# 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 $R d$ with Lebesgue measure dx. We denote by $S( R d )$ the space of complex-valued continuous functions on $R d$ rapidly decreasing at infinity. For any function $f: R d →C$, the translation and modulation operator are defined as $T x f(t)=f(t−x)$ and $M w f(t)= e 2 π i w t f(t)$ for $x,w∈ R d$, respectively. For $1≤p≤∞$, we write the Lebesgue spaces $( L p ( R d ), ∥ ⋅ ∥ p )$.

Let $〈x,t〉= ∑ i = 1 d x i t i$ be the usual scalar product on $R d$. The Fourier transform $f ˆ$ (or $Ff$) of $f∈ L 1 ( R d )$ is defined to be

$f ˆ (t)= ∫ R d f(x) e − 2 π i 〈 x , t 〉 dx.$

For a fixed nonzero $g∈S( R d )$ the short-time Fourier transform (STFT) of a function $f∈ S ′ ( R d )$ with respect to the window g is defined as

$V g f(x,w)=〈f, M w T x g〉= ∫ R d f(t) g ( t − x ) ¯ e − 2 π i t w dt,$

for $x,w∈ R d$. Then the localization operator $A a φ 1 , φ 2$ with symbol a and windows $φ 1$, $φ 2$ is defined to be

$A a φ 1 , φ 2 f(t)= ∫ R 2 d a(x,w) V φ 1 f(x,w) M w T x φ 2 dxdw.$

If $a∈ S ′ ( R d )$ and $φ 1 , φ 2 ∈S( R d )$, then the localization operator is a well-defined continuous operator from $S( R d )$ to $S ′ ( R d )$. Moreover, it is to be interpreted in a weak sense as

$〈 A a φ 1 , φ 2 f , g 〉 =〈a V φ 1 f, V φ 2 g〉=〈a, V φ 1 f ¯ V φ 2 g〉$

for $f,g∈S( R d )$, [1, 2].

Fix a nonzero window $g∈S( R d )$ and $1≤p,q≤∞$. Then the modulation space $M p , q ( R d )$ consists of all tempered distributions $f∈ S ′ ( R d )$ such that the short-time Fourier transform $V g f$ is in the mixed-norm space $L p , q ( R 2 d )$. The norm on $M p , q ( R d )$ is $∥ f ∥ M p , q = ∥ V g f ∥ L p , q$. If $p=q$, then we write $M p ( R d )$ instead of $M p , p ( R d )$. 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 $(X,μ)$. The measure μ is assumed to be nonnegative. We assume that the functions f are finite valued a.e. and some $y>0$, $μ( E y )<∞$, where $E y = E y [f]={x∈X∣|f(x)|>y}$. Then, for $y>0$,

$λ f (y)=μ( E y )=μ ( { x ∈ X ∣ | f ( x ) | > y } )$

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

$f ∗ (t)=inf { y > 0 ∣ λ f ( y ) ≤ t } =sup { y > 0 ∣ λ f ( y ) > t }$

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

$f ∗ ∗ (x)= 1 x ∫ 0 x f ∗ (t)dt.$

Note that $λ f$, $f ∗$, and $f ∗ ∗$ are nonincreasing and right continuous functions on $(0,∞)$. If $λ f (y)$ is continuous and strictly decreasing then $f ∗ (t)$ is the inverse function of $λ f (y)$. The most important property of $f ∗$ is that it has the same distribution function as f. It follows that

$( ∫ X | f ( x ) | p d μ ( x ) ) 1 p = ( ∫ 0 ∞ [ f ∗ ( t ) ] p d t ) 1 p .$
(1.1)

The Lorentz space denoted by $L(p,q)(X,μ)$ (shortly $L(p,q)$) is defined to be vector space of all (equivalence classes) of measurable functions f such that $∥ f ∥ p q ∗ <∞$, where

$∥ f ∥ p q ∗ = { ( q p ∫ 0 ∞ t q p − 1 [ f ∗ ( t ) ] q d t ) 1 q , 0 < p , q < ∞ , sup t > 0 t 1 p f ∗ ( t ) , 0 < p ≤ q = ∞ .$

