Boundedness of localization operators on Lorentz mixed-normed modulation spaces

Abstract

In this work we study certain boundedness properties for localization operators on Lorentz mixed-normed modulation spaces, when the operator symbols belong to appropriate modulation spaces, Wiener amalgam spaces, and Lorentz spaces with mixed norms.

1 Introduction

In this paper we will work on ${\mathbb{R}}^d$ with Lebesgue measure dx. We denote by $\mathcal{S}({\mathbb{R}}^d)$ the space of complex-valued continuous functions on ${\mathbb{R}}^d$ rapidly decreasing at infinity. For any function $f:{\mathbb{R}}^d\to \mathbb{C}$, the translation and modulation operator are defined as $T_{x}f(t)=f(t-x)$ and $M_{w}f(t)=e^{2\pi iwt}f(t)$ for $x,w\in {\mathbb{R}}^d$, respectively. For $1\le p\le \mathrm{\infty }$, we write the Lebesgue spaces $(L^p({\mathbb{R}}^d),{\parallel \cdot \parallel }_p)$.

Let $〈x,t〉={\sum }_{i=1}^d{x}_i{t}_i$ be the usual scalar product on ${\mathbb{R}}^d$. The Fourier transform $\hat{f}$ (or $\mathcal{F}f$) of $f\in L^1({\mathbb{R}}^d)$ is defined to be

$$\hat{f}(t)={\int }_{{\mathbb{R}}^d}f(x)e^{-2\pi i〈x,t〉}dx.$$

For a fixed nonzero $g\in \mathcal{S}({\mathbb{R}}^d)$ the short-time Fourier transform (STFT) of a function $f\in {\mathcal{S}}^{\prime }({\mathbb{R}}^d)$ with respect to the window g is defined as

$$V_{g}f(x,w)=〈f,M_w{T}_{x}g〉={\int }_{{\mathbb{R}}^d}f(t)\overline{g(t-x)}{e}^{-2\pi itw}dt,$$

for $x,w\in {\mathbb{R}}^d$. Then the localization operator $A_a^{{\phi }_1,{\phi }_2}$ with symbol a and windows ${\phi }_1$, ${\phi }_2$ is defined to be

$$A_a^{{\phi }_1,{\phi }_2}f(t)={\int }_{{\mathbb{R}}^{2d}}a(x,w)V_{{\phi }_1}f(x,w)M_w{T}_x{\phi }_{2}dxdw.$$

If $a\in {\mathcal{S}}^{\prime }({\mathbb{R}}^d)$ and ${\phi }_1,{\phi }_2\in \mathcal{S}({\mathbb{R}}^d)$, then the localization operator is a well-defined continuous operator from $\mathcal{S}({\mathbb{R}}^d)$ to ${\mathcal{S}}^{\prime }({\mathbb{R}}^d)$. Moreover, it is to be interpreted in a weak sense as

$$〈A_a^{{\phi }_1,{\phi }_2}f,g〉=〈a{V}_{{\phi }_1}f,V_{{\phi }_2}g〉=〈a,\overline{V_{{\phi }_1}f}{V}_{{\phi }_2}g〉$$

for $f,g\in \mathcal{S}({\mathbb{R}}^d)$, [1, 2].

Fix a nonzero window $g\in \mathcal{S}({\mathbb{R}}^d)$ and $1\le p,q\le \mathrm{\infty }$. Then the modulation space $M^{p,q}({\mathbb{R}}^d)$ consists of all tempered distributions $f\in {\mathcal{S}}^{\prime }({\mathbb{R}}^d)$ such that the short-time Fourier transform $V_{g}f$ is in the mixed-norm space $L^{p,q}({\mathbb{R}}^{2d})$. The norm on $M^{p,q}({\mathbb{R}}^d)$ is ${\parallel f\parallel }_{M^{p,q}}={\parallel V_{g}f\parallel }_{L^{p,q}}$. If $p=q$, then we write $M^p({\mathbb{R}}^d)$ instead of $M^{p,p}({\mathbb{R}}^d)$. Modulation spaces are Banach spaces whose definitions are independent of the choice of the window g (see [2, 3]).

$L(p,q)$ spaces are function spaces that are closely related to $L^p$ spaces. We consider complex-valued measurable functions f defined on a measure space $(X,\mu )$. The measure μ is assumed to be nonnegative. We assume that the functions f are finite valued a.e. and some $y>0$, $\mu (E_y)<\mathrm{\infty }$, where $E_y=E_y[f]=\{x\in X\mid |f(x)|>y\}$. Then, for $y>0$,

$${\lambda }_f(y)=\mu (E_y)=\mu \left(\{x\in X\mid |f(x)|>y\}\right)$$

is the distribution function of f. The rearrangement of f is given by

$$f^{\ast }(t)=inf\{y>0\mid {\lambda }_f(y)\le t\}=sup\{y>0\mid {\lambda }_f(y)>t\}$$

for $t>0$. The average function of f is also defined by

$$f^{\ast \ast }(x)=\frac{1}{x}{\int }_0^x{f}^{\ast }(t)dt.$$

Note that ${\lambda }_f$, $f^{\ast }$, and $f^{\ast \ast }$ are nonincreasing and right continuous functions on $(0,\mathrm{\infty })$. If ${\lambda }_f(y)$ is continuous and strictly decreasing then $f^{\ast }(t)$ is the inverse function of ${\lambda }_f(y)$. The most important property of $f^{\ast }$ is that it has the same distribution function as f. It follows that

$${\left({\int }_X\right|f(x){|}^{p}d\mu (x))}^{\frac{1}{p}}={\left({\int }_0^{\mathrm{\infty }}{[f^{\ast }(t)]}^{p}dt\right)}^{\frac{1}{p}}.$$

(1.1)

