Non-local approximations of the gradient
Résumé
We revisit the proofs of a few basic results concerning non-local approximations of the gradient. A typical such result asserts that,
if $(\rho_\varepsilon)$ is a radial approximation to the identity in ${\mathbb R}^N$ and $u$ belongs to a homogeneous Sobolev space $\dot W^{1,p}$, then
\begin{equation*}
V_\varepsilon(x):=N\int\limits_{{\mathbb R}^N}\frac{u(x+h)-u(x)}{|h|} \frac{h}{|h|}\rho_\varepsilon(h)\, dh, \ x\in{\mathbb R}^N,
\end{equation*}
converges in $L^p$ to the distributional gradient $\nabla u$ as $\varepsilon\to 0$.
We highlight the crucial role played by the representation formula $V_\varepsilon=(\nabla u)\ast F_\varepsilon$, where $F_\varepsilon$ is an approximation to the identity defined {\it via} $\rho_\varepsilon$. This formula allows to unify the proofs of a significant number of results in the literature, by reducing them to standard properties of the approximations to the identity.
We also highlight the effectiveness of a symmetric nonlocal integration by parts formula.
Relaxations of the assumptions on $u$ and $\rho_\varepsilon$, allowing, e.g., heavy tails kernels or a distributional definition of $V_\varepsilon$, are also discussed. In particular, we show that heavy tails kernels may be treated as perturbations of approximations to the identity.
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