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

We prove that if the exponent function $p((.))$ satisfies log-Holder continuity conditions locally and at infinity, then the fractional maximal operator $M_\alpha$, $0<\alpha <n%, maps $L^{p(.)}$ to $L^{q(.)}$, where $1/p(x) - 1/q(x) = \alpha /n$. We also prove a weak-type inequality corresponding to the weak $(1, n/(n - a))$ inequality for $M_\alpha$. We build upon earlier work on the Hardy-Littlewood maximal operator by Cruz-Uribe, Fiorenza and Neugebauer. 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.