Sobolev's inequality for Riesz potentials with variable exponent satisfying a log-Hölder condition at infinity |
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Authors: | Yoshihiro Mizuta |
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Institution: | a The Division of Mathematical and Information Sciences, Faculty of Integrated Arts and Sciences, Hiroshima University, Higashi-Hiroshima 739-8521, Japan b Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan |
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Abstract: | Our aim in this paper is to deal with the boundedness of maximal functions in generalized Lebesgue spaces Lp(⋅) when p(⋅) satisfies a log-Hölder condition at infinity that is weaker than that of Cruz-Uribe, Fiorenza and Neugebauer D. Cruz-Uribe, A. Fiorenza, C.J. Neugebauer, The maximal function on variable Lp spaces, Ann. Acad. Sci. Fenn. Math. 28 (2003) 223-238; 29 (2004) 247-249]. Our result extends the recent work of Diening L. Diening, Maximal functions on generalized Lp(⋅) spaces, Math. Inequal. Appl. 7 (2004) 245-254] and the authors Futamura and Mizuta T. Futamura, Y. Mizuta, Sobolev embeddings for Riesz potential space of variable exponent, preprint]. As an application of the boundedness of maximal functions, we show Sobolev's inequality for Riesz potentials with variable exponent. |
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Keywords: | Riesz potentials Maximal functions Sobolev's embedding theorem of variable exponent |
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