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

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.

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