The Maximal Operator Associated to a Nonsymmetric Ornstein–Uhlenbeck Semigroup

Abstract

Let (ℋ t ) t≥0 be the Ornstein–Uhlenbeck semigroup on ℝ d with covariance matrix I and drift matrix λ(R−I), where λ>0 and R is a skew-adjoint matrix, and denote by γ ∞ the invariant measure for (ℋ t ) t≥0. Semigroups of this form are the basic building blocks of Ornstein–Uhlenbeck semigroups which are normal on L 2(γ ∞). We prove that if the matrix R generates a one-parameter group of periodic rotations, then the maximal operator ℋ* f(x)=sup t≥o |ℋ t f(x)| is of weak type 1 with respect to the invariant measure γ ∞. We also prove that the maximal operator associated to an arbitrary normal Ornstein–Uhlenbeck semigroup is bounded on L p (γ ∞) if and only if 1<p≤∞.