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# [

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###### Abstract

Let be the class of functions bounded away from one and infinity and such that the Hardy-Littlewood maximal function is bounded on the variable Lebesgue space . We denote by the class of variable exponents for which with some , , and . Rabinovich and Samko [13] observed that each globally log-Hölder continuous exponent belongs to . We show that the class contains many interesting exponents beyond the class of globally log-Hölder continuous exponents.

On an Interesting Class of Variable Exponents] On an Interesting Class of Variable Exponents A. Yu. Karlovich]Alexei Yu. Karlovich

I. M. Spitkovsky]Ilya M. Spitkovsky

\@mkboth\shortauthors\shorttitle

[

To Professor Stefan Samko on the occasion of his 70th birthday

Mathematics Subject Classification (2000). Primary 42B25; Secondary 46E30, 26A16.

Keywords. Variable Lebesgue space, variable exponent, globally log-Hölder continuous function, Hardy-Littlewood maximal operator..

\@xsect

Let be a measurable a.e. finite function. By we denote the set of all complex-valued functions on such that

 Ip(⋅)(f/λ):=∫Rn|f(x)/λ|p(x)dx<∞

for some . This set becomes a Banach space when equipped with the norm

 ∥f∥p(⋅):=inf{λ>0:Ip(⋅)(f/λ)≤1}.

It is easy to see that if is constant, then is nothing but the standard Lebesgue space . The space is referred to as a variable Lebesgue space. We will always suppose that

 1

Under these conditions, the space is separable and reflexive, and its dual is isomorphic to , where

 1/p(x)+1/p′(x)=1(x∈Rn)

(see e.g. [5, Chap. 3]).

Given , the Hardy-Littlewood maximal operator is defined by

 Mf(x):=supQ∋x1|Q|∫Q|f(y)|dy

where the supremum is taken over all cubes containing (here, and throughout, cubes will be assumed to have their sides parallel to the coordinate axes). By denote the set of all measurable functions such that (1.1) holds and the Hardy-Littlewood maximal operator is bounded on .

Assume that (1.1) is fulfilled. L. Diening [4] proved that if satisfies

 |p(x)−p(y)|≤clog(e+1/|x−y|)(x,y∈Rn) (1.2)

and is constant outside some ball, then . Further, the behavior of at infinity was relaxed by D. Cruz-Uribe, A. Fiorenza, and C. Neugebauer [2, 3], who showed that if satisfies (1.2) and there exists a such that

 |p(x)−p∞|≤clog(e+|x|)(x∈Rn), (1.3)

then . Following [5, Section 4.1], we will say that if conditions (1.2)–(1.3) are fulfilled, then is globally log-Hölder continuous.

A. Nekvinda [11, 12] relaxed condition (1.3). To formulate his results, we will need the notion of iterated logarithms. Put

 e0:=1,ek+1:=exp(ek)fork∈Z+:={0,1,2,…}.

The function is defined on the interval by

For and , put

 bk,α(x):=−1αddx(log−αkx)(x≥ek).

We say that a measurable function belongs to the Nekvinda class if conditions (1.1)–(1.2) are fulfilled and there exists a monotone function satisfying

 1

and such that for some , , ,

 ∣∣∣dsdx(x)∣∣∣≤Kbk,α(x)forx≥ek, (1.5)

and

 ∫{x∈Rn:p(x)≠s(|x|)}c1/|p(x)−s(|x|)|dx<∞ (1.6)

for some . According to [12, Theorem 2.2], . In particular, all locally log-Hölder continuous (that is, satisfying (1.2)) radially monotone exponents with monotone satisfying (1.4)–(1.5) belong to .

Observe, however, that A. Lerner [10] (see also [5, Example 5.1.8]) constructed exponents discontinuous at zero or at infinity and such that, nevertheless, belong to . Thus neither (1.2) nor (1.3) is necessary for . For more informastion on the class we refer to [5, Chaps. 4–5].

Finally, we note that V. Kokilashvili and S. Samko [8, 9]; V. Kokilashvili, N. Samko, and S. Samko [7]; D. Cruz-Uribe, L. Diening, and P. Hästö [1] studied the boundedness of the Hardy-Littlewood maximal operator on variable Lebesgue spaces with weights under assumptions (1.1)–(1.3) or their analogues in the case of metric measure spaces.

