# On Scales of Sobolev spaces associated to generalized Hardy operators

## TL;DR

This paper investigates the Sobolev spaces linked to the fractional Laplacian with Hardy potential, extending previous $L^2$ results using harmonic analysis tools, despite limitations with negative coupling constants.

## Contribution

It introduces a scale of Sobolev spaces for the fractional Laplacian with Hardy potential and develops a Littlewood--Paley theory using a H"ormander multiplier theorem.

## Key findings

- Extended Sobolev space analysis beyond $L^2$
- Developed a Littlewood--Paley framework for the operator
- Identified limitations with negative coupling constants due to heat kernel decay

## Abstract

We consider the fractional Laplacian with Hardy potential and study the scale of homogeneous $L^p$ Sobolev spaces generated by this operator. Besides generalized and reversed Hardy inequalities, the analysis relies on a H\"ormander multiplier theorem which is crucial to construct a basic Littlewood--Paley theory. The results extend those obtained recently in $L^2$ but do not cover negative coupling constants in general due to the slow decay of the associated heat kernel.