Littlewood–Paley functions associated with general Ornstein–Uhlenbeck semigroups

Abstract

In this paper, \(L^p(\mathbb {R}^d,\gamma _\infty )\)-boundedness properties for Littlewood–Paley g-functions involving time and spatial derivatives of Ornstein–Uhlenbeck semigroups are established. Here, \(\gamma _\infty\) denotes the invariant measure. 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 Littlewood–Paley g-functions. By the way \(L^p(\mathbb {R}^d,\gamma _\infty )\)-boundedness properties for maximal and variation operators for Ornstein–Uhlenbeck semigroups are proved.