A Local Estimate for the Maximal Function in Lebesgue Spaces with EXP-Type Exponents

It is proven that if 1 ≤ p(⋅) < ∞ in a bounded domain Ω ⊂ Rn and if p(⋅) ∈ EXP a (Ω) for some a > 0, then given f ∈ L(Ω), the Hardy-Littlewood maximal function of f,Mf, is such that p(⋅)log(Mf) ∈ EXP a/(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(⋅) ∈ EXP a (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.

The Hardy-Littlewood maximal operator of a function  ∈  1 loc (R  ) is defined by where the supremum is taken over all cubes  ⊂ R  that contain  and whose sides are parallel to the coordinate axes (the symbol ⨏  denotes 1/||, and for a Lebesgue measurable  ⊂ R  , the symbol || denotes its Lebesgue measure).If  ∈  1 (Ω) (and, in particular, also if  ∈  (⋅) (Ω); see [1, Proposition 2.41 page 36]), then by , we mean the maximal operator computed on the extension of  by zero outside Ω; in this case, it is known (see, e.g., [2, 8.15 page 43] or [3]) that 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][5][6][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 (⋅) is such that  + = ess sup Ω (⋅) < ∞.It is well-known that  is not necessarily bounded in  (⋅) (Ω) even if, in addition,  − = ess inf Ω (⋅) > 1 and (⋅) 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 (⋅) such that  − > 1 is expressed in terms of the so-called log-Hölder continuity, which may be required to the function 1/(⋅) (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 In fact, by (3), for 0 <  < 1, we have Note that from (5), one gets that, for 0 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 <  < 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 (⋅) is contained in a Lebesgue space with any finite, constant exponent.For instance, if () =  − + 1 on (0, 1),  > 0, then () = (1/√) 1/( − +1) ,  ∈ (0, 1), is such that  ∈  (⋅) (0, 1) but for every  > 0, it is Therefore, for any 0 <  < 1, For other results regarding the maximal operator that are specific to unbounded exponents, see [16].

The Local Estimate
In the following, three prerequisites are necessary.The first is the well-known Fefferman-Stein inequality ( [17]; see also [18]) which is true for  > 1 (here  is constant).The second is a well-known extrapolation characterisation (see, e.g., [19][20][21][22][23]) of EXP  (Ω),  > 0, the Orlicz space of the functions, which can be characterised in one of the two following equivalent ways: We remark that the growth of integrability of the functions in EXP  (Ω), 0 <  < 1, is somehow connected with the  ∞ property for Young functions; see [24].Finally, the third tool is an elementary inequality.For  ≥ 1,  ≥ 1, and a positive constant (as usual, subscripts indicate the dependence of the constant on the variables involved, which may change from line to line), it is This inequality is a direct consequence of the concavity of the th power of the logarithm in (exp( − 1), ∞).
We are now ready for the proof of the following theorem.

Theorem 1.
Let Ω be a bounded domain in R  , and let (⋅) ∈   (Ω) for some  > 0 be such that On the other hand, for any  > 1 and  > 1 such that  <  − , the following inequalities hold (we are going to apply (13) with the choices  = 1 and  = ()  ; note that because we work on the set in which () > exp(), it is also true that ()  > exp()): and because  1/ ,1 = /, by again using the fact that (⋅) ∈ EXP  (Ω) and by applying (10) where  > 0 is a constant that depends on all parameters and  ∈  (⋅) (Ω) but is independent of  ≥ 1.
The heart of Theorem 1 can be stated as follows: if (⋅) belongs to some EXP  (Ω), then for any  ∈  (⋅) (Ω) also (⋅) log() does.From this perspective, our result is optimal: if one assumes that (⋅) belongs to any Orlicz space that contains all of the EXP  (Ω)'s, it is possible to construct an exponent that is not in any EXP  (Ω) and a function  ∈  (⋅) (0, 1) such that (⋅) log() is also not in any EXP  (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]).
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. , R   ()   ()  ⋅  {∈Ω:()>exp()} () )  ⋅  ( ()   {∈Ω:()>exp()} ()) ) Ω  () /(−1) )