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Let $D\subseteq\mathbb{R}^d$ be a nice but not smooth domain, somewhere between Lipschitz and $C^2$. I am looking for a reference on the theory of mollifiers and regularization for functions on $\partial D$, but I have absolutely no experience with manifolds and I do not know where to start from. I would also appreciate any intuitions and comments on the differences and obstacles in comparison with the situation on $\mathbb{R}^d$.

I would hope to have kernels $k_\epsilon\colon \partial D\times \partial D \to \mathbb{R}$ of mass 1 for each $x$, supported in $\{(x,y)\in \partial D\times \partial D: y\in B(x,\epsilon)\}$ (but is the ball in $\mathbb{R}^d$ or $\partial D$?), compatible with each other and smooth in some sense (perhaps Lipschitz). My guess is that a function from $L^p(\partial D)$, $p\in [1,\infty)$, regularized with $k$, that is $$f_\epsilon(x) = \int_{B(x,\epsilon)} f(y) k_\epsilon(x,y)\, dy,$$ should be Lipschitz (which is good enough for my purposes) and that there is convergence in $L^p$ as $\epsilon\to 0^+$. The biggest problem for me is the strict definition of all the objects, but this seems like something which should be covered well somewhere in the literature.

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    $\begingroup$ I don't have a good reference, but the standard approach in such situations would be to just localize and flatten the boundary. I.e. if $\partial D$ locally looks like $(x,h(x))$ for some Lipschitz $h:\mathbb{R}^{n-1} \to \mathbb{R}$, then just localize with a partition of unity, mollify $f \circ (id,h)$ classically in $\mathbb{R}^{n-1}$ and push the result back to $\partial D$. All the estimates are then obtained form the classical estimates and the area formula (or chain rule, if you want derivatives). $\endgroup$
    – mlk
    Mar 29, 2021 at 11:58
  • $\begingroup$ Yes, that's one way to do it. However, I aim to work with certain Sobolev-type spaces and that approach would require a (new) extension result for them. This would not necessarily be difficult, but rather lengthy, and a mollifier on the boundary could save all this work. $\endgroup$
    – ajr
    Mar 29, 2021 at 12:55

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