Some remarks on essential self-adjointness and ultracontractivity of a class of singular elliptic operators

✍ Scribed by Michael M. H. Pang

• Publisher: John Wiley and Sons
• Year: 2008
• Tongue: English
• Weight: 143 KB
• Volume: 281
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.200810660

✦ Synopsis

Abstract

We study the properties of essential self‐adjointness on C^∞^~c~ (ℝ^N^ ) and semigroup ultracontractivity of a class of singular second order elliptic operators

equation image

defined in L^2^ (ℝ^N^ , σ^–a –N^ (x) dx) with Dirichlet boundary conditions, where a, b ∈ ℝ and σ: ℝ^N^ → (0, ∞) is a C^∞^‐function satisfying

c^‐1^(1 + |x |) ≤ σ (x) ≤ c (1 + |x |) (x ∈ ℝ__N__).

We also obtain sharp short time upper and lower diagonal bounds on the heat kernel of e –^Ht^. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)