By (1.1), it follows that $∥ f ∥ p p ∗ = ∥ f ∥ p$ and so $L(p,p)= L p$. Also, $L(p,q)(X,μ)$ is a normed space with the norm

$∥ f ∥ p q = { ( q p ∫ 0 ∞ t q p − 1 [ f ∗ ∗ ( t ) ] q d t ) 1 q , 0 < p , q < ∞ , sup t > 0 t 1 p f ∗ ∗ ( t ) , 0 < p ≤ q = ∞ .$

For any one of the cases $p=q=1$; $p=q=∞$ or $1 and $1≤q≤∞$, the Lorentz space $L(p,q)(X,μ)$ is a Banach space with respect to the norm $∥ ⋅ ∥ p q$. It is also well known that if $1, $1≤q≤∞$ we have

$∥ ⋅ ∥ p q ∗ ≤ ∥ ⋅ ∥ p q ≤ p p − 1 ∥ ⋅ ∥ p q ∗$

(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 $(X×Y,μ×ν)$, $1, and $1≤Q=( q 1 , q 2 )≤∞$. The Lorentz mixed norm space $L(P,Q)=L(P,Q)(X×Y)$ is defined by

$L(P,Q)=L( p 2 , q 2 ) [ L ( p 1 , q 1 ) ] = { f : ∥ f ∥ P Q = ∥ f ∥ L ( p 2 , q 2 ) ( L ( p 1 , q 1 ) ) = ∥ ∥ f ∥ p 1 q 1 ∥ p 2 q 2 < ∞ } .$

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 $∥ ⋅ ∥ P Q$ (see [6, 7]).

Fix a window function $g∈S( R d )∖{0}$, $1≤P=( p 1 , p 2 )<∞$, and $1≤Q=( q 1 , q 2 )≤∞$. We let $M(P,Q)( R d )$ denote the subspace of tempered distributions $S ′ ( R d )$ consisting of $f∈ S ′ ( R d )$ such that the Gabor transform $V g f$ of f is in the Lorentz mixed norm space $L(P,Q)( R 2 d )$. We endow it with the norm $∥ f ∥ M ( P , Q ) = ∥ V g f ∥ P Q$, where $∥ ⋅ ∥ P Q$ is the norm of the Lorentz mixed norm space. It is well known that $M(P,Q)( R d )$ 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 $M(P,Q)( R d )$ is the standard modulation space $M p , q ( R d )$, and if $P=p$ and $Q=q$, in this case $M(P,Q)( R d )=M(p,q)( R d )$ (see [8, 9]), where the space $M(p,q)( R d )$ is Lorentz type modulation space (see [10]). Furthermore, the space $M(p,q)( R d )$ was generalized to $M(p,q,w)( R d )$ 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 $1≤r,s≤∞$. Fix a compact $Q⊂ R d$ with nonempty interior. Then the Wiener amalgam space $W( L r , L s )( R d )$ with local component $L r ( R d )$ and global component $L s ( R d )$ is defined as the space of all measurable functions $f: R d →C$ such that $f χ K ∈ L r ( R d )$ for each compact subset $K⊂ R d$, for which the norm

$∥ f ∥ W ( L r , L s ) = ∥ F f ∥ s = ∥ ∥ f χ Q + x ∥ r ∥ s$

is finite, where $χ K$ is the characteristic function of K and

$F f (x)= ∥ f χ Q + x ∥ r ∈ L s ( R d ) .$

It is known that if $r 1 ≥ r 2$ and $s 1 ≤ s 2$ then $W( L r 1 , L s 1 )( R d )⊂W( L r 2 , L s 2 )( R d )$. If $r=s$ then $W( L r , L r )( R d )= L r ( R d )$ (see [1315]).

