Splitting of closed ideals in $({\rm DFN})$-algebras of entire functions and the property $({\rm DN})$

Abstract

For a plurisubharmonic weight function $p$ on ${{\mathbf {C}}^n}$ let ${A_p}({{\mathbf {C}}^n})$ denote the (DFN)-algebra of all entire functions on ${{\mathbf {C}}^n}$ which do not grow faster than a power of $\exp (p)$. We prove that the splitting of many finitely generated closed ideals of a certain type in ${A_p}({{\mathbf {C}}^n})$, the splitting of a weighted $\overline \partial$-complex related with $p$, and the linear topological invariant (DN) of the strong dual of ${A_p}({{\mathbf {C}}^n})$ are equivalent. Moreover, we show that these equivalences can be characterized by convexity properties of $p$, phrased in terms of greatest plurisubharmonic minorants. For radial weight functions $p$, this characterization reduces to a covexity property of the inverse of $p$. Using these criteria, we present a wide range of examples of weights $p$ for which the equivalences stated above hold and also where they fail.