A note on the boundedness of sublinear operators on grand variable Herz spaces

R E S E A R C H

Open Access

A note on the boundedness of sublinear

operators on grand variable Herz spaces

Hammad Nafis

_{, Humberto Rafeiro}

_{and Muhammad Asad Zaighum}

*_{Correspondence:}

[email protected]

2_{College of Sciences, Department of}

Mathematical Sciences, United Arab Emirates University, Al Ain, United Arab Emirates

Full list of author information is available at the end of the article

Abstract

In this paper, we introduce grand variable Herz type spaces using discrete grand spaces and prove the boundedness of sublinear operators on these spaces.

MSC: 46E30

Keywords: Sublinear operators; Herz spaces; Variable exponent analysis; Grand spaces

1 Introduction

The Herz spacesK˙qα,p(Rn) andK α,p

q (Rn) were introduced in [10] being known under the

names of homogeneous and non-homogeneous Herz spaces and they are defined by the norms

fK˙qα,p:=

k∈Z

2kαp

R₂k_–1,2k

f(x)qdx

p/q1/p

, (1)

f_Kα,p

q :=fLq(B(0.1))+

k∈N

2kαp

R₂k_–1,2k

f(x)qdx

p/q1/p

, (2)

respectively, whereRt,τ stands for the annulusRt,τ :=B(0,τ)\B(0,t). These spaces were

studied in many papers; see for instance [5,7,9,12–15,22] and the references therein. Last two decades, under the influence of some applications revealed in [32], there was a vast boom of research in the so called variable exponent spaces (see e.g. [30]). For the time being, the theory of such variable exponent Lebesgue, Orlicz, Lorentz, and Sobolev func-tion spaces is widely developed, we refer to the books [2–4,20,21]. Herz spaces with vari-ables exponent have been recently introduced in [1,12,13]. Samko in [33] used variable exponent Herz spaces (with variable parameters), known as continual Herz spaces. An-other approach regarding variable smoothness and integrability to study Herz type Hardy spaces was used in [29].

Grand Lebesgue spaces on bounded sets have been widely studied. They were intro-duced in [8,11], cf. [2]. Grand spaces proved to be useful in application to partial differen-tial equations. Various operators of harmonic analysis were intensively studied in the last years, cf. [6,16–20,27,28] and the references therein.

Grand Lebesgue sequence spaces were introduced in [31], where various operators of harmonic analysis were studied in these spaces, e.g. maximal, convolutions, Hardy, Hilbert, and fractional operators, among others.

The aim of this paper is to introduce grand variable Herz spaces K˙qα,p),θ(Rn) and

ob-tain the boundedness of sublinear operators onK˙qα,p),θ(Rn). The present article has three

sections apart from the Introduction. Section2deals with some basic notions regarding grand Lebesgue sequence spaces. In Sect.3we give the definition of grand variable Herz spaces and prove Hölder’s inequality. In Sect.4we discuss boundedness of sublinear op-erators on grand variable Herz spaces. Throughout the paper, constants (often different constant in the same series of inequalities) will mainly be denoted bycorC.fgmeans thatf ≤Cgandf≈gmeans thatf gf.

2 Preliminaries

2.1 Lebesgue space with variable exponent

For the current section we refer to [4,23] unless and until stated otherwise. LetX⊆Rn

be an open set andp(·) be a real-valued measurable function onXwith values in [1,∞). We suppose that

1≤p–(X)≤p+(X) <∞, (3)

wherep–(X) :=ess infx∈Xp(x) andp+(X) :=ess supx∈Xp(x). ByLp(·)(X) we denote the space

of measurable functionf onXsuch that

Ip(·)(f) =

X

f(x)p(x)dx<∞.

It is a Banach space equipped with the norm (see e.g. [4]):

f_Lp(·)_(X)=inf

η> 0 :Ip(·)

f η

≤1

.

