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.
Best Answer
Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities, by E. Hebey is probably what you need.