Heat Kernel Estimates of Fractional Schrödinger Operators with Negative Hardy Potential

  • Tomasz JakubowskiEmail author
  • Jian Wang
Open Access


We obtain two-sided estimates for the heat kernel (or the fundamental function) associated with the following fractional Schrödinger operator with negative Hardy potential Δα/2λ|x|α
$$ {\Delta}^{\alpha/2} -\lambda |x|^{-\alpha} $$
on Open image in new window , where α ∈ (0, d ∧ 2) and λ > 0. The proof is purely analytical and elementary. In particular, for upper bounds of heat kernel we use the Chapman-Kolmogorov equation and adopt self-improving argument.


Fractional Laplacian Hardy potential Heat kernel The Chapman-Kolmogorov equation The Feynman-Kac formula Duhamel’s formula 

Mathematics Subject Classification (2010)

60G51 60G52 60J25 60J75 



We would like to thank Krzysztof Bogdan and Kamil Kaleta for interesting discussions and helpful comments. We also would like to thank the referee for his/her careful reading and numerous corrections.


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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Faculty of Pure and Applied MathematicsWrocław University of Science and TechnologyWrocławPoland
  2. 2.College of Mathematics and Informatics & Fujian Key Laboratory of Mathematical Analysis and Applications (FJKLMAA)Fujian Normal UniversityFuzhouPeople’s Republic of China

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