I was reading a paper where I came across the following argument :

For any x in M and for a geodesic ball B(x; epsilon) in a compact Riemannian manifold M with injectivity radius bigger than or equal to epsilon, and for any smooth eigenfunction f of Laplacian on M, we have :

the square of f(x) is <= C times ( the square of L^2 norm of f over B(x;epsilon) + square of L^2 norm of L(f) over B(x:epsilon)),

where L(f)= Laplacian of f, where C is independent of the Riemannian metric on M.

I was unable to see, with my limited Analysis knowledge, why this is true, but they mentioned that it follows from Sobolev's and Garding's inequality, for which they referred to S. Agmon's "Lectures on Elliptic boundary value problems"... still it is unclear to me.

N.B.: ihe injectivity radius of a manifold is the smallest of all numbers r such that I can have a geodesic ball of radius r around each point of M. e.g. injectivity radius of the sphere of radius 1 with standard metric is pi, injectivity radius of R^n is infinity etc.

Any help ? Thanks in adavance !