Maximal operator, Littlewood-Paley functions and variation operators associated with nonsymmetric Ornstein-Uhlenbeck operators

TL;DR

This paper proves $L^p$ boundedness for maximal, Littlewood-Paley, and variation operators linked to nonsymmetric Ornstein-Uhlenbeck semigroups, extending harmonic analysis tools to this class of operators.

Contribution

It establishes $L^p$ boundedness for these operators associated with nonsymmetric Ornstein-Uhlenbeck operators, a novel extension beyond symmetric cases.

Findings

Proved $L^p$ boundedness for maximal operators

Established boundedness for Littlewood-Paley functions

Demonstrated bounded variation operators for nonsymmetric Ornstein-Uhlenbeck semigroups

Abstract

In this paper we establish $L^p$ boundedness properties for maximal operators, Littlewood-Paley functions and variation operators involving Poisson semigroups and resolvent operators associated with nonsymmetric Ornstein-Uhlenbeck operators. We consider the Ornstein-Uhlenbeck operators defined by the identity as the covariance matrix and having a drift given by the matrix $-\lambda (I+R)$, being $\lambda >0$ and $R$ a skew-adjoint matrix. The semigroup associated with these Ornstein-Uhlenbeck operators are the basic building blocks of all normal Ornstein-Uhlenbeck semigroups.