Log--Sobolev Inequalities and Regions with Exterior Exponential Cusps

Details Log--Sobolev Inequalities and Regions with Exterior Exponential Cusps Log--Sobolev Inequalities and Regions with Exterior Exponential Cusps

2001-07-12

arxiv.org/abs/math/0107088v1

Mathematics

Spectral Theory

20 pages. Submitted to J. Funct. Anal

Scientific paper

We begin by studying semigroup estimates that are more singular than those implied by a Sobolev embedding theorem but which are equivalent to certain logarithmic Sobolev inequalities. We then give a method for showing such log--Sobolev inequalities hold for Euclidean regions that satisfy a certain Hardy-type inequality. Our main application is to show that domains with exterior exponential cusps, and hence no Sobolev embedding theorem, satisfy such heat kernel bounds provided the cusp is not too sharp. Finally we consider a rotationally invariant domain with an exponentially sharp cusp and show that ultracontractivity does break down after a certain point.

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