Parabolic weighted norm inequalities and partial differential equations

TL;DR

This paper characterizes weighted norm inequalities for parabolic maximal operators, introduces a Jones type factorization for parabolic Muckenhoupt weights, and links parabolic BMO to these weights and maximal functions.

Contribution

It provides a comprehensive characterization of weighted inequalities and a new factorization and BMO characterization in the context of nonlinear parabolic PDEs.

Findings

Full characterization of weak and strong type weighted inequalities.

A Jones type factorization for parabolic Muckenhoupt weights.

A Coifman-Rochberg type BMO characterization via parabolic weights.

Abstract

We investigate parabolic Muckenhoupt weights and functions of bounded mean oscillation (BMO) related to nonlinear parabolic partial differential equations. The main result gives a full characterization of weak and strong type weighted norm inequalities for parabolic forward in time maximal operators. In addition, we give a Jones type factorization result for the parabolic Muckenhoupt weights and a Coifman-Rochberg type characterization of the parabolic BMO from Moser's seminal paper through parabolic Muckenhoupt weights and maximal functions.