Littlewood-Paley functions associated with general Ornstein-Uhlenbeck semigroups

TL;DR

This paper proves boundedness properties of Littlewood-Paley functions, maximal, and variation operators associated with Ornstein-Uhlenbeck semigroups in $L^p$ spaces, using advanced harmonic analysis techniques.

Contribution

It establishes new $L^p$-boundedness results for square functions, maximal, and variation operators related to Ornstein-Uhlenbeck semigroups, including weak and strong type estimates.

Findings

Proved $L^p$-boundedness for square functions involving derivatives of Ornstein-Uhlenbeck semigroups.

Established weak type (1,1) bounds by analyzing global and local operators.

Demonstrated $L^p$-boundedness for maximal and variation operators associated with these semigroups.

Abstract

In this paper we establish $L^p(\mathbb{R}^d,\gamma_\infty)$-boundedness properties for square functions involving time and spatial derivatives of Ornstein-Uhlenbeck semigroups. Here $\gamma_\infty$ denotes the invariant measure. In order to prove the strong type results for $1<p<\infty$ we use $R$-boundedness. The weak type (1,1) property is established by studying separately global and local operators defined for the square Littlewood-Paley functions. By the way we prove $L^p(\mathbb{R}^d,\gamma_\infty)$-boundedness properties for maximal and variation operators for Ornstein-Uhlenbeck semigroups.