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CHOQUET INTEGRALS, HAUSDORFF CONTENT AND FRACTIONAL OPERATORS

Part of: Harmonic analysis in several variables

Published online by Cambridge University Press:  19 March 2024

NAOYA HATANO ,
RYOTA KAWASUMI ,
HIROKI SAITO  and
HITOSHI TANAKA
NAOYA HATANO*
Affiliation:
Department of Mathematics, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
RYOTA KAWASUMI
Affiliation:
Graduate School of Economics, Hitotsubashi University, 2-1 Naka, Kunitachi, Tokyo 186-8601, Japan e-mail: a241010y@r.hit-u.ac.jp
HIROKI SAITO
Affiliation:
College of Science and Technology, Nihon University, Narashinodai 7-24-1, Funabashi City, Chiba 274-8501, Japan e-mail: saitou.hiroki@nihon-u.ac.jp
HITOSHI TANAKA
Affiliation:
Research and Support Center on Higher Education for the Hearing and Visually Impaired, National University Corporation Tsukuba University of Technology, Kasuga 4-12-7, Tsukuba City, Ibaraki 305-8521, Japan e-mail: htanaka@k.tsukuba-tech.ac.jp
*
e-mail: n.hatano.chuo@gmail.com
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Abstract

We show that the fractional integral operator $I_{\alpha }$, $0<\alpha <n$, and the fractional maximal operator $M_{\alpha }$, $0\le \alpha <n$, are bounded on weak Choquet spaces with respect to Hausdorff content. We also investigate these operators on Choquet–Morrey spaces. The results for the fractional maximal operator $M_\alpha $ are extensions of the work of Tang [‘Choquet integrals, weighted Hausdorff content and maximal operators’, Georgian Math. J. 18(3) (2011), 587–596] and earlier work of Adams and Orobitg and Verdera. The results for the fractional integral operator $I_{\alpha }$ are essentially new.

Keywords

Choquet–Morrey spacesfractional integral operatorfractional maximal operatorHausdorff contentweak Choquet spaces

MSC classification

Primary: 42B25: Maximal functions, Littlewood-Paley theory
Secondary: 42B35: Function spaces arising in harmonic analysis
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 2 , October 2024 , pp. 355 - 366
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

N.H. is financially supported by a Foundation of Research Fellows, The Mathematical Society of Japan. H.S. is supported by Grant-in-Aid for Young Scientists (19K14577), the Japan Society for the Promotion of Science. H.T. is supported by Grant-in-Aid for Scientific Research (C) (15K04918 and 19K03510), the Japan Society for the Promotion of Science.

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