The Brascamp--Lieb inequality on compact Lie groups and its extinction on homogeneous Lie groups
Michael G. Cowling, Ji Li, Chong-Wei Liang

TL;DR
This paper investigates the Brascamp--Lieb inequalities on various Lie groups, establishing when the constants are finite or infinite, and relating group inequalities to their Lie algebra counterparts, with specific results for homogeneous and compact Lie groups.
Contribution
It extends Brascamp--Lieb inequalities to nonabelian Lie groups, characterizes the constants on homogeneous and compact groups, and relates group inequalities to algebraic structures.
Findings
On homogeneous Lie groups, the Brascamp--Lieb constant equals that of the Lie algebra.
For Heisenberg-like groups, only multilinear H"older inequalities occur.
Necessary and sufficient conditions for finiteness of the constant on compact Lie groups.
Abstract
We study the Brascamp--Lieb inequalities on locally compact nonabelian groups and the Brascamp--Lieb constants associated to a Brascamp--Lieb datum: locally compact groups and , a family of homomorphisms and Lebesgue indices . We focus on homogeneous Lie groups and compact Lie groups. For homogeneous Lie groups , we show that the constant is equal to the constant , where is the Lie algebra of and is the differential of . For Heisenberg-like groups , we show that the only inequalities that can occur are multilinear H\"older inequalities. For compact Lie groups, we find necessary and sufficient conditions for finiteness…
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Taxonomy
TopicsGeometric Analysis and Curvature Flows · Advanced Mathematical Physics Problems · Advanced Operator Algebra Research
