maximal, potential and singular type operators on herz spaces with variable exponents

J. Math. Anal. Appl. 394 (2012) 781–795

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Maximal, potential and singular type operators on Herz spaces withvariable exponentsAlexandre Almeida a,∗, Douadi Drihem b

a Center for Research and Development in Mathematics and Applications, Department of Mathematics, University of Aveiro, 3810-322 Aveiro, Portugalb Department of Mathematics, Laboratory of Mathematics Pure and Applied, M’Sila University, P.O. Box 166, M’Sila 28000, Algeria

a r t i c l e i n f o

Article history:Received 10 February 2012Available online 30 April 2012Submitted by Maria J. Carro

Keywords:Herz spaceVariable exponentEmbeddingsSublinear operator

a b s t r a c t

We investigate both homogeneous and inhomogeneous Herz spaces where the two mainindices are variable exponents. Under natural regularity assumptions on the exponentfunctions, we prove the boundedness of awide class of sublinear operators on these spaces,which includes maximal, potential and Calderón–Zygmund operators.

© 2012 Elsevier Inc. All rights reserved.

1. Introduction

Function spaces with variable exponents have been intensively studied in the recent years by a significant number ofauthors. The motivation for the increasing interest in such spaces comes not only from theoretical purposes, but also fromapplications to fluid dynamics [1], image restoration [2] and PDE with non-standard growth conditions. A comprehensiveoverviewon existence and regularity results for PDE in the variable exponent setting, including an extensive list of referenceson this subject, is given in the recent survey [3]. In all these applications the Lebesgue and Sobolev spaces with variableintegrability, Lp(·) and W k,p(·), seem to appear in a natural way.

Lebesgue spaceswith variable exponent have been explicitly studied in [4], but the systematic study of the spaces Lp(·) andW k,p(·) started in the paper [5]. Since then various other function spaces and classical operators of Harmonic Analysis havebeen investigated in the variable exponent setting, remarkably after the boundedness of the Hardy–Littlewood maximaloperator has been proved in [6]. We only refer to the survey [7] and to the monograph [8] for further details and referenceson recent developments on this field.

It is well known that Herz spaces play an important role in Harmonic Analysis. After they have been introduced in [9],the theory of these spaces had a remarkable development in part due to its usefulness in applications. For instance, theyappear in the characterization of multipliers on Hardy spaces [10], in the summability of Fourier transforms [11] and in theregularity theory for elliptic and parabolic equations in divergence form [12,13].

Herz spaces Kα(·)

p(·),q and Kα(·)

p(·),q with variable exponent p but fixed α ∈ R and q ∈ (0, ∞] were recently studied by Izuki

[14,15]. Themain purpose of this paper is to consider Herz spaces Kα(·)

p(·),q and Kα(·)

p(·),q, where the exponent α is variable as well,and give boundedness results for a wide class of classical operators acting on such spaces. Allowing α to vary from pointto point will raise extra difficulties which, in general, are overcome by imposing regularity assumptions on this exponent,either at the origin or at infinity.

∗ Corresponding author.E-mail addresses: [email protected] (A. Almeida), [email protected] (D. Drihem).

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782 A. Almeida, D. Drihem / J. Math. Anal. Appl. 394 (2012) 781–795

The paper is organized as follows. First we give some preliminaries where we fix some notations and recall some basicfacts on function spaces with variable integrability. In the preliminary section, we also give some key technical lemmasneeded in the proofs of the main statements. To make the presentation more clear, we give their proofs later in Section 6.The variable exponent homogeneous and inhomogeneous Herz spaces are introduced in Section 3. In this section, wealso prove some embedding results involving these spaces. The main statements are formulated in Section 4, where wegive boundedness results for certain sublinear operators on variable Herz spaces from the corresponding boundedness onvariable Lebesgue spaces. Maximal operators, fractional integral operators and Calderón–Zygmund operators are includedin our discussion. In particular, we establish Hardy–Littlewood–Sobolev theorems for fractional integrals on variable Herzspaces. The proofs of the main statements are then given in Section 5. Finally, in Section 7 we discuss the optimality of someconditions assumed in the main theorems.

2. Preliminaries

As usual, we denote by Rn the n-dimensional real Euclidean space, N the collection of all natural numbers and N0 =

N ∪ {0}. The symbol Z stands for the set of all integer numbers. We write B(x, r) for the open ball in Rn centered at x ∈ Rn

and radius r > 0. We use c as a generic positive constant, i.e. a constant whose value may change from appearance toappearance. The expression f . g means that f 6 c g for some independent constant c (and non-negative functions f andg), and f ≈ g means f . g . f .

The notation X ↩→ Y stands for continuous embeddings from X to Y , where X and Y are quasi-normed spaces. If E ⊂ Rn

is a measurable set, then |E| stands for the (Lebesgue) measure of E and χE denotes its characteristic function. By suppf wedenote the support of the function f .

By ℓq, q ∈ (0, ∞], we denote the discrete Lebesgue space equipped with the usual quasinorm. Mostly we will deal withsequences defined either on N or Z.

The Hardy–Littlewood maximal operator M is defined on locally integrable functions by

Mf (x) = supr>0

1|B(x, r)|

B(x,r)

|f (y)| dy.

Spaces of variable integrability

We denote by P(Rn) the set of all measurable functions p : Rn→ [1, ∞) (called variable exponents). For E ⊂ Rn and

p ∈ P(Rn), we use the notation

p+

E = ess supE

p(x), p−

E = ess infE

p(x), p+= p+

Rn , p−= p−

Rn .

Everywhere below we shall consider bounded exponents.The variable exponent Lebesgue space Lp(·)(Rn) is the class of all measurable functions f on Rn such that the modular

ϱp(·)(f ) :=

Rn

|f (x)|p(x) dx

is finite. This is a Banach function space equipped with the norm

∥f ∥p(·) := inf

µ > 0 : ϱp(·)

1µf

6 1

.

