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Parabolic Muckenhoupt weights characterized by parabolic fractional maximal and integral operators with time lag

Part of: Parabolic equations and systems General theory of linear operators Harmonic analysis in several variables

Published online by Cambridge University Press:  17 March 2025

Weiyi Kong ,
Dachun Yang Open the ORCID record for Dachun Yang [Opens in a new window] ,
Wen Yuan  and
Chenfeng Zhu
Weiyi Kong
Affiliation:
Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, The People’s Republic of China e-mail: weiyikong@mail.bnu.edu.cn wenyuan@bnu.edu.cn
Dachun Yang*
Affiliation:
Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, The People’s Republic of China e-mail: weiyikong@mail.bnu.edu.cn wenyuan@bnu.edu.cn
Wen Yuan
Affiliation:
Laboratory of Mathematics and Complex Systems (Ministry of Education of China), School of Mathematical Sciences, Beijing Normal University, Beijing 100875, The People’s Republic of China e-mail: weiyikong@mail.bnu.edu.cn wenyuan@bnu.edu.cn
Chenfeng Zhu
Affiliation:
School of Mathematical Sciences, Zhejiang University of Technology, Hangzhou 310023, The People’s Republic of China e-mail: chenfengzhu@zjut.edu.cn
*
e-mail: dcyang@bnu.edu.cn
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Abstract

In this article, motivated by the regularity theory of the solutions of doubly nonlinear parabolic partial differential equations, the authors introduce the off-diagonal two-weight version of the parabolic Muckenhoupt class with time lag. Then the authors introduce the uncentered parabolic fractional maximal operator with time lag and characterize its two-weighted boundedness (including the endpoint case) in terms of these weights under an additional mild assumption (which is not necessary for one-weight case). The most novelty of this article exists in that the authors further introduce a new parabolic shaped domain and its corresponding parabolic fractional integral with time lag and, moreover, applying the aforementioned (two-)weighted boundedness of the parabolic fractional maximal operator with time lag, the authors characterize the (two-)weighted boundedness (including the endpoint case) of these parabolic fractional integrals in terms of the off-diagonal (two-weight) parabolic Muckenhoupt class with time lag; as applications, the authors further establish a parabolic weighted Sobolev embedding and a priori estimate for the solution of the heat equation. The key tools to achieve these include the parabolic Calderón–Zygmund-type decomposition, the chaining argument, and the parabolic Welland inequality, which is obtained by making the utmost of the geometrical relation between the parabolic shaped domain and the parabolic rectangle.

Keywords

Parabolic Muckenhoupt weighttime lagparabolic fractional maximal operatorparabolic fractional integralparabolic Welland inequality

MSC classification

Primary: 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)
Secondary: 47A30: Norms (inequalities, more than one norm, etc.) 42B25: Maximal functions, Littlewood-Paley theory 42B35: Function spaces arising in harmonic analysis 42B37: Harmonic analysis and PDE 35K05: Heat equation

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Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 54
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This project is partially supported by the National Key Research and Development Program of China (Grant No. 2020YFA0712900), the National Natural Science Foundation of China (Grant Nos. 12431006 and 12371093), and the Fundamental Research Funds for the Central Universities (Grant No. 2233300008).

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