The Littlewood-Paley theory for a symmetric diffusion semigroup , as developed by Stein, is here generalized to deal with the tensor extensions of these operators on the Bochner spaces , where is a Banach space. The -functions in this situation are formulated as expectations of vector-valued stochastic integrals with respect to a Brownian motion. A two-sided -function estimate is then shown to be equivalent to the UMD property of . As in the classical context, such estimates are used to prove the boundedness of various operators derived from the semigroup , such as the imaginary powers of the generator.
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Tuomas Hytönen, Littlewood-Paley-Stein theory for semigroups in UMD spaces. Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 973–1009DOI 10.4171/RMI/521