The fractional maximal operator and fractional integrals on variable $L^p$ spaces

Abstract

We prove that if the exponent function $p(\cdot )$ satisfies log-Hölder continuity conditions locally and at infinity, then the fractional maximal operator $M_{\alpha }$, $0<\alpha <n$, maps $L^{p(\cdot )}$ to $L^{q(\cdot )}$, where $\frac{1}{p(x)}-\frac{1}{q(x)}=\frac{\alpha }{n}$. We also prove a weak-type inequality corresponding to the weak $(1,n/(n-\alpha ))$ inequality for $M_{\alpha }$. We build upon earlier work on the Hardy-Littlewood maximal operator by Cruz-Uribe, Fiorenza and Neugebauer [The maximal function on variable $L^p$ spaces. Ann. Acad. Sci. Fenn. Math. 28 (2003), 223-238]. As a consequence of these results for $M_{\alpha }$, we show that the fractional integral operator $I_{\alpha }$ satisfies the same norm inequalities. These in turn yield a generalization of the Sobolev embedding theorem to variable $L^p$ spaces.