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Potential Analysis

, Volume 18, Issue 4, pp 359–390 | Cite as

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

  • Vitali Liskevich
  • Zeev Sobol
Article

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∇)+b1⋅∇+∇⋅b2+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.

heat kernel estimates second-order elliptic operators with measurable coefficients 

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Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Vitali Liskevich
    • 1
  • Zeev Sobol
    • 1
  1. 1.School of MathematicsUniversity of BristolBristolUK

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