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FRACTIONAL TYPE MARCINKIEWICZ INTEGRAL AND ITS COMMUTATOR ON NONHOMOGENEOUS SPACES

Part of: Analysis on metric spaces Harmonic analysis in several variables

Published online by Cambridge University Press:  03 June 2022

GUANGHUI LU Open the ORCID record for GUANGHUI LU [Opens in a new window]
GUANGHUI LU*
Affiliation:
College of Mathematics and Statistics Northwest Normal University Anning Road 967 Lanzhou 730070 P. R. China lghwmm1989@126.com
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Abstract

The aim of this paper is to establish the boundedness of fractional type Marcinkiewicz integral $\mathcal {M}_{\iota ,\rho ,m}$ and its commutator $\mathcal {M}_{\iota ,\rho ,m,b}$ on generalized Morrey spaces and on Morrey spaces over nonhomogeneous metric measure spaces which satisfy the upper doubling and geometrically doubling conditions. Under the assumption that the dominating function $\lambda $ satisfies $\epsilon $ -weak reverse doubling condition, the author proves that $\mathcal {M}_{\iota ,\rho ,m}$ is bounded on generalized Morrey space $L^{p,\phi }(\mu )$ and on Morrey space $M^{p}_{q}(\mu )$ . Furthermore, the boundedness of the commutator $\mathcal {M}_{\iota ,\rho ,m,b}$ generated by $\mathcal {M}_{\iota ,\rho ,m}$ and regularized $\mathrm {BMO}$ space with discrete coefficient (= $\widetilde {\mathrm {RBMO}}(\mu )$ ) on space $L^{p,\phi }(\mu )$ and on space $M^{p}_{q}(\mu )$ is also obtained.

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 42B35: Function spaces arising in harmonic analysis 30L99: None of the above, but in this section
Type
Article
Information
Nagoya Mathematical Journal , Volume 248 , December 2022 , pp. 801 - 822
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Copyright
© (2022) The Authors. The publishing rights in this article are licenced to Foundation Nagoya Mathematical Journal under an exclusive license

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Footnotes

This work is supported by the Young Teachers’ Scientific Research Ability Promotion Project of Northwest Normal University (NWNU-LKQN2020-07) and Innovation Fund Project for Higher Education of Gansu Province (2020A-010).

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