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HERZ–MORREY SPACES ON THE UNIT BALL WITH VARIABLE EXPONENT APPROACHING $1$ AND DOUBLE PHASE FUNCTIONALS

Part of: Harmonic analysis in several variables Linear function spaces and their duals

Published online by Cambridge University Press:  13 June 2019

YOSHIHIRO MIZUTA ,
TAKAO OHNO Open the ORCID record for TAKAO OHNO [Opens in a new window]  and
TETSU SHIMOMURA
YOSHIHIRO MIZUTA
Affiliation:
4-13-11 Hachi-Hom-Matsu-Minami, Higashi-Hiroshima 739-0144, Japan email yomizuta@hiroshima-u.ac.jp
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 deal with integrability of maximal functions for Herz–Morrey spaces on the unit ball with variable exponent $p_{1}(\cdot )$ approaching $1$ and for double phase functionals $\unicode[STIX]{x1D6F7}_{d}(x,t)=t^{p_{1}(x)}+a(x)t^{p_{2}}$, where $a(x)^{1/p_{2}}$ is nonnegative, bounded and Hölder continuous of order $\unicode[STIX]{x1D703}\in (0,1]$ and $1/p_{2}=1-\unicode[STIX]{x1D703}/N>0$. We also establish Sobolev type inequality for Riesz potentials on the unit ball.

MSC classification

Primary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 42B25: Maximal functions, Littlewood-Paley theory

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Type
Article
Information
Nagoya Mathematical Journal , Volume 242 , June 2021 , pp. 1 - 34
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Copyright
© 2019 Foundation Nagoya Mathematical Journal

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