2.27, (2.30) and (2.18) we have

$$\begin{aligned}&\Vert B_m(f_1,f_2)\Vert _{M^{\Phi _3,\Psi _3}_\omega }\le \\&\quad \le \int \limits _{\mathbb {R} ^{d}}\Vert (\tau _{\alpha t}f_1)(\tau _{\beta t}f_2)\Vert _{M^{\Phi _3,\Psi _3}_\omega }d|\mu |(t) \\&\quad \le C\int \limits _{\mathbb {R} ^{d}} \Vert \tau _{\alpha t} f_1\Vert _{M^{\Phi _1,\Psi _1}_{\sqrt{\omega _1}\otimes \omega _2}} \Vert \tau _{\beta t}f_2\Vert _{M^{\Phi _2,\Psi _2}_{\sqrt{\omega _1}\otimes \omega _2}}d|\mu |(t) \\&\quad \le C\int \limits _{\mathbb {R} ^{d}} N_{\Phi _1,\Psi _1,{\sqrt{\omega _1}\otimes \omega _2}}\Big (\tau _{(\alpha t,0)} V_\phi (f_1)\Big ) N_{\Phi _2,\Psi _2,{\sqrt{\omega _1}\otimes \omega _2}}\Big (\tau _{(\beta t,0)} V_\phi (f_2)\Big )d|\mu |(t) \\&\quad \le C\int \limits _{\mathbb {R} ^{d}}\sqrt{\omega _1(\alpha t) \omega _1(\beta t)}\Vert f_1\Vert _{M^{\Phi _1,\Psi _1}_{\sqrt{\omega _1}\otimes \omega _2}}\Vert f_2\Vert _{M^{\Phi _2,\Psi _2}_{\sqrt{\omega _1}\otimes \omega _2}}d|\mu |(t) \\&\quad \le C\Vert f_1\Vert _{M^{\Phi _1,\Psi _1}_{\sqrt{\omega _1}\otimes \omega _2}}\Vert f_2\Vert _{M^{\Phi _2,\Psi _2}_{\sqrt{\omega _1}\otimes \omega _2}}\int \limits _{\mathbb {R} ^{d}}\Big (\max \{\omega _1(\alpha t),\omega _1(\beta t)\}\Big ) d|\mu |(t). \end{aligned}$$

$\square$

Another approach to get easy examples of bilinear multipliers is the following.

Theorem 3.8

Let ${\varvec{\Phi }},{\varvec{\Psi }}$ be triples of Young functions as above and assume

$$\begin{aligned} \Phi _1^{-1}(t)\Phi _2^{-1}(t)\le t\Phi _3^{-1}(t), \quad t\ge 0\end{aligned}$$

(3.11)

and

$$\begin{aligned} \Psi _1^{-1}(t)\Psi _2^{-1}(t)\le \Psi _3^{-1}(t), \quad t\ge 0.\end{aligned}$$

(3.12)

Let $\omega \in {\mathcal {W}}_{2d}$ and $\textbf{w}=(\omega _1\otimes \sqrt{\omega _2}, \omega _1\otimes \sqrt{\omega _2},\omega )$. If $K\in L_{\sqrt{\omega _2}}^1(\mathbb {R} ^{d})$ and $m(\xi ,\eta )=K(\xi -\eta )$ then $m \in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$ and there exists $C>0$ such that

$$\begin{aligned} \Vert m\Vert _{\mathcal{B}\mathcal{M}}\le C\int _{\mathbb {R} ^{d}}\sqrt{\omega _2(\xi )}|K(\xi )| d\xi . \end{aligned}$$

Proof

Let $f_1,f_2 \in {\mathcal {S}}(\mathbb {R} ^{d})$ write for $m(\xi ,\eta )=K(\xi -\eta )$ and we can use the formula (see [3, 4])

$$\begin{aligned}B_m(f_1,f_2)(x)=\int _{\mathbb {R} ^{d}} (f_1* M_\xi f_2)(2x)K(\xi )e^{2\pi i\xi x} d\xi .\end{aligned}$$

