# On the maximal operator of a general Ornstein-Uhlenbeck semigroup

## TL;DR

This paper proves that the maximal operator associated with a general Ornstein-Uhlenbeck semigroup is of weak type (1,1), using a geometric approach and the forbidden zones method, extending understanding of its boundedness properties.

## Contribution

It establishes the weak type (1,1) boundedness of the maximal operator for a broad class of Ornstein-Uhlenbeck semigroups, employing a novel geometric proof technique.

## Key findings

- Maximal operator is of weak type (1,1) with respect to the invariant measure.
- The proof utilizes the forbidden zones method for geometric analysis.
- Results extend previous boundedness results to more general Ornstein-Uhlenbeck semigroups.

## Abstract

If $Q$ is a real, symmetric and positive definite $n\times n$ matrix, and $B$ a real $n\times n$ matrix whose eigenvalues have negative real parts, we consider the Ornstein--Uhlenbeck semigroup on $\mathbb{R}^n$ with covariance $Q$ and drift matrix $B$. Our main result says that the associated maximal operator is of weak type $(1,1)$ with respect to the invariant measure. The proof has a geometric gist and hinges on the "forbidden zones method" previously introduced by the third author.