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

Abstract

We investigate the asymptotic behavior, as $\epsilon \to 0$, of nonlocal functionals $${\mathcal{F}}_{\epsilon }(u)={\iint }_{{\mathbb{R}}^N\times {\mathbb{R}}^N}{\rho }_{\epsilon }(y-x)|u(x)-u(y){|}^{p}dxdy,u\in L^p({\mathbb{R}}^N),1⩽p<\infty ,$$ associated with a general family of nonnegative measurable kernels $\{{\rho }_{\epsilon }{\}}_{\epsilon >0}$. Our primary aim is to single out the weakest moment-type assumptions on the family $\{{\rho }_{\epsilon }{\}}_{\epsilon >0}$ that are necessary and sufficient for the pointwise convergence $$\underset{\epsilon \to 0}{\lim }{\mathcal{F}}_{\epsilon }(u)=2\Vert u{\Vert }_{L^p}^p$$ to hold for every $u$ in a prescribed subspace of $L^p({\mathbb{R}}^N)$. In the canonical smooth regime of compactly supported functions ($u\in C_c^{\infty }({\mathbb{R}}^N)$) 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 $\epsilon \to 0$ 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 ($u\in W^{1,p}({\mathbb{R}}^N)$). Finally, by adapting the method to fractional Sobolev spaces $W^{s,p}({\mathbb{R}}^N)$ with $s\in (0,1)$, we recover the Maz'ya-Shaposhnikova formula and extend it under analogous abstract conditions on the family $\{{\rho }_{\epsilon }{\}}_{\epsilon >0}$.