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SOBOLEV’S INEQUALITY FOR MUSIELAK–ORLICZ–MORREY SPACES OVER METRIC MEASURE SPACES

Part of: Linear function spaces and their duals

Published online by Cambridge University Press:  02 October 2019

TAKAO OHNO Open the ORCID record for TAKAO OHNO [Opens in a new window]  and
TETSU SHIMOMURA
TAKAO OHNO*
Affiliation:
Faculty of Education, Oita University, Dannoharu, Oita-city870-1192, Japan
TETSU SHIMOMURA
Affiliation:
Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima739-8524, Japan e-mail: tshimo@hiroshima-u.ac.jp
*
e-mail: t-ohno@oita-u.ac.jp
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Abstract

Our aim in this paper is to establish a generalization of Sobolev’s inequality for Riesz potentials $J_{\unicode[STIX]{x1D6FC}(\cdot )}^{\unicode[STIX]{x1D70E}}f$ of functions $f$ in Musielak–Orlicz–Morrey spaces $L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705}}(X)$. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

Keywords

Riesz potentialsMusielak–Orlicz–Morrey spacesSobolev’s inequalitymetric measure spacesPoincaré inequalitydouble phase functionals

MSC classification

Primary: 46E35: Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Secondary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 110 , Issue 3 , June 2021 , pp. 371 - 385
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Copyright
© 2019 Australian Mathematical Publishing Association Inc.

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Footnotes

Communicated by C. Meaney

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