2001/07/12 by C. Mason, Mason, C.
Computer Science · Mathematics · #35J25 #35P15 #47A75 #47B25 #47D99 #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Nonlinear Partial Differential Equations #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics #math.SP #msc:35J25 #msc:35P15 #msc:47A75 #msc:47B25 #msc:47D99
paper · pdf · doi:10.48550/arxiv.math/0107088
20 pages. Submitted to J. Funct. Anal
arxiv created 2001/07/12 · openalex publication_date 2001/07/12 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.