Poisson Problems involving fractional Hardy operators and measures

TL;DR

This paper investigates the Poisson problem with fractional Hardy operators and measures, addressing the challenges posed by nonlocality and singular potentials through advanced functional analysis and establishing key solvability and regularity results.

Contribution

It introduces a novel framework for analyzing Poisson problems involving fractional Hardy operators and measures, including dual operator analysis and new regularity results.

Findings

Established solvability and a priori estimates for the problem

Proved variants of Kato's inequality in this context

Derived regularity results for solutions

Abstract

In this paper, we study the Poisson problem involving a fractional Hardy operator and a measure source. The complex interplay between the nonlocal nature of the operator, the peculiar effect of the singular potential and the measure source induces several new fundamental difficulties in comparison with the local case. To overcome these difficulties, we perform a careful analysis of the dual operator in the weighted distributional sense and establish fine properties of the associated function spaces, which in turn allow us to formulate the Poisson problem in an appropriate framework. In light of the close connection between the Poisson problem and its dual problem, we are able to establish various aspects of the theory for the Poisson problem including the solvability, a priori estimates, variants of Kato's inequality and regularity results.