The Lorentz space denoted by $L(p,q)(X,\mu )$ (shortly $L(p,q)$) is defined to be vector space of all (equivalence classes) of measurable functions f such that ${\parallel f\parallel }_{pq}^{\ast }<\mathrm{\infty }$, where

$${\parallel f\parallel }_{pq}^{\ast }=\{\begin{array}{ll}{(\frac{q}{p}{\int }_0^{\mathrm{\infty }}{t}^{\frac{q}{p}-1}{[f^{\ast }(t)]}^{q}dt)}^{\frac{1}{q}},& 0<p,q<\mathrm{\infty },\\ {sup}_{t>0}{t}^{\frac{1}{p}}{f}^{\ast }(t),& 0<p\le q=\mathrm{\infty }.\end{array}$$

By (1.1), it follows that ${\parallel f\parallel }_{pp}^{\ast }={\parallel f\parallel }_p$ and so $L(p,p)=L^p$. Also, $L(p,q)(X,\mu )$ is a normed space with the norm

$${\parallel f\parallel }_{pq}=\{\begin{array}{ll}{(\frac{q}{p}{\int }_0^{\mathrm{\infty }}{t}^{\frac{q}{p}-1}{[f^{\ast \ast }(t)]}^{q}dt)}^{\frac{1}{q}},& 0<p,q<\mathrm{\infty },\\ {sup}_{t>0}{t}^{\frac{1}{p}}{f}^{\ast \ast }(t),& 0<p\le q=\mathrm{\infty }.\end{array}$$

For any one of the cases $p=q=1$; $p=q=\mathrm{\infty }$ or $1<p<\mathrm{\infty }$ and $1\le q\le \mathrm{\infty }$, the Lorentz space $L(p,q)(X,\mu )$ is a Banach space with respect to the norm ${\parallel \cdot \parallel }_{pq}$. It is also well known that if $1<p<\mathrm{\infty }$, $1\le q\le \mathrm{\infty }$ we have

$${\parallel \cdot \parallel }_{pq}^{\ast }\le {\parallel \cdot \parallel }_{pq}\le \frac{p}{p-1}{\parallel \cdot \parallel }_{pq}^{\ast }$$

(see [4, 5]).

Let X and Y be two measure spaces with σ-finite measures μ and ν, respectively, and let f be a complex-valued measurable function on $(X\times Y,\mu \times \nu )$, $1<P=(p_1,p_2)<\mathrm{\infty }$, and $1\le Q=(q_1,q_2)\le \mathrm{\infty }$. The Lorentz mixed norm space $L(P,Q)=L(P,Q)(X\times Y)$ is defined by

$$L(P,Q)=L(p_2,q_2)[L(p_1,q_1)]=\{f:{\parallel f\parallel }_{PQ}={\parallel f\parallel }_{L(p_2,q_2)(L(p_1,q_1))}={\parallel {\parallel f\parallel }_{p_1{q}_1}\parallel }_{p_2{q}_2}<\mathrm{\infty }\}.$$

Thus, $L(P,Q)$ occurs by taking an $L(p_1,q_1)$-norm with respect to the first variable and an $L(p_2,q_2)$-norm with respect to the second variable. The $L(P,Q)$ space is a Banach space under the norm ${\parallel \cdot \parallel }_{PQ}$ (see [6, 7]).

Fix a window function $g\in \mathcal{S}({\mathbb{R}}^d)\mathrm{\setminus }\{0\}$, $1\le P=(p_1,p_2)<\mathrm{\infty }$, and $1\le Q=(q_1,q_2)\le \mathrm{\infty }$. We let $M(P,Q)({\mathbb{R}}^d)$ denote the subspace of tempered distributions ${\mathcal{S}}^{\prime }({\mathbb{R}}^d)$ consisting of $f\in {\mathcal{S}}^{\prime }({\mathbb{R}}^d)$ such that the Gabor transform $V_{g}f$ of f is in the Lorentz mixed norm space $L(P,Q)({\mathbb{R}}^{2d})$. We endow it with the norm ${\parallel f\parallel }_{M(P,Q)}={\parallel V_{g}f\parallel }_{PQ}$, where ${\parallel \cdot \parallel }_{PQ}$ is the norm of the Lorentz mixed norm space. It is well known that $M(P,Q)({\mathbb{R}}^d)$ is a Banach space and different windows yield equivalent norms. If $p_1=q_1=p$ and $p_2=q_2=q$, then the space $M(P,Q)({\mathbb{R}}^d)$ is the standard modulation space $M^{p,q}({\mathbb{R}}^d)$, and if $P=p$ and $Q=q$, in this case $M(P,Q)({\mathbb{R}}^d)=M(p,q)({\mathbb{R}}^d)$ (see [8, 9]), where the space $M(p,q)({\mathbb{R}}^d)$ is Lorentz type modulation space (see [10]). Furthermore, the space $M(p,q)({\mathbb{R}}^d)$ was generalized to $M(p,q,w)({\mathbb{R}}^d)$ by taking weighted Lorentz space rather than Lorentz space (see [11, 12]).

In this paper, we will denote the Lorentz space by $L(p,q)$, the Lorentz mixed norm space by $L(P,Q)$, the standard modulation space by $M^{p,q}$, the Lorentz type modulation space by $M(p,q)$, and the Lorentz mixed-normed modulation space by $M(P,Q)$.

Let $1\le r,s\le \mathrm{\infty }$. Fix a compact $Q\subset {\mathbb{R}}^d$ with nonempty interior. Then the Wiener amalgam space $W(L^r,L^s)({\mathbb{R}}^d)$ with local component $L^r({\mathbb{R}}^d)$ and global component $L^s({\mathbb{R}}^d)$ is defined as the space of all measurable functions $f:{\mathbb{R}}^d\to \mathbb{C}$ such that $f{\chi }_K\in L^r({\mathbb{R}}^d)$ for each compact subset $K\subset {\mathbb{R}}^d$, for which the norm

$${\parallel f\parallel }_{W(L^r,L^s)}={\parallel F_f\parallel }_s={\parallel {\parallel f{\chi }_{Q+x}\parallel }_r\parallel }_s$$

is finite, where ${\chi }_K$ is the characteristic function of K and

$$F_f(x)={\parallel f{\chi }_{Q+x}\parallel }_r\in L^s\left({\mathbb{R}}^d\right).$$

It is known that if $r_1\ge r_2$ and $s_1\le s_2$ then $W(L^{r_1},L^{s_1})({\mathbb{R}}^d)\subset W(L^{r_2},L^{s_2})({\mathbb{R}}^d)$. If $r=s$ then $W(L^r,L^r)({\mathbb{R}}^d)=L^r({\mathbb{R}}^d)$ (see [13–15]).

