Preprint

Inserted: 26 sep 2025
Last Updated: 30 sep 2025

Year: 2025

Abstract:

We investigate the asymptotic behavior, as $\varepsilon \rightarrow 0$, of nonlocal functionals \[ \mathcal{F}_{\varepsilon} (u) = \iint_{\mathbb{R}^N \times \mathbb{R}^N} \rho_{\varepsilon} (y - x) \,
u (x) - u (y)
^p \hspace{0.17em} \mathrm{d} x \, \mathrm{d} y, \quad u \in L^p (\mathbb{R}^N ), \quad 1 \leqslant p < \infty, \] associated with a general family of non-negative measurable kernels $\{ \rho_{\varepsilon} \}_{\varepsilon > 0}$. Our primary aim is to single out the weakest moment-type assumptions on the family of kernels $\{ \rho_{\varepsilon} \}_{\varepsilon > 0}$ that are necessary and sufficient for the pointwise convergence \[ \underset{\varepsilon \rightarrow 0}{\lim } \mathcal{F}_{\varepsilon} (u) = 2 \
u\
^p_{L^p} \] to hold for every $u$ in a prescribed subspace of $L^p (\mathbb{R}^N)$. In the canonical smooth regime of compactly supported functions ($u \in C_c^{\infty} \left( \mathbb{R}^N \right)$), 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 $\varepsilon \to 0$ 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. Furthermore, 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 of kernels $\{ \rho_{\varepsilon} \}_{\varepsilon > 0}$.

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