ix I and drift matrix λ(R − I), where λ > 0 and R is a skew-adjoint matrix and denote by γ∞ the invariant measure for (Ht)t≥0 . Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L2(γ∞). We prove that if the matrix R gen- erates a one-parameter group of periodic rotations then the maximal operator H∗f(x) = supt≥o |Htf(x)| is of weak type 1 with respect to the invariant mea- sure γ∞ . We also prove that the maximal operator associated to an arbitrary normal Ornstein-Uhlenbeck semigroup is bounded on Lp(γ∞) if and only if 1 < p ≤ ∞.
The maximal operator associated to a nonsymmetric Ornstein-Uhlenbeck semigroup
MAUCERI, GIANCARLO;
2009-01-01
Abstract
ix I and drift matrix λ(R − I), where λ > 0 and R is a skew-adjoint matrix and denote by γ∞ the invariant measure for (Ht)t≥0 . Semigroups of this form are the basic building blocks of Ornstein-Uhlenbeck semigroups which are normal on L2(γ∞). We prove that if the matrix R gen- erates a one-parameter group of periodic rotations then the maximal operator H∗f(x) = supt≥o |Htf(x)| is of weak type 1 with respect to the invariant mea- sure γ∞ . We also prove that the maximal operator associated to an arbitrary normal Ornstein-Uhlenbeck semigroup is bounded on Lp(γ∞) if and only if 1 < p ≤ ∞.
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