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Sobolev’s Inequality for Musielak–Orlicz–Sobolev Functions

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Abstract

Our aim in this paper is to deal with Sobolev’s inequality for Musielak–Orlicz–Sobolev functions in \(W_0^{1, \Phi }(\textbf{R}^N)\). As an application we discuss double phase functionals \(\Phi (x,t) = t^{p(x)} + a(x) t^{q(x)}\) with variable exponents. Here p and q are variable exponents satisfying natural continuity conditions. Also the case when p attains the value 1 in some parts of the domain is included in the results. For this purpose we first study weak type inequality for variable Riesz potentials.

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Authors and Affiliations

  1. Department of Mathematics, Graduate School of Advanced Science and Engineering Hiroshima University, Higashi-Hiroshima, 739-8521, Japan

    Yoshihiro Mizuta

  2. Faculty of Education, Oita University, Dannoharu Oita-city, 870-1192, Japan

    Takao Ohno

  3. Department of Mathematics, Graduate School of Humanities and Social Sciences Hiroshima University, Higashi-Hiroshima, 739-8524, Japan

    Tetsu Shimomura

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  1. Yoshihiro Mizuta
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  2. Takao Ohno
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  3. Tetsu Shimomura
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Mizuta, Y., Ohno, T. & Shimomura, T. Sobolev’s Inequality for Musielak–Orlicz–Sobolev Functions. Results Math 78, 90 (2023). https://doi.org/10.1007/s00025-023-01858-x

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