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An algebraic theorem of E. Fischer, and the holomorphic Goursat problem. (English) Zbl 0706.35034
The basic problem discussed in this paper is whether the differential operator \(f\mapsto P(D)(Qf)\) is injective, surjective or both, as a map from \(E_ n\) to itself. Here \(E_ n\) denotes the space of entire functions of n complex variables. P and Q are polynomials on n letters with complex coefficients and \(D=(D_ 1,...,D_ n)\), \(D_ k=\partial /\partial z_ k\). Theorem 1 shows, using a classical algebraic theorem of E. Fischer, that this map is bijective when P and Q are homogeneous and \(P=Q^*\) where \(Q^*\) denotes the polynomial derived from Q by conjugating its coefficients. Theorem 3 shows that bijectivity persists when Q is homogeneous and \(P=Q^*-1.\)
In a corollary to this theorem a proof of bijectivity is sketched in the case P homogeneous and \(Q=P^*-1\). As remarked on p. 536 the hypothesis of homogeneity can be weakened to generalized homogeneity based on positive weights; this idea also is found in the work of E. Fischer. Relations of the indicated results to real initial value problems are indicated, and to eigenfunction expansions in \(E_ n\). Many open questions are stated.
Reviewer: H.S.Shapiro

MSC:
35E15 Initial value problems for PDEs and systems of PDEs with constant coefficients
35A10 Cauchy-Kovalevskaya theorems
35A20 Analyticity in context of PDEs
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