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 loc 1 function and there exists a separating set W ⊂ X ' such that 〈 f , x ' 〉 is holomorphic (harmonic) for all x ' ∈ 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.