Abstract
AbstractIf Q is a real, symmetric and positive definite $$n\times n$$ n × n matrix, and B a real $$n\times n$$ n × n matrix whose eigenvalues have negative real parts, we consider the Ornstein–Uhlenbeck semigroup on $$\mathbb {R}^n$$ R n with covariance Q and drift matrix B. Our main result says that the associated maximal operator is of weak type (1, 1) with respect to the invariant measure. The proof has a geometric gist and hinges on the “forbidden zones method” previously introduced by the third author.
Highlights
In this paper we prove a weak type (1, 1) theorem for the maximal operator associated to a general Ornstein–Uhlenbeck semigroup
We proved that the associated maximal operator is of weak type (1, 1) with respect to the invariant measure γ∞
In this paper we focus our attention on the Ornstein–Uhlenbeck semigroup in Rn
Summary
In this paper we prove a weak type (1, 1) theorem for the maximal operator associated to a general Ornstein–Uhlenbeck semigroup. In [4] the present authors recently considered a normal Ornstein–Uhlenbeck semigroup in Rn, that is, we assumed that Ht is for each t > 0 a normal operator on L2(γ∞) Under this extra assumption, we proved that the associated maximal operator is of weak type (1, 1) with respect to the invariant measure γ∞. Corollary 1.2 The Ornstein–Uhlenbeck maximal operator H∗ is bounded on L p(γ∞) for all p > 1 This result improves Theorem 4.2 in [9], where the L p boundedness of H∗ is proved for all p > 1 in the normal framework, under the additional assumption that the infinitesimal generator of Ht t>0 is a sectorial operator of angle less than π/2. If A is an n × n matrix, we write A for its operator norm on Rn with the Euclidean norm | · |
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