Vector-valued Littlewood-Paley-Stein Theory for Semigroups II

Inspired by a recent work of Hytonen and Naor, we solve a problem left open in our previous work joint with Martinez and Torrea on the vector-valued Littlewood-Paley-Stein theory for symmetric diffusion semigroups. We prove a similar result in the discrete case, namely, for any T which is the square of a symmetric diffusion Markovian operator on a measure space (Omega, mu). Moreover, we show that T circle times Id(x) extends to an analytic contraction on L-p (Omega; X) for any 1 < p < infinity and any uniformly convex Banach space X.