On Sobolev norms involving Hardy operators in a half-space

TL;DR

This paper compares Sobolev spaces generated by Hardy operators and fractional Laplacians in a half-space, extending previous results from the whole space and using heat kernel bounds.

Contribution

It establishes the comparability of Sobolev spaces generated by Hardy operators and fractional Laplacians in a half-space, under certain conditions on the coupling constant.

Findings

Sobolev spaces generated by Hardy operators and fractional Laplacians are comparable in a half-space.

Results extend known Euclidean space results to half-space settings.

Utilizes recent heat kernel bounds to establish these comparisons.

Abstract

We consider Hardy operators on the half-space, that is, ordinary and fractional Schr\"odinger operators with potentials given by the appropriate power of the distance to the boundary. We show that the scales of homogeneous Sobolev spaces generated by the Hardy operators and by the fractional Laplacian are comparable with each other when the coupling constant is not too large in a quantitative sense. Our results extend those in the whole Euclidean space and rely on recent heat kernel bounds.