Bourgain-Brezis-Mironescu formula for Riesz Potentials

TL;DR

This paper establishes a pointwise limit formula connecting nonlocal potential operators with classical Riesz potentials, extending the Bourgain-Brezis-Mironescu formula to fractional differential operators.

Contribution

It identifies the pointwise limit of a nonlocal potential operator involving fractional derivatives, linking it to the classical Riesz potential of the gradient, and extends the result to broader function spaces.

Findings

Limit of nonlocal potential operator equals classical Riesz potential of gradient

Result holds for smooth functions and extends to W^{1,1} functions

Almost everywhere convergence along subsequences

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.