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
, 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.