Cauchy formula for vector-valued holomorphic functions and the Cauchy-Kovalevskaja theorem

We establish a version of the Cauchy integral formula for holomorphic functions on a polidisc, with values in a complex Fréchet space $\mathfrak{X}$. We prove a vector-valued analog to Weierstrass’s theorem on sequences of holomorphic functions $f_\nu \in \mathcal{O} (\Omega, \mathfrak{X})$ converging uniformly on compact subsets of $\Omega \subset \mathbb{C}^n$. We obtain a Cauchy-Kovalevskaja-type theorem, i.e., prove existence of $\mathfrak{X}$-valued $C^\omega$ solutions to the Cauchy problem $P(x, D)u = f$ on $U\subset \Omega$ and $(\partial^j u / \partial x_n^j)\mid_{x_n = 0} = \varphi_j$ on $U_0 = U \cap \{ x_n = 0\ }$, $j \in \{0, 1, \dots m − 1 \}$, where $P(x, D) \equiv \sum_{|\alpha | \leq m} a_\alpha (x) D^\alpha$ and $a_\alpha \in C^\omega (\Omega)$, $f\in C^\omega (\Omega , \mathfrak{X})$ and $\varphi_j \in C^\omega (\Omega_0 , \mathfrak{X})$, with $0\in \Omega \subset \mathbb{R}^n$ open and $\Omega_0 = \Omega \cap \{x_n = 0\}$. The existence of an open set $0 \in U \subset \Omega$ and of a solution $u \in C^\omega (U , \mathfrak{X})$ to the Cauchy problem requires the structural condition $a_{(0, \dots , 0,m)} (0) \neq 0$ and relies, as well as in the classical case of scalar valued solutions, on the Cauchy integral formula and on Weierstrass’ theorem for $\mathfrak{X}$-valued holomorphic functions.