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

TL;DR

This paper characterizes embeddings of vector-valued holomorphic function spaces using Banach space type and cotype, and applies these results to Littlewood-Paley-Stein theory to obtain new $L^p$ estimates.

Contribution

It provides a new characterization of embeddings involving vector-valued holomorphic functions based on Banach space geometry, and applies this to Littlewood-Paley-Stein theory.

Findings

Characterization of embeddings $L^p(X) o \, ext{gamma}(X) o L^q(X)$ using type and cotype.

$L^p$-estimates for vector-valued Littlewood-Paley-Stein $g$-functions.

Embedding results for interpolation spaces under geometric conditions.

Abstract

In this paper 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 $q$ for 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.