1. Parabolic Muckenhoupt Weights Characterized by Parabolic Fractional Maximal and Integral Operators with Time Lag
- Author
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Kong, Weiyi, Yang, Dachun, Yuan, Wen, and Zhu, Chenfeng
- Subjects
Mathematics - Analysis of PDEs ,Mathematics - Classical Analysis and ODEs ,Mathematics - Functional Analysis ,Primary 42B20, Secondary 47A30, 42B25, 42B35, 42B37, 35K05 - 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) via these weights under an extra 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 uncentered 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 Calder\'on--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.
- Published
- 2024