Vector-valued Littlewood-Paley-Stein theory for semigroups

By José L. Torrea, Chao Zhang. 2006

69 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/ef37c18d0acf6199256ffd0a605aa6db7986dbf6

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

Littlewood-Paley-Stein theory for semigroups in UMD spaces (2007, methodology): “The same phenomenon is present in the vector-valued extension of Martínez, Torrea and Xu [14]: Also in their case, the square-function estimates in Theorem 1.”