Boundedness of localization operators on Lorentz mixed normed modulation spaces

R E S E A R C H

Open Access

Boundedness of localization operators on

Lorentz mixed-normed modulation spaces

Ay¸se Sandıkçı

Dedicated to Professor Ravi P Agarwal

*_{Correspondence:}

[email protected] Department of Mathematics, Faculty of Arts and Sciences, Ondokuz Mayıs University, Samsun, Turkey

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.

Keywords: localization operator; Lorentz spaces; Lorentz mixed normed spaces; Lorentz mixed-normed modulation spaces; Wiener amalgam spaces

1 Introduction

In this paper we will work onRd _{with Lebesgue measuredx. We denote byS(R}d_{) the}

space of complex-valued continuous functions onRdrapidly decreasing at infinity. For

any functionf :Rd_{→C, the translation and modulation operator are defined asT} xf(t) =

f(t–x) andMwf(t) =eπiwtf(t) forx,w∈Rd, respectively. For ≤p≤ ∞, we write the

Lebesgue spaces (Lp(Rd), · p).

Letx,t=d_i=xitibe the usual scalar product onRd. The Fourier transformfˆ(orFf)

off ∈L_(Rd_{) is defined to be}

ˆ f(t) =

Rd

f(x)e–πix,tdx.

For a fixed nonzerog∈S(Rd) the short-time Fourier transform (STFT) of a function f∈S(Rd_{) with respect to the windowgis defined as}

Vgf(x,w) =f,MwTxg=

Rdf(t)g(t–x)e

–πitw_dt,

forx,w∈Rd_{. Then the localization operatorA}ϕ,ϕ

a with symbolaand windowsϕ,ϕis

defined to be

Aϕ,ϕ

a f(t) =

Rd

a(x,w)Vϕf(x,w)MwTxϕdx dw.

Ifa∈S(Rd_{) andϕ}

,ϕ∈S(Rd), then the localization operator is a well-defined

continu-ous operator fromS(Rd_{) toS(R}d_{). Moreover, it is to be interpreted in a weak sense as}

Aϕ,ϕ

a f,g

=aVϕf,Vϕg=a,Vϕf Vϕg

forf,g∈S(Rd), [, ].

Fix a nonzero windowg∈S(Rd_{) and ≤p,q≤ ∞. Then the modulation spaceM}p,q_(Rd₎

consists of all tempered distributionsf ∈S(Rd_{) such that the short-time Fourier}

trans-form Vgf is in the mixed-norm space Lp,q(Rd). The norm on Mp,q(Rd) is fMp,q =

VgfLp,q. Ifp=q, then we writeMp(Rd) instead ofMp,p(Rd). Modulation spaces are

Ba-nach spaces whose definitions are independent of the choice of the windowg(see [, ]).

L(p,q) spaces are function spaces that are closely related toLp _{spaces. We consider}

complex-valued measurable functionsf defined on a measure space (X,μ). The measure

μis assumed to be nonnegative. We assume that the functionsf are finite valued a.e. and somey> ,μ(Ey) <∞, whereEy=Ey[f] ={x∈X| |f(x)|>y}. Then, fory> ,

λf(y) =μ(Ey) =μ

x∈X|f(x)>y

is the distribution function off. The rearrangement off is given by

f∗(t) =infy> |λf(y)≤t =sup

y> |λf(y) >t

fort> . The average function off is also defined by

f∗∗(x) =  x

x



f∗(t)dt.

Note thatλf,f∗, andf∗∗are nonincreasing and right continuous functions on (,∞). If λf(y) is continuous and strictly decreasing thenf∗(t) is the inverse function ofλf(y). The

most important property off∗is that it has the same distribution function asf. It follows that

X

f(x)pdμ(x)



p =

∞



f∗(t)pdt



p

. (.)

The Lorentz space denoted byL(p,q)(X,μ) (shortlyL(p,q)) is defined to be vector space of all (equivalence classes) of measurable functionsf such thatf∗_pq<∞, where

f∗_pq=

(q_p∞tqp–_[f∗_(t)]q_dt)q_{,  <p,q<}∞_, sup_t>tp_f∗_{(t),  <p}≤_q=∞_.

