## A new proof of the boundedness of maximal operators on variable Lebesgue spaces.(English)Zbl 1207.42011

Given a measurable function $$p(\cdot):\mathbb{R}^n \to [1, \infty]$$, let $$\Omega_{\infty, p(\cdot)}= \{ x \in \mathbb{R}^n : p(x) = \infty \}$$. We define the variable Lebesgue space $$L^{p(\cdot)}$$ to be the set of functions such that for some $$\lambda>0$$, $\rho_{p(\cdot)}(f/\lambda):=\int_{\mathbb{R}^n \setminus \Omega_{\infty, p(\cdot)}} \left(\frac{| f(x) |}{\lambda}\right)^{p(x)}dx+\lambda^{-1}\| f \|_{L^{\infty}(\Omega_{\infty, p(\cdot)})} < \infty.$ $$L^{p(\cdot)}$$ is a Banach space when equipped with the norm $\| f \|_{p(\cdot)}:=\inf \{ \lambda : \rho_{p(\cdot)}(f/\lambda) \leq 1 \}.$ Let \begin{aligned} p_{-} := \text{essinf}_{x \in \mathbb{R}^n} p(x), \\ p_{+} := \text{esssup}_{x \in \mathbb{R}^n} p(x).\end{aligned} Define the exponent function $$q(\cdot)$$ by $\frac{1}{p(x)} - \frac{1}{q(x)} = \frac{a}{n},$ where we let $$1/\infty =0$$. Given $$a, 0 \leq a <n$$, we define $M_a f(x) :=\sup_{Q \ni x}\frac{1}{| Q |^{1- a/n}}\int_Q | f(y) | \,dy.$ The authors prove the following. If $$p(\cdot)$$ satisfies local and global log-Hölder conditions and $$1 < p_{-} \leq p_{+} \leq n/a$$, then $\| M_a f \|_{q(\cdot)}\leq C\| f \|_{p(\cdot)}.$ This theorem was proved with the assumption that $$p_{+}< \infty$$ when $$a=0$$ or $$p_{+}< n/a$$ when $$a>0$$ by D. Cruz-Uribe, A. Fiorenza and C. J. Neugebauer [Ann. Acad. Sci. Fenn., Math. 28, No. 1, 223-238 (2003; Zbl 1037.42023)] and C. Capone, D. Cruz-Uribe and A. Fiorenza [Rev. Mat. Iberoam. 23, No. 3, 743–770 (2007; Zbl 1213.42063)]. They prove this theorem by using Calderón-Zygmund decomposition and their proof gives a unified treatment of the Hardy-Littlewood maximal operator and the fractional maximal operator. The previous proofs for the case $$a>0$$ required first proving that the Hardy-Littlewood maximal operator is bounded on $$l^{p(\cdot)}$$. They also give a new proof of a weak type inequality that extends to the endpoint case $$p_{-}=1$$.

### MSC:

 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.) 42B25 Maximal functions, Littlewood-Paley theory

### Citations:

Zbl 1037.42023; Zbl 1213.42063