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 s –harmonic functions. Furthermore, we give sufficient conditions for distributional solutions to the fractional Poisson’s equation ( − Δ ) s u = T (-Δ )^su=T on R d R^d to be of the form u ( x ) = ∫ R d T ( y ) | x − y | d − 2 s d y + l , l ∈ R . {equation*} u(x)=∫ R^d {T(y)}{|x-y|ᵈ⁻²ˢ}dy+l, l∈ R. {equation*}