We denote by the collection of all variable exponents for which there exist constants , , and a variable exponent such that

 1p(x)=θp0+1−θp1(x) (1.7)

for almost all .

This class implicitly appeared in V. Rabinovich and S. Samko’s paper [13] (see also [14]). Its introduction is motivated by the fact that the boundedness of the Hardy-Littlewood maximal operator on implies the boundedness of many important linear operators on (see e.g. [5, Chap. 6]). If such a linear operator is also compact on the standard Lebesgue space , then, by a Krasnoselskii type interpolation theorem for variable Lebesgue spaces, it is compact on the variable Lebesgue space as well.

In [13, Theorem 5.1], the boundedness of the pseudodifferential operators with symbols in the Hörmander class on the variable Lebesgue spaces was established, provided that satisfies (1.1)–(1.3). Then the above interpolation argument was used in the proof of [13, Theorem 6.1] to study the Fredholmness of pseudodifferential operators with slowly oscillating symbols on . In particular, the following is implicitly contained in the proof of [13, Theorem 6.1].

###### Theorem 1.1 (V. Rabinovich-S. Samko).

If satisfies (1.1)–(1.3), then belongs to .

Recently we generalized [13, Theorem 5.1] and proved that the pseudodifferential operators with symbols in the Hörmander class , where and , are bounded on variable Lebesgue spaces whenever (see [6, Theorem 1.2]). Further, [6, Theorem 1.3] delivers a sufficient condition for the Fredholmness of pseudodifferential operators with slowly oscillating symbols in the Hörmander class under the assumption that . The proof follows the same lines as V. Rabinovich and S. Samko’s proof of [13, Theorem 6.1] for exponents satisfying (1.1)–(1.3) and is based on the above mentioned interpolation argument.

The aim of this paper is to show that the class is much larger than the class of globally log-Hölder continuous exponents. Our first result says that all Nekvinda’s exponents belong to .

###### Theorem 1.2.

We have .

Modifying A. Lerner’s example [10], we further prove that there are exponents in that do not satisfy (1.3).

###### Theorem 1.3.

There exists a sufficiently small such that for every satisfying the function

 p(x)=2+α+βsin(log(log|x|)χ{x∈Rn:|x|≥e}(x))(x∈Rn)

belongs to .

The paper is organized as follows. For completeness, we give a proof of Theorem \@setrefth:RS in Section \@setrefsec:RS. Further, in Section \@setrefsec:Nekvinda we prove Theorem \@setrefth:Nekvinda. Section \@setrefsec:Lerner contains A. Lerner’s sufficient condition for in terms of mean oscillations of a function . In Section \@setrefsec:proof we show that Theorem \@setrefth:example follows from the results of Section \@setrefsec:Lerner.

\@xsect\@xsect

In this subsection we give a proof of Theorem \@setrefth:RS. A part of this proof will be used in the proof of Theorem \@setrefth:Nekvinda in the next subsection.

###### Proof of Theorem \@setrefth:RS.

Suppose satisfies (1.1)–(1.3). Let , , and be such that (1.7) holds. Then

 p1(x)=p0(1−θ)p(x)p0−θp(x). (2.1)

If we choose , then for ,

 1

Therefore

 (1−θ)p−≤(1−θ)p(x)≤p1(x)≤p0(1−θ)p(x)≤p0(1−θ)p+.

Hence and . If we choose such that , then and thus satisfies (1.1).

From (2.1) it follows that

 p1(x)−p1(y)=p20(1−θ)(p(x)−p(y))(p0−θp(x))(p0−θp(y))(x,y∈Rn).

Then, taking into account (2.2), we get

 |p1(x)−p1(y)|≤p20(1−θ)|p(x)−p(y)|(x,y∈Rn). (2.3)

Now put

 (p1)∞:=p0(1−θ)p∞p0−θp∞,

where is the constant from (1.3). Then

 p1(x)−(p1)∞=p20(1−θ)(p(x)−p∞)(p0−θp(x))(p0−θp∞)(x∈Rn). (2.4)

From (1.3) it follows that

 p∞=lim|x|→∞p(x).

