Necessary and Sufficient Conditions for the Maz'ya-Shaposhnikova Formula in (Fractional) Sobolev Spaces
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908562720030720 |
|---|---|
| author | Davoli, Elisa Di Fratta, Giovanni Giorgio, Rossella Pinamonti, Andrea |
| author_facet | Davoli, Elisa Di Fratta, Giovanni Giorgio, Rossella Pinamonti, Andrea |
| contents | We investigate the asymptotic behavior, as $\varepsilon \to 0$, of nonlocal functionals $$ \mathcal{F}_{\varepsilon}(u) = \iint_{\mathbb{R}^N\times\mathbb{R}^N}
ρ_{\varepsilon}(y-x)\,|u(x)-u(y)|^p\,dx\,dy,\qquad u\in L^p(\mathbb{R}^N),\quad 1\leqslant p<\infty, $$ associated with a general family of nonnegative measurable kernels $\{ρ_{\varepsilon}\}_{\varepsilon>0}$. Our primary aim is to single out the weakest moment-type assumptions on the family $\{ρ_{\varepsilon}\}_{\varepsilon>0}$ 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 $\{ρ_{\varepsilon}\}_{\varepsilon>0}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_23226 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Necessary and Sufficient Conditions for the Maz'ya-Shaposhnikova Formula in (Fractional) Sobolev Spaces Davoli, Elisa Di Fratta, Giovanni Giorgio, Rossella Pinamonti, Andrea Analysis of PDEs Mathematical Physics Classical Analysis and ODEs Functional Analysis Primary: 46E35, 26A33, Secondary: 46B20 We investigate the asymptotic behavior, as $\varepsilon \to 0$, of nonlocal functionals $$ \mathcal{F}_{\varepsilon}(u) = \iint_{\mathbb{R}^N\times\mathbb{R}^N} ρ_{\varepsilon}(y-x)\,|u(x)-u(y)|^p\,dx\,dy,\qquad u\in L^p(\mathbb{R}^N),\quad 1\leqslant p<\infty, $$ associated with a general family of nonnegative measurable kernels $\{ρ_{\varepsilon}\}_{\varepsilon>0}$. Our primary aim is to single out the weakest moment-type assumptions on the family $\{ρ_{\varepsilon}\}_{\varepsilon>0}$ 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 $\{ρ_{\varepsilon}\}_{\varepsilon>0}$. |
| title | Necessary and Sufficient Conditions for the Maz'ya-Shaposhnikova Formula in (Fractional) Sobolev Spaces |
| topic | Analysis of PDEs Mathematical Physics Classical Analysis and ODEs Functional Analysis Primary: 46E35, 26A33, Secondary: 46B20 |
| url | https://arxiv.org/abs/2509.23226 |