Rates of convergence of mollifiers with Sobolev norms on manifold

Question

Let $M$ be a smooth compact Riemannian manifold of dimension $n$, and let $H^s_p(M)$ for $s\in \mathbb{R}$ be the fractional Sobolev space of order $s$ on the manifold (defined for instance through the Laplace-Beltrami operator). Let $k:\mathbb{R}\to \mathbb{R}$ be a radial kernel and let $k_h=h^{-n}k(\cdot/h)$ for $h>0$. For $f:M\to \mathbb{R}$, one define $k_h*f$ by $$ k_h*f(x) = \int_{T_x M} k_h(|v|)f(\exp_x(v)) d v.$$ Consider the following assertion: For $s<r\in \mathbb{R}$, one can choose $k$ (with sufficiently zero moments) such that there exists a constant $C$ depending on the manifold, $r$ and $s$ with, $\forall f\in H_p^r(M)$, $$\|f-k_h*f\|_{H_p^s(M)} \leq Ch^{r-s} \|f\|_{H_p^r(M)}.$$

If one replace $M$ by $\mathbb{R}^n$, a simple proof of this assertion is possible by using the Fourier characterization of the Sobolev spaces. Such a construction does not generalize well to manifolds. I suspect this result to be standard. Any reference would be helpful.

Answer 1

For $s>0$ the easiest way to obtain convolution estimates on manifolds is described below.

First of all, by the Whitney embedding theorem (or by Nash theorem if you want to preserve the Riemannian metric) we may assume that $M$ is $n$-dimensional submimanifold of $\mathbb{R}^k$ for some $k>n$.

The trace operator is bounded as $T:H^{s+1/p}_p(\mathbb{R}^k)\to H^{s}_p(\mathbb{R}^{k-1})$ and there is an extension operator $E:H^{s}_p(\mathbb{R}^{k-1})\to H^{s+1/p}_p(\mathbb{R}^k)$. The same applies to compact submanifolds of codimension $1$. By induction, if $M\subset \mathbb{R}^k$ s a minifold of dimension $n$, there is a trace operator $T:H^{s+(k-n)/p}_p(\mathbb{R}^k)\to H^s_p(M)$ and the extension operator $E:H^s_p(M)\to H^{s+(k-n)/p}_p(\mathbb{R}^k)$. Now for $f\in H^s_p(M)$ we define convolution on $M$ as follows $$ K_h f=T(k_h*(Ef)). $$ That is, we extend $f$ to $\mathbb{R}^k$, we apply this desired convolution on $\mathbb{R}^k$, and then we restrict the resulting function back to $M$. That usually gives all estimates you want.

Assuming that the convolution in $\mathbb{R}^k$ satisfies your estimate, the same estimate will hold on $M$. Indeed, \begin{equation*} \begin{split} &\Vert f-K_h f\Vert_{H^s_p(M)}= \Vert T(Ef-K_h*(Ef))\Vert_{H^s_p(M)} \leq C_1\Vert Ef-K_h*(Ef)\Vert_{H^{s+(k-n)/p}(\mathbb{R}^k)}\\ &\leq C_2 h^{r-s}\Vert Ef\Vert_{H^{r+(k-n)/p}_p(\mathbb{R}^k)}\leq C_3 h^{r-s}\Vert f\Vert_{H^r_p(M)}. \end{split} \end{equation*}

Answer 2

Since your manifold is compact, $H^s_{loc}$ regularity will be equivalent to $H^s$ regularity. To check the local regularity, you can use cutoff functions and the coordinate charts.