Estimates for vector-valued holomorphic functions and Littlewood–Paley–Stein theory
Volume 228 / 2015
Studia Mathematica 228 (2015), 73-99 MSC: Primary 46B09; Secondary 42B25, 46B70, 46E40, 46B20, 47D07. DOI: 10.4064/sm228-1-7
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 \[ L^p(X)\subseteq \gamma (X) \subseteq L^q(X), \] 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.