# Elliptic problems and holomorphic functions in Banach spaces

## TL;DR

This paper characterizes vector-valued holomorphic and harmonic functions in Banach spaces, extending classical elliptic PDE results to the vector-valued setting and linking them to the UMD property.

## Contribution

It provides new characterizations of vector-valued holomorphic functions and connects elliptic PDE theory with Banach space properties like UMD.

## Key findings

- Holomorphicity characterized by domination and separating sets.
- Extension of classical elliptic PDE results to vector-valued functions.
- Equivalence of $L^p$ elliptic results with the UMD property.

## Abstract

In the first part we show that a vector-valued almost separably valued function $f$ is holomorphic (harmonic) if and only if it is dominated by an $L^1_\mathrm{loc}$ function and there exists a separating set $W\subset X'$ such that $\langle f,x'\rangle$ is holomorphic (harmonic) for all $x'\in W$. 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.