Hence . Therefore

 1

From (2.4) and (2.5) we obtain

 |p1(x)−(p1)∞|≤p20(1−θ)|p(x)−p∞|. (2.6)

From estimates (2.3), (2.6) and (1.2), (1.3) for the exponent we obtain that the exponent satisfies (1.2) and (1.3). Therefore and thus belongs to . ∎

\@xsect

Suppose . Let , , and be such that (1.7) holds. In the previous subsection we proved that if and , then satisfies (1.1)–(1.2).

Since , there exists a monotone function such that (1.4)–(1.6) are fulfilled. Let

 s1(x):=p0(1−θ)s(x)p0−θs(x)(x≥0). (2.7)

Put

 s−:=infx∈[0,∞)s(x),s+:=supx∈[0,∞)s(x). (2.8)

We will choose and subject to

 p0≥max{p+,s+},θ∈(0,min{1−1/p−,1−1/s−}).

Then, for ,

 1

Therefore, for ,

 (1−θ)s−≤(1−θ)s(x)≤s1(x)≤p0(1−θ)s(x)≤p0(1−θ)s+

and

 1<(1−θ)s−≤(s1)−,(s1)+≤p0(1−θ)s+<∞, (2.10)

where and are defined by (2.8) with in place of .

If for , then

 p0−θs(x)≥p0−θs(y),p0(1−θ)s(x)≤p0(1−θ)s(y).

Thus

 s1(x)=p0(1−θ)s(x)p0−θs(x)≤p0(1−θ)s(y)p0−θs(y)=s1(y),

that is, is monotone.

It is easy to see that for almost all ,

 ds1dx(x)=p20(1−θ)(p0−θs(x))2dsdx(x).

Taking into account (2.9), we obtain

 ∣∣∣ds1dx(x)∣∣∣≤p20∣∣∣dsdx(x)∣∣∣(x>0). (2.11)

From (2.1) and (2.7) we get

 E:={x∈Rn:p(x)≠s(|x|)}={x∈Rn:p1(x)≠s1(|x|)}

and for ,

 p1(x)−s1(|x|)=p20(1−θ)(p(x)−s(|x|))(p0−θp(x))(p0−θs(|x|)).

From this equality and inequalities (2.2) and (2.9) we get for ,

 (1−θ)|p(x)−s(|x|)|≤|p1(x)−s1(|x|)|≤p20(1−θ)|p(x)−s(|x|)|.

Therefore, there is a constant such that

 ∫Ec1/|p1(x)−s1(|x|)|dx≤M∫Ec1/|p(x)−s(|x|)|dx. (2.12)

Since satisfies (1.4)–(1.6), from (2.10)–(2.12) it follows that satisfies (1.4)–(1.6), too. Thus . By [12, Theorem 2.2], , which finishes the proof of . ∎

\@xsect\@xsect

Let . For a cube , put

 fQ:=1|Q|∫Qf(x)dx.

We recall that the mean oscillation of over a cube is given by

 Ω(f,Q):=1|Q|∫Q|f(x)−fQ|dx.
###### Lemma 3.1.

If is a Lipschitz function with the Lipschitz constant and is a real-valued function, then for every ,

 Ω(F∘f,Q)≤2cΩ(f,Q).
###### Proof.

It is easy to see that

 Ω(f,Q)≤1|Q|2∫Q∫Q|f(x)−f(y)|dxdy≤2Ω(f,Q).

From this estimate we immediately get the statement. ∎

Given any cube , let

 ℓ(Q):=log(e+max{|Q|,|Q|−1,|cenQ|}),

where is the center of .

###### Lemma 3.2 (see [10, Proposition 4.2]).

If

then

 supQ⊂Rnℓ(Q)Ω(L,Q)<∞.
###### Theorem 3.3 (see [10, Theorem 1.2]).

There is a positive constant , depending only on , such that for any measurable function with

 0

we have .

\@xsect

Let the function be as in Lemma \@setrefle:double-logarithm. Suppose and put

 F(x):=α+βsinx(x∈R)

and

 q(y):=F(L(y)),p(y):=2+q(y)(y∈Rn).

