Revista Matemática Iberoamericana

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Volume 23, Issue 3, 2007, pp. 743–770
DOI: 10.4171/RMI/511

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

Claudia Capone[1], David Cruz-Uribe[2] and Alberto Fiorenza[3]

(1) CNR - Istituto per le Applicazioni del Calcolo 'M. Picone', Consiglio Nazionale delle Ricerche, via P. Castellino 111, 80131, NAPOLI, ITALY
(2) Department of Mathematics, Trinity College, CT 06106-3100, HARTFORD, UNITED STATES
(3) Dip. di Costruzioni e Metodi Matematici in Archite, Università degli Studi di Napoli Federico II, Via Monteoliveto 3, 80134, NAPOLI, ITALY

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.

Keywords: Fractional maximal operator, fractional integral operator, Sobolev embedding theorem, variable Lebesgue space

Capone Claudia, Cruz-Uribe David, Fiorenza Alberto: The fractional maximal operator and fractional integrals on variable $L^p$ spaces. Rev. Mat. Iberoamericana 23 (2007), 743-770. doi: 10.4171/RMI/511