scholarly journals Logarithmic Sobolev Inequality on Free Loop Groups for Heat Kernel Measures Associated with the General Sobolev Spaces

2001 ◽  
Vol 179 (1) ◽  
pp. 170-213 ◽  
Author(s):  
Yuzuru Inahama
Keyword(s):  
Heat Kernel ◽  
Sobolev Spaces ◽  
Sobolev Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Loop Groups

LOGARITHMIC SOBOLEV INEQUALITY FOR $H_0^s$-METRIC ON PINNED LOOP GROUPS

2004 ◽  
Vol 07 (01) ◽  
pp. 1-26 ◽  
Author(s):  
YUZURU INAHAMA
Keyword(s):  
Asymptotic Behavior ◽  
Lie Groups ◽  
Sobolev Inequality ◽  
Ricci Tensor ◽  
Logarithmic Sobolev Inequality ◽  
Loop Groups ◽  
Compact Lie Groups

In this paper we prove the logarithmic Sobolev inequality on pinned loop groups over compact Lie groups equipped with [Formula: see text]-metric (s>1/2). We also compute the Ricci tensor with respect to [Formula: see text]-metric and investigate its asymptotic behavior as s decreases to 1/2.

scholarly journals A Generalization of a Logarithmic Sobolev Inequality to the Hölder Class

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
H. Ibrahim
Keyword(s):  
Sobolev Spaces ◽  
Sobolev Inequality ◽  
Critical Case ◽  
Type Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Continuous Functions ◽  
Hölder Continuous ◽  
Hölder Continuous Functions ◽  
Logarithmic Type ◽  
Class Of Functions

In a recent work of the author, a parabolic extension of the elliptic Ogawa type inequality has been established. This inequality is originated from the Brézis-Gallouët-Wainger logarithmic type inequalities revealing Sobolev embeddings in the critical case. In this paper, we improve the parabolic version of Ogawa inequality by allowing it to cover not only the class of functions from Sobolev spaces, but also the wider class of Hölder continuous functions.

scholarly journals Functional Inequalities for Two-Level Concentration

2021 ◽  
Author(s):  
Franck Barthe ◽  
Michał Strzelecki
Keyword(s):  
Sobolev Inequality ◽  
Poincaré Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Probability Measures ◽  
Exponential Rate ◽  
Functional Inequalities ◽  
Concentration Inequality ◽  
Poincare Inequality ◽  
Free Concentration ◽  
Concentration Property

AbstractProbability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension-free concentration property, known as two-level concentration. We show that a similar phenomenon occurs for the Latała–Oleszkiewicz inequalities, which were devised to uncover dimension-free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.

Equivalence between logarithmic Sobolev inequality and hypercontractivity in a probability gage space

2020 ◽  
pp. 1-13
Author(s):  
Zhang Lunchuan
Keyword(s):  
Dirichlet Form ◽  
Sobolev Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Markov Semigroup ◽  
Quantum Markov Semigroup

Abstract In this paper, we prove the equivalence between logarithmic Sobolev inequality and hypercontractivity of a class of quantum Markov semigroup and its associated Dirichlet form based on a probability gage space.

scholarly journals Application of Initial Boundary Value Problem on Logarithmic Wave Equation in Dynamics

2018 ◽  
Author(s):  
Shkelqim Hajrulla ◽  
Leonard Bezati ◽  
Fatmir Hoxha
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Boundary Value Problem ◽  
Sobolev Inequality ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Logarithmic Sobolev Inequality ◽  
Initial Boundary Value ◽  
The Galerkin Method

We introduce a class of logarithmic wave equation. We study the global existence of week solution for this class of equation. We deal with the initial boundary value problem of this class. Using the Galerkin method and the Gross logarithmic Sobolev inequality we establish the main theorem of existence of week solution for this class of equation arising from Q-Ball Dynamic in particular.

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