If p(x) ≡ p is constant, then Lp(·)(Rn) = Lp(Rn) is the classical Lebesgue space.A useful property is that ϱp(·)(f ) 6 1 if and only if ∥f ∥p(·) 6 1 (unit ball property). This property is clear for constant

exponents due to the obvious relation between the norm and the modular in that case.For variable exponents, Hölder’s inequality takes the form

∥fg∥s(·) 6 2 ∥f ∥p(·)∥g∥q(·)

where s is defined pointwise by 1s(x) =

1p(x) +

1q(x) . Often we use the particular case s(x) ≡ 1 corresponding to the situation

when q = p′ is the conjugate exponent of p.We say that a function g : Rn

→ R is locally log-Hölder continuous, if there exists a constant clog > 0 such that

|g(x) − g(y)| 6clog

log(e + 1/|x − y|)

for all x, y ∈ Rn. If

|g(x) − g(0)| 6clog

log(e + 1/|x|)

A. Almeida, D. Drihem / J. Math. Anal. Appl. 394 (2012) 781–795 783

for all x ∈ Rn, then we say that g is log-Hölder continuous at the origin (or has a log decay at the origin). If, for some g∞ ∈ Rand clog > 0, there holds

|g(x) − g∞| 6clog

log(e + |x|)

for all x ∈ Rn, then we say that g is log-Hölder continuous at infinity (or has a log decay at infinity).By P log

0 (Rn) and P log∞ (Rn) we denote the class of all exponents p ∈ P(Rn) which have a log decay at the origin and

at infinity, respectively. The notation P log(Rn) is used for all those exponents p ∈ P(Rn) which are locally log-Höldercontinuous and have a log decay at infinity, with p∞ := lim|x|→∞ p(x). Obviously we have P log(Rn) ⊂ P log

0 (Rn) ∩ P log∞ (Rn).

Note that p ∈ P log(Rn) if and only if p′∈ P log(Rn), and since (p′)∞ = (p∞)′ we write only p′

∞for any of these quantities.

We refer the reader to the recent monograph [8] for further details, historical remarks and more references on variableexponent spaces.

Some technical lemmas

Various important results have been proved in the space Lp(·)(Rn) under the assumption p ∈ P log(Rn). The locallog-Hölder continuity is crucial in the study of classical operators in variable exponent spaces, since it implies that

|B|p−

B ≈ |B|p+

B

for balls B with small radius. This fact was first realized by Diening [6] with the proof of the boundedness of the maximaloperator in Lp(·) spaces on bounded domains. This statement was then extended to the unbounded case by Cruz-Uribeet al. [16] with the introduction, in addition, of the logarithmic decay condition. Technically, the log condition at infinityis useful when one deals with large balls.

The definition of Herz spaces typically requires a construction over annulus centered at the origin (see Definition 3.1).The next auxiliary results show that the log decay conditions, either at the origin or at infinity, are sufficient to deal withexponents defined on such sets.

Lemma 2.1. Let α ∈ L∞(Rn) and r1 > 0. If α is log-Hölder continuous both at the origin and at infinity, then

rα(x)1 . rα(y)

2 ×

r1r2

α+

if 0 < r2 6r12

1 ifr12

< r2 6 2r1r1r2

α−

if r2 > 2r1

for any x ∈ B(0, r1) \ B(0, r12 ) and y ∈ B(0, r2) \ B(0, r2

2 ), with the implicit constant not depending on x, y, r1 and r2.

Very often we have to deal with the norm of characteristic functions on balls (or cubes) when studying the behavior ofvarious operators in Harmonic Analysis. In classical Lp spaces the norm of such functions is easily calculated, but this is notthe case when we consider variable exponents. Nevertheless, it is known that for p ∈ P log(Rn) there holds

∥χB∥p(·) ≈ |B|1

p(x) , x ∈ B,

for small balls B ⊂ Rn, and

∥χB∥p(·) ≈ |B|1

p∞

for large balls, with constants only depending on the log-Hölder constant of p (see, for example, [8, Section 4.5]).For characteristic functions defined on (dyadic) annulus we have similar norm estimates, without requiring the

log-Hölder continuity at every point.

Lemma 2.2. Let p ∈ P log∞ (Rn) and let R = B(0, r) \ B(0, r

2 ). If |R| > 2−n, then

∥χR∥p(·) ≈ |R|1

p(x) ≈ |R|1

p∞

with the implicit constants independent of r and x ∈ R.The left-hand side equivalence remains true for every |R| > 0 if we assume, additionally, p ∈ P log

0 (Rn) ∩ P log∞ (Rn).

We shall prove these lemmas later in Section 6.We end this section with one more lemma, which is basically a consequence of Young’s inequality in the sequence

Lebesgue space ℓq.


Lemma 2.3. Let 0 < a < 1 and 0 < q 6 ∞. Let {εk}k∈Z be a sequence of positive real numbers, such that{εk}k∈Z

ℓq

= I < ∞.

Then the sequencesδk : δk =

j6k a

k−jεjk∈Z

andηk : ηk =

j>k a

j−kεjk∈Z

belong to ℓq, and{δk}k∈Z

ℓq

+{ηk}k∈Z

ℓq

6 c I,

with c > 0 only depending on a and q.

3. Variable exponent Herz spaces

For convenience, we set

Bk := B(0, 2k), Rk := Bk \ Bk−1 and χk = χRk , k ∈ Z.

Definition 3.1. Let 0 < q 6 ∞, p ∈ P(Rn) and α : Rn→ R with α ∈ L∞(Rn). The inhomogeneous Herz space Kα(·)

p(·),q (Rn)

consists of all f ∈ Lp(·)loc (Rn) such that

∥f ∥Kα(·)p(·),q

:= ∥f χB0∥p(·) +

k>1

∥2kα(·)f χk∥qp(·)

1/q

< ∞. (3.2)

The homogeneous Herz space Kα(·)

p(·),q (Rn) is defined as the set of all f ∈ Lp(·)loc (Rn\ {0}) such that

∥f ∥Kα(·)p(·),q

:=

k∈Z

∥2kα(·)f χk∥qp(·)

1/q

< ∞ (3.3)

(with the usual modifications when q = ∞).

If α and p are constants, then Kα(·)

p(·),q (Rn) = Kαp,q (Rn) and Kα(·)

p(·),q (Rn) = Kαp,q (Rn) are classical Herz spaces. As in the

constant index case, (3.2) and (3.3) are quasinorms (norms if p ∈ P(Rn) and q > 1).The following statement gives some basic embeddings between Herz spaces.

Proposition 3.4. Let α, α0, α1 ∈ L∞(Rn), p ∈ P(Rn) and q0, q1 ∈ (0, ∞].

