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arXiv · 1904.03088

Elliptic problems and holomorphic functions in Banach spaces

Wolfgang Arendt·Manuel Bernhard·Marcel Kreuter
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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.

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Wolfgang Arendt, Manuel Bernhard, Marcel Kreuter. 2019-04-05. Elliptic problems and holomorphic functions in Banach spaces. https://doi.org/10.1215/00192082-8591584

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