It is proven that if 1≤p(·)<∞ in a bounded domain Ω⊂Rn and if p(·)∈EXPa(Ω) for some a>0, then given f∈Lp(·)(Ω), the Hardy-Littlewood maximal function of f, Mf, is such that p(·)log(Mf)∈EXPa/(a+1)(Ω). Because a/(a+1)<1, the thesis is slightly weaker than (Mf)λp(·)∈L1(Ω) for some λ>0. The assumption that p(·)∈EXPa(Ω) for some a>0 is proven to be optimal in the framework of the Orlicz spaces to obtain p(·)log(Mf) in the same class of spaces.

1. Introduction

Let Ω⊂Rn be a bounded domain. By p(·), we denote a finite exponent function on Ω, that is, a Lebesgue measurable function p(·):Ω→[1,∞[. The space Lp(·)(Ω) is defined as the set of Lebesgue measurable functions f on Ω such that (1)∫Ωμfxpxdx<∞,for some μ>0.

The Hardy-Littlewood maximal operator of a function f∈Lloc1(Rn) is defined by (2)Mfx=supQ∋x⨏Qfydy,x∈Rn,where the supremum is taken over all cubes Q⊂Rn that contain x and whose sides are parallel to the coordinate axes (the symbol ⨏Q denotes 1/|Q|, and for a Lebesgue measurable E⊂Rn, the symbol |E| denotes its Lebesgue measure). If f∈L1(Ω) (and, in particular, also if f∈Lp(·)(Ω); see [1, Proposition 2.41 page 36]), then by Mf, we mean the maximal operator computed on the extension of f by zero outside Ω; in this case, it is known (see, e.g., [2, 8.15 page 43] or [3]) that(3)Mfq∈L1E∀0<q<1,∀E⊂Rn,E<∞,(4)∫EMfxqdx≤cqE1-q∫Ωfxdxq.The main classical result regarding the maximal operator is that it is bounded in every Lebesgue space with (constant) exponent greater that 1 (see, e.g., [2, 4–7] and references therein for the main results regarding the maximal operator). The extension of this result in the framework of variable Lebesgue spaces theory has been intensively studied (see, e.g., [1, Chapter 3] or [8, Chapter 4], [9, Chapter 1] for a collection of results regarding this topic; see also the survey [10]).

Assume that p(·) is such that p+=esssupΩp(·)<∞. It is well-known that M is not necessarily bounded in Lp(·)(Ω) even if, in addition, p-=essinfΩp(·)>1 and p(·) is continuous (see [11]; the same example is also presented in [1, Example 4.43 page 160]). A well-known sufficient condition for boundedness of the maximal operator for exponents p(·) such that p->1 is expressed in terms of the so-called log-Hölder continuity, which may be required to the function 1/p(·) (see [12] and references therein for details). However, sufficient conditions that do not require continuity do exist (see, e.g., [13, 14]). In [15], another class of exponents that ensures the boundedness of the maximal operator for bounded domains has been presented.

However, we can assert that(5)p+<∞,f∈Lp·Ω⟹∃λ>0:Mfλp·∈L1Ω.In fact, by (3), for 0<q<1, we have (6)f∈Lp·Ω⟹f∈L1Ω⟹Mfq=Mfp+q/p+∈L1Ω⟹Mfp·q/p+∈L1Ω.Note that from (5), one gets that, for 0<β<1, if p+<∞,(7)f∈Lp·Ω⟹∃λ>0:expλp·logMfβ∈L1Ω.For unbounded (and possibly discontinuous) exponents, (7) is not generically true (see the example presented below). In this note, we prove that, for some 0<β<1, (7) holds true under an assumption of exponential integrability of the exponent. We observe that some type of high integrability of the exponent is needed; in fact, the result is not true if it is only assumed that p(·) is contained in a Lebesgue space with any finite, constant exponent. For instance, if p(x)=x-b+1 on (0,1), b>0, then f(x)=(1/x)1/(x-b+1), x∈(0,1), is such that f∈Lp(·)(0,1) but for every λ>0, it is (8)1<∫01fxdxλ<∞.Therefore, for any 0<β<1, (9)∫01expλpxlogMfxβdx≥∫01expλx-b+1log∫01fβdx≥cb,β∫01λx-b+1log∫01fβ1/bβdx=cb,β∫01λ1/bx-b+11/blog1/b∫01fdx=∞.For other results regarding the maximal operator that are specific to unbounded exponents, see [16].