In this paper, we consider boundedness properties for localization operators acting on Lorentz mixed-normed modulation spaces for the symbols in appropriate function spaces like modulation spaces, Wiener amalgam spaces, and Lorentz spaces with mixed norms. Our results extend some results in [1, 12] to the Lorentz mixed-normed modulation spaces.

2 Boundedness of localization operators on Lorentz mixed normed modulation spaces

We start with the following lemma, which will be used later on.

Lemma 2.1 Let $\frac{1}{P}+\frac{1}{P^{\prime }}=1$, $\frac{1}{Q_1}+\frac{1}{Q_2}\ge 1$, $f\in L(P,Q_1)({\mathbb{R}}^{2d})$, $h\in L(P^{\prime },Q_2)({\mathbb{R}}^{2d})$. Then $f\ast h\in L^{\mathrm{\infty }}({\mathbb{R}}^{2d})$ and

$$L(P,Q_1)\left({\mathbb{R}}^{2d}\right)\ast L(P^{\prime },Q_2)\left({\mathbb{R}}^{2d}\right)\hookrightarrow L^{\mathrm{\infty }}\left({\mathbb{R}}^{2d}\right)$$

(2.1)

with the norm inequality

$${\parallel f\ast h\parallel }_{\mathrm{\infty }}\le {\parallel f\parallel }_{P{Q}_1}{\parallel h\parallel }_{P^{\prime }{Q}_2},$$

(2.2)

where $P=(p_1,p_2)$, $Q_1=(Q_1^1,Q_1^2)$, $Q_2=(Q_2^1,Q_2^2)$.

Proof It is well known that there are $L(p,q_1)\ast L(p^{\prime },q_2)\hookrightarrow L^{\mathrm{\infty }}$ convolution relations between Lorentz spaces and

$${\parallel f\ast h\parallel }_{\mathrm{\infty }}\le {\parallel f\parallel }_{p{q}_1}{\parallel h\parallel }_{p^{\prime }{q}_2},$$

where $\frac{1}{p}+\frac{1}{p^{\prime }}=1$, $\frac{1}{q_1}+\frac{1}{q_2}\ge 1$, by Theorem 3.6 in [5]. Then (2.1) and (2.2) can easily be verified by using iteration and the one variable proofs given in [5]. □

Let $g\in \mathcal{D}({\mathbb{R}}^{2d})$ be a test function such that ${\sum }_{x\in {\mathbb{Z}}^{2d}}{T}_{x}g\equiv 1$. Let $X({\mathbb{R}}^{2d})$ be a translation invariant Banach space of functions with the property that $\mathcal{D}\cdot X\subset X$. In the spirit of [13, 16], the Wiener amalgam space $W(X,L(P,Q))$ with local component X and global component $L(P,Q)$ is defined as the space of all functions or distributions for which the norm

$${\parallel f\parallel }_{W(X,L(P,Q))}={\parallel {\parallel f\cdot T_{(z_1,z_2)}\overline{g}\parallel }_X\parallel }_{PQ}$$

is finite, where $1\le P<\mathrm{\infty }$, $1\le Q\le \mathrm{\infty }$. Moreover, different choices of $g\in \mathcal{D}$ yield equivalent norms and give the same space.

The boundedness of $A_{M_{\zeta }a}^{{\phi }_1,{\phi }_2}$ for $a\in M^{\mathrm{\infty }}$ is established by our next theorem. The proof is similar to Lemma 4.1 in [1] but let us provide the details anyway, for completeness’ sake.

Theorem 2.1

• (i) Let $1<P<\mathrm{\infty }$, $1\le Q<\mathrm{\infty }$. If $f\in M(P,Q)({\mathbb{R}}^d)$ and $g\in M^1({\mathbb{R}}^d)$, then $V_{g}f\in W(\mathcal{F}{L}^1,L(P,Q))({\mathbb{R}}^{2d})$ with

  $${\parallel V_{g}f\parallel }_{W(\mathcal{F}{L}^1,L(P,Q))}\le {\parallel f\parallel }_{M(P,Q)}{\parallel g\parallel }_{M^1}.$$

• (ii) Let $\frac{1}{P}+\frac{1}{P^{\prime }}=1$, $\frac{1}{Q_1}+\frac{1}{Q_2}\ge 1$. If $f\in M(P,Q_1)({\mathbb{R}}^d)$ and $g\in M(P^{\prime },Q_2)({\mathbb{R}}^d)$, then $V_{g}f\in W(\mathcal{F}{L}^1,L^{\mathrm{\infty }})({\mathbb{R}}^{2d})$ with

  $${\parallel V_{g}f\parallel }_{W(\mathcal{F}{L}^1,L^{\mathrm{\infty }})}\le {\parallel f\parallel }_{M(P,Q_1)}{\parallel g\parallel }_{M(P^{\prime },Q_2)}.$$

Proof (i) Let $\phi \in \mathcal{S}({\mathbb{R}}^d)\mathrm{\setminus }\{0\}$ and set $\mathrm{\Phi }=V_{\phi }\phi \in \mathcal{S}({\mathbb{R}}^{2d})$. By using the equality $V_{g}f(x,w)={(f\cdot T_x\overline{g})}^{\wedge }(w)$, we write

$$\begin{array}{rcl}{\parallel V_{g}f\cdot T_{(z_1,z_2)}\overline{\mathrm{\Phi }}\parallel }_{\mathcal{F}{L}^1}& =& {\int }_{{\mathbb{R}}^{2d}}|{(V_{g}f\cdot T_{(z_1,z_2)}\overline{\mathrm{\Phi }})}^{\wedge }(t)|dt\\ =& {\int }_{{\mathbb{R}}^{2d}}|V_{\mathrm{\Phi }}{V}_{g}f(z_1,z_2,t_1,t_2)|d{t}_{1}d{t}_2\\ =& {\int }_{{\mathbb{R}}^{2d}}|V_{\phi }g(-z_1-t_2,t_1)V_{\phi }f(-t_2,z_2+t_1)|d{t}_{1}d{t}_2\\ =& {\int }_{{\mathbb{R}}^{2d}}|V_{\phi }f(u_1,u_2)||V_{\phi }g(u_1-z_1,u_2-z_2)|d{u}_{1}d{u}_2\\ =& |V_{\phi }f|\ast |V_{\phi }g{|}^{\sim }(z_1,z_2),\end{array}$$

