Variational Inequalities for the Ornstein–Uhlenbeck Semigroup: The Higher-Dimensional Case

We study the ϱ-th order variation seminorm of a general Ornstein–Uhlenbeck semigroup Htt>0 in Rn, taken with respect to t. We prove that this seminorm defines an operator of weak type (1, 1) with respect to the invariant measure when ϱ>2. For large t, one has an enhanced version of the standar...

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Bibliographic Details
Published in:The Journal of geometric analysis 2025-01, Vol.35 (1)
Format: Article
Language:English
Subjects:
Derivatives
Integrals
Invariants
Operators (mathematics)
Semigroups
Online Access:Get full text
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Summary:We study the ϱ-th order variation seminorm of a general Ornstein–Uhlenbeck semigroup Htt>0 in Rn, taken with respect to t. We prove that this seminorm defines an operator of weak type (1, 1) with respect to the invariant measure when ϱ>2. For large t, one has an enhanced version of the standard weak-type (1, 1) bound. For small t, the proof hinges on vector-valued Calderón–Zygmund techniques in the local region, and on the fact that the t derivative of the integral kernel of Ht in the global region has a bounded number of zeros in (0, 1]. A counterexample is given for ϱ=2; in fact, we prove that the second-order variation seminorm of Htt>0, and therefore also the ϱ-th order variation seminorm for any ϱ∈[1,2), is not of strong nor weak type (p, p) for any p∈[1,∞) with respect to the invariant measure.
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-024-01859-4