Estimates of Integral Kernels for Semigroups Associated with Second-Order Elliptic Operators with Singular Coefficients

Abstract

In this paper we obtain pointwise two-sided estimates for the integral kernel of the semigroup associated with second-order elliptic differential operators −∇⋅(a∇)+b 1⋅∇+∇⋅b 2+V with real measurable (singular) coefficients, on an open set Ω⊂R N . The assumptions we impose on the lower-order terms allow for the case when the semigroup exists on L p (Ω) for p only from an interval in [1,∞), neither enjoys a standard Gaussian estimate nor is ultracontractive in the scale L p (Ω). We show however that the semigroup is ultracontractive in the scale of weighted spaces L p (Ω,ϕ2 dx) with a suitable weight ϕ and derive an upper and lower bound on its integral kernel.