By (.), it follows thatf∗_pp=f_p and soL(p,p) =Lp_{. Also,L(p,q)(X,μ) is a normed}

space with the norm

fpq=

For any one of the casesp=q= ;p=q=∞or  <p<∞and ≤q≤ ∞, the Lorentz spaceL(p,q)(X,μ) is a Banach space with respect to the norm · pq. It is also well known

that if  <p<∞, ≤q≤ ∞we have

·∗

pq≤ ·pq≤

p

p– ·

∗ pq

(see [, ]).

LetXandY be two measure spaces withσ-finite measuresμandν, respectively, and

letf be a complex-valued measurable function on (X×Y,μ×ν),  <P= (p,p) <∞, and

≤Q= (q,q)≤ ∞. The Lorentz mixed norm spaceL(P,Q) =L(P,Q)(X×Y) is defined

by

L(P,Q) =L(p,q)

L(p,q)

=f :fPQ=fL(p,q)(L(p,q))=fpqpq<∞ . Thus,L(P,Q) occurs by taking anL(p,q)-norm with respect to the first variable and an

L(p,q)-norm with respect to the second variable. TheL(P,Q) space is a Banach space

under the norm · PQ(see [, ]).

Fix a window functiong∈S(Rd_{)\{}, ≤P= (p}

,p) <∞, and ≤Q= (q,q)≤ ∞.

We letM(P,Q)(Rd_{) denote the subspace of tempered distributionsS(R}d_{) consisting of}

f ∈S(Rd_{) such that the Gabor transformV}

gf off is in the Lorentz mixed norm space

L(P,Q)(Rd). We endow it with the normfM(P,Q)=VgfPQ, where · PQis the norm

of the Lorentz mixed norm space. It is well known thatM(P,Q)(Rd) is a Banach space

and different windows yield equivalent norms. If p=q =pandp=q=q, then the

spaceM(P,Q)(Rd_{) is the standard modulation spaceM}p,q_(Rd_{), and ifP=pandQ=q, in}

this caseM(P,Q)(Rd_{) =M(p,q)(R}d_{) (see [, ]), where the spaceM(p,q)(R}d_{) is Lorentz}

type modulation space (see []). Furthermore, the spaceM(p,q)(Rd_{) was generalized to}

M(p,q,w)(Rd) by taking weighted Lorentz space rather than Lorentz space (see [, ]). In this paper, we will denote the Lorentz space byL(p,q), the Lorentz mixed norm space

byL(P,Q), the standard modulation space byMp,q_{, the Lorentz type modulation space by}

M(p,q), and the Lorentz mixed-normed modulation space byM(P,Q).

Let ≤r,s≤ ∞. Fix a compactQ⊂Rd_{with nonempty interior. Then the Wiener}

amal-gam space W(Lr_,Ls_)(Rd_{) with local componentL}r_(Rd_{) and global componentL}s_(Rd_{) is}

defined as the space of all measurable functionsf :Rd→Csuch thatfχK ∈Lr(Rd) for

each compact subsetK⊂Rd_{, for which the norm}

fW(Lr_,Ls₎=F_fs=fχ_Q+xr

s

is finite, whereχKis the characteristic function ofKand

Ff(x) =fχQ+xr∈Ls

Rd_.

It is known that ifr≥rands≤sthenW(Lr,Ls)(Rd)⊂W(Lr,Ls)(Rd). Ifr=sthen

W(Lr_,Lr_)(Rd_{) =L}r_(Rd_{) (see [–]).}

norms. Our results extend some results in [, ] to the Lorentz mixed-normed modula-tion 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 . Let 

P+



P = ,



Q+



Q ≥,f ∈L(P,Q)(R

d_),h∈L(P,Q

)(Rd).Then f∗h∈

L∞(Rd_)and

L(P,Q)

Rd_∗LP,Q



Rd_→L∞_Rd _(.)

with the norm inequality

f∗h∞≤ fPQhPQ, (.)

where P= (p,p),Q= (Q,Q),Q= (Q,Q).

Proof It is well known that there areL(p,q)∗L(p,q)→L∞convolution relations

be-tween Lorentz spaces and

f∗h∞≤ fpqhpq,

where 

p +



p = ,



q +



q ≥, by Theorem . in []. Then (.) and (.) can easily be

verified by using iteration and the one variable proofs given in [].

Letg∈D(Rd_{) be a test function such that}

x∈ZdTxg≡. LetX(Rd) be a translation

invariant Banach space of functions with the property thatD·X⊂X. In the spirit of [,

], the Wiener amalgam spaceW(X,L(P,Q)) with local componentXand global

compo-nentL(P,Q) is defined as the space of all functions or distributions for which the norm

fW(X,L(P,Q))=f·T(z,z)gXPQ

is finite, where ≤P<∞, ≤Q≤ ∞. Moreover, different choices ofg∈Dyield equiva-lent norms and give the same space.

