Compactness and Density Estimates for Weighted Fractional Heat Semigroups

Jian Wang ()
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Jian Wang: Fujian Normal University

Journal of Theoretical Probability, 2019, vol. 32, issue 4, 2066-2087

Abstract: Abstract We prove that the operator $$L_0=-(1+|x|)^\beta (-\Delta )^{\alpha /2}$$ L 0 = - ( 1 + | x | ) β ( - Δ ) α / 2 with $$\alpha \in (0,2)$$ α ∈ ( 0 , 2 ) , $$d>\alpha $$ d > α and $$\beta \ge 0$$ β ≥ 0 generates a compact semigroup or resolvent on $$L^2(\mathbb {R}^d;(1+|x|)^{-\beta }\,\mathrm{d}x)$$ L 2 ( R d ; ( 1 + | x | ) - β d x ) , if and only if $$\beta >\alpha $$ β > α . When $$\beta >\alpha $$ β > α , we obtain two-sided asymptotic estimates for high-order eigenvalues, and sharp bounds for the corresponding heat kernel.

Keywords: Weighted fractional Laplacian operator; Compactness; Heat kernel; (Intrinsic) Super Poincaré inequality; 60G51; 60G52; 60J25; 60J75 (search for similar items in EconPapers)
Date: 2019
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DOI: 10.1007/s10959-018-0838-9

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