# Extending Sobolev function on Riemannian manifold

Let $$(M, \mu, d)$$ be a geodesically complete non-compact Riemannian manifold such that measure $$\mu$$ is volume doubling, i.e. $$$$\label{VD}\mu(B(x, 2r))\leq C\mu(B(x, r))$$$$ for some constant $$C>0$$ and also $$M$$ satisfying the $$L^{2}$$-Poincare inequality $$\frac{1}{\mu(B(x, r))}\int_{B(x, r)}{|u-u_{B(x, r)}|^{2}d\mu}\leq cr^{2}\frac{1}{\mu(B(x, \delta r))}\int_{B(x, \delta r)}{|\nabla u|^{2}d\mu}$$ for all $$u\in W^{1, 2}(B(x, r))$$ with $$\delta>1$$ and $$\nabla u$$ being the weak gradient of $$u$$.

For a fixed point $$x_{0}\in M$$ and $$\alpha, \beta\in \mathbb{R}_{+}$$ consider the annuli $$P_{\alpha, \beta}=\{x\in M:\alpha

Question: Is there a statement known, such that under these assumptions (or even stronger), any function $$u\in L_{loc}^{2}(P_{\alpha, \beta})$$ with $$\int_{P_{\alpha, \beta}}{|\nabla u|^{2}d\mu}<\infty$$ can be extended to a function $$\widetilde{u}\in L_{loc}^{2}(M)$$ such that $$(*)\int_{M}{|\nabla \widetilde{u}|^{2}d\mu}\leq C\int_{P_{\alpha, \beta}}{|\nabla u|^{2}d\mu}?$$

The closest statement I found is from "On extensions of Sobolev functions defined on regular subsets of metric measure spaces" by P. Shvartsman. In this paper he proofs that considering a regular set $$S$$, i.e. a set such that there are constants $$\theta_{S}\geq 1$$ and $$\delta_{S}>0$$ such that for every $$x\in S$$ and $$0 $$\mu(B(x, r))\leq \theta_{S}\mu(B(x, r)\cap S),$$ then any function $$u\in L^{2}(S)$$ such that $$u_{1, S}^{\#}\in L^{2}(S)$$ where $$u_{1, S}^{\#}(x):=\sup_{r>0}\frac{r^{-1}}{\mu(B(x, r))}\int_{B(x, r)\cap S}{|u-u_{B(x, r)\cap S}|d\mu}$$ can be extended to a function $$\widetilde{u}\in CW^{1, 2}(M)$$ such that $$\|\widetilde{u}\|_{CW^{1, 2}(M)}\leq C(\|u\|_{L^{2}(S)}+\|u_{1, S}^{\#}\|_{L^{2}(S)}),$$ where $$CW^{1, 2}(M)$$ is the Calderon-Sobolev space which coincides with the classical Sobolev space $$W^{1, 2}(M)$$ if one assumes volume doubling and $$M$$ satisfying the $$L^{2}$$-Poincare inequality.

• If the constant $C$ is allowed to depend on the manifold $M$ and the annulus $P_{\alpha,\beta}$, then, since $\overline{P}_{\alpha,\beta}$ is compact, it suffices to do this locally. Therefore, an extension theorem for a domain in $\mathbb{R}^n$ suffices. Stein's book, Singular Integrals and Differentiability Properties of Functions, contains such a theorem, Aug 6 '20 at 17:18
Then a smooth function on the annulus can approach to $$0$$ from one side of the common part of the boundary and to $$1$$ from the other side. Such a function has no Sobolev extension at all.