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Let $M$ be an $n$-dimensional compact Riemannian manifold without boundary and $B(r)$ a geodesic ball of radius $r$. Then for $u\in W^{1,p}(B(r))$, the Poincare and Sobolev-Poincare inequalities are satisfied $$ \int_{B(r)} |u-u_{B(r)}|^p\leq Cr\int_{B(r)} |\nabla u|^p $$ and $$ \left(\int_{B(r)}|u-u_{B(r)}|^{\frac{np}{n-p}}\right)^{\frac{n-p}{np}} \leq C \left(\int_{B(r)}|\nabla u|^{p}\right)^{\frac{1}{p}} \quad \text{if $1\leq p<n$.} $$ This result is well known. It is easy to prove if $r$ is small, because, then $B(r)$ is contained in a local coordinate system and we can deduce the result from the Euclidean one. However, for large $r$, the geometry of a ball can be very complicated - the ball may have nasty self intersections. The only argument I can think of is to use the fact that geodesic balls are so called John domains and then one can use arguments that are typical in analysis on metric spaces, like e.g. the proof of Theorem 9.7 in this paper. That is pretty unsatisfactory and not accessible to those who are not in the field of John domains or analysis on metric spaces. Thus my questions is:

Is there a good reference for elementary and self-contained proofs of the inequalities listed above?

I believe such a reference would be very useful for the mathematical community. For a related post see here.

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    $\begingroup$ For any $r$. If $r\ge {\rm diam}\, M$, then $B(r)=M$ and the inequality is still true. $\endgroup$ – Piotr Hajlasz Mar 26 '18 at 12:53
  • $\begingroup$ Could you give a reference to show the second local estimate, please ? $\endgroup$ – DLIN Jan 16 at 13:02
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Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, by E. Hebey is probably what you need.

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  • $\begingroup$ Hebey indeed discusses inequalities on balls under the assumption of the lower bound for the Ricci curvature and he refers to the work of Buser and Maheux & Saloff-Coste. This is a very difficult argument and therefore not satisfactory. The approach mentioned in my question is much more direct, but I am not sure if it is written explicitly in any source except papers on analysis on metric spaces. $\endgroup$ – Piotr Hajlasz Mar 27 '18 at 18:13

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