O N A S Y M M E T R IC P L U R IC O M P L E X G R E E N F U N C T IO N

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1 NEW ZEALAND JOURNAL OF MATHEMATICS Volume 27 (1998), O N A S Y M M E T R IC P L U R IC O M P L E X G R E E N F U N C T IO N A r m e n E d i g a r i a n a n d W l o d z i m i e r z Z w o n e k (Received December 1996) Abstract. We give a new definition of a symmetric pluricomplex Green function, which turns out to be equal to a function defined by U. Cegrell. In our paper we give a new approach to a symmetric pluricomplex Green function, proposed by U. Cegrell (see [2]), based on a result of [8]. As in [2] the symmetric Green function is defined as supremum of some family of separately plurisubharmonic functions but in contrast to Cegrell s definition we do not require the subharmonicity of the functions involved 1. Moreover, due to the way we construct our function we are able to prove many nice properties of it. For the convenience of the reader we allow the plurisubharmonic functions to be identically oo. We denote also by PSHs(D x D ) the set of all functions u : D x D > [ oo, oo) such that for any w,z D u(w, ) and u(-, z) are plurisubharmonic on D, and we call them separately plurisubharmonic. Let E denote the unit disk in C. Let D b e a domain in Cn. For w, z G D we put k*d(w,z) := inf{m(ai, A2) : Ai, A2 E, there is 0 : E >D holomorphic with </>(Ai) = w, 0(A2) = z}, gniw, z) sup{u(z) : u PSH (D ) with u < 0, (C) log - HI = 0(1) (C -> w)}> c*d(w,z) := sup{m (f(w ),f(z )) : / : D >E is holomorphic}, where m(ai,a2) := -^f^-, Ai,A2 G. ^ 11 Ai A2 1 We call k*d (respectively go, c*d) the Lempert function (respectively, the pluricomplex Green function, the Caratheodory pseudodistance). For the basic properties of these functions the reader is asked to consult [4] and [6]. In [2] a symmetric version of the pluricomplex Green function W d was introduced. Namely, for w, z D we put WD(w, z) := sup{u (w,z)}, where the supremum is taken over all functions u G SH (D x D, [ oo, 0)) with the following properties: (i) for each w e D the function D 3 z >u(w,z) G [ oo,0) is plurisubharmonic, (ii) for each z e D the function D 3 w u(w, z) e [ oo, 0) is plurisubharmonic, (iii) u(w, z) < log lit; z\\ logmax{dist(u;, Cn \ D), dist(z, Cn \ D )} A M S Mathematics Subject Classification: 32H15. The work was supported by KBN grant No 2 P03A Professor Peter Pflug has pointed out to the authors that in the definition of U. Cegrell one need not assume the subharmonicity of the considered functions see [1] and [7].

2 36 ARMEN EDIGARIAN AND WLODZIMIERZ ZWONEK The main purpose of the paper is the following theorem. Theorem 1. There is a function go : D x D > [ oo, 0), which is < log k*d and such that go G PSHs(D x D ); moreover, for any function v G PSHs(Z) x D ) such that v < log k*d we have that v < go- Additionally, the following properties hold: go is upper semicontinuous with respect to both variables, go is symmetric, 9d (w,z) < m in {gd{w, z), gd(z,w )}, 9 d 9 d if and only if go is symmetric. (1) (2) (3) (4) Some of the properties of the function go are listed also in Proposition 5. The crucial role in the proof of Theorem 1 is played by a construction based on a theorem of Poletsky. Let us recall that theorem. T h eorem 2 ([8]). Let f be an upper semicontinuous function on D. Then is plurisubharmonic in D ; furthermore, u is the supremum of all plurisubharmonic functions v < f. In our construction the role of / will be played by the function loga;^ and its variations (more precisely by these functions with one of the variables fixed). P ro o f o f T heorem 1. At first we define inductively a sequence of functions, whose limit will be the function we are looking for. Put g D log k*d and define inductively for m = 1, 2,... gg " 1(w,z) := inf 0iT 2 {w,ip{elt))dr : ip e 0{ E, D),ip(0) = z^, g2dm(w,z) := g2 -\<P(eie),z)d e : 0 G 0(E,D),<f>( 0) = w }, w,z G D. For the sake of completeness and correctness of the above definition we prove the upper semicontinuity of g7^ as a function defined on D x D. P r o o f o f the U pper Sem icontinuity o f g >. We know that g D = log k*d is upper semicontinuous. Assume that the same holds for g for some m > 1. Fix two points a,b 6 D. Take any i e l such that g%+1(a,b) < A. Below we may assume that m is even (the case when m is odd may be proved exactly the same). Definition of g^ +1 implies that there is ip 0 (E,D ) such that ip(0) = b and

