Università degli Studi di Genova

The maximal operator associated to a nonsymmetric Ornstein-Uhlenbeck semigroup

Abstract:
Let $(\cH_t)_{t\ge 0}$ be the Ornstein-Uhlenbeck semigroup on $\BR^d$ with covariance matrix $I$ and drift matrix $\lambda(R-I)$, 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 prove that if the matrix $R$ generates a one-parameter group of periodic rotations then the maximal operator $\cH_*f(x)=\sup_{t\ge o}\mod{\cH_tf(x)}$ is of weak type $1$ with respect to the invariant measure $\gamma_\infty$. We also prove that the maximal operator associated to an arbitrary normal Ornstein-Uhlenbeck semigroup is bounded on $L^p(\gamma_\infty)$ if and only if $1<p\le \infty$.

Keywords:
Ornstein-Uhlenbeck semigroup, maximal operator, weak type

MSC:
42, 47

Pubblicato su: The Journal of Fourier Analysis and Applications Vol. 15 (2009) N. 2 Pag. 179-200