(i) If q0 ≤ q1, then

Kα(·)

p(·),q0(Rn) ↩→ Kα(·)

p(·),q1(Rn) and Kα(·)

p(·),q0(Rn) ↩→ Kα(·)

p(·),q1(Rn).

(ii) If (α0 − α1)− > 0, then

Kα0(·)p(·),q0

(Rn) ↩→ Kα1(·)p(·),q1

(Rn).

Proof. The embeddings in (i) are immediate consequences of the embedding ℓq0 ↩→ ℓq1 for 0 < q0 6 q1 6 ∞.By (i), to prove the embedding in (ii), it suffices to consider q1 finite. Then we have

∥ ∥2kα1(·)f χk ∥p(·) ∥ℓq1 (N) 6 c1 ∥ ∥2kα0(·)f χk ∥p(·) ∥ℓ∞(N) 6 c1 ∥ ∥2kα0(·)f χk ∥p(·) ∥ℓq0 (N)

with cq11 =

k>1 2−kq1(α0−α1)

−

< ∞. �

The next proposition gives more embedding results, now in the case when the main integration index is changed.

Proposition 3.5. Let α ∈ L∞(Rn), p0, p1 ∈ P(Rn) and q ∈ (0, ∞]. If p0 6 p1 and 1p0

−1p1

is log-Hölder continuous at infinity,then

Kα(·)+n/p0(·)−n/p1(·)p1(·),q

(Rn) ↩→ Kα(·)

p0(·),q(Rn).

Additionally, if 1p0

−1p1

has a log decay at the origin, then

Kα(·)+n/p0(·)−n/p1(·)p1(·),q

(Rn) ↩→ Kα(·)

p0(·),q(Rn).

Proof. Let 1t(x) =

1p0(x)

−1

p1(x). Then Hölder’s inequality yields

∥2kα(·)f χk∥p0(·) 6 2 ∥2kα(·)f χk∥p1(·) ∥χk∥t(·) = 2 ∥2kα(·)f χk ∥χk ∥t(·) ∥p1(·)


for any k ∈ Z. By Lemma 2.2, we get the equivalence

2kα(x)|f (x)| ∥χk∥t(·) ≈ 2kα(x)

|f (x)| |Rk|1

t(x) , k ∈ Z, x ∈ Rk,

where the constants are independent of k and x. Since |Rk|1

t(x) ≈ 2knt(x) , we obtain the estimate

∥2kα(·)f χk∥p0(·) . ∥2kα(·)+kn(1/p0(·)−1/p1(·))f χk∥p1(·)

from which both embeddings follow after taking the corresponding ℓq-quasinorm. �

Corollary 3.6. Let p0, p1 ∈ P log∞ (Rn). If p0 6 p1, then

L∞(Rn) ↩→ K−n/p1(·)p1(·),∞

(Rn) ↩→ K−n/p0(·)p0(·),∞

(Rn) ↩→ K−n1,∞(Rn)

and

K n∞,1(R

n) ↩→ Kn/p′

1(·)

p1(·),1(Rn) ↩→ K

n/p′0(·)

p0(·),1(Rn) ↩→ K 0

1,1(Rn) = L1(Rn).

Similar embeddings hold in the homogeneous case if p0, p1 ∈ P log0 (Rn) ∩ P log

∞ (Rn).

Proof. Take constant exponents r, s such that 1 6 r 6 p−

0 6 p+

1 6 s < ∞. Since p0, p1 have a log decay at infinity, so theydo 1

p1−

1s ,

1r −

1p0

and 1p0

−1p1. By Proposition 3.5 we have

K−n/ss,∞ (Rn) ↩→ K−n/p1(·)

p1(·),∞(Rn) ↩→ K−n/p0(·)

p0(·),∞(Rn) ↩→ K−n/r

r,∞ (Rn).

Hence the first chain of embeddings follows taking into account the classical assertions

L∞(Rn) = K 0∞,∞(Rn) ↩→ K−n/s

s,∞ (Rn) and K−n/rr,∞ (Rn) ↩→ K−n

1,∞(Rn).

The second chain of embeddings can be proved in a similar way. �

Remark 3.7. From the beginning, we have excluded from our study the situation when the exponent p is unbounded. Thischoice was made in virtue of the main results we are pursuing (see Section 4). Nevertheless, making natural modifications,one could show that the statements given in Propositions 3.4, 3.5 and Corollary 3.6 remain valid in the unbounded case. Insuch case the proof of Corollary 3.6 would be simpler by applying Proposition 3.5.

Let us denote

∥{gk}∥ℓq>(Lp(·)) :=

k>0

∥gk∥qp(·)

1/q

and ∥{gk}∥ℓq<(Lp(·)) :=

k<0

∥gk∥qp(·)

1/q

for sequences {gk}k∈Z of measurable functions (with the usual modification when q = ∞).

Proposition 3.8. Let α ∈ L∞(Rn), p ∈ P(Rn) and q ∈ (0, ∞]. If α is log-Hölder continuous at infinity, then

Kα(·)

p(·),q

Rn

= Kα∞

p(·),q

Rn .

Additionally, if α has a log decay at the origin, then

∥f ∥Kα(·)p(·),q

≈ ∥{2kα(0)f χk}∥ℓq<(Lp(·)) + ∥{2kα∞ f χk}∥ℓ

q>(Lp(·)).

Proof. If α has logarithmic decay at infinity, then for k > 0 and x ∈ Rk we have

k|α(x) − α∞| .k

log(e + |x|). 1.

Therefore 2kα(x)≈ 2kα∞ with constants independent of k and x, and hence

∥2kα(·)f χk∥p(·) ≈ 2kα∞∥f χk∥p(·).

If, in addition, α has a log decay at the origin, then we also have 2kα(x)≈ 2kα(0) for k < 0 and x ∈ Rk. Thus

∥2kα(·)f χk∥p(·) ≈ 2kα(0)∥f χk∥p(·)

and hence the result follows. �


4. Main results

In this section, we present the main results. Their proofs will be given later in Section 5.We consider sublinear operators satisfying the size condition

|Tf (x)| .

Rn

|f (y)||x − y|n

dy, x ∈ supp f , (4.1)

for integrable and compactly supported functions f . Condition (4.1) was first considered in [17] and it is satisfied by severalclassical operators in Harmonic Analysis, such as Calderón–Zygmund operators, the Carleson maximal operator and theHardy–Littlewood maximal operator (see [17,18]).

