On the global Gaussian Lipschitz space

TL;DR

This paper introduces a Lipschitz space within the Ornstein-Uhlenbeck framework, characterized by gradient bounds and continuity conditions, revealing functions with at most logarithmic growth at infinity.

Contribution

It defines and characterizes a new Lipschitz space in the Ornstein-Uhlenbeck setting, extending previous bounded function spaces with a focus on growth and regularity.

Findings

Functions in the space have at most logarithmic growth at infinity

The space is characterized via a Lipschitz-type continuity condition

Extension of previous bounded Lipschitz space to unbounded functions

Abstract

A Lipschitz space is defined in the Ornstein-Uhlenbeck setting, by means of a bound for the gradient of the Ornstein-Uhlenbeck Poisson integral. This space is then characterized with a Lipschitz-type continuity condition. These functions turn out to have at most logarithmic growth at infinity. The analogous Lipschitz space containing only bounded functions was introduced by Gatto and Urbina and has been characterized by the authors in \cite{LS}.