The maximal operator associated to a non-symmetric Ornstein-Uhlenbeck semigroup

TL;DR

This paper investigates the maximal operator linked to a class of Ornstein-Uhlenbeck semigroups, establishing weak type (1,1) bounds for certain cases and L^p boundedness criteria for normal semigroups.

Contribution

It proves weak type (1,1) bounds for the maximal operator when the drift matrix generates periodic rotations and characterizes L^p boundedness for normal Ornstein-Uhlenbeck semigroups.

Findings

Maximal operator is of weak type 1 for semigroups with periodic rotation generators.

Maximal operator is bounded on L^p for 1<p≤∞ if the semigroup is normal.

Provides a characterization of L^p boundedness for normal Ornstein-Uhlenbeck semigroups.

Abstract

Let (H_t) be the Ornstein-Uhlenbeck semigroup on R^d with covariance matrix I and drift matrix \lambda(R-I), where \lambda>0 and R is a skew-adjoint matrix and denote by \gamma_\infty the invariant measure for (H_t). Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L^2(\gamma_\infty). We prove that if the matrix R generates a one-parameter group of periodic rotations then the maximal operator associated to the semigroup is of weak type 1 with respect to the invariant measure. We also prove that the maximal operator associated to an arbitrary normal Ornstein-Uhlenbeck semigroup is bounded on L^p(\gamma_\infty) if and only if 1<p\le \infty.