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Sobolev’s Inequality for Riesz Potentials of Functions in Musielak–Orlicz–Morrey Spaces Over Non-doubling Metric Measure Spaces

Part of: Linear function spaces and their duals

Published online by Cambridge University Press:  06 September 2019

Takao Ohno  and
Tetsu Shimomura
Takao Ohno
Affiliation:
Faculty of Education, Oita University, Dannoharu Oita-city 870-1192, Japan Email: t-ohno@oita-u.ac.jp
Tetsu Shimomura
Affiliation:
Department of Mathematics, Graduate School of Education, Hiroshima University, Higashi-Hiroshima 739-8524, Japan Email: tshimo@hiroshima-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 $I_{\unicode[STIX]{x1D6FC}(\,\cdot \,),\unicode[STIX]{x1D70F}}f$ of order $\unicode[STIX]{x1D6FC}(\,\cdot \,)$ with $f\in L^{\unicode[STIX]{x1D6F7},\unicode[STIX]{x1D705},\unicode[STIX]{x1D703}}(X)$ over bounded non-doubling metric measure spaces. As a corollary we obtain Sobolev’s inequality for double phase functionals with variable exponents.

Keywords

maximal functionRiesz potentialMusielak–Orlicz–Morrey spaceSobolev’s inequalitymetric measure spacenon-doubling measuredouble phase functional

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
Article
Information
Canadian Mathematical Bulletin , Volume 63 , Issue 2 , June 2020 , pp. 287 - 303
Copyright
© Canadian Mathematical Society 2019

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