Byp(x) =p(x)/(p(x) – 1), we denote the conjugate exponent ofp(·). For the following

lemma we refer to e.g. [3].

Lemma 2.1(Generalized Hölder’s inequality) Let X be a measurable subset ofRn_.Suppose that1≤p–(X)≤p+(X) <∞.Then

fg_Lr(·)_(X)≤cf_Lp(·)_(X)g_Lq(·)_(X)

holds,where f ∈Lp(·)_(X),g∈Lq(·)_(X)and 1 r(x)=

1 p(x)+

1

q(x) for every x∈X.

In the sequel we use the well known log-condition

_{q(x) –q(y)≤} A

–ln|x–y|, |x–y| ≤

1

2,x,y∈X, (4)

whereA=A(q) > 0 does not depend onx,y, and the decay condition: there exists a number

q∞∈(1,∞), such that

q(x) –q∞≤ A

and also the decay condition

q(x) –q0≤ A

ln|x|, |x| ≤

1

2, (6)

holds for someq0∈(1,∞) in the case of homogeneous Herz spaces.

With respect to classes of variable exponents used in this paper, we adopt the following notations:

(i) Given a functionf ∈L1

loc(X), the Hardy–Littlewood maximal operatorMis defined

by

Mf(x) :=sup

r>0 r–n

B(x,r)

f(y)dy (x∈X),

where

B(x,r) := y∈X:|x–y|<r.

(ii) Lq_loc(·)(X) :={f :f ∈Lq(·)_{(K)for all compact subsetsK⊂X}.} (iii) The setP(X)consists of allq(·)satisfyingq–> 1andq+<∞.

(iv) Plog_=Plog_{(X)is the class of functionsq∈P(X)satisfying the conditions (3)}

and (4).

(v) In the caseXis unbounded,P∞(X)andP0,∞(X)are subsets of exponents inP(X)

with values in[1,∞)which satisfy condition (5) and the two conditions (5) and (6),

respectively.

(vi) B(X)is the set ofq(·)∈P(X)satisfying the condition thatMis bounded onLq(·)_(X).

The following lemma appears in [33].

Lemma 2.2 Let D> 1and q∈P0,∞(Rn).Then

1

c0

rqn(0) ≤ _χ

Rr,Drq(·)≤c0r n

q(0)_{, for0 <r}≤_{1, (7)}

and

1

c∞r

n q∞ ≤ _χR

r,Drq(·)≤c∞r n

q∞_{, for r}≥_{1, (8)}

respectively,where c0≥1and c∞≥1depend on D,but do not depend on r.

2.2 Herz spaces with variable exponent

In this subsection, we introduce variable exponent Herz spaces. In what follows, we denote

χk=χRk,Rk=Bk\Bk–1andBk={x∈R

n_{:|x| ≤2}k_{}for allk∈Z.}

Definition 2.1 Let 1 <p<∞,α ∈Randq(·)∈P(Rn). The homogeneous Herz space ˙

K_qα₍,_·p₎(Rn_{) is defined by}

˙

K_qα₍,_·p₎Rn= f∈Lq_loc(·)Rn\ {0}:fK˙_qα₍,_·p₎(Rn₎<∞

where

fK˙_qα₍,_·p₎(Rn₎=

_∞

k=–∞

2kαfχk p Lq(·)

1

p

.

Definition 2.2 Let 1 <p<∞,α∈Randq(·)∈P(Rn). The non-homogeneous Herz space

K_qα₍,_·p₎(Rn) is defined by

K_qα₍,_·p₎Rn= f∈L_locq(·)Rn:f_Kα,p q(·)(Rn)<∞

,

where

f_Kα,p q(·)(Rn)=

_∞

k=1

2kαfχk p Lq(·)

1

p

+f_Lq(·)_B(0,1).

2.3 Grand Lebesgue sequence space

In this subsection we introduce grand Lebesgue sequence space. For the following defini-tions and statements, see [31]. The letterXstands for one of the setsZn,Z,NandN0.