We have the following results.

Theorem 4.2. Let 0 < q 6 ∞, p ∈ P log∞ (Rn)with 1 < p− 6 p+ < ∞, and let α ∈ L∞(Rn) be log-Hölder continuous at infinity

with

−np∞

< α∞ <np′

∞

. (4.3)

Suppose that T is a sublinear operator satisfying estimate (4.1). If T is bounded on Lp(·)(Rn), then T is bounded on Kα(·)

p(·),q(Rn).

For homogeneous spaces we have the following statement.

Theorem 4.4. Let 0 < q 6 ∞, p ∈ P log0 (Rn) ∩ P log

∞ (Rn) with 1 < p− 6 p+ < ∞, and let α ∈ L∞(Rn) be log-Höldercontinuous, both at the origin and at infinity, such that

−np+

< α− 6 α+ < n1 −

1p−

. (4.5)

Then every sublinear operator T satisfying (4.1) which is bounded on Lp(·)(Rn) is also bounded on Kα(·)

p(·),q(Rn).

Remark 4.6. Corresponding statements to Theorems 4.2 and 4.4 were proved by Izuki [15], with α constant, under theassumption that the maximal operator M is bounded on Lp(·)(Rn) (both in the homogeneous and the inhomogeneoussituation). Here we are requiring the log-Hölder continuity at two points only (zero and infinity). We also note that ourconditions (4.3) and (4.5) are more explicit than those used in [15], hence allowing a better comparison with the alreadyknown constant exponent setting.

Since theHardy–Littlewoodmaximal operator M is sublinear, satisfies the size condition (4.1) and is bounded on Lp(·)(Rn)if p ∈ P log(Rn) and 1 < p− 6 p+ 6 ∞ (see [8, Theorem 4.3.8]), from Theorems 4.2 and 4.4 we immediately arrive at thefollowing result.

Corollary 4.7. Let 0 < q 6 ∞, p ∈ P log(Rn) with 1 < p− 6 p+ < ∞, and α ∈ L∞(Rn).

(i) If (4.3) holds and α satisfies the log decay condition at infinity, then M is bounded on Kα(·)

p(·),q(Rn).

(ii) If (4.5) holds and α has a log decay both at the origin and at infinity, then M is bounded on Kα(·)

p(·),q(Rn).

Remark 4.8. The boundedness of the maximal operator in variable exponent Lebesgue spaces was first proved byDiening [6]. Further details and references on subsequent contributions, including necessary conditions, can be found in[8, Chapter 4]. In particular, we note that the assumption p− > 1 is necessary to the boundedness of M on Lp(·)(Rn). In asense, the condition p ∈ P log(Rn) is also (close to) necessary.

Let us consider a more general class of operators in order to include, for instance, fractional integral operators. As in [19],we consider sublinear operators Tλ, 0 6 λ < n, satisfying the size estimate

|Tλf (x)| .

Rn

|f (y)||x − y|n−λ

dy, x ∈ supp f , (4.9)

for every integrable and compactly supported functions f . Since the limiting case λ = 0 was already discussed, we considernow λ ∈ (0, n). Observe that for such values of λ the Riesz potential operator

Iλf (x) :=

Rn

f (y)|x − y|n−λ

dy

and the fractional maximal function

Mλf (x) := supr>0

1

|B(x, r)|1−λn

B(x,r)

|f (y)| dy

satisfy both the size condition (4.9).


Let p∗ be the Sobolev exponent defined by

1p∗(x)

:=1

p(x)−

λ

n, x ∈ Rn.

We note that if p ∈ P(Rn) and 0 < λ < n/p+ then p∗∈ P(Rn) with

1 <np−

n − λp−= (p∗)− 6 (p∗)+ =

np+

n − λp+< ∞.

Moreover,we can easily show that the assumption p ∈ P log(Rn) implies p∗∈ P log(Rn). In particular,wehave (p∗)∞ = (p∞)∗

so that we write only p∗∞

for simplicity.

Theorem 4.10. Let 0 < λ < n, 0 < q0 6 q1 6 ∞, p ∈ P log∞ (Rn) with 1 < p− 6 p+ < n

λ, and let α ∈ L∞(Rn) be log-Hölder

continuous at infinity. If

λ −np∞

< α∞ <np′

∞

, (4.11)

then every sublinear operator Tλ satisfying (4.9) and bounded from Lp(·)(Rn) into Lp∗(·)(Rn) is also bounded from Kα(·)

p(·),q0(Rn) into

Kα(·)

p∗(·),q1(Rn).

The counterpart for homogeneous Herz spaces runs as follows.

Theorem 4.12. Let λ, q0, q1 be as in Theorem 4.10. Let p ∈ P log0 (Rn) ∩ P log

∞ (Rn) and α ∈ L∞(Rn) be log-Hölder continuous,both at the origin and at infinity, such that 1 < p− 6 p+ < n

λand

λ −np+

< α− 6 α+ < n1 −

1p−

. (4.13)

Then every sublinear operator Tλ satisfying (4.9) and bounded from Lp(·)(Rn) into Lp∗(·)(Rn) is also bounded from Kα(·)

p(·),q0(Rn) into

Kα(·)

p∗(·),q1(Rn).

In view of the well-known pointwise estimate

Mλf (x) . Iλ(|f |)(x),

and thep(·) → p∗(·)

- boundedness of Iλ for exponents p ∈ P log(Rn) with 1 < p− 6 p+ < n

λ(see [8, Theorem 6.1.9]),

from Theorems 4.10 and 4.12 we get the following results.

Corollary 4.14. Let 0 < λ < n, 0 < q0 6 q1 6 ∞, p ∈ P log(Rn) with 1 < p− 6 p+ < nλ, and α ∈ L∞(Rn).

(i) If (4.11) holds and α satisfies the log decay condition at infinity, then Iλ and Mλ are bounded from Kα(·)

p(·),q0(Rn) into

Kα(·)

p∗(·),q1(Rn).

(ii) If (4.13) holds and α has a log decay, both at the origin and at infinity, then Iλ and Mλ are bounded from Kα(·)

p(·),q0(Rn) into

Kα(·)

p∗(·),q1(Rn).