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 $1 P + 1 P ′ =1$, $1 Q 1 + 1 Q 2 ≥1$, $f∈L(P, Q 1 )( R 2 d )$, $h∈L( P ′ , Q 2 )( R 2 d )$. Then $f∗h∈ L ∞ ( R 2 d )$ and

$L(P, Q 1 ) ( R 2 d ) ∗L ( P ′ , Q 2 ) ( R 2 d ) ↪ L ∞ ( R 2 d )$
(2.1)

with the norm inequality

$∥ f ∗ h ∥ ∞ ≤ ∥ f ∥ P Q 1 ∥ h ∥ P ′ Q 2 ,$
(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 $L(p, q 1 )∗L( p ′ , q 2 )↪ L ∞$ convolution relations between Lorentz spaces and

$∥ f ∗ h ∥ ∞ ≤ ∥ f ∥ p q 1 ∥ h ∥ p ′ q 2 ,$

where $1 p + 1 p ′ =1$, $1 q 1 + 1 q 2 ≥1$, 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 $g∈D( R 2 d )$ be a test function such that $∑ x ∈ Z 2 d T x g≡1$. Let $X( R 2 d )$ be a translation invariant Banach space of functions with the property that $D⋅X⊂X$. 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

$∥ f ∥ W ( X , L ( P , Q ) ) = ∥ ∥ f ⋅ T ( z 1 , z 2 ) g ¯ ∥ X ∥ P Q$

is finite, where $1≤P<∞$, $1≤Q≤∞$. Moreover, different choices of $g∈D$ yield equivalent norms and give the same space.

The boundedness of $A M ζ a φ 1 , φ 2$ for $a∈ M ∞$ 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

1. (i)

Let $1, $1≤Q<∞$. If $f∈M(P,Q)( R d )$ and $g∈ M 1 ( R d )$, then $V g f∈W(F L 1 ,L(P,Q))( R 2 d )$ with

$∥ V g f ∥ W ( F L 1 , L ( P , Q ) ) ≤ ∥ f ∥ M ( P , Q ) ∥ g ∥ M 1 .$
2. (ii)

Let $1 P + 1 P ′ =1$, $1 Q 1 + 1 Q 2 ≥1$. If $f∈M(P, Q 1 )( R d )$ and $g∈M( P ′ , Q 2 )( R d )$, then $V g f∈W(F L 1 , L ∞ )( R 2 d )$ with

$∥ V g f ∥ W ( F L 1 , L ∞ ) ≤ ∥ f ∥ M ( P , Q 1 ) ∥ g ∥ M ( P ′ , Q 2 ) .$

Proof (i) Let $φ∈S( R d )∖{0}$ and set $Φ= V φ φ∈S( R 2 d )$. By using the equality $V g f(x,w)= ( f ⋅ T x g ¯ ) ∧ (w)$, we write

$∥ V g f ⋅ T ( z 1 , z 2 ) Φ ¯ ∥ F L 1 = ∫ R 2 d | ( V g f ⋅ T ( z 1 , z 2 ) Φ ¯ ) ∧ ( t ) | d t = ∫ R 2 d | V Φ V g f ( z 1 , z 2 , t 1 , t 2 ) | d t 1 d t 2 = ∫ R 2 d | V φ g ( − z 1 − t 2 , t 1 ) V φ f ( − t 2 , z 2 + t 1 ) | d t 1 d t 2 = ∫ R 2 d | V φ f ( u 1 , u 2 ) | | V φ g ( u 1 − z 1 , u 2 − z 2 ) | d u 1 d u 2 = | V φ f | ∗ | V φ g | ∼ ( z 1 , z 2 ) ,$
(2.3)

for $f,g∈S( R d )$, where $( V φ g ) ∼ (z)=( V φ g ¯ )(−z)$, $z∈ R 2 d$. Since $f,g∈S( R d )$, then $f∈M(P,Q)( R d )$ and $g∈ M 1 ( R d )$ by Proposition 2 in [8]. So $V φ f∈L(P,Q)( R 2 d )$ and $V φ g∈ L 1 ( R 2 d )$. Then, by Proposition 4 in [8], we obtain

$∥ V g f ∥ W ( F L 1 , L ( P , Q ) ) = ∥ ∥ V g f ⋅ T ( z 1 , z 2 ) Φ ¯ ∥ F L 1 ∥ P Q = ∥ | V φ f | ∗ | V φ g | ∼ ∥ P Q ≤ ∥ V φ f ∥ P Q ∥ V φ g ∥ 1 = ∥ f ∥ M ( P , Q ) ∥ g ∥ M 1 .$
(2.4)

This completes the proof.

