标题： Maz'ya-Shaposhnikova 公式在（分数）Sobolev 空间中的必要充分条件 显示英文标题

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

摘要： 我们研究当$\varepsilon \to 0$时与一般非负可测核族$\{\rho_{\varepsilon}\}_{\varepsilon>0}$相关的非局部泛函$$ \mathcal{F}_{\varepsilon}(u) = \iint_{\mathbb{R}^N\times\mathbb{R}^N} \rho_{\varepsilon}(y-x)\,|u(x)-u(y)|^p\,dx\,dy,\qquad u\in L^p(\mathbb{R}^N),\quad 1\leqslant p<\infty, $$的渐近行为。我们的主要目的是明确家族$\{\rho_{\varepsilon}\}_{\varepsilon>0}$上最弱的矩型假设，这些假设对于每个$u$在$L^p(\mathbb{R}^N)$的指定子空间中的逐点收敛$$ \lim_{\varepsilon\to 0}\mathcal{F}_{\varepsilon}(u)=2\|u\|_{L^p}^p $$成立是必要且充分的。 在紧支撑函数的规范光滑区域（$u\in C_c^{\infty}(\mathbb{R}^N)$）中，我们证明当满足两个最优条件时会发生收敛：（i）质量逃逸条件，以及（ii）短程衰减效应，由核在$\varepsilon\to 0$时的$p$-矩在原点任意固定邻域内的消失来表达。 这个一般框架恢复了分数型核的Maz'ya--Shaposhnikova定理，并将收敛结果扩展到更广泛的相互作用轮廓类别，这些轮廓可能是非对称和非齐次的。 通过使用保留矩假设的密度论证，我们证明相同的必要和充分条件在整数阶Sobolev设置（$u\in W^{1,p}(\mathbb{R}^N)$）中仍然有效。 最后，通过将方法适应到分数Sobolev空间$W^{s,p}(\mathbb{R}^N)$（具有$s\in(0,1)$），我们恢复了Maz'ya-Shaposhnikova公式，并在族$\{\rho_{\varepsilon}\}_{\varepsilon>0}$上的类似抽象条件下对其进行了扩展。 显示英文摘要

摘要： We investigate the asymptotic behavior, as $\varepsilon \to 0$, of nonlocal functionals $$ \mathcal{F}_{\varepsilon}(u) = \iint_{\mathbb{R}^N\times\mathbb{R}^N} \rho_{\varepsilon}(y-x)\,|u(x)-u(y)|^p\,dx\,dy,\qquad u\in L^p(\mathbb{R}^N),\quad 1\leqslant p<\infty, $$ associated with a general family of nonnegative measurable kernels $\{\rho_{\varepsilon}\}_{\varepsilon>0}$. Our primary aim is to single out the weakest moment-type assumptions on the family $\{\rho_{\varepsilon}\}_{\varepsilon>0}$ that are necessary and sufficient for the pointwise convergence $$ \lim_{\varepsilon\to 0}\mathcal{F}_{\varepsilon}(u)=2\|u\|_{L^p}^p $$ 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 $\varepsilon\to 0$ 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. 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 $\{\rho_{\varepsilon}\}_{\varepsilon>0}$.