# Revista Matemática Iberoamericana

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**Volume 23, Issue 3, 2007, pp. 973–1009**

**DOI: 10.4171/RMI/521**

Published online: 2007-12-31

Littlewood-Paley-Stein theory for semigroups in UMD spaces

Tuomas P. Hytönen^{[1]}(1) University of Helsinki, Finland

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.

*Keywords: *Brownian motion, diffusion semigroup, functional calculus, stochastic integral, unconditional martingale differences

Hytönen Tuomas: Littlewood-Paley-Stein theory for semigroups in UMD spaces. *Rev. Mat. Iberoamericana* 23 (2007), 973-1009. doi: 10.4171/RMI/521