The Littlewood-Paley-Stein Inequality on an Infinite Dimensional Manifold

Let M be a compact Riemannian manifold. We consider on MZν a diffusion operator with Ising-type interaction. We characterize Lp-space with respect to the Gibbs measure by the Littlewood-Paley-Stein inequality and then discuss the equivalence of Sobolev norms.