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Strong hypercontractivity and logarithmic Sobolev inequalities on stratified complex Lie groups


Authors: Nathaniel Eldredge, Leonard Gross and Laurent Saloff-Coste
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 35R03, 35H20; Secondary 43A15, 32W30
DOI: https://doi.org/10.1090/tran/7200
Published electronically: September 15, 2017
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Abstract: We show that for a hypoelliptic Dirichlet form operator $ A$ on a stratified complex Lie group, if the logarithmic Sobolev inequality holds, then a holomorphic projection of $ A$ is strongly hypercontractive in the sense of Janson. This extends previous results of Gross to a setting in which the operator $ A$ is not holomorphic.


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Additional Information

Nathaniel Eldredge
Affiliation: School of Mathematical Sciences, University of Northern Colorado, 501 20th Street, Box 122, Greeley, Colorado 80639
Email: neldredge@unco.edu

Leonard Gross
Affiliation: Department of Mathematics, Cornell University, 301 Malott Hall, Ithaca, New York 14853
Email: gross@math.cornell.edu

Laurent Saloff-Coste
Affiliation: Department of Mathematics, Cornell University, 301 Malott Hall, Ithaca, New York 14853
Email: lsc@math.cornell.edu

DOI: https://doi.org/10.1090/tran/7200
Keywords: Stratified complex Lie group, hypoelliptic heat kernel, strong hypercontractivity, logarithmic Sobolev inequality, holomorphic $L^p$ space
Received by editor(s): January 12, 2016
Received by editor(s) in revised form: January 12, 2017
Published electronically: September 15, 2017
Additional Notes: The first author was supported in part by a grant from the Simons Foundation (#355659, Nathaniel Eldredge)
The third author was supported in part by National Science Foundation grant DMS 1404435
Article copyright: © Copyright 2017 American Mathematical Society