Littlewood-Paley-Stein theory for semigroups in UMD spaces

Hytönen, Tuomas P.

Rev. Mat. Iberoamericana, Tome 23 (2007) no. 1, p. 973-1009 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 2007-12-15
Classification:  Brownian motion,  diffusion semigroup,  functional calculus,  stochastic integral,  unconditional martingale differences,  42A61,  42B25,  46B09,  46B20