Remark 4.15. Estimates for potential type operators in variable Lp(·) spaces were first considered by Samko [20]. Fractionalmaximal operators were first studied in this setting by Kokilashvili and Samko [21], including the case of variable orderλ = λ(x). We refer to [8, Chapter 6] for further contributions and historical remarks in the study of fractional integraloperators in variable exponent spaces. Nevertheless, it is worth noting the recent paper [22], where the Calderón–Zygmunddecomposition technique was used to derive a new proof for the boundedness of Mλ which includes the limiting casep+

= n/λ, λ > 0.

Remark 4.16. Hardy–Littlewood–Sobolev type theorems on Herz spaces were used in [23] to study the Laplace and thewave equation in this setting.

5. Proofs of the main results

Our proofs use partially some decomposition techniques already used in [19] where the constant exponent case wasstudied. We consider only q ∈ (0, ∞). The arguments are similar in the case q = ∞.


Proof of Theorem 4.2. In view of Proposition 3.8 we use the equivalent quasinorm ∥ · ∥Kα∞p(·),q

. We split the operator into

|Tf (x)| 6 |TfχB−2

(x)| + |T

fχBk−2\B−2

(x)| + |T (fχRk)(x)| + |T

fχRn\Bk+2

(x)|,

whereRk :=x ∈ Rn

: 2k−2 6 |x| < 2k+2with k ∈ N.

Estimation of TfχB−2

.

For x ∈ Rk and y ∈ B−2 we have |x − y| > |x| − |y| > 2k−2. Then

|TfχB−2

(x)| . 2−kn

B−2

|f (y)| dy . 2−kn∥fχB−2∥p(·)∥χB−2∥p′(·)

by Hölder’s inequality. Therefore, we have

|TfχB−2

(x)| . 2−kn

∥fχB−2∥p(·), x ∈ Rk,

with a constant independent of k and x. Taking the Lp(·)-norm we get

2kα∞∥χk TfχB−2

∥p(·) . 2k

α∞−n+ n

p∞

∥fχB−2∥p(·),

where we used ∥χk∥p(·) ≈ |Rk|1

p∞ ≈ 2knp∞ by Lemma 2.2. Taking now the ℓq-quasinorm and observing that ∥fχB−2∥p(·) 6

∥fχB0∥p(·), we obtaink>1

2kα∞q∥χk T

fχB−2

∥qp(·)

1/q

. c0 ∥fχB0∥p(·) 6 c0 ∥f ∥Kα∞p(·),q

,

with cq0 =

k>1 2kq(α∞−n+ n

p∞ )< ∞ taking into account assumption (4.3).

Estimation of TfχBk−2\B−2

.

Let k ∈ N and x ∈ Rk. Then we write

|TfχBk−2\B−2

(x)| .

Bk−2\B−2

|x − y|−n|f (y)| dy =

−16j6k−2

Rj

|x − y|−n|f (y)| dy.

To estimate the last integral we note that |x − y| > |x| − |y| > 2k4 if x ∈ Rk, y ∈ Rj. Hence we arrive at the inequality

2kα∞ |TfχBk−2\B−2

(x)| .

−16j6k−2

2(k−j)α∞−kn2jα∞

Rn

|f (y)|χj(y) dy, x ∈ Rk.

Hölder’s inequality implies thatRn

|f (y)|χj(y) dy 6 2 ∥fχj∥p(·) ∥χj∥p′(·).

Taking the Lp(·)-norm (with respect to x) in the previous pointwise estimate, we get

2kα∞∥χkTfχBk−2\B−2

∥p(·) .

−16j6k−2

2(k−j)α∞−kn2jα∞∥fχj∥p(·) ∥χj∥p′(·) ∥χk∥p(·).

Now Lemma 2.2 yields

∥χj∥p′(·) ≈ |Rj|1

p′∞ and ∥χk∥p(·) ≈ |Rk|1

p∞ .

Hence, for every k ∈ N,

2kα∞∥χkTfχBk−2\B−2

∥p(·) .

−16j6k−2

2(k−j)δ 2jα∞∥fχj∥p(·)

where we put δ = α∞ − n +n

p∞for short. Taking the ℓq-(quasi)norm we get

k>1

2kα∞q∥χkT

fχBk−2\B−2

∥qp(·)

1/q

.

k>1

−16j6k−2

2(k−j)δ2jα∞ ∥fχj∥p(·)

q1/q


.

k>1

2(k+1)δ2−α∞∥fχ−1∥p(·) + 2kδ

∥fχ0∥p(·) +

16j6k

2(k−j)δ2jα∞ ∥fχj∥p(·)

q1/q

.

Since α is bounded and χ−1, χ0 6 χB0 , the right-hand side of the last inequality is bounded by

c0 ∥fχB0∥p(·) +

k>1

16j6k

2(k−j)δ 2jα∞ ∥fχj∥p(·)

q1/q

,

with cq0 =

k>1 2kδq < ∞ by assumption (4.3). Now Lemma 2.3 gives

k>1

2kα∞q∥χkT

fχBk−2\B−2

∥qp(·)

1/q

. ∥fχB0∥p(·) +

j>1

2jα∞q∥fχj∥

qp(·)

1/q

= ∥f ∥Kα∞p(·),q

.

Estimation of TfχRk.

Using the boundedness of T on Lp(·)(Rn) and the fact that χRk =2

j=−1 χk+j, we derive the estimate

∥TfχRkχk∥p(·) . ∥T

fχRk∥p(·) . ∥fχRk∥p(·) .

2j=−1

∥fχk+j∥p(·).

Taking the (quasi)norm in the sequence space and noting that α is bounded, we obtaink>1

2kα∞q∥TfχRkχk∥

qp(·)

1/q

.

k>1

−16j62

2(k+j)α∞q∥fχk+j∥

qp(·)

1/q

.

∥fχ0∥

qp(·) +

k>2

2(k−1)α∞q∥fχk−1∥

qp(·) +

06j62

l>1

2lα∞q∥fχl∥

qp(·)

1/q

. ∥fχB0∥p(·) +

l>1

2lα∞q∥fχl∥

qp(·)

1/q

= ∥f ∥Kα∞p(·),q

.

Estimation of TfχRn\Bk+2

.

The arguments here are quite similar to those used in the estimation of TfχBk−2\B−2

. Given k ∈ N and x ∈ Rk we write

|TfχRn\Bk+2

(x)| .

