Riesz transforms of a general Ornstein--Uhlenbeck semigroup

Abstract

We consider Riesz transforms of any order associated to an Ornstein--Uhlenbeck operator $\mathcal{L}$, with covariance $Q$ given by a real, symmetric and positive definite matrix, and with drift $B$ given by a real matrix whose eigenvalues have negative real parts. In this general Gaussian context, we prove that a Riesz transform is of weak type $(1,1)$ with respect to the invariant measure if and only if its order is at most $2$.

29 pages. Second draft includes a new section (section 3), dealing with the definition of the negative powers of $-\mathcal{L}$

References in corpus (1)

• On the orthogonality of generalized eigenspaces for the Ornstein--Uhlenbeck operator

Cited by in corpus (1)

• On the orthogonality of generalized eigenspaces for the Ornstein--Uhlenbeck operator