JournalsrmiVol. 23, No. 3pp. 973–1009

Littlewood-Paley-Stein theory for semigroups in UMD spaces

  • Tuomas Hytönen

    University of Helsinki, Finland
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Abstract

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

Cite this article

Tuomas Hytönen, Littlewood-Paley-Stein theory for semigroups in UMD spaces. Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 973–1009

DOI 10.4171/RMI/521