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

Abstract

For a plurisubharmonic weight function p on cn let Ap(Cn) denote the (DFN)-algebra of all entire functions on cn 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 Ap(Cn), the splitting of a weighted 0-complex related with p, and the linear topological invariant (DN) of the strong dual of Ap(Cn) 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. For p a nonnegative plurisubharmonic (psh) function on cn) let Ap(Cn) denote the algebra of all entire functions f such that If(Z)l 0 depending on f. Algebras of this type arise at various places in complex analysis and functional analysis, e.g. as Fourier transforms of certain convolution algebras. The structure of their closed ideals has been studied for a long time, primarily in the work of Schwartz [25], Ehrenpreis [9], Malgrange [17], and Palamodov [23] in connection with the existence and approximation of (systems of) convolution equations. The question whether a certain parameter dependence of the right-hand side of such an equation is shared also by its solutions, is closely related with the question of the existence of a continuous linear right inverse. The existence of such a right inverse is equivalent to the splitting of the closed ideal I associated to the corresponding equation. Also, since the quotient space Ap(Cn)/I is quite often identified with the space Ap(V) of holomorphic functions on the variety V of I which satisfy the restricted growth conditions, the latter question is equivalent to the existence of a linear extension operator from Ap(V) to Ap(Cn). Answers to these questions for various algebras have been given e.g. by Grothendieck (see lVeves [28]), Cohoon [7], Djakov and Mityagin [8], and Vogt [33]. The fact that for P(z) = IZl8, s > 1 all closed ideals in Ap(C) are cornplemented, was observed by Taylor [27]. Then Meise [19] extended Taylor's resultsX using a more functional analytic approach. He showed that the structural property (DN) of the strong dual Ap(Cn)b of Ap(Cn) implies that all slowly decreasing ideals Received by the editors July 29, 1986. 1980 Mathematics Subject Classification (1985 Reon). Primary 32E25, 46E25; Secondary 46A12, 32Al .