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 \(p_{-}=1\).


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