On the maximal operator of a general Ornstein–Uhlenbeck semigroup

If Q is a real, symmetric and positive definite n× n matrix, and B a real n× n matrix whose eigenvalues have negative real parts, we consider the Ornstein–Uhlenbeck semigroup on 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.