(2.3)

for $f,g\in \mathcal{S}({\mathbb{R}}^d)$, where ${(V_{\phi }g)}^{\sim }(z)=(\overline{V_{\phi }g})(-z)$, $z\in {\mathbb{R}}^{2d}$. Since $f,g\in \mathcal{S}({\mathbb{R}}^d)$, then $f\in M(P,Q)({\mathbb{R}}^d)$ and $g\in M^1({\mathbb{R}}^d)$ by Proposition 2 in [8]. So $V_{\phi }f\in L(P,Q)({\mathbb{R}}^{2d})$ and $V_{\phi }g\in L^1({\mathbb{R}}^{2d})$. Then, by Proposition 4 in [8], we obtain

$$\begin{array}{rcl}{\parallel V_{g}f\parallel }_{W(\mathcal{F}{L}^1,L(P,Q))}& =& {\parallel {\parallel V_{g}f\cdot T_{(z_1,z_2)}\overline{\mathrm{\Phi }}\parallel }_{\mathcal{F}{L}^1}\parallel }_{PQ}\\ =& {\parallel |V_{\phi }f|\ast |V_{\phi }g{|}^{\sim }\parallel }_{PQ}\\ \le & {\parallel V_{\phi }f\parallel }_{PQ}{\parallel V_{\phi }g\parallel }_1\\ =& {\parallel f\parallel }_{M(P,Q)}{\parallel g\parallel }_{M^1}.\end{array}$$

(2.4)

This completes the proof.

• (ii) Using Lemma 2.1 and (2.3), we have

  $${\parallel V_{g}f\parallel }_{W(\mathcal{F}{L}^1,L^{\mathrm{\infty }})}={\parallel |V_{\phi }f|\ast |V_{\phi }g{|}^{\sim }\parallel }_{\mathrm{\infty }}\le {\parallel V_{\phi }f\parallel }_{P{Q}_1}{\parallel V_{\phi }g\parallel }_{P^{\prime }{Q}_2}={\parallel f\parallel }_{M(P,Q_1)}{\parallel g\parallel }_{M(P^{\prime },Q_2)}.$$

 □

Theorem 2.2 Let $1<P<\mathrm{\infty }$, $1\le Q<\mathrm{\infty }$. If $a\in M^{\mathrm{\infty }}({\mathbb{R}}^{2d})$, ${\phi }_1,{\phi }_2\in M^1({\mathbb{R}}^d)$, then $A_{M_{\zeta }a}^{{\phi }_1,{\phi }_2}$ is bounded on $M(P,Q)({\mathbb{R}}^d)$ for every $\zeta \in {\mathbb{R}}^{2d}$ with

$${\parallel A_{M_{\zeta }a}^{{\phi }_1,{\phi }_2}\parallel }_{B(M(P,Q))}\le {\parallel a\parallel }_{M^{\mathrm{\infty }}}{\parallel {\phi }_1\parallel }_{M^1}{\parallel {\phi }_2\parallel }_{M^1}.$$

Proof Let $f\in M(P,Q)({\mathbb{R}}^d)$ and $g\in M(P^{\prime },Q^{\prime })({\mathbb{R}}^d)$, where $\frac{1}{P}+\frac{1}{P^{\prime }}=1$, $\frac{1}{Q}+\frac{1}{Q^{\prime }}=1$. Then we write $\overline{V_{{\phi }_1}f}\in W(\mathcal{F}{L}^1,L(P,Q))({\mathbb{R}}^{2d})$ and $V_{{\phi }_2}g\in W(\mathcal{F}{L}^1,L(P^{\prime },Q^{\prime }))({\mathbb{R}}^{2d})$ by above theorem. Moreover, since $M(1,1)({\mathbb{R}}^d)=M^1({\mathbb{R}}^d)$, we have $W(\mathcal{F}{L}^1,L^1)=M^1=M(1,1)$ by [16]. Hence using the Hölder inequalities for Wiener amalgam spaces [13] and (2.4) we obtain

$$\begin{array}{rcl}{\parallel \overline{V_{{\phi }_1}f}\cdot V_{{\phi }_2}g\parallel }_{M^1}& =& {\parallel \overline{V_{{\phi }_1}f}\cdot V_{{\phi }_2}g\parallel }_{W(\mathcal{F}{L}^1,L^1)}\\ \le & {\parallel V_{{\phi }_1}f\parallel }_{W(\mathcal{F}{L}^1,L(P,Q))}{\parallel V_{{\phi }_2}g\parallel }_{W(\mathcal{F}{L}^1,L(P^{\prime },Q^{\prime }))}\\ ⪯& {\parallel {\phi }_1\parallel }_{M^1}{\parallel {\phi }_2\parallel }_{M^1}{\parallel f\parallel }_{M(P,Q)}{\parallel g\parallel }_{M(P^{\prime },Q^{\prime })}.\end{array}$$

(2.5)

Thus by using (2.5) we have

$$\begin{array}{rl}\left|〈A_{M_{\zeta }a}^{{\phi }_1,{\phi }_2}f,g〉\right|& =|〈M_{\zeta }a,\overline{V_{{\phi }_1}f}\cdot V_{{\phi }_2}g〉|\le {\parallel M_{\zeta }a\parallel }_{M(\mathrm{\infty },\mathrm{\infty })}{\parallel \overline{V_{{\phi }_1}f}\cdot V_{{\phi }_2}g\parallel }_{M(1,1)}\\ \le {\parallel a\parallel }_{M^{\mathrm{\infty }}}{\parallel {\phi }_1\parallel }_{M^1}{\parallel {\phi }_2\parallel }_{M^1}{\parallel f\parallel }_{M(P,Q)}{\parallel g\parallel }_{M(P^{\prime },Q^{\prime })}.\end{array}$$

