Is it possible to get a continuous embedding from $L^p(M)\hookrightarrow L^p(M\times M)$ where $p>1$ and $(M,g)$ is a compact Riemannian manifold ?
I was trying to prove the inequality for some $u\in L^p(M)$ that $$\iint_{M\times M}|u(x-y)|^p\,dv_g(x)dv_g(y)\leq C\int_M|u(x)|^p\,dv_g(x)$$ for some $C>0$. But I am not too sure whether this will exist. Any help is appreciated.
P.S. : Can we get this for (non-compact) Euclidean cases as well? Say for $M=\mathbb{R}$?