Littlewood--Paley--Stein inequalities on $\textup{RCD}(K,\infty)$ spaces

TL;DR

This paper proves $L^p$ boundedness of Littlewood--Paley square functions for heat flows on $ extup{RCD}(K, olinebreak\infty)$ spaces, using analytical and probabilistic methods depending on the range of p.

Contribution

It establishes the boundedness of Littlewood--Paley square functions on $ extup{RCD}(K, olinebreak\infty)$ spaces, combining Stein's analytical method and a recent probabilistic approach.

Findings

Proves $L^p$ boundedness for $1<p extless 2$ using Stein's method.

Proves $L^p$ boundedness for $2<p extless \infty$ using probabilistic techniques.

Extends Littlewood--Paley theory to non-smooth metric measure spaces.

Abstract

The $L^p$ boundedness on vertical Littlewood--Paley square functions for heat flows on $\textup{RCD}(K,\infty)$ spaces with $K\in\mathbb{R}$ is proved. With regards to the proof, for $1<p\leq 2$, Stein's analytical method is applied, while for $2<p<\infty$, the probabilistic approach in the sense of Ba\~{n}uelos--Bogdan--Luks introduced recently is employed.