Advanced Studies in Pure Mathematics

Maximal functions, Riesz potentials and Sobolev's inequality in generalized Lebesgue spaces

Yoshihiro Mizuta and Tetsu Shimomura

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Abstract

Our aim in this paper is to deal with the boundedness of maximal functions in Lebesgue spaces with variable exponent. Our result extends the recent work of Diening [4], Cruz-Uribe, Fiorenza and Neugebauer [3] and the authors [8]. As an application of the boundedness of maximal functions, we show Sobolev's inequality for Riesz potentials with variable exponent.

Article information

Source
Potential Theory in Matsue, H. Aikawa, T. Kumagai, Y. Mizuta and N. Suzuki, eds. (Tokyo: Mathematical Society of Japan, 2006), 255-281

Dates
Received: 22 January 2005
Revised: 12 April 2005
First available in Project Euclid: 16 December 2018

Permanent link to this document
https://projecteuclid.org/ euclid.aspm/1544999696

Digital Object Identifier
doi:10.2969/aspm/04410255

Mathematical Reviews number (MathSciNet)
MR2277839

Zentralblatt MATH identifier
1125.31001

Subjects
Primary: 42B25: Maximal functions, Littlewood-Paley theory 46E30: Spaces of measurable functions (Lp-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 31B15: Potentials and capacities, extremal length

Keywords
Lebesgue spaces with variable exponent maximal functions Riesz potentials Sobolev's inequality Hardy's inequality

Citation

Mizuta, Yoshihiro; Shimomura, Tetsu. Maximal functions, Riesz potentials and Sobolev's inequality in generalized Lebesgue spaces. Potential Theory in Matsue, 255--281, Mathematical Society of Japan, Tokyo, Japan, 2006. doi:10.2969/aspm/04410255. https://projecteuclid.org/euclid.aspm/1544999696


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