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Compactness and Density Estimates for Weighted Fractional Heat Semigroups

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Abstract

We prove that the operator \(L_0=-(1+|x|)^\beta (-\Delta )^{\alpha /2}\) with \(\alpha \in (0,2)\), \(d>\alpha \) and \(\beta \ge 0\) generates a compact semigroup or resolvent on \(L^2(\mathbb {R}^d;(1+|x|)^{-\beta }\,\mathrm{d}x)\), if and only if \(\beta >\alpha \). When \(\beta >\alpha \), 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 

Mathematics Subject Classification (2010)

60G51 60G52 60J25 60J75 

Notes

Acknowledgements

The author would like to think the referee for helpful comments and careful corrections. The research is supported by National Natural Science Foundation of China (No. 11522106), the Fok Ying Tung Education Foundation (No. 151002) and the Program for Probability and Statistics: Theory and Application (No. IRTL1704).

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.College of Mathematics and Informatics & Fujian Provincial Key Laboratory of Mathematical Analysis and its Applications (FJKLMAA)Fujian Normal UniversityFuzhouPeople’s Republic of China

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