In the first part, we show that a Banach space-valued function is holomorphic (harmonic) if and only if it is dominated by an function and there exists a separating set such that is holomorphic (harmonic) for all . This improves a known result which requires to be locally bounded. In the second part, we consider classical results in the theory for elliptic differential operators of second order. In the vector-valued setting, these results are shown to be equivalent to the UMD property.
"Elliptic problems and holomorphic functions in Banach spaces." Illinois J. Math. 64 (3) 331 - 347, September 2020. https://doi.org/10.1215/00192082-8591584