The boundedness ofAϕ,ϕ

Mζa fora∈M

∞_{is established by our next theorem. The proof is}

similar to Lemma . in [] but let us provide the details anyway, for completeness’ sake.

Theorem .

(i) Let <P<∞,≤Q<∞.Iff∈M(P,Q)(Rd_)andg∈M_(Rd_),then

Vgf∈W(FL,L(P,Q))(Rd)with

VgfW(FL_,L(P,Q))≤ f_M(P,Q)g_M.

(ii) Let 

P+



P = ,



Q +



Q ≥.Iff∈M(P,Q)(R

d_{)andg∈M(P,Q}

)(Rd),then

Vgf∈W(FL,L∞)(Rd)with

Proof (i) Letϕ∈S(Rd_{)\{}and set=Vϕϕ∈S(R}d_{). By using the equalityV}

gf(x,w) =

(f ·Txg)∧(w), we write

Vgf ·T(z,z)FL=

Rd

(Vgf ·T(z,z))

∧_(t)dt

=

Rd

VVgf(z,z,t,t)dtdt

=

Rd

Vϕg(–z–t,t)Vϕf(–t,z+t)dtdt

=

Rd

Vϕf(u,u)Vϕg(u–z,u–z)dudu

=|Vϕf| ∗ |Vϕg|∼(z,z), (.)

for f,g ∈ S(Rd), where (Vϕg)∼(z) = (Vϕg)(–z), z ∈Rd. Since f,g ∈S(Rd), then f ∈

M(P,Q)(Rd) andg∈M(Rd) by Proposition  in []. SoVϕf ∈L(P,Q)(Rd) andVϕg∈

L(Rd). Then, by Proposition  in [], we obtain

VgfW(FL_,L(P,Q))=Vgf ·T(z,z)FL_PQ =|Vϕf| ∗ |Vϕg|∼_PQ

≤ VϕfPQVϕg

=fM(P,Q)gM. (.)

This completes the proof.

(ii) Using Lemma . and (.), we have

VgfW(FL_,L∞₎=|Vϕf| ∗ |Vϕg|∼_∞≤ VϕfPQVϕgPQ=fM(P,Q)gM(P,Q).

Theorem . Let <P<∞, ≤Q<∞.If a∈M∞(Rd_),ϕ

,ϕ∈M(Rd),then AϕMζ,ϕa is

bounded on M(P,Q)(Rd_{)for everyζ∈R}d_with

_Aϕ,ϕ

MζaB(M(P,Q))≤ aM∞ϕMϕ_M.

Proof Letf ∈M(P,Q)(Rd_{) andg∈M(P,Q)(R}d_{), where} 

P+



P = ,



Q+



Q = . Then we

writeVϕf ∈W(FL,L(P,Q))(Rd) andVϕg∈W(FL,L(P,Q))(Rd) by above theorem. Moreover, sinceM(, )(Rd_{) =M}_(Rd_{), we haveW(FL}_,L_{) =M}_{=M(, ) by []. Hence}

using the Hölder inequalities for Wiener amalgam spaces [] and (.) we obtain

Vϕf ·VϕgM =Vϕf·VϕgW(FL_,L₎

≤ VϕfW(FL_,L(P,Q))VϕgW(FL_,L(P,Q))

Thus by using (.) we have

Aϕ,ϕ

Mζaf,g=Mζa,Vϕf·Vϕg≤ MζaM(∞,∞)Vϕf·VϕgM(,) ≤ aM∞ϕMϕMfM(P,Q)gM(P,Q).

Hence we get

Aϕ,ϕ

MζaB(M(P,Q))≤ aM∞ϕMϕM.

Theorem . Letϕ∈S(Rd_{)\{}be a window function.If <P,Q<∞,t ∈(,∞),s≤t ≤}

r and a∈W(Lr_,Ls_),then

Aϕ_M,ϕ

ζa:M(tP,tQ)

Rd_→MtP,tQRd

is bounded for everyζ ∈Rd_,where 

P+



P = ,



Q+



Q = ,and



t +



t = ,and the operator

norm satisfies the estimate

Aϕ_M,ϕ

ζa≤ aW(Lr,Ls).