1. (ii)

Using Lemma 2.1 and (2.3), we have

$∥ V g f ∥ W ( F L 1 , L ∞ ) = ∥ | V φ f | ∗ | V φ g | ∼ ∥ ∞ ≤ ∥ V φ f ∥ P Q 1 ∥ V φ g ∥ P ′ Q 2 = ∥ f ∥ M ( P , Q 1 ) ∥ g ∥ M ( P ′ , Q 2 ) .$

□

Theorem 2.2 Let $1, $1≤Q<∞$. If $a∈ M ∞ ( R 2 d )$, $φ 1 , φ 2 ∈ M 1 ( R d )$, then $A M ζ a φ 1 , φ 2$ is bounded on $M(P,Q)( R d )$ for every $ζ∈ R 2 d$ with

$∥ A M ζ a φ 1 , φ 2 ∥ B ( M ( P , Q ) ) ≤ ∥ a ∥ M ∞ ∥ φ 1 ∥ M 1 ∥ φ 2 ∥ M 1 .$

Proof Let $f∈M(P,Q)( R d )$ and $g∈M( P ′ , Q ′ )( R d )$, where $1 P + 1 P ′ =1$, $1 Q + 1 Q ′ =1$. Then we write $V φ 1 f ¯ ∈W(F L 1 ,L(P,Q))( R 2 d )$ and $V φ 2 g∈W(F L 1 ,L( P ′ , Q ′ ))( R 2 d )$ by above theorem. Moreover, since $M(1,1)( R d )= M 1 ( R d )$, we have $W(F L 1 , L 1 )= M 1 =M(1,1)$ by [16]. Hence using the Hölder inequalities for Wiener amalgam spaces [13] and (2.4) we obtain

$∥ V φ 1 f ¯ ⋅ V φ 2 g ∥ M 1 = ∥ V φ 1 f ¯ ⋅ V φ 2 g ∥ W ( F L 1 , L 1 ) ≤ ∥ V φ 1 f ∥ W ( F L 1 , L ( P , Q ) ) ∥ V φ 2 g ∥ W ( F L 1 , L ( P ′ , Q ′ ) ) ⪯ ∥ φ 1 ∥ M 1 ∥ φ 2 ∥ M 1 ∥ f ∥ M ( P , Q ) ∥ g ∥ M ( P ′ , Q ′ ) .$
(2.5)

Thus by using (2.5) we have

$| 〈 A M ζ a φ 1 , φ 2 f , g 〉 | = | 〈 M ζ a , V φ 1 f ¯ ⋅ V φ 2 g 〉 | ≤ ∥ M ζ a ∥ M ( ∞ , ∞ ) ∥ V φ 1 f ¯ ⋅ V φ 2 g ∥ M ( 1 , 1 ) ≤ ∥ a ∥ M ∞ ∥ φ 1 ∥ M 1 ∥ φ 2 ∥ M 1 ∥ f ∥ M ( P , Q ) ∥ g ∥ M ( P ′ , Q ′ ) .$

Hence we get

$∥ A M ζ a φ 1 , φ 2 ∥ B ( M ( P , Q ) ) ≤ ∥ a ∥ M ∞ ∥ φ 1 ∥ M 1 ∥ φ 2 ∥ M 1 .$

□

Theorem 2.3 Let $φ∈S( R d )∖{0}$ be a window function. If $1, $t ′ ∈(1,∞)$, $s≤ t ′ ≤r$ and $a∈W( L r , L s )$, then

