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

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 $(-\mathrm{\Delta }{)}^{s}u=T$ on ${\mathbb{R}}^d$ to be of the form $$u(x)={\int }_{{\mathbb{R}}^d}\frac{T(y)}{\mathrm{\mid }x-y{\mathrm{\mid }}^{d-2s}}dy+l,l\in \mathbb{R}\mathrm{.}$$