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Decembar, 2007 Littlewood-Paley-Stein theory for semigroups in UMD spaces
Tuomas P. Hytönen
Rev. Mat. Iberoamericana 23(3): 973-1009 (Decembar, 2007).


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.


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Tuomas P. Hytönen. "Littlewood-Paley-Stein theory for semigroups in UMD spaces." Rev. Mat. Iberoamericana 23 (3) 973 - 1009, Decembar, 2007.


Published: Decembar, 2007
First available in Project Euclid: 27 February 2008

zbMATH: 1213.42012
MathSciNet: MR2414500

Primary: 42A61
Secondary: 42B25 , 46B09 , 46B20

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

Rights: Copyright © 2007 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.23 • No. 3 • Decembar, 2007
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