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 $L^p(\mu,X)$, 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.
"Littlewood-Paley-Stein theory for semigroups in UMD spaces." Rev. Mat. Iberoamericana 23 (3) 973 - 1009, Decembar, 2007.