Elementary Brezis-Browder type results and Representation formulae for s-harmonic functions

TL;DR

This paper establishes Brezis--Browder type results and representation formulas for s-harmonic functions in fractional Sobolev spaces, providing new insights into solutions of fractional Poisson equations.

Contribution

It introduces new Brezis--Browder type theorems and explicit representation formulas for solutions to fractional Poisson equations in Sobolev spaces.

Findings

Proved Brezis--Browder type results for fractional Sobolev spaces.

Derived quantitative estimates for s-harmonic functions.

Provided conditions for solutions to be represented as integral formulas.

Abstract

We prove Brezis--Browder type results for fractional Sobolev spaces and quantitative type estimates for $s$-harmonic functions. Furthermore, we give sufficient conditions for distributional solutions to the fractional Poisson's equation $(-\Delta)^su=T$ on $\mathbb{R}^d$ to be of the form $$u(x)=\int_{\mathbb{R}^d}\frac{T(y)}{|x-y|^{d-2s}}dy+l,\quad l\in \mathbb{R}.$$