This is an announcement for the paper "Vector-valued Littlewood-Paley-Stein theory for semigroups" by Teresa Martinez, Jose L. Torrea and Quanhua Xu.
Abstract: We develop a generalized Littlewood-Paley theory for semigroups acting on $L^p$-spaces of functions with values in uniformly convex or smooth Banach spaces. We characterize, in the vector-valued setting, the validity of the one-sided inequalities concerning the generalized Littlewood-Paley-Stein $g$-function associated with a subordinated Poisson symmetric diffusion semigroup by the martingale cotype and type properties of the underlying Banach space. We show that in the case of the usual Poisson semigroup and the Poisson semigroup subordinated to the Ornstein-Uhlenbeck semigroup on ${\mathbb R}^n$, this general theory becomes more satisfactory (and easier to be handled) in virtue of the theory of vector-valued Calder'on-Zygmund singular integral operators.
Archive classification: Functional Analysis
Mathematics Subject Classification: 46B20; 42B25, 42A61
Remarks: To appear in Adv. Math
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Submitted from: qx@math.univ-fcomte.fr
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