Therefore, using Minkowski’s inequality again, Theorem 3.3 and (2.31), we can write

$$\begin{aligned} \Vert B_m(f_1,f_2)\Vert _{M^{\Phi _3,\Psi _3}_\omega }\le & {} \Vert D_2\Vert _{M^{\Phi _3,\Psi _3}_\omega \rightarrow M^{\Phi _3,\Psi _3}_\omega }\int _{\mathbb {R} ^{d}} \Vert (f_1* M_\xi f_2)\Vert _{M^{\Phi _3,\Psi _3}_\omega }|K(\xi )| d\xi \\\le & {} C\int _{\mathbb {R} ^{d}} \Vert f_1\Vert _{M^{\Phi _1,\Psi _1}_{\omega _1\otimes \sqrt{\omega _2}}}\Vert M_\xi f_2\Vert _{M^{\Phi _2,\Psi _2}_{\omega _1\otimes \sqrt{\omega _2}}}|K(\xi )| d\xi \\\le & {} C\int _{\mathbb {R} ^{d}} \Vert f_1\Vert _{M^{\Phi _1,\Psi _1}_{\omega _1\otimes \sqrt{\omega _2}}}\Vert f_2\Vert _{M^{\Phi _2,\Psi _2}_{\omega _1\otimes \sqrt{\omega _2}}}\sqrt{\omega _2(\xi )}|K(\xi )| d\xi . \end{aligned}$$

$\square$

Now we give some elementary properties of the bilinear multipliers by using translations and modulations.

Theorem 3.9

Let ${\varvec{\Phi }},{\varvec{\Psi }}$ triples of Young functions as above and $\textbf{w}=(\omega ,{\tilde{\omega }}, v)$ and let $m\in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$. Then

• (1) $\tau _{(\xi _0,\eta _0)}m\in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$ for each $(\xi _0,\eta _0) \in \mathbb {R} ^{2d}$ and

  $$\begin{aligned} \Vert \tau _{(\xi _0,\eta _0)}m\Vert _{\mathcal{B}\mathcal{M}}\le v_2(\xi _o+\eta _0)\omega _2(-\xi _0){\tilde{\omega }}_2(-\eta _0)\Vert m\Vert _{\mathcal{B}\mathcal{M}}. \end{aligned}$$

• (2) $M_{(x_0,y_0)}m\in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$ for each $(x_0,y_0) \in \mathbb {R} ^{2d}$ and

  $$\begin{aligned} \Vert M_{(x_0,y_0)}m\Vert _{\mathcal{B}\mathcal{M}}\le \omega _1(-x_0){\tilde{\omega }}_1(-y_0)\Vert m\Vert _{\mathcal{B}\mathcal{M}}. \end{aligned}$$

Proof

(1) Let $f_1,f_2\in \mathcal {S}(\mathbb {R} ^{d})$. Using

$$\begin{aligned} B_{\tau _{(\xi _0,\eta _0)}m}(f_1,f_2)(x)=e^{2\pi i \langle \xi _0 +\eta _0,x\rangle }B_m(M_{-\xi _0}f_1,M_{-\eta _0}f_2)(x) \end{aligned}$$

(3.13)