Definition 2.3 Let 1≤p<∞andθ> 0. The grand Lebesgue sequence spacelp),θ_is

de-fined by the norm

_{xk}k∈X_lp),θ_(X)=xlp),θ_(X)

:=sup

ε>0

εθ k∈X

|xk|p(1+ε)

1

p(1+ε)

=sup

ε>0

ε

θ

p(1+ε)_x lp(1+ε)_(X),

where x ={xk}k∈X.

Note that the following nesting properties hold:

lp(1–ε)→lp→lp),θ1→_lp),θ2→_lp(1+δ) ₍₉₎

for 0 <ε<1

p,δ> 0 and 0 <θ1≤θ2.

3 Grand variable Herz space

In this section, we introduce grand variable Herz space in a natural way using the discrete space from Definition2.3.

Definition 3.1 Letα∈R, 1≤p<∞,q:Rn_{→[1,∞),θ> 0. We define the homogeneous}

grand variable Herz space by

˙

K_qα₍,_·p₎),θRn= f∈Lq_loc(·)Rn\ {0}:f_˙

K_qα₍,_·p₎),θ(Rn₎<∞

where

f_K˙α,p),θ

q(·) (Rn)

=sup

ε>0

εθ k∈Z

2kαp(1+ε)fχkp(1+ ε) Lq(·)

1

p(1+ε)

=sup

ε>0

ε

1

p(1+ε)_f

˙

K_qα₍,_·p₎(1+ε)(Rn₎.

In a similar way, non-homogeneous grand variable Herz spaces can be introduced. In the following theorem, we prove that Herz space is contained in grand variable Herz space.

Theorem 3.1 For p> 1,we haveK˙_qα₍,_·p₎(Rn_{)⊂ ˙K}α,p),θ

q(·) (Rn),θ> 0.

Proof Letf ∈ ˙K_qα₍,_·p₎(Rn_{). Then}

f_K˙α,p),θ

q(·) (Rn)

=sup

ε>0

εθ k∈Z

2kαp(1+ε)_fχ kp(1+

ε) Lq(·)

1

p(1+ε)

=2kα_fχ kLq(·)

lp),θ ≤C2kα_fχ

kLq(·) lp=C

k∈Z

2kαp_fχ kp_Lq(·)

1

p

=CfK˙_qα₍,_·p₎(Rn₎,

where we used (9).

Now we prove the Hölder inequality for a grand variable Herz space.

Theorem 3.2 If 0 < pi ≤ ∞, 1≤ q– ≤q+ <∞, –∞<αi <∞, i= 1, 2, 1_p = _p1₁ + _p1₂,

1 =_q1_(·)+_q1_(·)andα=α1+α2.Then

fg_˙

K1α,p),θ(Rn)≤ fK˙

α1,p1),θ

q(·) (Rn) g_˙

K_qα2,_(·p₎2),θ(Rn₎.

Proof We have

fg_˙

K1α,p),θ(Rn)= sup

ε>0

εθ

k∈Z

2kαp(1+ε)fgχkp_L(1+1_(Rεn)₎

1

p(1+ε)

=sup

ε>0

εθ

k∈Z

2kαp(1+ε) 2k+1

2k |fg|

p(1+ε)_p(1+1_ε)

.

By using Hölder’s inequality

≤Csup

ε>0

εθ k∈Z

2k(α1+α2)p(1+ε)_fχ kp(1+

ε) Lq(·)

gχkp(1+ ε) Lq(·)

1

p(1+ε)

=Csup

ε>0

εθ k∈Z

2kα1_fχ kLq(·)

p(1+ε)

2kα2_gχ k_Lq(·)

p(1+ε) 1

p(1+ε)

by generalized Hölder’s inequality

≤Csup

ε>0

εθ

k∈Z

2kα1_fχ kLq(·)

p1(1+ε)1/p1(1+ε)

×sup

ε>0

εθ

k∈Z

2kα2_gχ k_Lq(·)

p2(1+ε)1/p2(1+ε)

=Cf_˙

K_qα₍1,_·)p1),θ(Rn₎g_K˙α2,p2),θ

q(·) (Rn)

.

