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Decembar, 2007 The fractional maximal operator and fractional integrals on variable $L^p$ spaces
Claudia Capone , David Cruz-Uribe, SFO , Alberto Fiorenza
Rev. Mat. Iberoamericana 23(3): 743-770 (Decembar, 2007).

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.

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Claudia Capone . David Cruz-Uribe, SFO . Alberto Fiorenza . "The fractional maximal operator and fractional integrals on variable $L^p$ spaces." Rev. Mat. Iberoamericana 23 (3) 743 - 770, Decembar, 2007.

Information

Published: Decembar, 2007
First available in Project Euclid: 27 February 2008

zbMATH: 1213.42063
MathSciNet: MR2414490

Subjects:
Primary: 42B25, 42B35

Rights: Copyright © 2007 Departamento de Matemáticas, Universidad Autónoma de Madrid

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Vol.23 • No. 3 • Decembar, 2007
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