$A M ζ a φ , φ :M(tP,tQ) ( R d ) →M ( ( t P ′ ) ′ , ( t Q ′ ) ′ ) ( R d )$

is bounded for every $ζ∈ R 2 d$, where $1 P + 1 P ′ =1$, $1 Q + 1 Q ′ =1$, and $1 t + 1 t ′ =1$, and the operator norm satisfies the estimate

$∥ A M ζ a φ , φ ∥ ≤ ∥ a ∥ W ( L r , L s ) .$

Proof Let $t<∞$, $f∈M(tP,tQ)( R d )$, and $h∈M(t P ′ ,t Q ′ )( R d )$. Then we have $V φ f∈L(tP,tQ)( R 2 d )$ and $V φ h∈L(t P ′ ,t Q ′ )( R 2 d )$. Since $V φ f∈L(tP,tQ)( R 2 d )$, then $∥ V φ f ∥ ( t P ) ( t Q ) ∗ <∞$. By using the equality (3.6) in [12], we get

$∥ V φ f ∥ ( t P ) ( t Q ) ∗ = ∥ ∥ V φ f ∥ ( t p 1 ) ( t q 1 ) ∗ ∥ ( t p 2 ) ( t q 2 ) ∗ = ∥ ( ∥ | V φ f | t ∥ p 1 q 1 ∗ ) 1 t ∥ ( t p 2 ) ( t q 2 ) ∗ = ( ∥ | ( ∥ | V φ f | t ∥ p 1 q 1 ∗ ) 1 t | t ∥ p 2 q 2 ∗ ) 1 t = ( ∥ ∥ | V φ f | t ∥ p 1 q 1 ∗ ∥ p 2 q 2 ∗ ) 1 t = ( ∥ | V φ f | t ∥ P Q ∗ ) 1 t .$
(2.6)

Hence we have $| V φ f | t ∈L(P,Q)( R 2 d )$. Similarly, $| V φ h | t ∈L( P ′ , Q ′ )( R 2 d )$. By the Hölder inequality for Lorentz spaces with mixed norm and (2.6) we have

$∥ V φ f ⋅ V φ h ∥ t t = ∥ | V φ f | t | V φ h | t ∥ 1 ≤ ∥ | V φ f | t ∥ P Q ∥ | V φ h | t ∥ P ′ Q ′ = ∥ V φ f ∥ ( t P ) ( t Q ) t ∥ V φ h ∥ ( t P ′ ) ( t Q ′ ) t .$
(2.7)

Since $a∈W( L r , L s )$, then $M ζ a∈W( L r , L s )$ for every $ζ∈ R 2 d$. Also since $W( L r , L s )⊂W( L t ′ , L t ′ )= L t ′ ( R 2 d )$, then we have

$∥ a ∥ t ′ = ∥ M ζ a ∥ t ′ ≤ ∥ M ζ a ∥ W ( L r , L s ) = ∥ a ∥ W ( L r , L s ) .$
(2.8)

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

$| 〈 A M ζ a φ , φ f , h 〉 | = | 〈 M ζ a V φ f , V φ h 〉 | ≤ ∬ R 2 d | M ζ a ( x , w ) | | ( V φ f ⋅ V φ h ) ( x , w ) | d x d w ≤ ∥ M ζ a ∥ t ′ ∥ V φ f ⋅ V φ h ∥ t ≤ ∥ a ∥ t ′ ∥ V φ f ∥ ( t P ) ( t Q ) ∥ V φ h ∥ ( t P ′ ) ( t Q ′ ) ≤ ∥ a ∥ W ( L r , L s ) ∥ f ∥ M ( t P , t Q ) ∥ h ∥ M ( t P ′ , t Q ′ ) .$
(2.9)

If $( t p ′ ) ′ , ( t q ′ ) ′ ≠∞$, then $( M ( ( t P ′ ) ′ , ( t Q ′ ) ′ ) ( R d ) ) ∗ =M(t P ′ ,t Q ′ )( R d )$ by Theorem 8 in [8]. Thus we have from (2.9) that