4 Boundedness of sublinear operators

In this section, we show that sublinear operators are bounded onK˙_qα₍,_·p₎),θ(Rn_{). Hernandez,}

Li, Lu and Yang [9,24,26] have proved that if a sublinear operatorTis bounded onLp_(Rn₎

and satisfies the size condition

Tf(x)≤C

Rn|x–y|

–n_{f(y)dy, x∈/sptf (10)}

for allf ∈L1_(Rn_{) with compact support thenT is bounded on the homogeneous Herz}

spaceK˙qα,pand on the non-homogeneous Herz spaceKqα,p. The condition (11) is satisfied

by many interesting operators in harmonic analysis, such as Calderón–Zygmund opera-tors, Carleson’s maximal operator, the Hardy–Littlewood maximal operator, Fefferman’s singular multipliers, Fefferman’s singular integrals, Ricci–Stein’s oscillatory singular inte-grals, and the Bochner–Riesz means (for details see [25,34]).

Theorem 4.1 Let1 <p<∞,q(·)∈P0,∞(Rn)such that–n/q(0) <α<n/q(0)and–n/q∞<

α<n/q_∞.Suppose that T is a sublinear operator and bounded on Lq(·)(Rn)satisfying the size condition(10).Then T is bounded onK˙_qα₍,_·p₎),θ(Rn).

Proof SinceTis sublinear, we have for everyf ∈ ˙K_qα₍,_·p₎),θ(Rn₎

Tf_˙ K_qα₍,_·p₎),θ(Rn₎

=sup

ε>0

εθ k∈Z

2kαp(1+ε)χkTf_Lp(1+q(·)₍ε_R)n₎

1

p(1+ε)

≤sup

ε>0

εθ

k∈Z 2kαp(1+ε)

_∞

l=–∞

χkT(fχl) p(1+ε) Lq(·)_(Rn₎

1

p(1+ε)

≤csup

ε>0

εθ

k∈Z

2kαp(1+ε) _k–2

l=–∞

χkT(fχl)_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

+csup

ε>0

εθ

k∈Z 2kαp(1+ε)

_k+1

l=k–1

χkT(fχl)_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

+csup

ε>0

εθ

k∈Z

2kαp(1+ε) _∞

l=k+2

χkT(fχl)_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

ForE2, using theLq(·)(Rn) boundedness ofT we obtain

E2≤csup

ε>0

εθ

k∈Z 2kαp(1+ε)

_k+1

l=k–1

T(fχl)_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

≤csup

ε>0

εθ

k∈Z

2kαp(1+ε) _k+1

l=k–1

fχlLq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

≤csup

ε>0

εθ k∈Z

2kαp(1+ε)fχk_Lp(1+q_(·)(ε_R)n₎

1

p(1+ε)

=cf_˙ K_qα₍,_·p₎),θ(Rn₎.

ForE1, we use the facts that, for eachk∈Zandl≤k– 2 and a.e.x∈Rk, size condition

(10) and Hölder’s inequality imply

T(fχl)(x)≤C

Rl

|x–y|–nf(y)dy

≤c2–kn

Rl

_f(y)dy≤c2–kn_fχ

lLq(·)_(Rn₎χl_Lq(·)_(Rn₎. (11)

Moreover, splittingE1by means of Minkowski’s inequality we have

E1≤csup

ε>0

εθ

–1

k=–∞ 2kαp(1+ε)

_k–2

l=–∞

χkT(fχl)_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

+csup

ε>0

εθ

∞

k=0

2kαp(1+ε) _k–2

l=–∞

χkT(fχl)_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

:=E11+E12.