3 ON A SYMMETRIC PLURICOMPLEX GREEN FUNCTION 37 There is a continuous mapping defined on de such that h(elt) > grg (a, ^(e*t)) and 1 f 2n / h(elt)dt < A. J o Put "06(A) := b b + tp(a), A G E. For sufficiently small open U D {a } x {6} x de we may define a mapping Certainly V : D xd x8edu3 (a,b,eit) -> ^ ( «5^ ( e ir)) - /i(eir). The upper semicontinuity of ^ implies that for a (respectively b) close enough to a (respectively b) we have 1 p2tt -i /*27r 2tt Jo 9D { ^ M ett))dt ^ 2^ J Q h(elt)d r < A. This completes the inductive step and, consequently, finishes the proof of upper semicontinuity of g^. Before we go on to the Proof of Theorem 1 let us state and prove some simple lemma. Lem m a 3. g ld = gd. P r o o f o f Lem m a 3. We know that any plurisubharmonic function u(z) < logk*d(w,z) for fixed w G D is smaller than or equal to go{w,z). In fact, note that the following property holds log k*d(w, z) < M + \og\\w z\\ for 2: close to w and for some M > 0. This, in connection with Theorem 2, upper semicontinuity of k*d{w, ) and the definition of go completes the proof. As the constant functions may be applied in the definition of g^ we have that nm > nm d 9 d Because of the upper semicontinuity of g^ we have in view of Theorem 2 that for any w,z G D g2dm(, z) PSH( >),m = 1,2,..., g2^+1(w, ) e PSH( >), m = 0,1... Consequently, the sequence g^ tends to some function go, so go = limm_>oo g7^ = limm_>00 g limto-^oo g^n+1, which implies that go is in PSHs(D x D ) (remember that the functions g^n+1(w, ), ^^TO(-, z) G PSH(D) and the suitable sequences are decreasing). Now, take any function v G PSHS(D x D), which is < logk*d. By Theorem 2 (applied to the mapping log kp(w, ) for any w G D) we have that v next applying Theorem 2 once more (this time to gld(-,z)) we get that v < 92d- Repeating this procedure we get that v < g1^ for any m. Consequently, v < go- This gives us the existence of the function claimed in the theorem. To check the properties of gp notice that (1) is obvious as go is a limit of a decreasing sequence of upper semicontinuous functions. To get symmetry consider the mapping go(w, z) := m&x{gd(w, z),gd {z, «;)}, we have certainly that go G PSHs(.D x D) and go < 9 % = log k*d. But this implies, because of the maximality of go that go = 9d, which gives the symmetry of go-