Then

 q−=α−β>0,∥q∥L∞(Rn)=α+β<∞. (3.1)

From Lemma \@setrefle:double-logarithm we know that

 CL:=supQ⊂Rnℓ(Q)Ω(L,Q)<∞. (3.2)

Since is a Lipschitz function with the Lipschitz constant equal to , we obtain from Lemma \@setrefle:Lipschitz-oscillation that

 supQ⊂Rnℓ(Q)Ω(q,Q) =supQ⊂Rnℓ(Q)Ω(F∘L,Q) ≤2βsupQ⊂Rnℓ(Q)Ω(L,Q)=2βCL. (3.3)

From (3.1), (3.3), and it follows that

 ∥q∥L∞(Rn)+supQ⊂Rnℓ(Q)Ω(q,Q)≤α+β+2βCL<2α(1+CL). (3.4)

Fix and take the function such that

 12+F(x)=θ2+1−θ2+G(x)(x∈R).

Then

 12+G(x)=11−θ(12+F(x)−θ2)=2−θ(2+F(x))2(1−θ)(2+F(x)).

Therefore

 G(x) =2(1−θ)(2+F(x))2−θ(2+F(x))−2 =2(1−θ)(2+F(x))−4+2θ(2+F(x))2−θ(2+F(x)) =2(2+F(x))−42−θ(2+F(x)) =2F(x)2−θ(2+F(x))

and

 G′(x)=2F′(x)(2−2θ−θF(x))+2θF(x)F′(x)[2−θ(2+F(x))]2=4(1−θ)F′(x)[2−θ(2+F(x))]2

Since , we have for and then

 2−2θ−θF(x)<2. (3.5)

Hence

 G(x)=2F(x)2−2θ−θF(x)>F(x).

If we take , then . Therefore . Hence

 G(x)=2F(x)2−2θ−θF(x)≤2F(x) (3.6)

and

 |G′(x)|=4(1−θ)|F′(x)|[2−2θ−θF(x)]2≤4(1−θ)|F′(x)|<4|F′(x)|. (3.7)

Thus is Lipschitz with the Lipschitz constant equal to .

Put for . Then from (3.5)–(3.6) it follows that

 (q1)−≥α−β>0,∥q1∥L∞(Rn)≤2(α+β). (3.8)

Further, from (3.7) and Lemma \@setrefle:Lipschitz-oscillation we obtain

 supQ⊂Rnℓ(Q)Ω(q1,Q)=supQ⊂Rnℓ(Q)Ω(G∘L,Q)≤8βCL. (3.9)

From (3.8)–(3.9) and we deduce

 ∥q1∥L∞(Rn)+supQ⊂Rnℓ(Q)Ω(q1,Ω)≤2(α+β)+8βCL<8α(1+CL). (3.10)

Let be the constant from Theorem \@setrefth:Lerner. Put

 ε:=μn8(1+CL).

If , then from (3.1), (3.4) and (3.8), (3.10) it follows that , , and

 ∥q∥L∞(Rn)+supQ⊂Rnℓ(Q)Ω(q,Ω)≤μn,∥q1∥L∞(Rn)+supQ⊂Rnℓ(Q)Ω(q1,Ω)≤μn.

By Theorem \@setrefth:Lerner, and . Hence, (1.7) holds with , and . Thus . ∎

## References

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• [10] A. K. Lerner, Some remarks on the Hardy-Littlewood maximal function on variable spaces. Math. Z. 251 (2005), no. 3, 509-521.
• [11] A. Nekvinda, Hardy-Littlewood maximal operator on . Math. Inequal. Appl. 7 (2004), 255–265.
• [12] A. Nekvinda, Maximal operator on variable Lebesgue spaces for almost monotone radial exponent. J. Math. Anal. Appl. 337 (2008), 1345-1365.
• [13] V. Rabinovich and S. Samko, Boundedness and Fredholmness of pseudodifferential operators in variable exponent spaces. Integral Equations Operator Theory 60 (2008), 507–537.
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Departamento de Matemática
Quinta da Torre
2829–516 Caparica
Portugal

e-mail: oyk@fct.unl.pt

Department of Mathematics
College of William & Mary
Williamsburg, VA, 23187-8795
U.S.A.

e-mail: ilya@math.wm.edu

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