Fractional Laplacian with Hardy potential

TL;DR

This paper provides precise estimates for the semigroup of the fractional Laplacian combined with Hardy potential on Euclidean space, addressing the critical case using advanced analytical techniques.

Contribution

It introduces a novel integral analysis method for Duhamel's formula and sharp two-sided estimates for the semigroup with Hardy potential, including the critical constant case.

Findings

Sharp two-sided estimates for the semigroup are established.

The method combines Davies' approach with a new integral analysis technique.

Results include the critical Hardy constant case.

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.