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

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