Question1
I want to find a manifold such that the Sobolev inequality on M of the form $\lVert f \rVert_{n/(n-1)} \leq C\lVert \bigtriangledown f \rVert_1$, where $C=C(n)$, implies that $vol(B(r)) \geq cr^n$ for some constant c. (That is: at-least-Euclidean volume growth)
I can't find the existence of such manifold, does everyone have such example?
Question2
Let $M^{(n)}$ be a compact Riemannian manifold. $f:M\longrightarrow \mathbb{R}$ a smooth function. Is $f$ Lipschitz implies that $f \in L_1^p(M)$ for all $p \geq 1$?
I am appreciate for your answers. Thank you very much!