p-Summing weighted holomorphic lipschitz mappings

Abstract

Let \(U\) be an open subset of a complex Banach space \(E\), let \(v\) be a weight on \(U\) that separates the points of \(\mathcal {H}_v(U)\), and let \(F\) be a complex Banach space. We introduce and study the class \(\Pi _p^{\textrm{lip}_v}(U,F)\) of \(p\)-summing weighted holomorphic Lipschitz mappings from \(U\) into \(F\). We prove that this class, modulo constant functions, forms an injective Banach ideal of weighted holomorphic mappings. Using the Lipschitz-free space \(\mathcal {F}_v(U)\) over \((U,d_v)\), we establish a linearization theorem identifying these mappings with \(p\)-summing linear operators from \(\mathcal {F}_v(U)\) into \(F\). Variants of the Pietsch Domination Theorem, Pietsch Factorization Theorem, and Maurey Extrapolation Theorem are presented. Furthermore, we introduce the space of \(F\)-valued \(\mathcal {H}_v\)-Lipschitz molecules and prove that its dual is isometrically isomorphic to \(\Pi _p^{\textrm{lip}_v}(U,F^*)\) under a suitable version of the Chevet-Saphar tensor norm.