Bourgain-Brezis-Mironescu formula for Riesz Potentials
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2026
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| _version_ | 1866911597205651456 |
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| author | Claros, Alejandro Pérez, Carlos |
| author_facet | Claros, Alejandro Pérez, Carlos |
| contents | We identify the Bourgain-Brezis-Mironescu pointwise limit of the nonlocal potential operator $(1-α)\, I_α(\mathcal D^αf)$, $0<α<1$, where $I_α$ denotes the Riesz potential and $\mathcal D^α$ a nonlinear fractional differential operator. Specifically, for every $f\in C_c^\infty(\mathbb R^n)$ and every $x\in \mathbb R^n$, we show that \begin{equation*}
\lim_{α\to 1^-} (1-α)\, I_α(\mathcal D^αf)(x) = K_n\, I_1(|\nabla f|)(x), \end{equation*} where $K_n$ is the geometric constant appearing in the well-known Bourgain-Brezis-Mironescu formula [BBM02]. By a density argument, we further extend this result to every $f\in W^{1,1}(\mathbb R^n)$, obtaining almost everywhere convergence along subsequences. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_06827 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bourgain-Brezis-Mironescu formula for Riesz Potentials Claros, Alejandro Pérez, Carlos Analysis of PDEs Classical Analysis and ODEs Functional Analysis We identify the Bourgain-Brezis-Mironescu pointwise limit of the nonlocal potential operator $(1-α)\, I_α(\mathcal D^αf)$, $0<α<1$, where $I_α$ denotes the Riesz potential and $\mathcal D^α$ a nonlinear fractional differential operator. Specifically, for every $f\in C_c^\infty(\mathbb R^n)$ and every $x\in \mathbb R^n$, we show that \begin{equation*} \lim_{α\to 1^-} (1-α)\, I_α(\mathcal D^αf)(x) = K_n\, I_1(|\nabla f|)(x), \end{equation*} where $K_n$ is the geometric constant appearing in the well-known Bourgain-Brezis-Mironescu formula [BBM02]. By a density argument, we further extend this result to every $f\in W^{1,1}(\mathbb R^n)$, obtaining almost everywhere convergence along subsequences. |
| title | Bourgain-Brezis-Mironescu formula for Riesz Potentials |
| topic | Analysis of PDEs Classical Analysis and ODEs Functional Analysis |
| url | https://arxiv.org/abs/2604.06827 |