Two-weight inequality for the heat flow and solvability of Hardy-H\'enon parabolic equation

TL;DR

This paper establishes two-weight inequalities for heat flow using sparse domination, extending results to general weights in Muckenhoupt classes, and applies these to prove existence results for Hardy-Hénon parabolic equations.

Contribution

It introduces a novel approach using sparse domination to handle general weights in Muckenhoupt classes for heat flow inequalities and applies this to Hardy-Hénon equations.

Findings

Established two-weight inequalities for heat flow with general weights.

Extended previous results from power weights to Muckenhoupt class weights.

Proved local and global existence of solutions for Hardy-Hénon parabolic equations.

Abstract

In this article, we provide two-weight inequalities for the heat flow on the whole space by applying the sparse domination. For power weights, such inequalities were given by several authors. Owing to the sparse domination, we can treat general weights in Muckenhoupt classes. As a application, we present the local and global existence results for the Hardy-H\'enon parabolic equation, which has a potential belonging to a Muckenhoupt class.