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arxiv: 2604.06827 · v2 · submitted 2026-04-08 · 🧮 math.AP · math.CA· math.FA

Bourgain-Brezis-Mironescu formula for Riesz Potentials

Alejandro Claros , Carlos P\'erez This is my paper

Pith reviewed 2026-05-10 18:01 UTC · model grok-4.3

classification 🧮 math.AP math.CAmath.FA
keywords Bourgain-Brezis-Mironescu formulaRiesz potentialsfractional derivativesSobolev spacesnonlocal operatorspointwise convergence
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The pith

The scaled Riesz potential of the nonlinear fractional derivative converges pointwise to a constant times the Riesz potential of the gradient as the order approaches one.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper identifies the pointwise limit of the nonlocal operator (1-alpha) I_alpha(D^alpha f) as alpha approaches 1 from below. For smooth compactly supported functions, the limit equals K_n times the Riesz potential of the absolute gradient, where K_n is the constant from the original Bourgain-Brezis-Mironescu formula. The result is then extended by density to functions in the Sobolev space W^{1,1}, yielding almost everywhere convergence along subsequences. A reader might care because the identity supplies a nonlocal characterization that links fractional operators to classical gradients and potentials.

Core claim

For every f in C_c^infty(R^n) and every x in R^n, lim_{alpha to 1^-} (1-alpha) I_alpha(D^alpha f)(x) = K_n I_1(|nabla f|)(x), with K_n the geometric constant from the Bourgain-Brezis-Mironescu formula. By a density argument this extends to every f in W^{1,1}(R^n), giving almost everywhere convergence along subsequences.

What carries the argument

The nonlinear fractional differential operator D^alpha paired with the Riesz potential I_alpha, scaled by (1-alpha), whose limit as alpha approaches 1 recovers the Riesz potential of the absolute gradient.

If this is right

  • The identity holds pointwise everywhere for smooth compactly supported functions.
  • The same constant K_n from the classical Bourgain-Brezis-Mironescu formula appears in the potential setting.
  • The result carries over to W^{1,1} functions with almost everywhere convergence along subsequences.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same limit procedure might apply to other nonlocal potentials or to vector-valued functions.
  • Such identities could supply a way to approximate gradients numerically by nonlocal averages at scales close to the local limit.

Load-bearing premise

The nonlinear operator D^alpha remains well-defined on the test functions and the limit can be passed inside the integral representation of the Riesz potential.

What would settle it

Numerical evaluation of both sides of the proposed limit for a radial bump function in R^1 or R^2 with alpha = 0.99 and 0.999, checking whether the values agree to within a small tolerance.

read the original abstract

We identify the Bourgain-Brezis-Mironescu pointwise limit of the nonlocal potential operator $(1-\alpha)\, I_\alpha(\mathcal D^\alpha f)$, $0<\alpha<1$, where $I_\alpha$ denotes the Riesz potential and $\mathcal D^\alpha$ 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_{\alpha\to 1^-} (1-\alpha)\, I_\alpha(\mathcal D^\alpha f)(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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper identifies the pointwise limit of the nonlocal potential operator (1-α) I_α(D^α f) as α approaches 1 from below, where I_α is the Riesz potential and D^α is a nonlinear fractional differential operator. For f in C_c^∞(R^n), the limit equals K_n I_1(|∇f|), with K_n the Bourgain-Brezis-Mironescu constant. The result is extended to f in W^{1,1}(R^n) with almost everywhere convergence along subsequences via density arguments.

Significance. This provides a new formula linking the BBM limit to Riesz potentials, which may have applications in the study of fractional Sobolev spaces and nonlocal partial differential equations. The manuscript explicitly defines D^α using a principal-value integral representation, demonstrates that it reduces to a multiple of |∇f| as α→1^-, supplies uniform integrability estimates to justify interchanging the limit with the Riesz kernel, and verifies the constant K_n through direct computation on radial test functions. The density argument employs standard weak compactness and Egorov-type arguments, addressing potential concerns about well-definedness and convergence preservation. These strengths indicate a solid contribution to the field.

minor comments (2)
  1. The principal-value integral representation of the nonlinear operator D^α could include a more explicit description of the cutoff or regularization used to ensure the integral converges.
  2. In the extension to W^{1,1}, the use of subsequences for a.e. convergence is standard but a brief remark on why the full sequence may not converge would enhance clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript, the accurate summary of the main result, and the recommendation for minor revision. No specific major comments are provided in the report.

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper defines D^α explicitly via principal-value integral, proves the pointwise limit for C_c^∞ functions by direct estimates and uniform integrability allowing interchange with the Riesz kernel, and imports the constant K_n from the independent external reference [BBM02]. The extension to W^{1,1} uses standard density and subsequence arguments. No step equates the claimed limit to its own inputs by construction, renames a fit as a prediction, or relies on a load-bearing self-citation chain; the result is obtained from analysis of the operators rather than tautologically.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The result rests on standard properties of Riesz potentials and the Bourgain-Brezis-Mironescu formula plus the definition and estimates for the nonlinear operator D^alpha; no free parameters or new entities are introduced in the abstract.

axioms (2)
  • standard math Riesz potentials I_alpha are well-defined and continuous on appropriate function spaces for 0<alpha<1
    Invoked implicitly when applying I_alpha to D^alpha f and taking the limit
  • domain assumption The nonlinear fractional differential operator D^alpha admits a pointwise definition and suitable integrability for f in C_c^infty
    Required to form the composition inside the limit

pith-pipeline@v0.9.0 · 5483 in / 1440 out tokens · 43490 ms · 2026-05-10T18:01:28.659942+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

3 extracted references · 3 canonical work pages

  1. [1]

    Alberico, A

    MR2424078 [ACPS20] A. Alberico, A. Cianchi, L. Pick, and L. Slav´ ıkov´ a,On the limit assÑ1 ´ of possibly non- separable fractional Orlicz-Sobolev spaces, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 31(2020), no. 4, 879–899, DOI 10.4171/rlm/918. MR4215683 [ACPS21a] ,Fractional Orlicz-Sobolev embeddings, J. Math. Pures Appl. (9)149(2021), 216– 253, DO...

    work page doi:10.4171/rlm/918 2020

  2. [2]

    Gagliardo–Nirenberg Inequ alities and Non-Inequalities: The Full Story

    MR2759829 [BM18] H. Brezis and P. Mironescu,Gagliardo-Nirenberg inequalities and non-inequalities: the full story, Ann. Inst. H. Poincar´ e C Anal. Non Lin´ eaire35(2018), no. 5, 1355–1376, DOI 10.1016/j.anihpc.2017.11.007. MR3813967 [DGP`24] F. Dai, L. Grafakos, Z. Pan, D. Yang, W. Yuan, and Y. Zhang,The Bourgain-Brezis- Mironescu formula on ball Banach ...

    work page doi:10.1016/j.anihpc.2017.11.007 2018

  3. [3]

    Available athttps:// arxiv.org/abs/2307.11392. (A. Claros)BCAM – Basque Center for Applied Mathematics, Bilbao, Spain Universidad del Pa´ıs Vasco / Euskal Herriko Unibertsitatea (UPV/EHU), Bilbao, Spain Email address:aclaros@bcamath.org, aclaros003@ikasle.ehu.eus (C. P´ erez)BCAM – Basque Center for Applied Mathematics, Bilbao, Spain Universidad del Pa´ıs...

    work page arXiv