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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!

1 Answers 1

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The answer to Question 2 is yes. Since $f$ is smooth and has bounded derivative, it belongs to $L_1^p$ for all $p\ge 1$.

I'm not sure I understand Question 1. Does $M=\mathbb R^n$ work?