Parabolic Muckenhoupt Weights Characterized by Parabolic Fractional Maximal and Integral Operators with Time Lag

TL;DR

This paper introduces a new class of parabolic weights with time lag and characterizes their boundedness properties via fractional maximal and integral operators, with applications to parabolic PDE regularity and Sobolev embeddings.

Contribution

It develops a novel parabolic weighted theory involving off-diagonal two-weight classes with time lag and characterizes boundedness of fractional operators in this setting.

Findings

Characterization of two-weight boundedness of parabolic fractional maximal operators with time lag.

Introduction of a new parabolic shaped domain and associated fractional integral.

Establishment of weighted Sobolev embedding and a priori estimates for heat equation solutions.

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.