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Hytönen, Tuomas P.

Littlewood-Paley-Stein theory for semigroups in UMD spaces. (English) Zbl 1213.42012

Rev. Mat. Iberoam. 23, No. 3, 973-1009 (2007).
Summary: 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.

Cited in 2 Reviews
Cited in 21 Documents

MSC:

42A61 Probabilistic methods for one variable harmonic analysis
42B25 Maximal functions, Littlewood-Paley theory
46B09 Probabilistic methods in Banach space theory
46B20 Geometry and structure of normed linear spaces

Keywords:

Brownian motion; diffusion semigroup; functional calculus; stochastic integral; unconditional martingale differences
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\textit{T. P. Hytönen}, Rev. Mat. Iberoam. 23, No. 3, 973--1009 (2007; Zbl 1213.42012)
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References:

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