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 \[ \langle t\mapsto φ( f(\cdot,t) ) , μ\rangle= φ(z \mapsto \langle f(z, \cdot) , μ\rangle )\] whenever $\mu \in E^{\prime }$ and $\phi$ is a linear functional on $H^{\infty }(O)$ that is sequentially continuous with respect to bounded pointwise convergence in $H^{\infty }(O)$.