we can invoke Proposition 2.20 to obtain

$$\begin{aligned} \Vert B_{\tau _{(\xi _0,\eta _0)}m}(f_1,f_2)\Vert _{M^{\Phi _3,\Psi _3}_{v}}&= \Vert e^{2\pi i \langle \xi _0 +\eta _0,x\rangle }B_m(M_{-\xi _0}f_1,M_{-\eta _0}f_2)\Vert _{M^{\Phi _3,\Psi _3}_{v}}\\&\le v_2(\xi _o+\eta _0)\Vert B_m(M_{-\xi _0}f_1,M_{-\eta _0}f_2)\Vert _{M^{\Phi _3,\Psi _3}_{v}}\\&\le v_2(\xi _o+\eta _0) \Vert m\Vert _{\mathcal{B}\mathcal{M}} \Vert M_{-\xi _0}f_1\Vert _{M^{\Phi _1,\Psi _1}_{\omega }}\Vert M_{-\eta _0}f_2\Vert _{M^{\Phi _2,\Psi _2}_{{\tilde{\omega }}}}\\&\le v_2(\xi _o+\eta _0)\omega _2(-\xi _0){\tilde{\omega }}_2(-\eta _0)\Vert m\Vert _{\mathcal{B}\mathcal{M}}\Vert f_1\Vert _{M^{\Phi _1,\Psi _1}_{\omega }}\Vert f_2\Vert _{M^{\Phi _2,\Psi _2}_{{\tilde{\omega }}}}. \end{aligned}$$

Hence $\tau _{(\xi _0,\eta _0)}m\in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$ and we have the estimate of the norm.

(2) Let $f_1,f_2\in \mathcal {S}(\mathbb {R} ^{d})$. We can rewrite the $B_m(f_1,f_2)$

$$\begin{aligned} B_{M_{(x_0,y_0)}m}(f_1,f_2)(x)=B_m(\tau _{-x_0}f_1,\tau _{-y_0}f_2)(x). \end{aligned}$$

(3.14)

Therefore we obtain, using Proposition 2.20 again,

$$\begin{aligned} \Vert B_{M_{(x_0,y_0)}m}(f_1,f_2)\Vert _{M^{\Phi _3,\Psi _3}_{v}}= & {} \Vert B_m(\tau _{-x_0}f_1,\tau _{-y_0}f_2)\Vert _{M^{\Phi _3,\Psi _3}_{v}}\\\le & {} \Vert m\Vert _{\mathcal{B}\mathcal{M}}\Vert \tau _{-x_0}f_1\Vert _{M^{\Phi _1,\Psi _1}_{\omega }}\Vert \tau _{-y_0}f_2\Vert _{M^{\Phi _2,\Psi _2}_{{\tilde{\omega }}}}\\\le & {} \Vert m\Vert _{\mathcal{B}\mathcal{M}}\omega _1(-x_0){\tilde{\omega }}_1(-y_0) \Vert f_1\Vert _{M^{\Phi _1,\Psi _1}_{\omega }} \Vert f_2\Vert _{M^{\Phi _2,\Psi _2}_{\omega }}. \end{aligned}$$

Hence $M_{(x_0,y_0)}m\in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$ and we have the estimate of the norm.

$\square$

We now analyze the bilinear multipliers under dilations.

Theorem 3.10

Let ${\varvec{\Phi }},{\varvec{\Psi }}$ as above and let $m:\mathbb {R} ^{2d}\rightarrow \mathbb {C}$ be a bounded measurable function such that $m\in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }})$. Then $D_{\lambda } m\in \mathcal{B}\mathcal{M}({\varvec{\Phi }},\mathbf{\Psi })$ for any $\lambda >0$ and there exists $K_1>0$ such that

$$\begin{aligned} \Vert D_{\lambda } m\Vert _{\mathcal{B}\mathcal{M}}\le K_1 \frac{(1+ \lambda ^2)^{3d/2}}{\lambda ^{2d}}C_{({\varvec{\Phi }},{\varvec{\Psi }})}(\lambda )\Vert m\Vert _{\mathcal{B}\mathcal{M}}. \end{aligned}$$

where

$$\begin{aligned}C_{({\varvec{\Phi }},{\varvec{\Psi }})}(\lambda )=C_{\Phi _3,d}(\frac{1}{\lambda })C_{\Psi _3,d}(\lambda )\prod _{i=1}^2 C_{\Phi _i,d}(\lambda )C_{\Psi _i,d}(\frac{1}{\lambda }).\end{aligned}$$