Hence we get

$${\parallel A_{M_{\zeta }a}^{{\phi }_1,{\phi }_2}\parallel }_{B(M(P,Q))}\le {\parallel a\parallel }_{M^{\mathrm{\infty }}}{\parallel {\phi }_1\parallel }_{M^1}{\parallel {\phi }_2\parallel }_{M^1}.$$

 □

Theorem 2.3 Let $\phi \in \mathcal{S}({\mathbb{R}}^d)\mathrm{\setminus }\{0\}$ be a window function. If $1<P,Q<\mathrm{\infty }$, $t^{\prime }\in (1,\mathrm{\infty })$, $s\le t^{\prime }\le r$ and $a\in W(L^r,L^s)$, then

$$A_{M_{\zeta }a}^{\phi ,\phi }:M(tP,tQ)\left({\mathbb{R}}^d\right)\to M({\left(t{P}^{\prime }\right)}^{\prime },{\left(t{Q}^{\prime }\right)}^{\prime })\left({\mathbb{R}}^d\right)$$

is bounded for every $\zeta \in {\mathbb{R}}^{2d}$, where $\frac{1}{P}+\frac{1}{P^{\prime }}=1$, $\frac{1}{Q}+\frac{1}{Q^{\prime }}=1$, and $\frac{1}{t}+\frac{1}{t^{\prime }}=1$, and the operator norm satisfies the estimate

$$\parallel A_{M_{\zeta }a}^{\phi ,\phi }\parallel \le {\parallel a\parallel }_{W(L^r,L^s)}.$$

Proof Let $t<\mathrm{\infty }$, $f\in M(tP,tQ)({\mathbb{R}}^d)$, and $h\in M(t{P}^{\prime },t{Q}^{\prime })({\mathbb{R}}^d)$. Then we have $V_{\phi }f\in L(tP,tQ)({\mathbb{R}}^{2d})$ and $V_{\phi }h\in L(t{P}^{\prime },t{Q}^{\prime })({\mathbb{R}}^{2d})$. Since $V_{\phi }f\in L(tP,tQ)({\mathbb{R}}^{2d})$, then ${\parallel V_{\phi }f\parallel }_{(tP)(tQ)}^{\ast }<\mathrm{\infty }$. By using the equality (3.6) in [12], we get

$$\begin{array}{rcl}{\parallel V_{\phi }f\parallel }_{(tP)(tQ)}^{\ast }& =& {\parallel {\parallel V_{\phi }f\parallel }_{(t{p}_1)(t{q}_1)}^{\ast }\parallel }_{(t{p}_2)(t{q}_2)}^{\ast }={\parallel {\left({\parallel |V_{\phi }f{|}^t\parallel }_{p_1{q}_1}^{\ast }\right)}^{\frac{1}{t}}\parallel }_{(t{p}_2)(t{q}_2)}^{\ast }\\ =& {\left({\parallel |{\left({\parallel |V_{\phi }f{|}^t\parallel }_{p_1{q}_1}^{\ast }\right)}^{\frac{1}{t}}{|}^t\parallel }_{p_2{q}_2}^{\ast }\right)}^{\frac{1}{t}}={\left({\parallel {\parallel |V_{\phi }f{|}^t\parallel }_{p_1{q}_1}^{\ast }\parallel }_{p_2{q}_2}^{\ast }\right)}^{\frac{1}{t}}\\ =& {\left({\parallel |V_{\phi }f{|}^t\parallel }_{PQ}^{\ast }\right)}^{\frac{1}{t}}.\end{array}$$

(2.6)

Hence we have $|V_{\phi }f{|}^t\in L(P,Q)({\mathbb{R}}^{2d})$. Similarly, $|V_{\phi }h{|}^t\in L(P^{\prime },Q^{\prime })({\mathbb{R}}^{2d})$. By the Hölder inequality for Lorentz spaces with mixed norm and (2.6) we have

$$\begin{array}{rcl}{\parallel V_{\phi }f\cdot V_{\phi }h\parallel }_t^t& =& {\parallel |V_{\phi }f{|}^t|V_{\phi }h{|}^t\parallel }_1\le {\parallel |V_{\phi }f{|}^t\parallel }_{PQ}{\parallel |V_{\phi }h{|}^t\parallel }_{P^{\prime }{Q}^{\prime }}\\ =& {\parallel V_{\phi }f\parallel }_{(tP)(tQ)}^t{\parallel V_{\phi }h\parallel }_{(t{P}^{\prime })(t{Q}^{\prime })}^t.\end{array}$$

(2.7)

Since $a\in W(L^r,L^s)$, then $M_{\zeta }a\in W(L^r,L^s)$ for every $\zeta \in {\mathbb{R}}^{2d}$. Also since $W(L^r,L^s)\subset W(L^{t^{\prime }},L^{t^{\prime }})=L^{t^{\prime }}({\mathbb{R}}^{2d})$, then we have

$${\parallel a\parallel }_{t^{\prime }}={\parallel M_{\zeta }a\parallel }_{t^{\prime }}\le {\parallel M_{\zeta }a\parallel }_{W(L^r,L^s)}={\parallel a\parallel }_{W(L^r,L^s)}.$$

(2.8)

