ESI The Erwin S hr�odinger International Boltzmanngasse 9Institute for Mathemati al Physi s A-1090 Wien, AustriaQuantitative Estimates for the Green Fun tionand an Appli ation to the Bergman Metri Klas Diederi hGregor Herbort

Vienna, Preprint ESI 877 (2000) April 27, 2000Supported by Federal Ministry of S ien e and Transport, AustriaAvailable via http://www.esi.a .at

Quantitative estimates for the Greenfun tion and an appli ation to theBergman metri Klas Diederi h and Gregor HerbortAbstra tLet D � Cn be a bounded pseudo onvex domain that admits a hoelder ontinuous plurisubharmoni exhaustion fun tion. Let its pluri omplex Greenfun tion be denoted by GD(�; �). In this arti le we give for a ompa t subsetK � D a quantitative upper bound for the supremum supz2K jGD(z; w)j interms of the boundary distan e of K and w. This enables us to prove that, ona smooth bounded regular domain D (in the sense of Diederi h-Fornaess), theBergman di�erential metri BD(w;X) tends to in�nity, for X 2 Cn nf0g, whenw 2 D tends to a boundary point. Furthermore, we prove that the order ofgrowth of BD(w; �) under nontangential approa h of w 2 D to a point z0 2 �Dof �nite type, an be estimated from below by 1=N , where N denotes the orderof pseudo onvex extendability of �D at z0.AMS Classi� ation: 32 U 35, 32 F 45, 32 T 25, 32 T 27Keywords: Pluri omplex Green fun tion, Monge-Amp�ere equation, order of pseudo- onvex extendability, Bergman metri 1 Introdu tionThe pluri omplex Green fun tion GD(z;w) with a single pole w for bounded pseudo- onvex domains D �� Cn was introdu ed in [15℄. Its onne tion with the omplexMonge-Amp�ere operator and other properties were extensively studied in [5℄. Bya re ent result of M. Carlehed, U. Cegrell, F. Wikstr�om [4℄ one has the followingimportant fa t for the ase of hyper onvex domains:Theorem 1.1 Let D �� Cn be a bounded hyper onvex domain and let (wk) � Dbe a sequen e with limk!1 wk = w0 2 �D. Then there is a pluripolar set E � D su hthat for every z 2 D n E limk!1GD(z;wk) = 0Typeset by the TEX-prepro essor PreTEX

2 Klas Diederi h and Gregor HerbortSin e GD(�; w) has a logarithmi pole at w, it an be used as a weight in H�ormander's�-theory in order to onstru t holomorphi L2-fun tions with ertain pres ribed prop-erties. This has turned out to be a useful tool in studying the Bergman theory of su hdomains D, the most re ent result in this dire tion being the Bergman ompletenessof hyper onvex domains whi h was proved in [2℄ and [12℄ using Theorem 1.1.Considering Theorem 1.1 one might want to ask, whether the ex eptional pluripolarset E (whi h might even depend on the hoi e of the sequen e (wk) onverging to a�xed boundary point w0 2 �D) ne essarily appears. This seems to be unknown forgeneral bounded hyper onvex domains D �� Cn. However, under a slightly strongerhypothesis on D, the following statement was proved in [13℄:Theorem 1.2 Let D �� Cn be a bounded domain admitting a bounded plurisub-harmoni exhaustion fun tion whi h is H�older ontinuous on D. Then one has forany ompa t set K � D and any point w0 2 �Dlimw!w0supK jGD(�; w)j = 0 (1.1)Remark 1.3 Noti e, that, a ording to [6℄, all bounded pseudo onvex domains inCn with C2-smooth boundary satisfy the hypothesis of Theorem 1.2.It might be useful for future appli ations to go beyond the qualitative result of The-orem 1.2, by asking under whi h ir umstan es (1.1) an be put into a more quanti-tative form. An indi ation, that this might be possible an be seen in the result ofCarlehed who showed in [3℄ that one has on any strongly pseudo onvex domain Dwith smooth C2-boundary the inequalityjGD(z;w)j � C ÆD(z)ÆD(w)jz � wj4for all z;w 2 D, with a onstant C > 0. Here, as always in this arti le, ÆD(�) denotesthe boundary distan e in D.The �rst goal of this arti le now is to showTheorem 1.4 Assume that on the bounded pseudo onvex domain D �� Cn thereis a bounded plurisubharmoni exhaustion fun tion � : D ! [�1; 0), su h that forsome number 0 < � < 1 and onstants C1; C2 > 0 the inequalityC2Æ�D(z) � j�(z)j � C1Æ�D(z) (1.2)holds everywhere on D. Then for any �xed number 0 < t� 1 there exists a onstant~Ct � 1, su h that for all ompa t subsets K � D and all w 2 D satisfyingÆD(w) < ~C�1t ÆD(K)

