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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