A Quasi Maximum Principle for Holomorphic Solutions of Partial Differential Equations in Cn

We present a quasi maximum principle stating roughly that holomorphic solutions of a given partial differential equation with constant coefficents in Cn,P(D) u=0, (†)achieve essentially their maximal growth on a certain algebraic hypersurfaceΓrelated to the operator. We prove it in the case wherePis homogeneous andΓis the conjugate dual cone, and also in the case whereP(D)=D21+…+D2nandΓis the complexified real sphere. We obtain a weak (semi-local) variant of the quasi maximum principle for certain non-homogeneous operatorsP(D), in which caseΓis the conjugate dual cone related to the principal part of the operator. This weaker variant is closely intertwined with several other notions. One of them is a quasi balayage principle for solutions of (†), involving the “sweeping” of measures in CnontoΓ.