Bourgain-Brezis-Mironescu formula for Riesz Potentials

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 $f\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ 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.