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

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Abstract

In this paper we will prove the logarithmic Sobolev inequality on free loop groups for various heat kernel measures which P. Malliavin (1989, 1991, in "Diffusion Process and Related Problems in Analysis (M. A. Pinsley, Ed.), Vol. I, Birkhäuser, Basel) constructed. Those measures are associated with the Sobolev spaces of order s (s>1/2) of the free loops in the Lie algebra. We will equipp the free loop groups with those metrics and will show that a formula of Weitzenböck type holds, which enables us to apply the method of Driver and Lohrenz (1996, J. Funct. Anal.146, 381-448).

Original languageEnglish
Pages (from-to)170-213
Number of pages44
JournalJournal of Functional Analysis
Volume179
Issue number1
DOIs
Publication statusPublished - Jan 10 2001
Externally publishedYes

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Logarithmic Sobolev Inequality
Loop Groups
Heat Kernel
Free Group
Sobolev Spaces
Diffusion Process
Driver
Lie Algebra
Metric

All Science Journal Classification (ASJC) codes

  • Analysis

Cite this

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abstract = "In this paper we will prove the logarithmic Sobolev inequality on free loop groups for various heat kernel measures which P. Malliavin (1989, 1991, in {"}Diffusion Process and Related Problems in Analysis (M. A. Pinsley, Ed.), Vol. I, Birkh{\"a}user, Basel) constructed. Those measures are associated with the Sobolev spaces of order s (s>1/2) of the free loops in the Lie algebra. We will equipp the free loop groups with those metrics and will show that a formula of Weitzenb{\"o}ck type holds, which enables us to apply the method of Driver and Lohrenz (1996, J. Funct. Anal.146, 381-448).",
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