$∥ A M ζ a φ , φ f ∥ M ( ( t P ′ ) ′ , ( t Q ′ ) ′ ) = sup 0 ≠ h ∈ M ( t P ′ , t Q ′ ) | 〈 A M ζ a φ , φ f , h 〉 | ∥ h ∥ M ( t P ′ , t Q ′ ) ≤ ∥ a ∥ W ( L r , L s ) ∥ f ∥ M ( t P , t Q ) .$

Hence $A M ζ a φ , φ$ is bounded. Also we have

$∥ A M ζ a φ , φ ∥ = sup 0 ≠ f ∈ M ( t P , t Q ) ∥ A M ζ a φ , φ f ∥ M ( ( t P ′ ) ′ , ( t Q ′ ) ′ ) ∥ f ∥ M ( t P , t Q ) ≤ ∥ a ∥ W ( L r , L s ) .$

□

Theorem 2.4 Let $φ∈ ⋂ 1 ≤ R , S < ∞ M(R,S)( R d )$, where $R=( r 1 , r 2 )$, $S=( s 1 , s 2 )$. If $1≤s≤r≤∞$ and $a∈W( L r , L s )$ then

$A M ζ a φ , φ :M(P,Q) ( R d ) →M(P,Q) ( R d )$

is bounded for every $ζ∈ R 2 d$, with

$∥ A M ζ a φ , φ ∥ ≤C ∥ a ∥ W ( L r , L s )$

for some $C>0$.

Proof Since $a∈W( L r , L s )$, then $M ζ a∈W( L r , L s )$ for every $ζ∈ R 2 d$. Also since $s≤r$, there exists $1≤ t 0 ≤∞$ such that $s≤ t 0 ≤r$. Then $W( L r , L s )( R 2 d )⊂ L t 0 ( R 2 d )$ and

$∥ M ζ a ∥ t 0 = ∥ a ∥ t 0 ≤ ∥ a ∥ W ( L r , L s ) = ∥ M ζ a ∥ W ( L r , L s )$
(2.10)

for all $a∈W( L r , L s )( R 2 d )$. Let $B(M(P,Q)( R d ),M(P,Q)( R d ))$ be the space of the bounded linear operators from $M(P,Q)( R d )$ into $M(P,Q)( R d )$. Also let T be an operator from $L 1 ( R 2 d )$ into $B(M(P,Q)( R d ),M(P,Q)( R d ))$ by $T(a)= A M ζ a φ , φ$. Take any $f∈M(P,Q)( R d )$ and $h∈M( P ′ , Q ′ )( R d )$. Assume that $a∈W( L 1 , L 1 )( R 2 d )= L 1 ( R 2 d )$. By the Hölder inequality we get

$| 〈 T ( a ) f , h 〉 | = | 〈 A M ζ a φ , φ f , h 〉 | = | 〈 M ζ a V φ f , V φ h 〉 | ≤ ∬ R 2 d | M ζ a ( x , w ) | | V φ f ( x , w ) | | V φ h ( x , w ) | d x d w = ∬ R 2 d | a ( x , w ) | | 〈 f , M w T x φ 〉 | | 〈 h , M w T x φ 〉 | d x d w ≤ ∬ R 2 d | a ( x , w ) | ∥ f ∥ M ( P , Q ) ∥ M w T x φ ∥ M ( P ′ , Q ′ ) ∥ h ∥ M ( P ′ , Q ′ ) × ∥ M w T x φ ∥ M ( P , Q ) d x d w = ∥ f ∥ M ( P , Q ) ∥ φ ∥ M ( P ′ , Q ′ ) ∥ h ∥ M ( P ′ , Q ′ ) ∥ φ ∥ M ( P , Q ) ∥ a ∥ 1 .$
(2.11)

Hence by (2.11)

$∥ T ( a ) f ∥ M ( P , Q ) = ∥ A M ζ a φ , φ f ∥ M ( P , Q ) = sup 0 ≠ h ∈ M ( P ′ , Q ′ ) | 〈 A M ζ a φ , φ f , h 〉 | ∥ h ∥ M ( P ′ , Q ′ ) ≤ ∥ φ ∥ M ( P ′ , Q ′ ) ∥ φ ∥ M ( P , Q ) ∥ f ∥ M ( P , Q ) ∥ a ∥ 1 .$

