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\|²_H⁻ˢ(ᵈ)=∫ᵈ∫ᵈT(x)T(y)|x-y|ᵈ⁻²ˢdx dy.
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Damiano Greco (2024) studied this question.
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