Vector-Valued Holomorphic Functions and Abstract Fubini-Type Theorems

Abstract

Let $f = f(z,t)$ be a function holomorphic in $z \in O \subseteq {\mathbb C}^d$ for fixed $t\in \Omega$ and measurable in $t$ for fixed $z$ and such that$z \mapsto f(z,\cdot)$ is bounded with values in$E := L_{p}(\Omega)$, $1\le p \le \infty$. It is proved (among other things) that \[ \langle t\mapsto \varphi( f(\cdot,t) ) , \mu \rangle= \varphi(z \mapsto \langle f(z, \cdot) , \mu\rangle )\] whenever $\mu \in E'$ and $\varphi$ is a linear functional on $H^\infty(O)$ that is sequentially continuous with respect to bounded pointwise convergence in $H^\infty(O)$.