Then

$∥ T ( a ) ∥ = ∥ A M ζ a φ , φ ∥ = sup 0 ≠ f ∈ M ( P , Q ) ∥ A M ζ a φ , φ f ∥ M ( P , Q ) ∥ f ∥ M ( P , Q ) ≤ ∥ φ ∥ M ( P ′ , Q ′ ) ∥ φ ∥ M ( P , Q ) ∥ a ∥ 1 .$
(2.12)

Thus the operator

$T: L 1 ( R 2 d ) →B ( M ( P , Q ) ( R d ) , M ( P , Q ) ( R d ) )$
(2.13)

is bounded. Now let $a∈W( L ∞ , L ∞ )( R 2 d )= L ∞ ( R 2 d )$. Take any $f∈M(P,Q)( R d )$ and $h∈M( P ′ , Q ′ )( R d )$. Then $V φ f∈L(P,Q)( R 2 d )$, $V φ h∈L( P ′ , Q ′ )( R 2 d )$. Applying the Hölder inequality

$| 〈 T ( a ) f , h 〉 | = | 〈 A M ζ a φ , φ f , h 〉 | = | 〈 M ζ a V φ f , V φ h 〉 | ≤ ∬ R 2 d | M ζ a ( x , w ) | | V φ f ( x , w ) | | V φ h ( x , w ) | d x d w ≤ ∥ a ∥ ∞ ∬ R 2 d | V φ f ( x , w ) | | V φ h ( x , w ) | d x d w ≤ ∥ a ∥ ∞ ∥ V φ f ∥ P Q ∥ V φ h ∥ P ′ Q ′ .$
(2.14)

By using (2.14) we write

$∥ T ( a ) f ∥ M ( P , Q ) = ∥ A M ζ a φ , φ f ∥ M ( P , Q ) = sup 0 ≠ h ∈ M ( P ′ , Q ′ ) | 〈 A M ζ a φ , φ f , h 〉 | ∥ h ∥ M ( P ′ , Q ′ ) ≤ ∥ a ∥ ∞ ∥ f ∥ M ( P , Q ) .$
(2.15)

Hence by (2.15)

$∥ T ( a ) ∥ = ∥ A M ζ a φ , φ ∥ = sup 0 ≠ f ∈ M ( P , Q ) ∥ A M ζ a φ , φ f ∥ M ( P , Q ) ∥ f ∥ M ( P , Q ) ≤ ∥ a ∥ ∞ .$

That means the operator

$T: L ∞ ( R 2 d ) →B ( M ( P , Q ) ( R d ) , M ( P , Q ) ( R d ) )$
(2.16)

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

$T: L t ( R 2 d ) →B ( M ( P , Q ) ( R d ) , M ( P , Q ) ( R d ) )$

is bounded by interpolation theorem for $1≤t≤∞$. That means the localization operator

$A M ζ a φ , φ :M(P,Q) ( R d ) →M(P,Q) ( R d )$

is bounded for $1≤t≤∞$. Hence there exists $C>0$ such that

$∥ T ( a ) ∥ = ∥ A M ζ a φ , φ ∥ ≤C ∥ a ∥ t .$
(2.17)

This implies that it is also true for $1≤ t 0 ≤∞$. From (2.10) and (2.17) we write

$∥ T ( a ) ∥ = ∥ A M ζ a φ , φ ∥ ≤C ∥ a ∥ t 0 ≤C ∥ a ∥ W ( L r , L s ) .$

□

Proposition 2.1 Let $φ∈ ⋂ 1 ≤ R , S < ∞ M(R,S)( R d )$, where $R=( r 1 , r 2 )$, $S=( s 1 , s 2 )$. If $0 and $a∈W( L 1 , L s )( R 2 d )$ then

$A M ζ a φ , φ :M(P,Q) ( R d ) →M(P,Q) ( R d )$

is bounded.

