Revista Matemática Iberoamericana

Littlewood-Paley-Stein theory for semigroups in UMD spaces

Tuomas P. Hytönen

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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.

Article information

Rev. Mat. Iberoamericana, Volume 23, Number 3 (2007), 973-1009.

First available in Project Euclid: 27 February 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 42A61: Probabilistic methods
Secondary: 42B25: Maximal functions, Littlewood-Paley theory 46B09: Probabilistic methods in Banach space theory [See also 60Bxx] 46B20: Geometry and structure of normed linear spaces

Brownian motion diffusion semigroup functional calculus stochastic integral unconditional martingale differences


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

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