Brezis--Browder type results for Fractional Sobolev spaces and applications

In this paper we prove Brezis--Browder type results for homogeneous fractional Sobolev spaces Hˢ (ᵈ) and quantitive type estimates for s--harmonic functions. We further relate such outcomes with the question of whether a linear and continuous functional T defined on Hˢ (ᵈ) admits (up to a constant) an integral representation of its norm in terms of the Coulomb type energy \|T\|²₇^-ₒ (ᵈ) =㵧㵧T (x) T (y) |x-y|^{d-2s}dx dy.