Vector-valued holomorphic functions and abstract Fubini-type theorems

Let \(f = f(z,t)\) $f=f(z,t)$ be a function holomorphic in \(z \in O \subseteq {\mathbb {C}}^d\) $z\in O\subseteq {\mathbb{C}}^d$ for fixed \(t\in \Omega \) $t\in \mathrm{\Omega }$ and measurable in t for fixed z and such that \(z \mapsto f(z,\cdot )\) $z\mapsto f(z,·)$ is bounded with values in \(E:= \textrm{L}_{p}(\Omega )\) $E:={\text{L}}_p(\mathrm{\Omega })$ , \(1\le p \le \infty \) $1\le p\le \infty$ . It is proved (among other things) that \(\begin{aligned} \langle t\mapsto \varphi ( f(\cdot ,t)),\mu \rangle = \varphi (z \mapsto \langle f(z, \cdot ),\mu \rangle ) \end{aligned}\) $$\begin{array}{c}\hfill ⟨t\mapsto \phi (f(·,t)),\mu ⟩=\phi (z\mapsto ⟨f(z,·),\mu ⟩)\end{array}$$ whenever \(\mu \in E'\) $\mu \in E^{\prime }$ and \(\varphi \) $\phi$ is a bp-continuous linear functional on \(\textrm{H}^\infty (O)\) ${\text{H}}^{\infty }(O)$ .