A unification of hypercontractivities of the Ornstein–Uhlenbeck semigroup and its connection with Φ-entropy inequalities

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Abstract

Let γd be the d-dimensional standard Gaussian measure and Q={Qt}t≥0 the Ornstein–Uhlenbeck semigroup acting on L1d). The semigroup Q enjoys the hypercontractivity, which is also known to be equivalent to the exponential hypercontractivity. In this paper, by employing stochastic analysis, we derive a family of inequalities that unifies the exponential and original hypercontractivities; a generalization of the Gaussian logarithmic Sobolev inequality is obtained as a corollary. We then discuss a connection of those results with Φ-entropy inequalities in a general framework of Markov semigroups. A unification of the exponential hypercontractivity and the reverse hypercontractivity of the Ornstein–Uhlenbeck semigroup Q is also provided.

Original languageEnglish
Pages (from-to)2647-2683
Number of pages37
JournalJournal of Functional Analysis
Volume275
Issue number10
DOIs
Publication statusPublished - 2018 Nov 15

Keywords

  • Hypercontractivity
  • Logarithmic Sobolev inequality
  • Ornstein–Uhlenbeck semigroup
  • Φ-entropy

ASJC Scopus subject areas

  • Analysis

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