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Variable exponent Lebesgue spaces on metric spaces: the Hardy-Littlewood maximal operator. (English) Zbl 1072.42016

Let \((X, d, \mu)\) be a metric measure space, where \(X\) is a set, \(d\) is a metric on X and \(\mu\) is a non-negative Borel regular outer measure on \(X\) which is finite in every bounded set. A measurable function \(p: X\rightarrow [1,\infty)\) is called to be a variable exponent, and let \(p^+=\text{ ess}\sup_{x\in X}p(x)\) and \(p^-=\text{ ess}\inf_{x\in X}p(x)\). Define \(L^{p(\cdot)}(X)\) to be the set of all \(\mu\)-measurable functions \(u:\;X\to{\mathbb R}\) such that \[ \| u\| _{p(\cdot)}=\inf\bigg\{\lambda>0: \int_X \left| \frac {u(y)}\lambda\right| ^{p(y)}\,d\mu(y)\leq1\bigg\}<\infty. \] The authors first prove that (i) The space \(L^{p(\cdot)}(X)\) is a Banach space; (ii) If X is a locally compact doubling space and \(p^+<\infty\), then continuous functions with compact support are dense in \(L^{p(\cdot)}(X)\).
Let \(p\) be a log-Hölder continuous function which means that \(| p(x)-p(y)| \leq\frac c{-\log d(x,y)}\), when \(d(x,y)\leq1/2\); and let \(Mf\) be the central Hardy-Littlewood maximal function. The main result of this paper is: Let \(X\) be a bounded doubling space. Suppose that \(p\) is log-Hölder continuous with \(1<p^-\leq p^+<\infty\). Then \[ \| Mf\| _{p(\cdot)}\leq C\| f\| _{p(\cdot)}. \] The authors also show that in the metric spaces the maximal function can be bounded even though the variable exponent is not log-Hölder continuous, which is essentially the optimal condition for the maximal operator which is bounded on variable exponent Lebesgue spaces defined on Euclidean spaces. Moreover, the authors prove that if \(X\) is a doubling space and \(p^+<\infty\), then for all \(f\in L^{p(\cdot)}(X)\) and \(t>0\), \[ \mu(\{x\in X: Mf(x)>t\})\leq C\int_X\left(\frac{| f(y)| }{t}\right)^{p(y)}d\mu(y). \] As an application of this result, the authors further derive that if \(X\) is a locally compact doubling space and \(p^+<\infty\), then for every \(u\in L^{p(\cdot)}_{loc}(X)\) and almost every \(x\in X\), \[ \lim\sup_{r\to 0}\frac 1{\mu(B(x,r))} \int_{B(x,r)}| u(y)-u(x)| ^{p(y)}\, d\mu(y)=0. \]
Reviewer: Yang Dachun (Kiel)

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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