Littlewood-Paley-Stein theory for semigroups in UMD spaces

The Littlewood–Paley theory for a symmetric diffusion semigroup T t, as developed by Stein, is here generalized to deal with the tensor extensions of these operators on the Bochner spaces Lp(μ,X), where X is a Banach space. The g-functions in this situation are formulated as expectations of vector-valued stochastic integrals with respect to a Brownian motion. A two-sided g-function estimate is then shown to be equivalent to the UMD property of X. As in the classical context, such estimates are used to prove the boundedness of various operators derived from the semigroup T t, such as the imaginary powers of the generator.