Proof Let t < ∞, f ∈ M(tP,tQ)(Rd_{), and h ∈ M(tP,tQ)(R}d_{). Then we have V}

ϕf ∈

L(tP,tQ)(Rd_{) andV}

ϕh∈L(tP,tQ)(Rd). SinceVϕf ∈L(tP,tQ)(Rd), thenVϕf∗_(tP)(tQ)<

∞. By using the equality (.) in [], we get

Vϕf∗_(tP)(tQ)=Vϕf∗_{(tp)(tq)}∗_(tp

)(tq)=|Vϕf|

t∗ pq

 t∗

(tp)(tq) =|Vϕf|t

∗ pq

 tt∗

pq 

t ₌|_V

ϕf|t ∗ pq

∗ pq

 t

=|Vϕf|t∗_PQ



t_{. (.)}

Hence we have|Vϕf|t∈L(P,Q)(Rd). Similarly,|Vϕh|t∈L(P,Q)(Rd). By the Hölder

in-equality for Lorentz spaces with mixed norm and (.) we have

Vϕf ·Vϕht_t=|Vϕf|t|Vϕh|t_≤|Vϕf|t_PQ|Vϕh|t_PQ

=Vϕft_(tP)(tQ)Vϕht_(tP)(tQ). (.)

Since a∈W(Lr_,Ls_{), then M}

ζa∈W(Lr,Ls) for everyζ ∈Rd. Also since W(Lr,Ls)⊂

W(Lt,Lt) =Lt(Rd), then we have

at =Mζat ≤ MζaW(Lr_,Ls₎=a_W(Lr_,Ls₎. (.)

By using (.), (.), and applying again the Hölder inequality, we get

Aϕ_M,ϕ_ζaf,h=MζaVϕf,Vϕh

≤

Rd

Mζa(x,w)(Vϕf·Vϕh)(x,w)dx dw

≤ atVϕf(tP)(tQ)Vϕh(tP)(tQ)

≤ aW(Lr_,Ls₎f_M(tP,tQ)h_M(tP,tQ). (.)

If (tp), (tq) =∞, then (M((tP), (tQ))(Rd₎₎∗_=M(tP,tQ)(Rd_{) by Theorem  in []. Thus}

we have from (.) that

Aϕ_M,ϕ

ζafM((tP),(tQ))= sup =h∈M(tP,tQ)

|Aϕ_M,ϕ ζaf,h| hM(tP,tQ) ≤

aW(Lr_,Ls₎f_M(tP,tQ).

HenceAϕ_M,ϕ

ζais bounded. Also we have

Aϕ_M,ϕ

ζa= sup

=f∈M(tP,tQ)

Aϕ_M,ϕ_ζafM((tP),(tQ))

fM(tP,tQ) ≤

aW(Lr_,Ls₎.

Theorem . Letϕ∈_{≤R,S<∞}M(R,S)(Rd_{),where R= (r}

,r),S= (s,s).If≤s≤r≤ ∞

and a∈W(Lr_,Ls_)then

Aϕ_M,ϕ_ζa:M(P,Q)Rd→M(P,Q)Rd

is bounded for everyζ∈Rd,with

Aϕ_M,ϕ

ζa≤CaW(Lr,Ls)

for some C> .

Proof Sincea∈W(Lr_,Ls_{), thenM}

ζa∈W(Lr,Ls) for everyζ ∈Rd. Also sinces≤r, there

exists ≤t≤ ∞such thats≤t≤r. ThenW(Lr,Ls)(Rd)⊂Lt(Rd) and

Mζat=at≤ aW(Lr_,Ls₎=M_ζa_W(Lr_,Ls₎ (.)

for alla∈W(Lr_,Ls_)(Rd_{). LetB(M(P,Q)(R}d_),M(P,Q)(Rd_{)) be the space of the bounded}

linear operators from M(P,Q)(Rd_{) into M(P,Q)(R}d_{). Also let T be an operator from}

L_(Rd_{) intoB(M(P,Q)(R}d_),M(P,Q)(Rd_{)) byT(a) =A}ϕ,ϕ

Mζa. Take anyf ∈M(P,Q)(R

d_{) and}

h∈M(P,Q)(Rd). Assume thata∈W(L,L)(Rd) =L(Rd). By the Hölder inequality we get

T(a)f,h=Aϕ_M,ϕ_ζaf,h=MζaVϕf,Vϕh

≤

Rd

Mζa(x,w)Vϕf(x,w)Vϕh(x,w)dx dw

=

Rd

a(x,w)f,MwTxϕh,MwTxϕdx dw

≤

Rd

a(x,w)fM(P,Q)MwTxϕM(P,Q)hM(P,Q)

× MwTxϕM(P,Q)dx dw

Hence by (.)

