We prove that if the exponent function satisfies log-Hölder continuity conditions locally and at infinity, then the fractional maximal operator , , maps to , where . We also prove a weak-type inequality corresponding to the weak inequality for . We build upon earlier work on the Hardy-Littlewood maximal operator by Cruz-Uribe, Fiorenza and Neugebauer [The maximal function on variable spaces. Ann. Acad. Sci. Fenn. Math. 28 (2003), 223-238]. As a consequence of these results for , we show that the fractional integral operator satisfies the same norm inequalities. These in turn yield a generalization of the Sobolev embedding theorem to variable spaces.
Cite this article
Claudia Capone, David Cruz-Uribe, Alberto Fiorenza, The fractional maximal operator and fractional integrals on variable spaces. Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 743–770DOI 10.4171/RMI/511