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 $t\in \mathrm{\Omega }$ and measurable in $t$ for fixed $z$ and such that$z\mapsto f(z,\cdot )$ is bounded with values in$E:=L_p(\mathrm{\Omega })$, $1\le p\le \mathrm{\infty }$. It is proved (among other things) that $$⟨t\mapsto \phi (f(\cdot ,t)),\mu ⟩=\phi (z\mapsto ⟨f(z,\cdot ),\mu ⟩)$$ whenever $\mu \in E^{\mathrm{\prime }}$ and $\phi$ is a linear functional on $H^{\mathrm{\infty }}(O)$ that is sequentially continuous with respect to bounded pointwise convergence in $H^{\mathrm{\infty }}(O)$.