T(a)f_M(P,Q)=A_Mϕ,ϕ_ζaf_M(P,Q)= sup

=h∈M(P,Q)

|Aϕ_M,ϕ_ζaf,h| hM(P,Q)

≤ ϕM(P,Q)ϕM(P,Q)fM(P,Q)a.

Then

T(a)=Aϕ_M,ϕ

ζa= sup

=f∈M(P,Q)

Aϕ_M,ϕ

ζafM(P,Q) fM(P,Q) ≤

ϕM(P,Q)ϕM(P,Q)a. (.)

Thus the operator

T:LRd→BM(P,Q)Rd,M(P,Q)Rd (.)

is bounded. Now let a∈W(L∞,L∞)(Rd) =L∞(Rd). Take any f ∈ M(P,Q)(Rd) and h∈M(P,Q)(Rd). ThenVϕf ∈L(P,Q)(Rd),Vϕh∈L(P,Q)(Rd). Applying the Hölder

in-equality

T(a)f,h=Aϕ_M,ϕ_ζaf,h=MζaVϕf,Vϕh

≤

Rd

Mζa(x,w)Vϕf(x,w)Vϕh(x,w)dx dw

≤ a∞

Rd

Vϕf(x,w)Vϕh(x,w)dx dw

≤ a∞VϕfPQVϕhPQ. (.)

By using (.) we write

T(a)f_M(P,Q)=A_Mϕ,ϕ_ζaf_M(P,Q)= sup

=h∈M(P,Q)

|Aϕ_M,ϕ_ζaf,h| hM(P,Q) ≤

a∞fM(P,Q). (.)

Hence by (.)

T(a)=Aϕ_M,ϕ_ζa= sup

=f∈M(P,Q)

Aϕ_M,ϕ_ζafM(P,Q)

fM(P,Q) ≤

a∞.

That means the operator

T:L∞Rd→BM(P,Q)Rd,M(P,Q)Rd (.)

is bounded. Combining (.) and (.) we obtain

T:LtRd→BM(P,Q)Rd,M(P,Q)Rd

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

is bounded for ≤t≤ ∞. Hence there existsC>  such that

T(a)=Aϕ_M,ϕ_ζa≤Cat. (.)

This implies that it is also true for ≤t≤ ∞. From (.) and (.) we write

T(a)=Aϕ_M,ϕ

ζa≤Cat≤CaW(Lr,Ls).

Proposition . Letϕ∈_{≤R,S<∞}M(R,S)(Rd_{),where R= (r}

,r),S= (s,s).If <s≤

and a∈W(L,Ls)(Rd)then

Aϕ_M,ϕ_ζa:M(P,Q)Rd→M(P,Q)Rd

is bounded.

Proof Let  <s≤ and let a∈W(L_,Ls_)(Rd_{). Then M}

ζa∈W(L,Ls) for every ζ ∈

Rd_{. Since W(L}_,Ls_)(Rd_)⊂L_(Rd_{), there exists a numberC>  such that M} ζa ≤

CMζaW(L_,Ls₎. Hence by (.),

Aϕ_M,ϕ_ζa≤ ϕM(P,Q)ϕM(P,Q)Mζa

≤CϕM(P,Q)ϕM(P,Q)MζaW(L_,Ls₎ =CϕM(P,Q)ϕM(P,Q)aW(L_,Ls₎.

Then the localization operator from M(P,Q)(Rd_{) into M(P,Q)(R}d_{) is bounded for}

 <s≤.

Proposition . Letϕ∈_{≤R,S<∞}M(R,S)(Rd),where R= (r,r),S= (s,s).If≤P,Q<

∞and a∈L(P,Q)(Rd_{)then the localization operator}

Aϕ_M,ϕ

ζa:M(P,Q)

Rd_→M(P,Q)Rd

is bounded,where 

P+



P = ,



Q+



Q = .

Proof Let a ∈ L(P,Q)(Rd_{). Then M}

ζa ∈ L(P,Q)(Rd) for every ζ ∈ Rd with

MζaPQ =aPQ. Take anyf ∈M(P,Q)(Rd) andh∈M(P,Q)(Rd). Applying the Hölder

inequality we have by (.)

