Slowly decreasing functions and closed ideals.

Abstract

Given an algebra ${\cal A}$ of entire functions and finitely many functions in ${\cal A}$, we study the problem of determining when the functions generate a closed ideal in ${\cal A}$. Interest in this problem was started by the work of Ehrenpreis and Malgrange who showed that such "division problems" were, in many important cases, equivalent to the existence of solutions of systems of partial differential or convolution equations. We give a necessary condition for when a single function generates a closed ideal in the algebra $A\sb{p}$($C\sp{\rm n}$) of all entire functions f(z) satisfying $\vert f$(z)$\vert\ \leq\ Ae\sp{Bp(z)}$ for some A and B $\geq$ 0. Here p(z) is a continuous plurisubharmonic function on $C\sp{\rm n}$ satisfying some mild technical hypothesis. A necessary condition for an entire function $f \in\ A\sb{p}$($C\sp{\rm n}$) to generate a closed ideal is that the natural harmonic (plurisubharmonic if n $>$ 1) extension h(z) of the function p(z) from the boundary of the set $\{z \in\ C\sp{\rm n}$: $\vert f$(z)$\vert \leq\ \epsilon{e}\sp{-Cp(z)}\}$ satisfies the inequality h(z) $\leq$ Ap(z) + B for some A, B, $\epsilon$, C $>$ 0. We apply this condition to give concrete characterizations in specific new examples.