Fractional Laplacian with Hardy potential

Abstract

We give sharp two-sided estimates of the semigroup generated by the fractional Laplacian plus the Hardy potential on ${\mathbb{R}}^d$, including the case of the critical constant. We use Davies’ method back-to-back with a new method of integral analysis of Duhamel’s formula.

Keywords:

• Fractional Laplacian
• Hardy inequality
• heat kernel

2010 Mathematics Subject Classification:

• Primary 47D08
• 60J35
• Secondary 31C05
• 46E35

Disclosure statement

No potential conflict of interest was reported by the authors.

Acknowledgments

We thank Rupert Frank for detailed comments on [23]. We thank Rodrigo Bañuelos, Bartłomiej Dyda, Jerome Goldstein, Martin Hairer, Panki Kim, Giorgio Metafune, Edwin Perkins, and Yuli Semenov for helpful discussions.

Notes

1 Note that the exponent $(d-\alpha -\beta )/\alpha$ in the definition of f is denoted β in [3, Corollary 6].

2 The trivial factor $\frac{1}{2}$ is missing in [3, Proposition 5].

Additional information

Funding

The first and the second authors were partially supported by the Narodowe Centrum Nauki (NCN) NCN grant 2014/14/M/ST1/00600. The third author was supported by the NCN grant 2015/18/E/ST1/00239. The fourth author was supported by the NCN grant 2012/07/D/ST1/02095.