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

TL;DR

This work develops a parabolic weight theory with time lag by introducing the off-diagonal two-weight class $TA_{r,q}^+(\gamma)$ and characterizing the two-weight boundedness of the uncentered parabolic fractional maximal operator $M^{\gamma+}_\beta$ and the parabolic fractional integral $I^{\gamma+}_\beta$. It proves independence of the time lag, establishes a self-improving property, and derives a parabolic Welland inequality linking maximal operators to fractional integrals. The authors also introduce a parabolic domain and corresponding fractional integrals, achieving two-weight boundedness characterizations that lead to weighted parabolic Sobolev embeddings and a priori estimates for the heat equation. The results rely on a parabolic Calderón–Zygmund decomposition and chaining arguments, providing a robust framework for parabolic potential theory with time lag and advancing regularity theory for doubly nonlinear parabolic PDEs.

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 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.

Theorems & Definitions (53)

• Definition 1.1
• Definition 1.2
• Definition 2.1
• Remark 2.2
• Proposition 2.3
• Definition 2.4
• Proposition 2.5
• proof
• Proposition 2.6
• proof