Vector-valued holomorphic functions and abstract Fubini-type theorems

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 \(E:= \textrm{L}_{p}(\Omega )\), \(1\le p \le \infty \). It is proved (among other things) that

$$\begin{aligned} \langle t\mapsto \varphi ( f(\cdot ,t)),\mu \rangle = \varphi (z \mapsto \langle f(z, \cdot ),\mu \rangle ) \end{aligned}$$

whenever \(\mu \in E'\) and \(\varphi \) is a bp-continuous linear functional on \(\textrm{H}^\infty (O)\).