Rn\Bk+2

|x − y|−n|f (y)| dy =

j>k+3

Rj

|x − y|−n|f (y)| dy.

Noting that |x−y| > 2j−3 for x ∈ Rk and y ∈ Rj, as in the second case we use successively Hölder’s inequality and Lemma 2.2to get the estimate

2kα∞∥χkTfχRn\Bk+2

∥p(·) .

j>k+3

2(k−j)(α∞+n

p∞ ) 2jα∞∥fχj∥p(·).

Therefore, since α∞ +n

p∞> 0 by (4.3), we apply Lemma 2.3 and obtain

k>1

2kα∞q∥χkT

fχRn\Bk+2

∥qp(·)

1/q

.

k>1

j>k

2(k−j)α∞+

np∞

2jα∞ ∥fχj∥p(·)

q1/q

.

j>1

2jα∞q∥fχj∥

qp(·)

1/q

6 ∥f ∥Kα∞p(·),q

.

Recalling the definition of ∥ · ∥Kα∞p(·),q

, it remains to show that ∥(Tf )χB0∥p(·) . ∥f ∥Kα∞p(·),q

to complete the proof. Since

|Tf (x)| 6 |TfχB0

(x)| + |T (fχ1)(x)| + |T

fχRn\B1

(x)|,


we can deal with the three terms separately. By hypothesis T is bounded in Lp(·)(Rn), so that

∥TfχB0

χB0∥p(·) . ∥T

fχB0

∥p(·) . ∥fχB0∥p(·) 6 ∥f ∥Kα∞

p(·),q

and

∥Tfχ1

χB0∥p(·) . ∥fχ1∥p(·) . 2α∞∥fχ1∥p(·) . ∥f ∥Kα∞

p(·),q.

For the remaining term, we note that |x − y| > 2k−2, x ∈ B0, y ∈ Rk, and derive

|TfχRn\B1

(x)| .

k>2

2−knRk

|f (y)| dy.

An application of Hölder’s inequality and Lemma 2.2 gives

|TfχRn\B1

(x)| .

k>2

2−kα∞+

np∞

2kα∞∥fχk∥p(·) 6 c1 sup

j∈N2jα∞∥fχj∥p(·),

where c1 =

k>2 2−k(α∞+

np∞ )

< ∞ since α∞ +n

p∞> 0. Thus we get

∥TfχRn\B1

χB0∥p(·) . 2jα∞∥fχj∥p(·) ∥χB0∥p(·) . ∥f ∥Kα∞

p(·),q.

The proof is complete. �

Proof of Theorem 4.12. In view of Proposition 3.4(i), it is sufficient to consider the case q0 = q1(=q).Following the notation of the previous section, for k ∈ Z and x ∈ Rk we write

|Tλf (x)| 6 |Tλ

fχBk−2

(x)| + |Tλ(fχRk)(x)| + |Tλ

fχRn\Bk+2

(x)|.

Estimation of Tλ

fχBk−2

.

Using an appropriate discretization of the integral (as in the previous proof, second case) and the log-Hölder continuityof α at the origin for changing its value (cf. Lemma 2.1), we have

2kα(x)|Tλ

fχBk−2

(x)| .

j6k−2

2(k−j)α++k(λ−n)

Rj2jα(y)

|f (y)| dy.

After applying Hölder’s inequality to the last integral, we get

∥2kα(·) χkTλ

fχBk−2

∥p∗(·) .

j6k−2

2(k−j)α++k(λ−n)

∥2jα(·)fχj∥p(·) ∥χj∥p′(·) ∥χk∥p∗(·).

Since p ∈ P log0 (Rn) ∩ P log

∞ (Rn) implies p′, p∗∈ P log

0 (Rn) ∩ P log∞ (Rn), then Lemma 2.2 gives

∥χj∥p′(·) ≈ |Rj|

1p′(xj) , xj ∈ Rj, and ∥χk∥p∗(·) ≈ |Rk|

1p∗(xk) , xk ∈ Rk.

Hence the sum above can be rewritten asj6k−2

2(k−j)(α+−n)

|Rj|−

1p(xj) |Rk|

1p(xk)

2jα(·)fχjp(·) .

Now we can distinguish three cases as follows: 0 6 j 6 k − 2: by Lemma 2.2 we get

|Rj|−

1p(xj) |Rk|

1p(xk) ≈ |Rj|

−1

p∞ |Rk|1

p∞ ≈ 2(k−j) np∞ . 2

(k−j) np− .

j < 0 6 k − 2: in this case we obtain

|Rj|−

1p(xj) |Rk|

1p(xk) . |Rj|

−1p− |Rk|

1p− . 2

(k−j) np− .

j 6 k − 2 < 0: here we have

|Rj|−

1p(xj) |Rk|

1p(xk) ≈

|Rk ∥ Rj|

−1 1p(xj) |Rk|

1p(xk)

−1

p(xj) . 2(k−j) n

p− .


Indeed, since |xk| < 2k, |xj| < 2j < 2k we make use of local log-Hölder continuity of p at the origin and get, for k 6 0,

1p(xk)

−1

p(xj)

log 1|Rk|

.

log

12k

log

e +

12k

6 c

with c > 0 independent of k, j, xk, xj.Therefore, in all cases we have essentially the same bound and hence, combining the estimates above, we arrive at the

inequality2kα(·)χk Tλ

fχBk−2

p∗(·)

.j6k−2

2(k−j)

α+

−n+ np−

2jα(·)fχjp(·) .

Since α+− n +

np− < 0 by (4.13), we apply Lemma 2.3 as in the proof of Theorem 4.2 and get

Tλ

fχBk−2

Kα(·)

p∗(·),q=

k∈Z

2kα(·)χkTλ

fχBk−2

qp∗(·)

1/q

. ∥f ∥Kα(·)p(·),q

.

Estimation of Tλ

fχRk.

By Proposition 3.8 we have

∥{2kα(·)TfχRkχk}∥ℓq(Lp(·)) . ∥{2kα(0)T

fχRkχk}∥ℓ

q<(Lp(·)) + ∥{2kα∞T

fχRkχk}∥ℓ

q>(Lp(·)).