Proof

Assume that $\Vert m\Vert _{\mathcal{B}\mathcal{M}}=1$ and $f_1,f_2\in {\mathcal {S}}(\mathbb {R} ^{d})$ with $\Vert f_1\Vert _{M^{\Phi _1,\Psi _1}}=\Vert f_2\Vert _{M^{\Phi _2,\Psi _2}}=1$. We first recall (see [3, 4]) that

$$\begin{aligned} B_m(D_\lambda f_1, D_\lambda f_2)(x)= D_\lambda B_{D_\lambda m}(f_1,f_2).\end{aligned}$$

Therefore, using Proposition 2.26 and denoting

$$\begin{aligned}A_{(\Phi ,\Psi )}(\lambda )= (\frac{1+\lambda ^2}{\lambda ^2})^{d/2}C_{\Phi ,d}(\lambda )C_{\Psi ,d}(\frac{1}{\lambda })\end{aligned}$$

we can write

$$\begin{aligned} \Vert B_{D_\lambda m}(f_1,f_2)\Vert _{M^{\Phi _3,\Psi _3}}= & {} \Vert D_{1/\lambda }B_m(D_\lambda f_1, D_\lambda f_2)\Vert _{M^{\Phi _3,\Psi _3}}\\\le & {} K A_{(\Phi _3,\Psi _3)}(1/\lambda )\Vert B_{m}(D_\lambda f_1,D_\lambda f_2)\Vert _{M^{\Phi _3,\Psi _3}}\\\le & {} K A_{(\Phi _3,\Psi _3)}(1/\lambda )\Vert D_\lambda f_1\Vert _{M^{\Phi _1,\Psi _1}}\Vert D_\lambda f_2\Vert _{M^{\Phi _2,\Psi _2}}\\\le & {} K^3 A_{(\Phi _3,\Psi _3)}(1/\lambda )\prod _{i=1}^2A_{(\Phi _i,\Psi _i)}(\lambda )\\\le & {} K^3 \frac{(1+ \lambda ^2)^{3d/2}}{\lambda ^{2d}}C_{({\varvec{\Phi }},{\varvec{\Psi }})}(\lambda ) . \end{aligned}$$

The proof now follows from the above estimate. $\square$

Let us give a method to generate bilinear multipliers between weighted mixed normed Orlicz modulation spaces.

Theorem 3.11

Let ${\varvec{\Phi }},{\varvec{\Psi }},\textbf{w}$ as above with $\textbf{w}=(\omega , {\tilde{\omega }}, v)$, and let $m\in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$.

• (1) If $\phi ^* \in L^1_{W}(\mathbb {R} ^{2d})$ for $W(\xi ,\eta )=v_2(-(\xi +\eta ))\omega _2(\xi ){\tilde{\omega }}_2(\eta )$, then $\phi *m\in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$ and

  $$\begin{aligned} \Vert \phi *m\Vert _{\mathcal{B}\mathcal{M}}\le \Vert \phi ^*\Vert _{L^1_{W}}\Vert m\Vert _{\mathcal{B}\mathcal{M}}. \end{aligned}$$

• (2) If $\phi \in L^1_{\omega _1\otimes \omega _1}(\mathbb {R} ^{2d})$ then ${\hat{\phi }} m\in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\omega )$ and

  $$\begin{aligned} \Vert {\hat{\phi }} m\Vert _{\mathcal{B}\mathcal{M}}\le \Vert \phi \Vert _{L^1_{\omega _1\otimes \omega _1}}\Vert m\Vert _{\mathcal{B}\mathcal{M}}.\end{aligned}$$

Proof

(1) Let $f_1,f_2 \in \mathcal {S} (\mathbb {R} ^{d})$. Then we have that

$$\begin{aligned} B_{\phi *m}(f_1,f_2)(x)=\int \limits _{\mathbb {R} ^{d}}\int \limits _{\mathbb {R} ^{d}} \phi (\xi ,\eta )B_{\tau _{(\xi ,\eta )}m}(f_1,f_2)(x)d\xi d\eta . \end{aligned}$$

