Phragmén-Lindelöf principles on algebraic varieties

Abstract:

Estimates of Phragmén–Lindelöf (PL) type for plurisubharmonic functions on algebraic varieties in $\mathbb {C}^n$ have been of interest for a number of years because of their equivalence with certain properties of constant coefficient partial differential operators; e.g. surjectivity, continuation properties of solutions and existence of continuous linear right inverses. Besides intrinsic interest, their importance lies in the fact that, in many cases, verification of the relevant PL-condition is the only method to check whether a given operator has the property in question. In the present paper the property ${PL}({\mathbb R}^n,\omega )$ which characterizes the existence of continuous linear right inverses is investigated. It is also the one closest in spirit to the classical Phragmén–Lindelöf Theorem as various equivalent formulations for homogeneous varieties show. These also clarify the relation between ${PL}({\mathbb R}^n,\omega )$ and the PL-condition used by Hörmander to characterize the surjectivity of differential operators on real-analytic functions. We prove the property ${PL}({\mathbb R}^n,\omega )$ for an algebraic variety $V$ implies that $V_h$, the tangent cone of $V$ at infinity, also has this property. The converse implication fails in general. However, if $V_h$ is a manifold outside the origin, then $V$ satisfies ${PL}({\mathbb R}^n,\omega )$ if and only if the real points in $V_h$ have maximal dimension and if the distance of $z\in V$ to $V_h$ is bounded by $C\omega (|z|)$ as $z$ tends to infinity. In the general case, no geometric characterization of the algebraic varieties which satisfy ${PL}({\mathbb R}^n,\omega )$ is known, nor any of the other PL-conditions alluded to above. Besides these main results the paper contains several auxiliary necessary conditions and sufficient conditions which make it possible to treat interesting examples completely. Since it was submitted they have been applied by several authors to achieve further progress on questions left open here.