On the product property of the pluricomplex Green function

Abstract

We prove that the pluricomplex Green function has the product property 9D1 x D2 = max{g9D1 9D2 } for any domains D1 C C? and D2 C Cm. Let E denote the unit disc in C. For any domain G C C' define 9D(a,z) inf D{ 7 JAIord\(t -a)}, a,zCD, pEO(E, D) A1 (a) ~o(O)=z A~1a aGEo(E) where (?(E, D) denotes the set of all holomorphic mappings E -> D and ordA ((p a) denotes multiplicity of (o a at A. The function 9D is proposed by Poletsky (cf. [Pol]) and is called the pluricomplex Green function for D. We have that (see [Jar-Pfll], Chapter IV) (a) gD (a, z)z inf { J7J AIordx\(~p a)} a,zcD. a C ( E, D) A E ,(a) aGE9(E) Note that in the formula (a) we take only A c (p1 (a) such that A c E. (b) For any domains D1, D2 and any holomorphic mapping f: D1 D2 we have the following contractible property: 9D2 (f (z), f (w)) follows from the property (b). So, we have to prove NlKJv and J6j ... ( > NjIl>K . For, if I(, ... Qvl A), and it contradicts the minimality of v. If (v ?. < 161 .. (Lj, then we replace (P1 with the mapping ;1(A) = (p1(tA), where t : Then |j <1] 1). .. v (use (2)), and (t .. (t)=l-.(. Hence, we may assume that I(,.. (vlI = |(l ...(p C < N. Moreover, replacing (Pj (A) with Pi (e-61 A) and 02(2) with (P2 (e-i2 A), where 01, 02 are chosen such that e i2 8I ... e281 ?_ C and e iO2 ... eiO2 = C, we may assume that (I... (v= 1.H= C. We consider Blaschke products B1(A) = < j=1 J and B1(Av) = _ B1(A)O-B1(O) A A-wa A cE. I Note that mappings P1j and 02 are holorriorphic in some neighborhood of E, and the sets ypj 1(al) and o1 may contain points outside of E. 2For example, it is enough to change very little the mappings y01 and p2 by the formula (3) given below. This content downloaded from 157.55.39.170 on Tue, 26 Jul 2016 06:23:20 UTC All use subject to http://about.jstor.org/terms TI'HE PRODUCTI' PROPERTI'Y OF TI'HE PLURICOMvPLEX GREEN FUNCTI'ION 2857 We choose different wj, 1 < j < v, as close to wj as we want such that O C {w1,. .. ,w'}. Define Aw' Note that B7i1(-C) {?i,...,Q} We can find w',...,w' such that G71(-C) consists of v different points I, 1 ? j < V, as close to points ?j as we want. Let us replace the mapping pi with the mapping (3) (j) := (p(A)-a, ) += Clearly, when j, 1 < j < V, are sufficiently close to (j, ;i maps E into D1 (recall that ,oj maps some neighborhood of E into D1, hence 9p1(E) @ DI), and (71 (?) = (01 (?), I4 (?j K) = (01 (?j) Repeating this process for P2, we may assume that for Blaschke products B1 and B2 derivatives are not equal to 0 either on preimages of C or at points ?j or j respectively. Let A be the union of images of singular points under mappings B1 and B2. Note that neither 0 nor C is in A. Let ir be a holomorphic universal covering of E A by E with ir(0) = C. There are liftings Vl' and V)2 mapping E into E such that 7r = B, o 01 = B2 f({2 andj1(?) = 02(0) = O. If7r-1(0) = {711,j2,.... .}, then mappings (Pj o? 1 and (P2 o 02 map 0 into a2 and b2, and all points 71 into a, and b1 respectively. Note that ir has all radial limits either in aE or in A. Since A is finite, Xr is an inner function. By Theorem 2 of Ch. III in [Nos], every inner function which has no zero radial limits is a Blaschke product. Thus