Surjective partial differential operators on spaces of real analytic functions
by
Michael Langenbruch
University of Oldenburg

The lecture is concerned with the basic question when

P(D) : A(\Omega) --> A(\Omega)

is surjective. Here P(D) is a partial differential operator with constant coefficients, \Omega is an open subset of Rⁿ and A(\Omega) is the space of real analytic functions on \Omega.

Hörmander (1973) gave a characterization of the surjectivity for open convex sets \Omega by means of a Phragmen-Lindelof condition valid on the complex variety of P. Sufficient conditions have been given by Kawai (1972) and Kaneko (1985) using (Fourier) hyperfunctions. In the lecture we will present a new characterization which roughly states that surjectivity holds iff P(D) has (generalized) elementary solutions which are real analytic on arbitrary relatively compact open subsets of \Omega. For the sufficiency part, this extends corresponding results of Kawai (1972) and Kaneko (1985). For the necessity part, the characterization implies that hyperbolicity of the localizations P_{m, \theta} of the principal part P_m and local hyperbolicity of P_m are necessary for surjectivity in many cases.

Date received: February 27, 2001