Sobolev spaces on a Riemannian manifold and their equivalence (Q1210073)

The author studies the problem of equivalence of Sobolev spaces of sections of vector bundles over complete Riemannian manifolds, defined as extensions of classical spaces \((u\in L_ p,\) \((\nabla)^ ju\in L_ p)\) (here \((\nabla)^ j\) is the composition of \(j\) gradient operators) on the one hand as a natural extension of the classical space of Bessel potentials on the other hand. The conditions for the equivalence are described in terms of curvatures, weight function and potential. The construction is based on a new generalization of the Littlewood-Paley- Stein inequality. In particular, some stochastic methods are applied.