Characterization of surjective partial differential operators on spaces of real analytic functions
Let A(Ω) denote the real analytic functions defined on an open set Ω ⊂ R. We show that a partial differential operator P (D) with constant coefficients is surjective on A(Ω) if and only if for any relatively compact open ω ⊂ Ω, P (D) admits (shifted) hyperfunction elementary solutions on Ω which are real analytic on ω (and if the equation P (D)f = g, g ∈ A(Ω), may be solved on ω). The latter condition is redundant if the elementary solutions are defined on conv(Ω). This extends and improves previous results of Andersson, Kawai, Kaneko and Zampieri. For convex Ω, a different characterization of surjective operators P (D) on A(Ω) was given by Hörmander using a Phragmén–Lindelöf type condition, which cannot be extended to the case of noncovex Ω. The paper is based on a surjectivity criterion for exact sequences of projective (DFS)spectra which improves earlier results of Braun and Vogt, and Frerick and Wengenroth. This paper is concerned with the basic question when P (D) : A(Ω)→ A(Ω) is surjective. (0.1) Here P (D) is a partial differential operator with constant coefficients, Ω ⊂ Rn is an open set and A(Ω) is the space of real analytic functions on Ω. Several methods have been developed to solve this problem: The first counterexample to (0.1) is due to Piccinini [34] who showed that the heat equation is not surjective on A(R3) (see also the conjecture of De Giorgi and Cattabriga [10]). Then Hörmander [13] characterized (0.1) for convex sets Ω by means of a Phragmén–Lindelöf condition valid on the complex variety of P . Since then Hörmander’s method has been adapted by several authors for further studies on this problem (Miwa [31], Andreotti and Nacinovich [3], Zampieri [42], Braun [6] and the recent series of papers of Braun, Meise and Taylor [7, 8]). 2000 Mathematics Subject Classification: Primary 35E10, 35E05; Secondary 46F15.