ABSTRACT
In this chapter, we present Sobolev, log Sobolev, parabolic Harnack and some related inequalities on a Riemann manifold. Here are some notations to be used in this chapter. We use M or M to denote a compact Riemann manifold with a metric g, unless stated otherwise; d(x, y), dµ will denote the distance and volume element respectively; B(x, r), |B(x, r)| mean the geodesic ball centered at x with radius r and its volume respectively; ∇ stands for the gradient of a function, relative to the metric g.
