Università degli Studi di Genova


Riesz transforms for a non-symmetric Ornstein-Uhlenbeck semigroup

Let $(\cH_t)_{t\ge 0}$ be the Ornstein-Uhlenbeck semigroup on $\BR^d$ with covariance matrix $I$ and drift matrix $-\lambda(I+R)$, where $\lambda>0$ and $R$ is a skew-adjoint matrix and denote by $\gamma_\infty$ the invariant measure for $(\cH_t)_{t\ge 0}$. Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on $L^2(\gamma_\infty)$. We investigate the weak type $1$ estimate of the Riesz transforms for $(\cH_t)_{t\ge 0}$. We prove that if the matrix $R$ generates a one-parameter group of periodic rotations then the first order Riesz transforms are of weak type $1$ with respect to the invariant measure $\gamma_\infty$. We also prove that the Riesz transforms of any order associated to a general Ornstein-Uhlenbeck semigroup are bounded on $L^p(\gamma_\infty)$ if $1<p<\infty$.

Ornstein-Uhlenbeck, Riesz transform, weak type

42, 47

Pubblicato su: Semigroup Forum Vol. 77 (2008) N. 3 Pag. 380-398