Sobolev spaces on a Riemannian manifold and their equivalence

Abstract The Littlewood-Paley-Stein inequality, which plays a fundamental role in the analysis on the Euclidean space is extended to Riemannian manifold under a mild hypothesis on the bound of the curvature. The proof is carried out by running a Brownian motion in the manifold. As an application of the Littlewood-Paley-Stein inequality, we prove that two typical Sobolev-norms (one is defined in terms of iterations of covariant differentiations and another in terms of the fractional powers of the Laplacian) are equivalent.