Aϕ_M,ϕ_ζaf,h≤

Rd

Mζa(x,w)Vϕf(x,w)h,MwTxϕdx dw

≤

Rd

_{a(x,w)Vϕf(x,w)hM(P,Q)MwTx}ϕM(P,Q)dx dw

=hM(P,Q)ϕM(P,Q)

Rd

a(x,w)Vϕf(x,w)dx dw

Similarly to (.), we get

Aϕ_M,ϕ

ζa≤ ϕM(P,Q)aPQ.

Then the localization operatorAϕ_M,ϕ_ζafromM(P,Q)(Rd) intoM(P,Q)(Rd) is bounded.

Corollary . It is known by Propositionin[]thatS(Rd_)⊂M(R,S)(Rd_{)for≤R,S<}

∞.ThenS(Rd_)⊂

≤R,S<∞M(R,S)(Rd).So,Theorem.,Propositions.and.are still

true under the same hypotheses for them ifϕ∈S(Rd_).

Corollary . It is known[]that if P=p and Q=q,then Lorentz mixed-normed

modu-lation space M(P,Q)(Rd)is the Lorentz type modulation space M(p,q)(Rd).Therefore our

theorems hold for a Lorentz type modulation space rather than for a Lorentz mixed-normed modulation space.

Competing interests

The author declares that she has no competing interests.

Received: 21 August 2014 Accepted: 24 October 2014 Published:31 Oct 2014

References

1. Cordero, E, Gröchenig, K: Time-frequency analysis of localization operators. J. Funct. Anal.205(1), 107-131 (2003) 2. Gröchenig, K: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)

3. Feichtinger, HG: Modulation spaces on locally compact Abelian groups. Technical Report, University of Vienna (1983) 4. Hunt, RA: OnL(p,q) spaces. Enseign. Math.12(4), 249-276 (1966)

5. O’Neil, R: Convolution operators andL(p,q) spaces. Duke Math. J.30, 129-142 (1963)

6. Blozinski, AP: Multivariate rearrangements and Banach function spaces with mixed norms. Trans. Am. Math. Soc.1, 149-167 (1981)

7. Fernandez, DL: Lorentz spaces, with mixed norms. J. Funct. Anal.25, 128-146 (1977)

8. Sandıkçı, A: On Lorentz mixed normed modulation spaces. J. Pseud.-Differ. Oper. Appl.3, 263-281 (2012)

9. Sandıkçı, A: Continuity of Wigner-type operators on Lorentz spaces and Lorentz mixed normed modulation spaces. Turk. J. Math.38, 728-745 (2014). doi:10.3906/mat-1311-43

10. Gürkanli, AT: Time-frequency analysis and multipliers of the spacesM(p,q)(Rd_{) andS(p,q)(R}d_{). J. Math. Kyoto Univ.} 46(3), 595-616 (2006)

11. Sandıkçı, A, Gürkanlı, AT: Gabor analysis of the spacesM(p,q,w)(Rd_{) andS(p,q,r,w,ω)(R}d_{). Acta Math. Sci.31(1),}

141-158 (2011)

12. Sandıkçı, A, Gürkanlı, AT: Generalized Sobolev-Shubin spaces, boundedness and Schatten class properties of Toeplitz operators. Turk. J. Math.37, 676-692 (2013). doi:10.3906/mat-1203-5

13. Feichtinger, HG: Banach convolution algebras of Wiener type. In: Proc. Conf. on Functions, Series, Operators (Budapest 1980). Colloq. Math. Soc. Janos Bolyai, vol. 35, pp. 509-524. North Holland, Amsterdam (1983) 14. Heil, C: An introduction to weighted Wiener amalgams. In: Krishna, M, Radha, R, Thangavelu, S (eds.) Wavelets and

Their Applications (Chennai, January 2002), pp. 183-216. Allied Publishers, New Delhi (2003) 15. Holland, F: Harmonic analysis on amalgams ofLp_andq_{. J. Lond. Math. Soc.10(2), 295-305 (1975)}

16. Feichtinger, HG, Luef, F: Wiener amalgam spaces for the fundamental identity of Gabor analysis. Collect. Math.57, Vol. Extra, 233-253 (2006)

10.1186/1029-242X-2014-430

Cite this article as:Sandıkçı:Boundedness of localization operators on Lorentz mixed-normed modulation spaces.