By using (2.7), (2.8), and applying again the Hölder inequality, we get

$$\begin{array}{rcl}\left|〈A_{M_{\zeta }a}^{\phi ,\phi }f,h〉\right|& =& |〈M_{\zeta }a{V}_{\phi }f,V_{\phi }h〉|\\ \le & {\iint }_{{\mathbb{R}}^{2d}}|M_{\zeta }a(x,w)||(V_{\phi }f\cdot V_{\phi }h)(x,w)|dxdw\\ \le & {\parallel M_{\zeta }a\parallel }_{t^{\prime }}{\parallel V_{\phi }f\cdot V_{\phi }h\parallel }_t\\ \le & {\parallel a\parallel }_{t^{\prime }}{\parallel V_{\phi }f\parallel }_{(tP)(tQ)}{\parallel V_{\phi }h\parallel }_{(t{P}^{\prime })(t{Q}^{\prime })}\\ \le & {\parallel a\parallel }_{W(L^r,L^s)}{\parallel f\parallel }_{M(tP,tQ)}{\parallel h\parallel }_{M(t{P}^{\prime },t{Q}^{\prime })}.\end{array}$$

(2.9)

If ${(t{p}^{\prime })}^{\prime },{(t{q}^{\prime })}^{\prime }\ne \mathrm{\infty }$, then ${(M({(t{P}^{\prime })}^{\prime },{(t{Q}^{\prime })}^{\prime })({\mathbb{R}}^d))}^{\ast }=M(t{P}^{\prime },t{Q}^{\prime })({\mathbb{R}}^d)$ by Theorem 8 in [8]. Thus we have from (2.9) that

$${\parallel A_{M_{\zeta }a}^{\phi ,\phi }f\parallel }_{M({(t{P}^{\prime })}^{\prime },{(t{Q}^{\prime })}^{\prime })}=\underset{0\ne h\in M(t{P}^{\prime },t{Q}^{\prime })}{sup}\frac{|〈A_{M_{\zeta }a}^{\phi ,\phi }f,h〉|}{{\parallel h\parallel }_{M(t{P}^{\prime },t{Q}^{\prime })}}\le {\parallel a\parallel }_{W(L^r,L^s)}{\parallel f\parallel }_{M(tP,tQ)}.$$

Hence $A_{M_{\zeta }a}^{\phi ,\phi }$ is bounded. Also we have

$$\parallel A_{M_{\zeta }a}^{\phi ,\phi }\parallel =\underset{0\ne f\in M(tP,tQ)}{sup}\frac{{\parallel A_{M_{\zeta }a}^{\phi ,\phi }f\parallel }_{M({(t{P}^{\prime })}^{\prime },{(t{Q}^{\prime })}^{\prime })}}{{\parallel f\parallel }_{M(tP,tQ)}}\le {\parallel a\parallel }_{W(L^r,L^s)}.$$

 □

Theorem 2.4 Let $\phi \in {\bigcap }_{1\le R,S<\mathrm{\infty }}M(R,S)({\mathbb{R}}^d)$, where $R=(r_1,r_2)$, $S=(s_1,s_2)$. If $1\le s\le r\le \mathrm{\infty }$ and $a\in W(L^r,L^s)$ then

$$A_{M_{\zeta }a}^{\phi ,\phi }:M(P,Q)\left({\mathbb{R}}^d\right)\to M(P,Q)\left({\mathbb{R}}^d\right)$$

is bounded for every $\zeta \in {\mathbb{R}}^{2d}$, with

$$\parallel A_{M_{\zeta }a}^{\phi ,\phi }\parallel \le C{\parallel a\parallel }_{W(L^r,L^s)}$$

for some $C>0$.

Proof Since $a\in W(L^r,L^s)$, then $M_{\zeta }a\in W(L^r,L^s)$ for every $\zeta \in {\mathbb{R}}^{2d}$. Also since $s\le r$, there exists $1\le t_0\le \mathrm{\infty }$ such that $s\le t_0\le r$. Then $W(L^r,L^s)({\mathbb{R}}^{2d})\subset L^{t_0}({\mathbb{R}}^{2d})$ and

$${\parallel M_{\zeta }a\parallel }_{t_0}={\parallel a\parallel }_{t_0}\le {\parallel a\parallel }_{W(L^r,L^s)}={\parallel M_{\zeta }a\parallel }_{W(L^r,L^s)}$$

(2.10)

for all $a\in W(L^r,L^s)({\mathbb{R}}^{2d})$. Let $B(M(P,Q)({\mathbb{R}}^d),M(P,Q)({\mathbb{R}}^d))$ be the space of the bounded linear operators from $M(P,Q)({\mathbb{R}}^d)$ into $M(P,Q)({\mathbb{R}}^d)$. Also let T be an operator from $L^1({\mathbb{R}}^{2d})$ into $B(M(P,Q)({\mathbb{R}}^d),M(P,Q)({\mathbb{R}}^d))$ by $T(a)=A_{M_{\zeta }a}^{\phi ,\phi }$. Take any $f\in M(P,Q)({\mathbb{R}}^d)$ and $h\in M(P^{\prime },Q^{\prime })({\mathbb{R}}^d)$. Assume that $a\in W(L^1,L^1)({\mathbb{R}}^{2d})=L^1({\mathbb{R}}^{2d})$. By the Hölder inequality we get

$$\begin{array}{rcl}\left|〈T(a)f,h〉\right|& =& \left|〈A_{M_{\zeta }a}^{\phi ,\phi }f,h〉\right|=|〈M_{\zeta }a{V}_{\phi }f,V_{\phi }h〉|\\ \le & {\iint }_{{\mathbb{R}}^{2d}}|M_{\zeta }a(x,w)||V_{\phi }f(x,w)||V_{\phi }h(x,w)|dxdw\\ =& {\iint }_{{\mathbb{R}}^{2d}}|a(x,w)||〈f,M_w{T}_x\phi 〉||〈h,M_w{T}_x\phi 〉|dxdw\\ \le & {\iint }_{{\mathbb{R}}^{2d}}|a(x,w)|{\parallel f\parallel }_{M(P,Q)}{\parallel M_w{T}_x\phi \parallel }_{M(P^{\prime },Q^{\prime })}{\parallel h\parallel }_{M(P^{\prime },Q^{\prime })}\\ \times {\parallel M_w{T}_x\phi \parallel }_{M(P,Q)}dxdw\\ =& {\parallel f\parallel }_{M(P,Q)}{\parallel \phi \parallel }_{M(P^{\prime },Q^{\prime })}{\parallel h\parallel }_{M(P^{\prime },Q^{\prime })}{\parallel \phi \parallel }_{M(P,Q)}{\parallel a\parallel }_1.\end{array}$$