Using the sublinearity and thep(·) → p∗(·)

- boundedness of Tλ, the right-hand side is bounded by

∥{T2kα(0)fχRk}∥ℓ

q<(Lp(·)) + ∥{T

2kα∞ fχRk}∥ℓ

q>(Lp(·)) . ∥{2kα(0)fχRk}∥ℓ

q<(Lp(·)) + ∥{2kα∞ fχRk}∥ℓ

q>(Lp(·))

. ∥{2kα(·)fχRk}∥ℓq(Lp(·))

. ∥f ∥Kα(·)p(·),q

.

Estimation of Tλ

fχRn\Bk+2

.

Using a combination of the arguments used in the corresponding step of the proof of Theorem 4.2 and those used in thefirst step above, we arrive at the inequality2kα(·)χk Tλ

fχRn\Bk+2

p∗(·)

.j>k+3

2(k−j)

α−

−λ+np+

2jα(·)fχjp(·) .

Observing that α−− λ +

np+ > 0 by (4.13), an application of Lemma 2.3 yields the desired inequality, i.e.

Tλ

fχRn\Bk+2

Kα(·)

p∗(·),q=

k∈Z

2kα(·)χkTλ

fχRn\Bk+2

qp∗(·)

1/q

. ∥f ∥Kα(·)p(·),q

,

and hence the proof of Theorem 4.12 is complete. �

We omit the proofs of Theorems 4.4 and 4.10 since they are essentially similar to the proofs of Theorems 4.12 and 4.2,respectively.

6. Proof of the lemmas

Proof of Lemma 2.1. The case 0 < r2 6r12 . We consider first the case r2 > 1. We have

rα(x)1 6

r1r2

α+

rα(x)−α(y)2 rα(y)

2 .

Hence the claim follows if we show that there exists a constant c > 1 such that

c−1 6 rα(x)−α(y)2 6 c

for all x, ywith |x| >r12 and |y| >

r22 , which is equivalent to the inequality

|α(x) − α(y)| log r2 6 log c.


The latter follows from the logarithmic decay condition on α at infinity. In fact, we get

|α(x) − α(y)| log r2 6 |α(x) − α∞| log r2 + |α(y) − α∞| log r2

.log r2

log(e + |x|)+

log r2log(e + |y|)

.log r1

2

log

e +

r12

+log 2 + log r2

2

log

e +

r22

. 1.

Now let us consider r2 < 1. If r1 > 1 then

rα(x)1 =

r1r2

α+

rα(x)−α+

1 rα+

2 6

r1r2

α+

rα(x)−α+

1 rα(y)2 .

r1r2

α+

rα(y)2 .

If r1 < 1 we use the log-Hölder continuity of α at the origin to show that

|α(x) − α(y)| log1r1

. |α(x) − α(0)| log1r1

+ |α(y) − α(0)| log1r1

6log 1

r1

log

e +

1r1

6 c

for any x ∈ B(0, r1) and y ∈ B(0, r2), which means that rα(x)−α(y)1 ≈ 1 for such values of x, y. Then we have

rα(x)1 =

r1r2

α(y)


2 .

r1r2

α+

rα(y)2 .

The case r12 < r2 6 2r1. Since

rα(x)1 . rα(x)−α(y)

1 rα(y)2 ,

we get the result if we prove that rα(x)−α(y)1 ≈ 1. When r1 > 1 this equivalence follows immediately from the log decay

of α at infinity noting that |x| >r12 , |y| >

r22 . If r1 < 1 we use the log-Hölder continuity at the origin instead to prove the

equivalence as we did in the first case.The case r2 > 2r1. Again, if r1 > 1 then from the logarithmic decay condition at infinity we derive

rα(x)1 =

r1r2

α(y)


2 .

r1r2

α−

rα(y)2 .

If r1 < 1 and r2 < 1, we use the fact that α has a log decay at the origin to obtain rα(x)−α(y)2 ≈ 1 for x ∈ B(0, r1), y ∈ B(0, r2),

and consequently to get

rα(x)1 =

r1r2

α(x)


2 .

r1r2

α−

rα(y)2 .

If r1 < 1 and r2 > 1, then we have

rα(x)1 6 rα−

1 = rα−

1 r−α(y)2 rα(y)

2 6

r1r2

α−

rα(y)2 .

Hence the lemma is proved. �

Proof of Lemma 2.2. Step 1. We show that ∥χR∥p(·) ≈ |R|1

p(x) .

First we prove that|R|− 1

p(x) χR(·)

p(·)

. 1 (with constants independent of x and r). Since p+ < ∞, by the unit ball

property it is sufficient to show that ϱp(·)|R|−

1p(x) χR

6 c , which in turn holds if there exists c1 > 0 such that

|R|p(x)−p(y)

p(x) 6 c1, for all x, y ∈ R.


When |R| > 1 the latter is obtained from the inequalityp(x) − p(y)p(x)

log |R| 6 c2,

which is a consequence of the logarithmic decay condition of p at infinity, noting thatp(x) − p(y)p(x)

log |R| 6 |p(x) − p∞| log |R| + |p(y) − p∞| log |R|

and that |x|, |y| > r2 .

The case 2−n 6 |R| < 1 is obvious since then

|p(x) − p(y)| log1|R|

6 2np+ log 2.

In the case |R| < 2−n, we use the fact that |x| < r, |y| < r and the log-Hölder decay of p at the origin to get the estimate

|p(x) − p(y)| log1|R|

. 1.

Nowwe show that |R|1

p(x) . ∥χR∥p(·). This is a consequence of Hölder’s inequality and the estimate ∥χR∥p′(·) . |R|1

p′(x) whichwas already proved (recall that p′ inherits the log-Hölder assumptions from p). In fact, we have

|R|1

p(x) = |R|1

p(x) −1

RnχR(y) dy 6 2 |R|−

1p′(x) ∥χR∥p(·)∥χR∥p′(·) . ∥χR∥p(·).

Step 2. We show that |R|1

p(x) ≈ |R|1

p∞ when |R| > 2−n and p has log-decay.As argued before, this relation is equivalent to the inequalities 1

p(x)−

1p∞

log |R| . 1 when |R| > 1,

and 1p(x)

−1p∞

log 1|R|

. 1 when 2−n 6 |R| < 1,

which immediately follow from the logarithmic decay assumption on p at infinity.The proof is complete. �

7. Optimality of some conditions

In this section, we discuss the optimality of some conditions assumed in the main statements. In the calculations belowwe consider the 1-dimensional case for simplicity.