Hence from the vector-valued Minkowski inequality together with (1) in Theorem  3.9 we get

$$\begin{aligned}&\Vert B_{\phi *m}(f_1,f_2)\Vert _{M^{\Phi _3,\Psi _3}_{v}}\le \\&\quad \le \int \limits _{\mathbb {R} ^{d}}\int \limits _{\mathbb {R} ^{d}} |\phi (u,\xi )|\Vert B_{\tau _{(\xi ,\eta )}m}(f_1,f_2)\Vert _{M^{\Phi _3,\Psi _3}_{v}}d\xi d\eta \\&\quad \le \int \limits _{\mathbb {R} ^{d}}\int \limits _{\mathbb {R} ^{d}} |\phi (-\xi ,-\eta )|\Vert \tau _{(-\xi ,-\eta )}m\Vert _{\mathcal{B}\mathcal{M}}\Vert f_1\Vert _{M^{\Phi _1,\Psi _1}_\omega } \Vert f_2\Vert _{M^{\Phi _2,\Psi _2}_{{\tilde{\omega }}}}d\xi d\eta \\&\quad \le \Vert m\Vert _{\mathcal{B}\mathcal{M}}\Vert f_1\Vert _{M^{\Phi _1,\Psi _1}_{\omega }}\Vert f_2\Vert _{M^{\Phi _2,\Psi _2}_{{\tilde{\omega }}}}\int \limits _{\mathbb {R} ^{d}}\int \limits _{\mathbb {R} ^{d}} v_2(-\xi -\eta )\omega _2(\xi ){\tilde{\omega }}_2(\eta ) |\phi (-\xi ,-\eta )|d\xi d\eta \\&\quad = \Vert m\Vert _{\mathcal{B}\mathcal{M}}\Vert \phi ^*\Vert _{L^1_{W}}\Vert f_1\Vert _{M^{\Phi _1,\Psi _1}_{\omega }} \Vert f_2\Vert _{M^{\Phi _2,\Psi _2}_{{\tilde{\omega }}}}. \end{aligned}$$

This gives $\phi *m\in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$ and $\Vert \phi *m\Vert _{\mathcal{B}\mathcal{M}}\le \Vert \phi ^*\Vert _{L^1_{W}}\Vert m\Vert _{\mathcal{B}\mathcal{M}}.$

(2) This will follow similarly using (2) in Theorem  3.9 and the formula

$$\begin{aligned} B_{{\hat{\phi }} m}(f_1,f_2)(x)=\int \limits _{\mathbb {R} ^{d}}\int \limits _{\mathbb {R} ^{d}}\phi (u,v)B_{M_{(-u,-v)}m}(f_1,f_2)(x)dudv. \end{aligned}$$

which holds for functions in the Schwartz class.

$\square$

Corollary 3.12

Let ${\varvec{\Phi }},{\varvec{\Psi }}$ as above and set $\textbf{w}=(\omega ,\omega , 1)$ for some $\omega \in {\mathcal {W}}_{2d}$, and assume $m\in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$. If $Q_1,Q_2$ are bounded measurable sets in $\mathbb {R} ^{d}$ and, denoting $\omega _2(Q)=\int _Q \omega _2(t)dt$,

$$\begin{aligned} h(\xi ,\eta )=\frac{1}{\omega _2(Q_1)\omega _2(Q_2)}\int _{Q_1}\int _{Q_2}m(\xi +u,\eta +v)dudv \end{aligned}$$

then $h(\xi ,\eta )\in \mathcal{B}\mathcal{M}({\varvec{\Phi }},{\varvec{\Psi }},\textbf{w})$ and $\Vert h\Vert _{\mathcal{B}\mathcal{M}}\le \Vert m\Vert _{\mathcal{B}\mathcal{M}}.$

Proof

Since $h= \frac{\chi _{Q_1\times Q_2}}{\omega _2(Q_1)\omega _2(Q_2)}* m$ the result follows from (1) in Theorem 3.11 using that $W(u,v)=\omega (u)\omega (v)$. $\square$