Elliptic problems and holomorphic functions in Banach spaces

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 $L_{\mathrm{loc}}^1$ function and there exists a separating set $W\subset X^{\prime }$ such that $⟨f,x^{\prime }⟩$ is holomorphic (harmonic) for all $x^{\prime }\in W$. 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.

16 pages