A sharp form of Poincaré inequalities on balls and spheres

Let u be a real valued function on an n-dimensional Riemannian manifold M^n. We consider an inequality between the L^q-norm of u minus its mean value over M^n and the L^p-norm of the gradient of u. The best constant in such inequality is exhibited in the following cases: i) M^n is an open ball in IR^n and p=1, 0 < q \leq n/(n−1); ii) M^n is a sphere in IR^{n+1} and either p=1, 0 < q \leq n/(n−1) or p>n, q=∞.