Phragmén-Lindel(?)f Conditions On Algebraic Varieties And Its Perturbation

In 1970s, the Phragmen-Lindelof principle was firstly introduced by L.Hormander to characterize the surjectivity of the constant coefficient linear partial differential equations P(D) on the space of real analytic functions. He combined certain propertiesof the partial differential operator P(D) with some estimates of Phragmen-Lindelof condition for plurisubharmonic functions on the algebraic variety V (see [1]). Later. Meise.Taylor,Vogt and Braun began to investigate the existence of continuous linear right inverse for the linear partial operators P(D) using Phragmen-Lindelof estimate. Furthermore, they study about the Phragmen-Lindelof principles on the algebraic variety V and its equivalent conditions (see [7-9][14-20]). The perturbation of Phragmen-Lindelof condition is one important part of these researches.In this paper,we will further discuss the perturbation of Phragmen-Lindelof conditionbased on the work of these scholars, mainly study the perturbations for the Phragmen-Lindelof condition with one or two independent variates. We get the followingconclusions:Theorem 1 Let P∈C[z₁,z₂,…,z_n] be of degree m and assume that its principalpart P_m(z) is real. Let q(t)∈C[t]have degee k < m and non-real leading coefficientb. Set Q(z, t, s) = P(z) + q(t) + as with Im(b +a)≠0. If V(Q) satisfies PL(Rⁿ⁺²,ω)for a weight functionω, then t^1/m = O(ω(t)) as t tends to infinity.Theorem 2 Let P∈[z₁,z₂,…,z_n] be of degree m and assume that its principalpart P_m(z) is real. Let q(t)∈C[t] have degee k < m and non-real leading coefficientb, r(s)∈C[s] have degree l < m and non-real leading coefficient a, Im(b + a)≠0. SetQ(z,t,s) = P(z) + q(t) + r(s). If V(Q) satisfies PL(Rⁿ⁺²,ω) for some weight functionωand D= max{km,lm},thenωsatisfies t^kl/D=O(ω(t)) as t tends to infinity.Theorem 3 Let P∈C[z₁,z₂,…,z_n] be of degree m and assume that its principalpart P_m(z) is real. Set Q(z, z_n+1, z_n+2) = P(z) + bz_n+1z_n+2 with Imb≠0, m > 3.If V(Q) satisfies PL(Rⁿ⁺².ω) for some weight functionω, thenωsatisfies t^2/2m-3= O(ω(t)) as t tends to infinity.