Proof Let $0 and let $a∈W( L 1 , L s )( R 2 d )$. Then $M ζ a∈W( L 1 , L s )$ for every $ζ∈ R 2 d$. Since $W( L 1 , L s )( R 2 d )⊂ L 1 ( R 2 d )$, there exists a number $C>0$ such that $∥ M ζ a ∥ 1 ≤C ∥ M ζ a ∥ W ( L 1 , L s )$. Hence by (2.12),

$∥ A M ζ a φ , φ ∥ ≤ ∥ φ ∥ M ( P ′ , Q ′ ) ∥ φ ∥ M ( P , Q ) ∥ M ζ a ∥ 1 ≤ C ∥ φ ∥ M ( P ′ , Q ′ ) ∥ φ ∥ M ( P , Q ) ∥ M ζ a ∥ W ( L 1 , L s ) = C ∥ φ ∥ M ( P ′ , Q ′ ) ∥ φ ∥ M ( P , Q ) ∥ a ∥ W ( L 1 , L s ) .$

Then the localization operator from $M(P,Q)( R d )$ into $M(P,Q)( R d )$ is bounded for $0. □

Proposition 2.2 Let $φ∈ ⋂ 1 ≤ R , S < ∞ M(R,S)( R d )$, where $R=( r 1 , r 2 )$, $S=( s 1 , s 2 )$. If $1≤P,Q<∞$ and $a∈L( P ′ , Q ′ )( R 2 d )$ then the localization operator

$A M ζ a φ , φ :M(P,Q) ( R d ) →M(P,Q) ( R d )$

is bounded, where $1 P + 1 P ′ =1$, $1 Q + 1 Q ′ =1$.

Proof Let $a∈L( P ′ , Q ′ )( R 2 d )$. Then $M ζ a∈L( P ′ , Q ′ )( R 2 d )$ for every $ζ∈ R 2 d$ with $∥ M ζ a ∥ P ′ Q ′ = ∥ a ∥ P ′ Q ′$. Take any $f∈M(P,Q)( R d )$ and $h∈M( P ′ , Q ′ )( R d )$. Applying the Hölder inequality we have by (2.11)

$| 〈 A M ζ a φ , φ f , h 〉 | ≤ ∬ R 2 d | M ζ a ( x , w ) | | V φ f ( x , w ) | | 〈 h , M w T x φ 〉 | d x d w ≤ ∬ R 2 d | a ( x , w ) | | V φ f ( x , w ) | ∥ h ∥ M ( P ′ , Q ′ ) ∥ M w T x φ ∥ M ( P , Q ) d x d w = ∥ h ∥ M ( P ′ , Q ′ ) ∥ φ ∥ M ( P , Q ) ∬ R 2 d | a ( x , w ) | | V φ f ( x , w ) | d x d w ≤ ∥ h ∥ M ( P ′ , Q ′ ) ∥ φ ∥ M ( P , Q ) ∥ f ∥ M ( P , Q ) ∥ a ∥ P ′ Q ′ .$

Similarly to (2.12), we get

$∥ A M ζ a φ , φ ∥ ≤ ∥ φ ∥ M ( P , Q ) ∥ a ∥ P ′ Q ′ .$

Then the localization operator $A M ζ a φ , φ$ from $M(P,Q)( R d )$ into $M(P,Q)( R d )$ is bounded. □

Corollary 2.1 It is known by Proposition 2 in [8]that $S( R d )⊂M(R,S)( R d )$ for $1≤R,S<∞$. Then $S( R d )⊂ ⋂ 1 ≤ R , S < ∞ M(R,S)( R d )$. So, Theorem 2.4, Propositions 2.1 and 2.2 are still true under the same hypotheses for them if $φ∈S( R d )$.

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

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Sandıkçı, A. Boundedness of localization operators on Lorentz mixed-normed modulation spaces. J Inequal Appl 2014, 430 (2014). https://doi.org/10.1186/1029-242X-2014-430