2. The Local Estimate

In the following, three prerequisites are necessary. The first is the well-known Fefferman-Stein inequality ([17]; see also [18])(10)∫RnMfxqgxdx≤cq∫RnfxqMgxdx,f,g∈Lloc1Rn,which is true for q>1 (here q is constant). The second is a well-known extrapolation characterisation (see, e.g., [19–23]) of EXPa(Ω), a>0, the Orlicz space of the functions, which can be characterised in one of the two following equivalent ways:(11)limsupk→∞1k1/a⨏Ωhxkdx1/k<∞⟺∃μ>0:⨏Ωexpμhxadx<∞.We remark that the growth of integrability of the functions in EXPa(Ω), 0<a<1, is somehow connected with the A∞ property for Young functions; see [24]. Finally, the third tool is an elementary inequality. For k≥1, α≥1, and a positive constant (12)γk,α=kαk-1+1exp-kα,(as usual, subscripts indicate the dependence of the constant on the variables involved, which may change from line to line), it is(13)logkξ≤kαk+γk,αξ-expkα∀ξ≥expkα.This inequality is a direct consequence of the concavity of the kth power of the logarithm in (exp(k-1),∞).

We are now ready for the proof of the following theorem.

Theorem 1.

Let Ω be a bounded domain in Rn, and let p(·)∈EXPa(Ω) for some a>0 be such that 1≤p(·)<∞. If f∈Lp(·)(Ω), f≢0, then p·log(Mf)∈EXPa/(a+1)(Ω).

Proof.

Let us first assume that p->1. Considering that p(·)∈EXPa(Ω), for every k≥logfL1(Ω) we have (14)1Ω∫x∈Ω:Mfx≤expkpxklogMfxkdx1/k≤1Ω·∫x∈Ω:Mfx≤expkpxklogkexpkdx1/k=1Ω∫x∈Ω:Mfx≤expkpxkkkdx1/k≤k⨏Ωpxkdx1/k≤kcp·k1/a=cp·k1+1/a.On the other hand, for any q>1 and r>1 such that qr<p-, the following inequalities hold (we are going to apply (13) with the choices α=1 and ξ=Mf(x)q; note that because we work on the set in which Mf(x)>exp(k), it is also true that Mf(x)q>exp(k)):(15)1Ω∫x∈Ω:Mfx>expkpxklogMfxkdx1/k=1q1Ω∫x∈Ω:Mfx>expkpxk·logkMfxqdx1/k≤1q1Ω·∫x∈Ω:Mfx>expkpxk·kk+γk,1Mfxq-expkdx1/k=1q1Ω·∫x∈Ω:Mfx>expkpxkkkdx+1Ω·∫x∈Ω:Mfx>expkpxk·γk,1Mfxq-expkdx1/k≤1q1Ω·∫x∈Ω:Mfx>expkpxkkkdx1/k+1q1Ω·∫x∈Ω:Mfx>expkpxk·γk,1Mfxq-expkdx1/k,and because γk,11/k=k/e, by again using the fact that p(·)∈EXPa(Ω) and by applying (10),(16)≤kq⨏Ωpxkdx1/k+keq1Ω·∫x∈Ω:Mfx>expkpxkMfxqdx1/k≤kq·cp·k1/a+keq1Ω∫RnMfxqpxk·χx∈Ω:Mfx>expkxdx1/k≤cp·qk1+1/a+keqcqΩ∫Rnfxq·Mpxkχx∈Ω:Mfx>expkxdx1/k≤cp·q·k1+1/a+keqcqΩ∫ΩfxqMpxkdx1/k.By Hölder’s inequality and the boundedness of the maximal operator in Lr/(r-1)(Ω),(17)≤cp·qk1+1/a+keqcq1/k⨏Ωfxqrdx1/kr·⨏ΩMpxkr/r-1dxr-1/kr≤cp·qk1+1/a+cq1/kkeq⨏Ωfxqrdx1/qr·q/k·cr⨏Ωpxkr/r-1dxr-1/kr≤cp·qk1+1/a+cq1/kcrkeq⨏Ωfxp-dx1/p-·q/k·cp·krr-11/a≤cp·qk1+1/a+cacq1/kcr·cp·q⨏Ωfxp-dx1/p-·q/kk1+1/a≤Ck1+1/a,where C>0 is a constant that depends on all parameters and f∈Lp(·)(Ω) but is independent of k≥1.

Combining the estimates obtained above, we obtain (18)⨏ΩpxklogMfxkdx1/k≤Ck1+1/a.That is, using (11) (with a replaced by a/(a+1)), we obtain the assertion.

