Critical weak-$L^{p}$ differentiability of singular integrals
Luigi Ambrosio, Augusto C. Ponce, Rémy Rodiac
TL;DR
This work addresses the differentiability of the distributional gradient for functions $u$ with $\Delta u$ a finite measure, showing that $\nabla u$ has a second approximate derivative $D_{\mathrm{ap}}^{2}u$ almost everywhere and that $(\Delta u)_{\mathrm{a}}=\mathrm{tr}(D_{\mathrm{ap}}^{2}u)\,dx$. The authors develop a weak-$L^{\frac{N}{N-1}}$ differentiability theory for singular integrals of the form $K*\mu$ with $\mu$ a finite measure, via a refined Calderón–Zygmund analysis built upon Hajłasz's Lipschitz-type estimates and a uniformization principle. They establish a global weak-$L^{1}$ bound for the approximate derivative, and prove existence and identification results for ${D}_{\mathrm{ap}}^{2}u$ in terms of the a.c. part of $\Delta u$, including a density-point argument showing vanishing on level sets $\{u=\alpha\}$ and $\{\nabla u=e\}$. The paper then applies these findings to (i) level sets of subharmonic functions and (ii) limiting vorticities in the two-dimensional Ginzburg–Landau system, obtaining a precise decomposition of the limiting vorticity and a Schrödinger-type equation for the limiting magnetic field.
Abstract
We establish that for every function $u \in L^1_\mathrm{loc}(Ω)$ whose distributional Laplacian $Δu$ is a signed Borel measure in an open set $Ω$ in $\mathbb{R}^{N}$, the distributional gradient $\nabla u$ is differentiable almost everywhere in $Ω$ with respect to the weak-$L^{\frac{N}{N-1}}$ Marcinkiewicz norm. We show in addition that the absolutely continuous part of $Δu$ with respect to the Lebesgue measure equals zero almost everywhere on the level sets $\{u = α\}$ and $\{\nabla u = e\}$, for every $α\in \mathbb{R}$ and $e \in \mathbb{R}^N$. Our proofs rely on an adaptation of Calderón and Zygmund's singular-integral estimates inspired by subsequent work by Hajlasz.