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.
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