Claim I. The condition (4.11) (in Theorem 4.10) is necessary to the boundedness of Tλ from Kα(·)

p(·),q(Rn) to Kα(·)

p∗(·),q(Rn).

Proof. We only consider 0 < q < ∞. Let f = χB0 . Simple calculations show that f ∈ Kα(·)

p(·),q(R). Taking the Riesz potentialoperator Iλ, we derive

Iλf (x) =

B0

1|x − y|1−λ

dy & 2k(λ−1), x ∈ Rk.

Hence

∥Iλf ∥q

Kα(·)

p∗(·),q

& ∥Iλf ∥qKα∞

p∗(·),q&k>1

2k(λ−1)q 2kα∞q∥ χk∥

qp∗(·) ≈

k>1

2k(α∞−1+1/p∞)q

where we used ∥χk∥p∗(·) ≈ |Rk|1

p∗∞ (by Lemma 2.2). Therefore, if Iλ maps Kα(·)

p(·),q(R) into Kα(·)

p∗(·),q(R), then α∞ < 1 − 1/p∞

must necessarily hold.In order to prove the necessity of the left-hand side of (4.11), we now take f = χN (N ∈ N). We have

∥f ∥Kα(·)p(·),q

≈ ∥f ∥Kα∞p(·),q

≈ 2Nα∞ ∥χN∥p(·) ≈ 2N(α∞+1/p∞)


again by Lemma 2.2. On the other hand, since |x − y| 6 |x| + |y| < 2N+1 for any k 6 N, x ∈ Rk and y ∈ RN , we also have

Iλf (x) =

RN

dy|x − y|1−λ

& 2Nλ, x ∈ Rk, k ∈ {1, . . . ,N},

with the implicit constant independent of k, x and N . Therefore,

∥Iλf ∥q

Kα(·)

p∗(·),q

& ∥Iλf ∥qKα∞

p∗(·),q&

Nk=1

2kα∞q∥(IλχN)χk∥

qp∗(·)

& 2NλqN

k=1

2kα∞q∥χk∥

qp∗(·) ≈ 2Nλq

Nk=1

2k(α∞−λ+1/p∞)q

once again by Lemma 2.2. Hence, if Iλ is bounded from Kα(·)

p(·),q(R) into Kα(·)

p∗(·),q(R), we get

2NλqN

k=1

2k(α∞−λ+1/p∞)q . 2(α∞+1/p∞)Nq

for every N ∈ N or, equivalently,N

k=1

2(k−N)(α∞−λ+1/p∞)q . 1

for every N ∈ N. Thus α∞ > λ − 1/p∞ must necessarily hold. �

Claim II. The condition (4.3) is necessary to the boundedness of T from Kα(·)

p(·),q(Rn) to Kα(·)

p(·),q(Rn) (in Theorem 4.2).

Proof. We can use the same examples of the previous proof, but taking the maximal operator instead.For every k ∈ N and x ∈ Rk we have B0 ⊂ B(x, 2k+1). Hence

MχB0(x) = supr>0

1|B(x, r)|

B(x,r)

χB0(y) dy = supr>0

|B(x, r) ∩ B0|

|B(x, r)|

>|B(x, 2k+1) ∩ B0|

|B(x, 2k+1)|=

|B0|

|B(x, 2k+1)|≈ 2−k, x ∈ Rk.

As in the proof of Claim I, we get

∥MχB0∥q

Kα(·)p(·),q

&k>1

2k(α∞−1+1/p∞)q.

Therefore, MχB0 ∈ Kα(·)

p(·),q(R) implies α∞ < 1 − 1/p∞.We have seen that ∥χN∥Kα(·)

p(·),q≈ 2N(α∞+1/p∞) for every N ∈ N. Since RN ⊂ B(x, 2N+1) for all x ∈ Rk with k 6 N , we have

MχN(x) >|RN |

|B(x, 2N+1)|≈ 1, x ∈ Rk, k ∈ {1, . . . ,N},

and hence

∥MχN∥q

Kα(·)p(·),q

&

Nk=1

2k(α∞+1/p∞)q.

The boundedness of M in Kα(·)

p(·),q(R) implies α∞ > −1/p∞. We have then shown that the condition (4.3) is optimal. �

Claim III. The assumption q0 6 q1 is optimal in Theorems 4.10 and 4.12.

Proof. Take fN(x) = |x|−(α∞+1/p∞)χ{1<|x|<2N }(x) with N ∈ N. Simple calculations yield

∥fN∥q0Kα(·)p(·),q0

=

Nk=1

2kα(·)| · |

−(α∞+1/p∞)χkq0p(·) .

Nk=1

2−kq0/p∞ ∥2k(α(·)−α∞)χk∥q0p(·).

By Lemma 2.2 and the fact that 2kα(x)≈ 2kα∞ for x ∈ Rk, we conclude that

∥fN∥q0Kα(·)p(·),q0

.

Nk=1

1 = N.


On the other hand,

IλfN(x) >

Rk

|y|−(α∞+1/p∞)

|x − y|1−λdy & 2k(λ−α∞−1/p∞), x ∈ Rk, k ∈ {1, . . . ,N},

since |x − y| < 2k+1 for any x, y ∈ Rk with 1 6 k 6 N . Hence

∥IλfN∥q1Kα(·)

p∗(·),q1

&

Nk=1

∥2k(α(·)−α∞+λ−1/p∞)χk∥q1p∗(·) &

Nk=1

1 = N,

after using Lemma 2.2 once again. Therefore, if Iλ is bounded from Kα(·)

p(·),q0(R) into Kα(·)

p∗(·),q1(R), then we obtain N1/q1 . N1/q0 .

This gives the desired result. The same arguments can be used in the inhomogeneous case. �

Acknowledgments

The authors would like to thank Stefan Samko and the anonymous referee for their valuable comments to the originalversion of this paper.

First author’s work was partially supported by FEDER funds through COMPETE–Operational Programme Factors ofCompetitiveness (‘‘Programa Operacional Factores de Competitividade’’) and by Portuguese funds through the Center forResearch and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Scienceand Technology (‘‘FCT—Fundação para a Ciência e a Tecnologia’’), within project PEst-C/MAT/UI4106/2011 with COMPETEnumber FCOMP-01-0124-FEDER-022690.

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maximal, potential and singular type operators on herz spaces with variable exponents

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