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This article surveys results on the global surjectivity of linear partial differential operators with constant coefficients on the space of real analytic functions. Some new results are also included. 0. Introduction. In this talk, we first briefly review in §1 the old subject of general surjectivity of a linear partial differential operator P (D) with constant coefficients in various spaces of (generalized) functions on a given open set. Then in §2 we introduce results concerning the case of the space of real analytic functions in precise formulation. Finally, in §3 we discuss recent developments related to micro-local analysis. This is, so to speak, an application of singularity theory. We thus believe that this talk fits the general subject of this Semester. The author participated about 15 years ago in a meeting organized by L. Cattabriga at Trento in Italy and encountered many people by whose influences he undertook this research. The author would therefore like to record here his special memory of the late Prof. L. Cattabriga. 1. General surjectivity. When a fundamental solution E(x) was constructed for a general operator P (D) with constant coefficients by Hormander, the natural problem to be attacked next was that of solvability of the equation P (D)u = f for the given right-hand side f . But this contained another difficulty when f was a general function which may grow up near the boundary of the domain, because the simple formula

On the global solvability of linear partial differential equations with constant coefficients in the space of real analytic functions