Nearly Hyperbolic Varieties and Phragmén-Lindelöf Conditions

Abstract

By a classical estimate in function theory, each subharmonic function u(z) on the unit disk that is bounded above by 1 and bounded by 0 on the real axis must satisfy a bound of the form u(z) ≤ A|Imz| on smaller subdisks. When an analogous estimate holds for the plurisubharmonic functions in a neighborhood of a real point ξ in an analytic variety, the variety is said to satisfy the local Phragmén-Lindelöf condition at ξ. Interest in such conditions originated from a theorem of Hörmander who showed that the surjective constant coefficient linear partial differential operators on the space of real analytic functions on ℝⁿ are characterized in terms of these conditions. We give a new geometric condition on a local variety that is necessary in order that the local Phragmén-Lindelöf condition holds, and is sufficient in the case of varieties of dimension 1 or 2.