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

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

Keywords:

• Semigroup
• Covariance matrix
• Periodic graph (geometry)
• Invariant measure
• Pure mathematics
• Ornstein–Uhlenbeck process
• Matrix (mathematics)
• Operator (computer programming)
• Bounded function
• Mathematical analysis
• Mathematics
• Lambda

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