Now, let p(·)∈EXPa(Ω) for some a>0, and assume p-=1. If p+<∞, we already proved (see (7)) that the assertion is trivially true. If p+=∞, set(19)A=x∈Ω:1≤px≤2a.e.,B=x∈Ω:px>2a.e.,and observe that A is nonempty (because p-=1) and B is nonempty (because p+=∞). By (3), (Mf)2/3∈L1(A). Therefore, (Mf)p(·)/3∈L1(A), from which (20)expp·3logMfa/a+1∈L1A.On the other hand, since p(·)∈EXPa(B) and essinfBp(·)≥2, there exists μ>0 such that (21)expμp·logMfa/a+1∈L1B.In conclusion, since Ω=A∪B, exp((λp(·)|log(Mf)|)a/(a+1))∈L1(Ω) for λ=min{1/3,μ}.

The heart of Theorem 1 can be stated as follows: if p(·) belongs to some EXPa(Ω), then for any f∈Lp(·)(Ω) also p·log(Mf) does. From this perspective, our result is optimal: if one assumes that p(·) belongs to any Orlicz space that contains all of the EXPa(Ω)’s, it is possible to construct an exponent that is not in any EXPa(Ω) and a function f∈Lp(·)(0,1) such that p·log(Mf) is also not in any EXPa(Ω). This statement is the essence of the following result (for the definition of Young functions and the classical embedding theorem for Orlicz spaces; see, e.g., [25]).

Theorem 2.

Let Φ:[0,∞[→[0,∞[ be a Young function such that for all a∈]0,1](22)Φt≤expta∀t>ta.There exists p(·)∈LΦ(0,1), p(·)∉EXPa(0,1)∀a∈]0,1], p->1, and there exists f∈Lp(·)(0,1) such that(23)p·logMf∉EXPa0,1∀a∈0,1.

Proof.

Set A(t)=log(Φexpt) for t large. For every β∈]0,1], set Aβ(t)=exp(βt). Then, (22) reads as (24)Alogt-Aalogt≤0,or, equivalently,(25)A-1t≥Aa-1t=1alogt,for large t. Set (26)px=Φ-1Φ2x=expA-1logΦ2xx∈0,1.We observe that, trivially, p(·)∈LΦ(0,1) and p-=2>1.

We now prove that p(·)∉EXPa(0,1)∀a∈]0,1]. Let a∈]0,1] and k>0. Fix β∈]0,a[ (for instance, β=a/2 can be assumed in the following). For sufficiently small x, using (25) with a replaced by β, we see that(27)expkpxa=expkexpA-1logΦ2xa≥expkexp1βloglogΦ2xa=expklogΦ2x1/βa≥exp21/alogΦ2x1/aa=Φ22x.Therefore,(28)expkp·a∉L10,1∀a∈0,1,∀k>0.Let us now set (29)fx=1x1/pxx∈0,1,so that f∈Lp(·)(0,1). It remains to prove (23). Fix λ>0, set k=λlog(∫01|f(x)|dx), and note that k>0. Thus, (30)∫01expλpxlogMfxadx≥∫01expkpxadx=∞,where the last equality results from (28).

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

On January 27, 2011, Miroslav Krbec, who should have been the second author of this paper, sent me a message that contained an idea for a proof of a local boundedness-type result for the maximal operator in variable exponent Lebesgue spaces. At that time, I was writing a book [1] on variable Lebesgue spaces with David Cruz-Uribe. The collaboration with Mirek had begun several years before, but we never discussed questions related to variable exponents. Unfortunately, I only glanced at that message, and I replied with a short, evasive answer. I thought the idea was nothing special. On March 4, 2011 Mirek again asked me about the idea. On April 14, 2012, Mirek invited me to visit Prague in June, and he asked me once more to consider the idea. At this point, my opinion changed: I decided that the idea was a very good starting point for a project. With this new perspective, I planned to pack my enthusiasm in my luggage and announce to Mirek in June that a new research project concerning variable Lebesgue spaces should be initiated. However, Mirek died on June 17, 2012, one week before my arrival at Prague. For me, his death was a significant loss: I lost both a friend and a collaboration that had given me much joy from both the human and scientific perspectives. I was upset with myself. I had been lazy, and if I had spent time on this idea, we could have published one more paper together. However, my enthusiasm was too late, and the paper went unwritten. After Mirek’s death, I spent much effort on another project (again with Mirek as coauthor) that I was working on (the final goal of which was to prove that the small Lebesgue spaces are the correct setting in which to define the optimal dimensional-free gain of integrability for the Sobolev embedding theorem, when the domain is the unit cube in all dimensions). This other project yielded a publication ([26]; for a further development, see [27]). The overall result was that, after the original idea about the variable Lebesgue space question, much time passed, and I was again upset with myself. This paper tries to fill the gap left by the significant amount of time that has passed. This work will never make up for my laziness; however, it at last provides a venue for an idea originally proposed by Mirek.

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