(2.11)

Hence by (2.11)

$$\begin{array}{rcl}{\parallel T(a)f\parallel }_{M(P,Q)}& =& {\parallel A_{M_{\zeta }a}^{\phi ,\phi }f\parallel }_{M(P,Q)}=\underset{0\ne h\in M(P^{\prime },Q^{\prime })}{sup}\frac{|〈A_{M_{\zeta }a}^{\phi ,\phi }f,h〉|}{{\parallel h\parallel }_{M(P^{\prime },Q^{\prime })}}\\ \le & {\parallel \phi \parallel }_{M(P^{\prime },Q^{\prime })}{\parallel \phi \parallel }_{M(P,Q)}{\parallel f\parallel }_{M(P,Q)}{\parallel a\parallel }_1.\end{array}$$

Then

$$\parallel T(a)\parallel =\parallel A_{M_{\zeta }a}^{\phi ,\phi }\parallel =\underset{0\ne f\in M(P,Q)}{sup}\frac{{\parallel A_{M_{\zeta }a}^{\phi ,\phi }f\parallel }_{M(P,Q)}}{{\parallel f\parallel }_{M(P,Q)}}\le {\parallel \phi \parallel }_{M(P^{\prime },Q^{\prime })}{\parallel \phi \parallel }_{M(P,Q)}{\parallel a\parallel }_1.$$

(2.12)

Thus the operator

$$T:L^1\left({\mathbb{R}}^{2d}\right)\to B(M(P,Q)\left({\mathbb{R}}^d\right),M(P,Q)\left({\mathbb{R}}^d\right))$$

(2.13)

is bounded. Now let $a\in W(L^{\mathrm{\infty }},L^{\mathrm{\infty }})({\mathbb{R}}^{2d})=L^{\mathrm{\infty }}({\mathbb{R}}^{2d})$. Take any $f\in M(P,Q)({\mathbb{R}}^d)$ and $h\in M(P^{\prime },Q^{\prime })({\mathbb{R}}^d)$. Then $V_{\phi }f\in L(P,Q)({\mathbb{R}}^{2d})$, $V_{\phi }h\in L(P^{\prime },Q^{\prime })({\mathbb{R}}^{2d})$. Applying the Hölder inequality

$$\begin{array}{rcl}\left|〈T(a)f,h〉\right|& =& \left|〈A_{M_{\zeta }a}^{\phi ,\phi }f,h〉\right|=|〈M_{\zeta }a{V}_{\phi }f,V_{\phi }h〉|\\ \le & {\iint }_{{\mathbb{R}}^{2d}}|M_{\zeta }a(x,w)||V_{\phi }f(x,w)||V_{\phi }h(x,w)|dxdw\\ \le & {\parallel a\parallel }_{\mathrm{\infty }}{\iint }_{{\mathbb{R}}^{2d}}|V_{\phi }f(x,w)||V_{\phi }h(x,w)|dxdw\\ \le & {\parallel a\parallel }_{\mathrm{\infty }}{\parallel V_{\phi }f\parallel }_{PQ}{\parallel V_{\phi }h\parallel }_{P^{\prime }{Q}^{\prime }}.\end{array}$$

(2.14)

By using (2.14) we write

$${\parallel T(a)f\parallel }_{M(P,Q)}={\parallel A_{M_{\zeta }a}^{\phi ,\phi }f\parallel }_{M(P,Q)}=\underset{0\ne h\in M(P^{\prime },Q^{\prime })}{sup}\frac{|〈A_{M_{\zeta }a}^{\phi ,\phi }f,h〉|}{{\parallel h\parallel }_{M(P^{\prime },Q^{\prime })}}\le {\parallel a\parallel }_{\mathrm{\infty }}{\parallel f\parallel }_{M(P,Q)}.$$

(2.15)

Hence by (2.15)

$$\parallel T(a)\parallel =\parallel A_{M_{\zeta }a}^{\phi ,\phi }\parallel =\underset{0\ne f\in M(P,Q)}{sup}\frac{{\parallel A_{M_{\zeta }a}^{\phi ,\phi }f\parallel }_{M(P,Q)}}{{\parallel f\parallel }_{M(P,Q)}}\le {\parallel a\parallel }_{\mathrm{\infty }}.$$

That means the operator

$$T:L^{\mathrm{\infty }}\left({\mathbb{R}}^{2d}\right)\to B(M(P,Q)\left({\mathbb{R}}^d\right),M(P,Q)\left({\mathbb{R}}^d\right))$$

(2.16)

is bounded. Combining (2.13) and (2.16) we obtain

$$T:L^t\left({\mathbb{R}}^{2d}\right)\to B(M(P,Q)\left({\mathbb{R}}^d\right),M(P,Q)\left({\mathbb{R}}^d\right))$$

is bounded by interpolation theorem for $1\le t\le \mathrm{\infty }$. That means the localization operator

$$A_{M_{\zeta }a}^{\phi ,\phi }:M(P,Q)\left({\mathbb{R}}^d\right)\to M(P,Q)\left({\mathbb{R}}^d\right)$$

is bounded for $1\le t\le \mathrm{\infty }$. Hence there exists $C>0$ such that

$$\parallel T(a)\parallel =\parallel A_{M_{\zeta }a}^{\phi ,\phi }\parallel \le C{\parallel a\parallel }_t.$$

(2.17)

This implies that it is also true for $1\le t_0\le \mathrm{\infty }$. From (2.10) and (2.17) we write

$$\parallel T(a)\parallel =\parallel A_{M_{\zeta }a}^{\phi ,\phi }\parallel \le C{\parallel a\parallel }_{t_0}\le C{\parallel a\parallel }_{W(L^r,L^s)}.$$

 □

Proposition 2.1 Let $\phi \in {\bigcap }_{1\le R,S<\mathrm{\infty }}M(R,S)({\mathbb{R}}^d)$, where $R=(r_1,r_2)$, $S=(s_1,s_2)$. If $0<s\le 1$ and $a\in W(L^1,L^s)({\mathbb{R}}^{2d})$ then

$$A_{M_{\zeta }a}^{\phi ,\phi }:M(P,Q)\left({\mathbb{R}}^d\right)\to M(P,Q)\left({\mathbb{R}}^d\right)$$

is bounded.