ForE11by Lemma2.2we have

2–knχkLq(·)_(Rn₎χl_Lq(·)_(Rn₎≤c2–kn2

(_qkn₍₀₎)

2(

ln q(0))≤_c2

(l–k)n

q(0)_{. (12)}

Applying (11) and (12) toE11, we get

E11≤csup

ε>0

εθ

–1

k=–∞

2kαp(1+ε)

× _k–2

l=–∞

χkLq(·)_(Rn₎2–knfχlLq(·)_(Rn₎χl_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

≤csup

ε>0

εθ

–1

k=–∞ _k–2

l=–∞

2αlfχlLq(·)_(Rn₎2b(l–k)

p(1+ε)_p(1+1_ε)

whereb:=_qn₍₀₎–α> 0. Then we use Hölder’s inequality, Fubini’s theorem for series and

2–p(1+ε)_{< 2}–p_{to obtain}

≤csup

ε>0

εθ

–1

k=–∞

_k–2

l=–∞

2αp(1+ε)lfχl_Lpq(1+(·)₍ε_R)n₎2

bp(1+ε)(l–k)/2

× _k–2

l=–∞

2b(p(1+ε))(l–k)/2

p(1+ε)/(p(1+ε))_p(1+1_ε)

=csup

ε>0

εθ

–1

k=–∞ k–2

l=–∞

2αp(1+ε)lfχl_Lp(1+q(·)₍ε_R)n₎2

bp(1+ε)(l–k)/2

1

p(1+ε)

=csup

ε>0

εθ

–1

l=–∞

2αp(1+ε)l_fχ lp(1+

ε) Lq(·)_(Rn₎

–1

k=l+2

2bp(1+ε)(l–k)/2

1

p(1+ε)

<csup

ε>0

εθ

–1

l=–∞

2αp(1+ε)lfχl_Lp(1+q(·)ε_(R)n₎

–1

k=l+2

2bp(l–k)/2 1

p(1+ε)

≤csup

ε>0

εθ l∈Z

2αp(1+ε)lfχl_Lp(1+q(·)₍ε_R)n₎

1

p(1+ε)

≤cf_˙ K_qα₍,_·p₎),θ(Rn₎.

Now forE12using Minkowski’s inequality we have

E12≤csup

ε>0 εθ ∞ k=0

2kαp(1+ε) _–1

l=–∞

χkT(fχl)_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

+csup

ε>0 εθ ∞ k=0

2kαp(1+ε) _k–2

l=0

χkT(fχl)_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

:=A1+A2.

The estimate forA2follows in a similar manner toE11withq(0) replaced byq_∞and using

the fact that _qn_∞ –α> 0. ForA1using Lemma2.2we have

2–knχkLq(·)_(Rn₎χ_lLq(·)_(Rn₎≤c2–kn2( kn q∞)₂(

ln q(0))≤_c2(

–kn q_∞)₂(

ln

q(0))_{. (14)}

Now using (11) and (14) and the fact thatα–_qn_∞ < 0 we have

A1≤sup

ε>0 εθ ∞ k=0

2kαp(1+ε) _–1

l=–∞

χkTf(χl)_Lq(·)_(Rn₎

p(1+ε)1/p(1+ε)

≤csup

ε>0 εθ ∞ k=0

2kαp(1+ε)

× _–1

l=–∞

2–kn2(kn/q∞)2(ln/q(0))f(χl)_Lq(·)_(Rn₎

≤csup

ε>0

εθ

∞

k=0

2kαp(1+ε)

× _–1

l=–∞

2(–kn/q∞)₂(ln/q(0))_f(χ

l)_Lq(·)_(Rn₎

p(1+ε)1/p(1+ε)

≤csup

ε>0

εθ

∞

k=0

2(kα–kn/q_∞)p(1+ε)

× _–1

l=–∞

2(ln/q(0))f(χl)_Lq(·)_(Rn₎

p(1+ε)1/p(1+ε)

≤csup

ε>0

εθ

_–1

l=–∞

2(ln/q(0))f(χl)_Lq(·)_(Rn₎

p(1+ε)1/p(1+ε)

≤csup

ε>0

εθ

_–1

l=–∞

2lαf(χl)_Lq(·)_(Rn₎2(ln/q

_(0)–lα)

p(1+ε)1/p(1+ε)

.

