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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(tx) and M w f(t)= e 2 π i w t f(t) for x,w R d , respectively. For 1p, 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 gS( 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,gS( R d ), [1, 2].

Fix a nonzero window gS( R d ) and 1p,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]={xX|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<p< and 1q, 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<p<, 1q 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<P=( p 1 , p 2 )<, and 1Q=( 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 gS( R d ){0}, 1P=( p 1 , p 2 )<, and 1Q=( 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 1r,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, fL(P, Q 1 )( R 2 d ), hL( P , Q 2 )( R 2 d ). Then fh 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 gD( R 2 d ) be a test function such that x Z 2 d T x g1. Let X( R 2 d ) be a translation invariant Banach space of functions with the property that DXX. 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 1P<, 1Q. Moreover, different choices of gD 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<P<, 1Q<. If fM(P,Q)( R d ) and g M 1 ( R d ), then V g fW(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 fM(P, Q 1 )( R d ) and gM( P , Q 2 )( R d ), then V g fW(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 φS( R d ){0} and set Φ= V φ φS( 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,gS( R d ), where ( V φ g ) (z)=( V φ g ¯ )(z), z R 2 d . Since f,gS( R d ), then fM(P,Q)( R d ) and g M 1 ( R d ) by Proposition 2 in [8]. So V φ fL(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<P<, 1Q<. 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 fM(P,Q)( R d ) and gM( 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 gW(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 φS( R d ){0} be a window function. If 1<P,Q<, t (1,), s t r and aW( 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<, fM(tP,tQ)( R d ), and hM(t P ,t Q )( R d ). Then we have V φ fL(tP,tQ)( R 2 d ) and V φ hL(t P ,t Q )( R 2 d ). Since V φ fL(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 aW( L r , L s ), then M ζ aW( 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 1sr and aW( 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 aW( L r , L s ), then M ζ aW( L r , L s ) for every ζ R 2 d . Also since sr, 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 aW( 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 fM(P,Q)( R d ) and hM( P , Q )( R d ). Assume that aW( 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 aW( L , L )( R 2 d )= L ( R 2 d ). Take any fM(P,Q)( R d ) and hM( P , Q )( R d ). Then V φ fL(P,Q)( R 2 d ), V φ hL( 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 1t. That means the localization operator

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

is bounded for 1t. 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<s1 and aW( 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<s1 and let aW( L 1 , L s )( R 2 d ). Then M ζ aW( 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<s1. □

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 1P,Q< and aL( 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 aL( P , Q )( R 2 d ). Then M ζ aL( P , Q )( R 2 d ) for every ζ R 2 d with M ζ a P Q = a P Q . Take any fM(P,Q)( R d ) and hM( 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 1R,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 φS( 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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Correspondence to Ayşe Sandıkçı.

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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

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Keywords

  • localization operator
  • Lorentz spaces
  • Lorentz mixed normed spaces
  • Lorentz mixed-normed modulation spaces
  • Wiener amalgam spaces