Estimates of Integral Kernels for Semigroups Associated with Second-Order Elliptic Operators with Singular Coefficients |
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Authors: | Liskevich Vitali Sobol Zeev |
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Institution: | (1) School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK |
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Abstract: | In this paper we obtain pointwise two-sided estimates for the integral kernel of the semigroup associated with second-order elliptic differential operators –![nabla](/content/n15h7572g746423w/xxlarge8711.gif) (a )+b
1![sdot](/content/n15h7572g746423w/xxlarge8901.gif) +![nabla](/content/n15h7572g746423w/xxlarge8711.gif) b
2+V with real measurable (singular) coefficients, on an open set ![OHgr](/content/n15h7572g746423w/xxlarge937.gif) R
N
. The assumptions we impose on the lower-order terms allow for the case when the semigroup exists on L
p
( ) for p only from an interval in 1, ), neither enjoys a standard Gaussian estimate nor is ultracontractive in the scale L
p
( ). We show however that the semigroup is ultracontractive in the scale of weighted spaces L
p
( , 2 dx) with a suitable weight and derive an upper and lower bound on its integral kernel. |
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Keywords: | heat kernel estimates second-order elliptic operators with measurable coefficients |
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