Weighted Littlewood--Paley inequalities for heat flows in $\RCD$ spaces

TL;DR

This paper proves weighted Littlewood--Paley inequalities for heat flows in $ ext{RCD}^*(0,N)$ spaces, establishing sharp estimates and comparing different weight classes, extending harmonic analysis tools to metric measure spaces.

Contribution

It introduces weighted Littlewood--Paley inequalities for heat flows in $ ext{RCD}^*(0,N)$ spaces, including sharp estimates and weight class comparisons.

Findings

Inequalities hold for heat flows in $ ext{RCD}^*(0,N)$ spaces with volume growth conditions.

Sharp estimates are established for 2-heat and 2-Muckenhoupt weights.

Comparison of $p$-Muckenhoupt and $p$-heat weights for all $p ext{ in }(1, olinebreak ext{infinity})$.

Abstract

We establish inequalities on vertical Littlewood--Paley square functions for heat flows in the weighted $L^2$ space over metric measure spaces satisfying the $\RCD^\ast(0,N)$ condition with $N\in [1,\infty)$ and the maximum volume growth assumption. In the noncompact setting, the later assumption can be removed by showing that the volume of the ball growths at least linearly. The estimates are sharp on the growth of the 2-heat weight and the 2-Muckenhoupt weight considered. The $p$-Muckenhoupt weight and the $p$-heat weight are also compared for all $p\in(1,\infty)$.