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Estimates for vector-valued holomorphic functions and Littlewood-Paley-Stein theory

Mark VeraarLutz Weis — 2015

Studia Mathematica

We consider generalized square function norms of holomorphic functions with values in a Banach space. One of the main results is a characterization of embeddings of the form L p ( X ) γ ( X ) L q ( X ) , in terms of the type p and cotype q of the Banach space X. As an application we prove L p -estimates for vector-valued Littlewood-Paley-Stein g-functions and derive an embedding result for real and complex interpolation spaces under type and cotype conditions.

H calculus and dilatations

Andreas M. FröhlichLutz Weis — 2006

Bulletin de la Société Mathématique de France

We characterise the boundedness of the H calculus of a sectorial operator in terms of dilation theorems. We show e. g. that if - A generates a bounded analytic C 0 semigroup ( T t ) on a UMD space, then the H calculus of A is bounded if and only if ( T t ) has a dilation to a bounded group on L 2 ( [ 0 , 1 ] , X ) . This generalises a Hilbert space result of C.LeMerdy. If X is an L p space we can choose another L p space in place of L 2 ( [ 0 , 1 ] , X ) .

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