Now applying Hölder’s inequality and using the fact that_qn₍₀₎–α> 0 we have

A1≤csup

ε>0

εθ

_–1

l=–∞

2lαp(1+ε)_f(χ l)

p(1+ε) Lq(·)_(Rn₎

× _–1

l=–∞

2l(n/q(0)–α)(p(1+ε))

p(1+ε)/(p(1+ε))1/p(1+ε)

≤csup

ε>0

εθ

l∈Z

2lαp(1+ε)f(χl)p(1+ ε) Lq(·)_(Rn₎

1/p(1+ε)

≤cf_˙ K_qα₍,_·p₎),θ(Rn₎.

Next, we estimateE3. For eachk∈Zandl≥k+ 2 and a.e.x∈Rk; the size condition (11)

and Hölder’s inequality imply

T(fχl)(x)≤C

Rl

|x–y|–nf(y)dy

≤c2–ln

Rl

f(y)dy≤c2–lnfχlLq(·)_(Rn₎χl_Lq(·)_(Rn₎. (15)

Similar toE1, splittingE3by means of Minkowski’s inequality we have

E3≤csup

ε>0

εθ

–1

k=–∞

2kαp(1+ε) _∞

l=k+2

χkT(fχl)_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

+csup

ε>0

εθ

∞

k=0

2kαp(1+ε)

_∞

l=k+2

χkT(fχl)_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

ForE32Lemma2.2yields

2–lnχkLq(·)_(Rn₎χl_Lq(·)_(Rn₎≤c2–ln2( kn q∞)₂(

ln

q_∞)≤_c2(kq–∞l)n_{. (16)}

Using (15) and (16) forE32, we get

E32≤csup

ε>0 εθ ∞ k=0

2kαp(1+ε)

× _∞

l=k+2

χkLq(·)_(Rn₎2–lnfχlLq(·)_(Rn₎χl_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

≤csup

ε>0 εθ ∞ k=0 _∞

l=k+2

2αl_fχ

lLq(·)_(Rn₎2d(k–l)

p(1+ε)_p(1+1_ε)

,

whered:= _qn_∞ +α> 0. Then we use Hölder’s inequality, Fubini’s theorem for series and 2–p(1+ε)_{< 2}–p_{to obtain}

≤csup

ε>0 εθ ∞ k=0 _∞

l=k+2

2αp(1+ε)l_fχ lp(1+

ε) Lq(·)_(Rn₎2

dp(1+ε)(k–l)/2

× _∞

l=k+2

2d(p(1+ε))(k–l)/2

p(1+ε)/(p(1+ε))_p(1+1_ε)

=csup

ε>0 εθ ∞ k=0 ∞

l=k+2

2αp(1+ε)l_fχ lp(1+

ε) Lq(·)_(Rn₎2

dp(1+ε)(k–l)/2

1

p(1+ε)

=csup

ε>0 εθ ∞ l=0

2αp(1+ε)lfχlp(1+ ε) Lq(·)(Rn₎

l–2

k=0

2dp(1+ε)(k–l)/2 1

p(1+ε)

<csup

ε>0

εθ l∈Z

2αp(1+ε)l_fχ lp(1+

ε) Lq(·)_(Rn₎

l–2

k=–∞

2dp(k–l)/2 1

p(1+ε)

=csup

ε>0

εθ

l∈Z

2αp(1+ε)l_fχ lp(1+

ε) Lq(·)_(Rn₎

1

p(1+ε)

≤cf_˙ K_qα₍,_·p₎),θ(Rn₎.

