Littlewood-Paley functions associated with general Ornstein-Uhlenbeck 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.

This is a revised version of the preliminary one arXiv:2202.06136v1