4 38 ARMEN EDIGARIAN AND WLODZIMIERZ ZWONEK In view of Lemma 3 we have that gld = go- Now, the symmetry implies (3). To get (4) it is enough to prove the following fact If 9d is symmetric for some m > 1, then gld = g7^ for any I > m. To prove that it is enough to show that fact for I = m 4-1 and m even. Take any a,b G D and ip G 0 (E, D ) such that ip(0) = b. In view of symmetry and separate plurisubharmonicity of g^ we have 1 /*27r 2tt Jo 9D (a^ ( elt))dr - 2d ( M ), but from the construction of g7^ we have g%+1 < gp. This completes the proof of (4) and the whole theorem. From the proof of Theorem 1 we may conclude the following: R em ark 4. If v G PSHS(D x D ) and v < g^, then v < go- The well-known properties of k*d and go deliver us also much information about go- Below we list some of these properties: P rop osition 5. (a) The functions g7^ form, the family of contractible functions with respect to holomorphic mappings, i.e. gq (F(w), F (z)) < g^(w, z) for any F G 0 ( D, G) w,z G D and g^ = loga;^. Certainly, the same properties hold for go. In particular, logc*d <gd <gd< k*d. ( b ) If D j C D j+1 and U Dj = D, then g o j gd - (c) If P C D is a closed pluripolar set, then ^ d ( d \ p ) x ( d \ p ) = 9 d \p - (d) g o {w,z ) is the supremum of all v G PSHs(Z) x D ), v < 0 such that for any w G D there is C(w) such that v(w,z) < log w z\\ + C{w) for z close w. P r o o f o f P roposition 5. (a) is a simple consequence of the contractibility of k* and the definition of g^. To prove (b) put u := lim ^^., we have certainly that u > gu and u G PSHs(J9 x D). But gdj < logkp. > loga:^ so u < log k*d and now Theorem 1 applies. The property (c) is a consequence of the similar property of go, Theorem 1 and Remark 4. One gets (d) from definitons of go, gd and Remark 4. C orollary 6. The function go coincides with W p. P roof. It is enough to prove the subharmonicity of gp. But this is a simple consequence of the fact that plurisubharmonicity implies subharmonicity, upper semicontinuity of go and Fubini s theorem. We have already pointed out that due to [7] and [1] one can get rid of the proof of Corollary 6; nevertheless, we leave it as it is very simple and the whole paper remains more self-contained.

5 ON A SYMMETRIC PLURICOMPLEX GREEN FUNCTION 39 Exam ple 7. (a) Let a := (<*1,..., an) be such that all a3's are relatively prime positive integers, put D := {z = (z i,..., zn) G Cn : \za \< 1}. Then for any w,z G D gd(w,z) = log w z 1 wazc In fact, notice that the function defined above is separately plurisubharmonic and equal to m m {gd {w,z),gd {z,w )} (see [4]). In view of Theorem 1 (3) and the maximality of go this completes the proof. (b) Let 0 < a\ < a2 < 1, 0 < 61 < b2 < 1) Q ' = [«i, 0-2] x [^1^ 2] be such that (0,0) ^ Q and (int Q) fl {(f, t) : t G [0,1]} = 0. Put D := {(z 1,z 2) E x E : ( zi, \z2\) & Q}. Then (see [5]) 5d ((0, 0), (zi,z2)) = gd{(0, 0),{z i,z 2)) = max{log \zi\, log\z2\}. In fact, m ax{log 2i, log ^21} = 5r>((0,0), (z!,z2)) > gd((0,0), (zlt z2)) > 9ex e({0, 0), (^1,^2)) = 0jEx ((O,O),(zi,22)) = max{log ^i, log \z2\}. Acknow ledgem ents. The authors would like to thank Professors Marek Jarnicki and Peter Pflug for their precious remarks and their help. R eferences 1. M.V. Avanissan, Fonctions plurisousharmoniques et fonctions doublement sousharmoniques, Ann. Sc. Ec. Norm. Sup. 78 (1961), U. Cegrell, Capacities in Complex Analysis, Vieweg, A. Edigarian, On definitions of the pluricomplex Green function, Annales Polonici Mathematici 67 (3) (1997), M. Jarnicki and P. Pflug, Invariant Distances and Metrics in Complex Analysis, Walter de Gruyter, M. Jarnicki and P. Pflug, Remarks on the pluricomplex Green function, Indiana Univ. Math. J. 44 (1995), M. Klimek, Extremal plurisubharmonic functions and invariant pseudodistances, Bull. Soc. Math. France 113 (1985),

6 40 ARMEN EDIGARIAN AND WLODZIMIERZ ZWONEK 7. P. Lelong, Les fonctions plurisousharmoniques, Ann. Sc. Ec. Norm. Sup. 62 (1945). 8. E.A. Poletsky, Plurisubharmonic functions as solutions of variational problems, Proceedings of Symposia in Pure Mathematics 52 Part 1 (1991), Armen Edigarian Instytut Matematyki Uniwersytet Jagielloriski Reymonta Krakow P O LAN D edigaria@im.uj.edu.pl Wlodzimierz Zwonek Instytut Matematyki Uniwersytet Jagielloriski Reymonta Krakow P O LAN D zwonek@im.uj.edu.pl