# The maximal operator of a normal Ornstein--Uhlenbeck semigroup is of weak type $(1,1)$

## TL;DR

This paper proves that the maximal operator of a normal Ornstein-Uhlenbeck semigroup in ^n is of weak type (1,1) with respect to its invariant measure, extending previous results to more general covariance and drift matrices.

## Contribution

It extends the weak type (1,1) boundedness of the maximal operator to a broader class of Ornstein-Uhlenbeck semigroups with general covariance and drift matrices.

## Key findings

- Maximal operator is of weak type (1,1) for the considered semigroup.
- Extension of previous results to more general covariance and drift matrices.
- Proof utilizes a special case with identity covariance and diagonal drift matrix.

## Abstract

Consider a normal Ornstein--Uhlenbeck semigroup in $\Bbb{R}^n$, whose covariance is given by a positive definite matrix. The drift matrix is assumed to have eigenvalues only in the left half-plane. We prove that the associated maximal operator is of weak type $(1,1)$ with respect to the invariant measure. This extends earlier work by G. Mauceri and L. Noselli. The proof goes via the special case where the matrix defining the covariance is $I$ and the drift matrix is diagonal.