I want to verify the statement below :
Let $(M,g)$ be a compact Riemannian $N$-manifold, $p\in[1,\infty)$ and $s\in(0,1)$. Then $$||u||_{W_0^{s,p}(M)}\leq C||u||_{W^{1,p}(M)}$$ for some suitable positive constant $C=C(N,p,s)\geq1$. In particular $$W^{1,p}(M)\subseteq W_0^{s,p}(M)$$
We know that $$||u||^p_{W_0^{s,p}(M)}=\iint_{M\times M}\frac{|u(x)-u(y)|^p}{(d_g(x,y))^{N+ps}}$$ $$||u||^p_{W^{1,p}(M)}=||u||^p_{L^p(M)}+||\nabla u||^p_{L^p(M)}$$ $$||u||^p_{W^{s,p}(M)}=||u||^p_{L^p(M)}+||u||^p_{W_0^{s,p}(M)}$$
I have been able to see that for some geodesic $\gamma:[0,1]\to M$ joining $x,y\in M$ $$\int_M\int_{M\cap\{d_g(x,y)<1\}}\frac{|u(x)-u(y)|^p}{(d_g(x,y))^{N+ps}}\leq C(n,p,s)||\nabla u||_{L^p(M)}$$ using mean value theorem. Also $$\int_M\int_{M\cap\{d_g(x,y)\geq1\}}\frac{|u(x)-u(y)|^p}{(d_g(x,y))^{N+ps}}\leq C(n,p)||u||_{L^p(M)}$$ using Minkowski's inequality. Combining them gives the required embedding.
Is the above approach correct? Moreover, can we improve the (continuous) embedding to $W^{1,p}(M)\hookrightarrow W^{s,p}(M)$? Any help is appreciated.