Geometric Interpretation of the Phragmén–Lindelöf Estimates

In Section 3 we state a geometric version of Hörmanderʹs criterion (Invent. Math.21(1973), 151–182) for global existence on open convex domains in nof real analytic solutions to linear differential equations with constant coefficients. In Section 1 we discuss local hyperbolicity of analytic functions following the treatment given in Atiyahet al.(Acta Math.124(1970), 109–189;131(1973), 145–206). In Section 2 we give sufficient conditions for the Phragmén–Lindelöf implications to hold on the zero–sets of such functions. We also show that those conditions are necessary when each irreducible component of the zero–set has multiplicity 2. The results on existence of real analytic solutions have already been stated in Zampieri (J. Fac. Sci. Univ. Tokyo Sect. IA Math.31, No. 2 (1984), 372–390). The new contribution of the present paper consists in more accurate Phragmén– Lindelöf estimates and in a much shorter and more satisfactory proof. Also a new example is introduced; it explains in what sense our method is sharper than the others which have appeared in the literature (compare with Kawaï (J. Math. Soc. Japan24(1972), 481–517)). This was the motivation for our research in this area.