TY - JOUR
T1 - A unification of hypercontractivities of the Ornstein–Uhlenbeck semigroup and its connection with Φ-entropy inequalities
AU - Hariya, Yuu
N1 - Funding Information:
The author is grateful to an anonymous referee for his/her valuable comments and suggestions; in particular, the paper [3] and another proof of Lemma 3.2 , as well as the reasoning to observe the optimality of the inequality (1.6) , were notified of by the referee. His thanks also go to Professor P. Ivanisvili for sending him a note [9] explaining how the generalization (3.21) of the logarithmic Sobolev inequality is also derived from [10, Theorem 1] , which reasoning made the author aware of a link between (3.21) and Φ-entropy inequalities; most of the contents of Remark 3.4 is taken from that note. This work was partially supported by JSPS KAKENHI Grant Number 17K05288 .
Publisher Copyright:
© 2018 Elsevier Inc.
PY - 2018/11/15
Y1 - 2018/11/15
N2 - Let γd be the d-dimensional standard Gaussian measure and Q={Qt}t≥0 the Ornstein–Uhlenbeck semigroup acting on L1(γd). 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.
AB - Let γd be the d-dimensional standard Gaussian measure and Q={Qt}t≥0 the Ornstein–Uhlenbeck semigroup acting on L1(γd). 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.
KW - Hypercontractivity
KW - Logarithmic Sobolev inequality
KW - Ornstein–Uhlenbeck semigroup
KW - Φ-entropy
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U2 - 10.1016/j.jfa.2018.08.009
DO - 10.1016/j.jfa.2018.08.009
M3 - Article
AN - SCOPUS:85051999742
SN - 0022-1236
VL - 275
SP - 2647
EP - 2683
JO - Journal of Functional Analysis
JF - Journal of Functional Analysis
IS - 10
ER -