Now forE31using Minkowski’s inequality we have

E31≤csup

ε>0

εθ

–1

k=–∞

2kαp(1+ε) _–1

l=k+2

χkT(fχl)_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

+csup

ε>0

εθ

–1

k=–∞ 2kαp(1+ε)

_∞

l=0

χkT(fχl)_Lq(·)_(Rn₎

p(1+ε)_p(1+1_ε)

The estimate forB1follows in a similar manner toE32withq∞replaced byq(0) and using the fact that _q(0)n +α> 0. ForB2using Lemma2.2we have

2–lnχkLq(·)_(Rn₎χlLq(·)_(Rn₎≤c2–ln2

(_qkn₍₀₎)

2(

ln

q_∞)≤_c2(qkn(0))₂(q–∞ln)_{. (17)}

Now using (15) and (17) and the fact thatα+ n

q(0)> 0 we have

B2≤sup

ε>0

εθ

–1

k=–∞ 2kαp(1+ε)

_∞

l=0

χkTf(χl)_Lq(·)_(Rn₎

p(1+ε)1/p(1+ε)

≤csup

ε>0

εθ

–1

k=–∞

2kαp(1+ε)

× _∞

l=0

2–ln2(kn/q(0))2(ln/q∞))_f(χ

l)_Lq(·)_(Rn₎

p(1+ε)1/p(1+ε)

≤csup

ε>0

εθ

–1

k=–∞

2kαp(1+ε)

× _∞

l=0

2(kn/q(0))2–(ln/q∞)f(χl)_Lq(·)_(Rn₎

p(1+ε)1/p(1+ε)

≤csup

ε>0

εθ

–1

k=–∞

2k(α+n/q(0))p(1+ε)

× _∞

l=0

2–(ln/q∞)_f(χ

l)_Lq(·)_(Rn₎

p(1+ε)1/p(1+ε)

≤csup

ε>0

εθ

_∞

l=0

2–(ln/q∞)_f(χ

l)_Lq(·)_(Rn₎

p(1+ε)1/p(1+ε)

≤csup

ε>0

εθ

_∞

l=0

2lα_f(χ

l)_Lq(·)_(Rn₎2–l(n/q∞+α)

p(1+ε)1/p(1+ε)

.

Now applying Hölder’s inequality and using the fact that_qn_∞+α> 0 we have

B2≤csup

ε>0

εθ

_∞

l=0

2lαp(1+ε)_f(χ l)

p(1+ε) Lq(·)_(Rn₎

× _∞

l=0

2–l(n/q∞+α)(p(1+ε))

p(1+ε)/(p(1+ε))1/p(1+ε)

≤csup

ε>0

εθ

l∈Z

2lαp(1+ε)f(χl)p(1+ ε) Lq(·)_(Rn₎

1/p(1+ε)

Combining the estimates forE1,E2andE3yields

Tf_˙

K_qα₍,_·p₎),θ(Rn₎≤cf_K˙α,p),θ q(·) (Rn)

,

which ends the proof.

Acknowledgements Not applicable.

Funding

The research of H. Rafeiro was supported by a Research Start-Up Grant of United Arab Emirates University, Al Ain, United Arab Emirates via Grant No. G00002994.

Availability of data and materials Not applicable.

Competing interests

The authors declare that there are no competing interests.

Authors’ contributions

There was an equal amount of contributions from all three authors. All authors read and approved the final manuscript.

Author details

1_{Department of Mathematics and Statistics, Riphah International University, Islamabad, Pakistan.}2_{College of Sciences,}

Department of Mathematical Sciences, United Arab Emirates University, Al Ain, United Arab Emirates.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Received: 16 May 2019 Accepted: 6 December 2019

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