Proof Let $0<s\le 1$ and let $a\in W(L^1,L^s)({\mathbb{R}}^{2d})$. Then $M_{\zeta }a\in W(L^1,L^s)$ for every $\zeta \in {\mathbb{R}}^{2d}$. Since $W(L^1,L^s)({\mathbb{R}}^{2d})\subset L^1({\mathbb{R}}^{2d})$, there exists a number $C>0$ such that ${\parallel M_{\zeta }a\parallel }_1\le C{\parallel M_{\zeta }a\parallel }_{W(L^1,L^s)}$. Hence by (2.12),

$$\begin{array}{rcl}\parallel A_{M_{\zeta }a}^{\phi ,\phi }\parallel & \le & {\parallel \phi \parallel }_{M(P^{\prime },Q^{\prime })}{\parallel \phi \parallel }_{M(P,Q)}{\parallel M_{\zeta }a\parallel }_1\\ \le & C{\parallel \phi \parallel }_{M(P^{\prime },Q^{\prime })}{\parallel \phi \parallel }_{M(P,Q)}{\parallel M_{\zeta }a\parallel }_{W(L^1,L^s)}\\ =& C{\parallel \phi \parallel }_{M(P^{\prime },Q^{\prime })}{\parallel \phi \parallel }_{M(P,Q)}{\parallel a\parallel }_{W(L^1,L^s)}.\end{array}$$

Then the localization operator from $M(P,Q)({\mathbb{R}}^d)$ into $M(P,Q)({\mathbb{R}}^d)$ is bounded for $0<s\le 1$. □

Proposition 2.2 Let $\phi \in {\bigcap }_{1\le R,S<\mathrm{\infty }}M(R,S)({\mathbb{R}}^d)$, where $R=(r_1,r_2)$, $S=(s_1,s_2)$. If $1\le P,Q<\mathrm{\infty }$ and $a\in L(P^{\prime },Q^{\prime })({\mathbb{R}}^{2d})$ then the localization operator

$$A_{M_{\zeta }a}^{\phi ,\phi }:M(P,Q)\left({\mathbb{R}}^d\right)\to M(P,Q)\left({\mathbb{R}}^d\right)$$

is bounded, where $\frac{1}{P}+\frac{1}{P^{\prime }}=1$, $\frac{1}{Q}+\frac{1}{Q^{\prime }}=1$.

Proof Let $a\in L(P^{\prime },Q^{\prime })({\mathbb{R}}^{2d})$. Then $M_{\zeta }a\in L(P^{\prime },Q^{\prime })({\mathbb{R}}^{2d})$ for every $\zeta \in {\mathbb{R}}^{2d}$ with ${\parallel M_{\zeta }a\parallel }_{P^{\prime }{Q}^{\prime }}={\parallel a\parallel }_{P^{\prime }{Q}^{\prime }}$. Take any $f\in M(P,Q)({\mathbb{R}}^d)$ and $h\in M(P^{\prime },Q^{\prime })({\mathbb{R}}^d)$. Applying the Hölder inequality we have by (2.11)

$$\begin{array}{rcl}\left|〈A_{M_{\zeta }a}^{\phi ,\phi }f,h〉\right|& \le & {\iint }_{{\mathbb{R}}^{2d}}|M_{\zeta }a(x,w)||V_{\phi }f(x,w)||〈h,M_w{T}_x\phi 〉|dxdw\\ \le & {\iint }_{{\mathbb{R}}^{2d}}|a(x,w)||V_{\phi }f(x,w)|{\parallel h\parallel }_{M(P^{\prime },Q^{\prime })}{\parallel M_w{T}_x\phi \parallel }_{M(P,Q)}dxdw\\ =& {\parallel h\parallel }_{M(P^{\prime },Q^{\prime })}{\parallel \phi \parallel }_{M(P,Q)}{\iint }_{{\mathbb{R}}^{2d}}|a(x,w)||V_{\phi }f(x,w)|dxdw\\ \le & {\parallel h\parallel }_{M(P^{\prime },Q^{\prime })}{\parallel \phi \parallel }_{M(P,Q)}{\parallel f\parallel }_{M(P,Q)}{\parallel a\parallel }_{P^{\prime }{Q}^{\prime }}.\end{array}$$

Similarly to (2.12), we get

$$\parallel A_{M_{\zeta }a}^{\phi ,\phi }\parallel \le {\parallel \phi \parallel }_{M(P,Q)}{\parallel a\parallel }_{P^{\prime }{Q}^{\prime }}.$$

Then the localization operator $A_{M_{\zeta }a}^{\phi ,\phi }$ from $M(P,Q)({\mathbb{R}}^d)$ into $M(P,Q)({\mathbb{R}}^d)$ is bounded. □

Corollary 2.1 It is known by Proposition 2 in [8]that $\mathcal{S}({\mathbb{R}}^d)\subset M(R,S)({\mathbb{R}}^d)$ for $1\le R,S<\mathrm{\infty }$. Then $\mathcal{S}({\mathbb{R}}^d)\subset {\bigcap }_{1\le R,S<\mathrm{\infty }}M(R,S)({\mathbb{R}}^d)$. So, Theorem 2.4, Propositions 2.1 and 2.2 are still true under the same hypotheses for them if $\phi \in \mathcal{S}({\mathbb{R}}^d)$.

Corollary 2.2 It is known [8]that if $P=p$ and $Q=q$, then Lorentz mixed-normed modulation space $M(P,Q)({\mathbb{R}}^d)$ is the Lorentz type modulation space $M(p,q)({\mathbb{R}}^d)$. Therefore our theorems hold for a Lorentz type modulation space rather than for a Lorentz mixed-normed modulation space.