The maximal operator associated to a non-symmetric Ornstein-Uhlenbeck semigroup

Details The maximal operator associated to a non-symmetric Ornstein-Uhlenbeck semigroup The maximal operator associated to a non-symmetric Ornstein-Uhlenbeck semigroup

2009-01-11

arxiv.org/abs/0901.1455v1

Mathematics

Functional Analysis

20 pages, to appear in J Fourier Anal Appl, available on line at http://www.springerlink.com

Scientific paper

Let (H_t) be the Ornstein-Uhlenbeck semigroup on R^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 (H_t). 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 associated to the semigroup is of weak type 1 with respect to the invariant measure. 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

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