The geometry of analytic varieties satisfying the local Phragmen-Lindelof condition and a geometric characterization of the partial differential operators that are surjective on A(R4)

The local Phragmen-Lindelof condition for analytic subvarieties of C-n at real points plays a crucial role in complex analysis and in the theory of constant coefficient partial differential operators, as Hormander has shown. Here, necessary geometric conditions for this Phragmen-Lindelof condition are derived. They are shown to be sufficient in the case of curves in arbitrary dimension and of surfaces in C-3. The latter result leads to a geometric characterization of those constant coefficient partial differential operators which are surjective on the space of all real analytic functions on R-4.