Elliptic problems and holomorphic functions in Banach spaces

Abstract

In the first part, we show that a Banach space-valued function $f$ is holomorphic (harmonic) if and only if it is dominated by an $L_{\mathrm{loc}}^1$ function and there exists a separating set $W\subset X\text{'}$ such that $\langle f,x\text{'}\rangle$ is holomorphic (harmonic) for all $x\text{'}\in W$. This improves a known result which requires $f$ to be locally bounded. In the second part, we consider classical results in the $L^p$ theory for elliptic differential operators of second order. In the vector-valued setting, these results are shown to be equivalent to the UMD property.