Necessary and Sufficient Conditions for the Maz'ya-Shaposhnikova Formula in (Fractional) Sobolev Spaces

We investigate the asymptotic behavior, as ε → 0, of nonlocal functionals Fε(u) = RNN ρε(y-x)\,|u(x)-u(y)|ᵖ\,dx\,dy, u∈ Lᵖ(RN), 1 p<∞, associated with a general family of nonnegative measurable kernels \ρε\ε>0. Our primary aim is to single out the weakest moment-type assumptions on the family \ρε\ε>0 that are necessary and sufficient for the pointwise convergence limε→ 0Fε(u)=2\|u\|Lᵖᵖ to hold for every u in a prescribed subspace of Lᵖ(RN). In the canonical smooth regime of compactly supported functions (u∈ Cc∞(RN)) we show that convergence occurs when two optimal conditions are satisfied: (i) a mass-escape condition, and (ii) a short-range attenuation effect, expressed by the vanishing as ε→ 0 of the kernels' p-moments in any fixed neighborhood of the origin. This general framework recovers the classical Maz'ya--Shaposhnikova theorem for fractional-type kernels and extends the convergence result to a much broader class of interaction profiles, which may be non-symmetric and non-homogeneous. Using a density argument that preserves the moment assumptions, we prove that the same necessary and sufficient conditions remain valid in the integer-order Sobolev setting (u∈ W1,p(RN)). Finally, by adapting the method to fractional Sobolev spaces Ws,p(RN) with s∈(0,1), we recover the Maz'ya-Shaposhnikova formula and extend it under analogous abstract conditions on the family \ρε\ε>0.