Pluri omplex Green fun tion 3the pluri omplex Green fun tion GD of D an be estimated bysupK jGD(�; w)j � ~Ct �Æ1�2ntD (w)ÆD(K) ��=n�log 2RDÆD(w)�1=n + Æt�D (w)jlog ÆD (w)j!where RD denotes the diameter of D.Remark 1.5 Noti e, that the se ond term on the right hand side of this inequality is learly dominating for w very lose to �D. However, the �rst term be omes relevantnear the other extreme, namely near ÆD(w) = ~CtÆD(K).This theorem has a useful onsequen e for the sublevel sets of the Green fun tion,showing that they on entrate near the boundary with a ertain speed. Namely, onehasCorollary 1.6 Let D be as in Theorem (1.4). Then there is for any �xed number0 < t � 1 a number 0 < Æt � 1, su h that for all w 2 D with ÆD(w) < Ætthe sublevel set A(w;D) := fz 2 D : GD(z;w) < �1g is ontained in the ollarfz 2 D : ÆD(z) < Æ1�3ntD (w)g.In addition, for strongly pseudo onvex domains, we an state a slight improvementof Theorem 1.4, namely:Theorem 1.7 Suppose D �� Cn is a strongly pseudo onvex domain with smoothC2-boundary. Then there is a onstant C > 0, su h that one has for any ompa t setK � D and any point w 2 D with ÆD(w) < C�1ÆD(K) the inequalitysupK jGD(�; w)j � C ÆD(w)ÆD(K)In parti ular there exists a number C1 > 0 su h that for all w 2 D with ÆD(w) < C�11the relation fz 2 D : GD(z;w) < �1g � fz 2 D : ÆD(z) < C1ÆD(w)gholds.In [10℄ the Bergman distan e fun tion dB on pseudo onvex domainsD � Cn was stud-ied that admit a ontinuous plurisubharmoni exhaustion fun tion � whi h satis�esan estimate of the form CÆmD � j�j � C�Æ1=mDwith ertain onstants C;C�;m > 0. For these domains an e�e tive lower boundwas obtained for dB(�; z0) (with a �xed z0 2 D). Namely, there exist onstantsC;C 0 > 0 and Æ0 > 0 su h that dB(z; z0) � Cj log j logC 0ÆD(z)j j, whenever ÆD(z) < Æ0.This result applies for example to all those domains, whi h an be represented asinterse tions of �nitely many C2- smooth bounded pseudo onvex domains.We will show in this arti le two appli ations of Theorems 1.2 and Theorem 1.4 toquestions about the boundary behavior of the Bergman di�erential metri BD of D.We have at �rst:

4 Klas Diederi h and Gregor HerbortTheorem 1.8 Let D �� Cn be a pseudo onvex domain admitting a bounded uni-formly H�older ontinuous plurisubharmoni exhaustion fun tion and having the prop-erty, that any point q 2 �D is a peak point for the family P(D) := C0(D)\PSH(D)of ontinuous fun tions on D plurisubharmoni on D. Then we have for any q 2 �Dand any ve tor X 2 Cn n f0g limw!qBD(w;X) = +1Remark 1.9 Note, that no �nite type assumption is made in this theorem. For in-stan e, all regular domains (in the sense of [7℄) satisfy the hypothesis of the Theorem.Before stating the se ond result on the boundary behavior of BD we remind the readerof the de�nition of two invariants asso iated to boundary points of D:De�nition 1.10 Let D �� Cn be a pseudo onvex domain and w0 2 �D a boundarypoint su h that �D is C1-smooth near w0.a) The boundary �D is alled pseudo onvexly extendable of an order m � 2 atw0, if there is an open neighborhood U of w0 and a C3-fun tion � on U , su h that�(w0) = 0, the surfa e S := f� = 0g is smooth and pseudo onvex from the sidef� < 0g and for some onstant C > 0 the estimate�C�ÆD(z) + ��z �w0��2� � �(z) < ���z � w0��mholds on D \ U .b) The numberN��D;w0� := inf �m � 2 : �D is pseudo onv: ext: of order m at w0is alled the order of pseudo onvex extendability of �D at w0.An important analyti lo al invariant of �D at w0 isDe�nition 1.11 Let D;w0 be as in De�nition 1.10. Then the numberhD�w0� := sup�s : lim infz!w0 ÆsD(z)BD(z;X) =1 nontangentially;8X 2 Cn n f0g�is alled the growth exponent of the Bergman metri at w0.Remark 1.12 For further details and properties on erning the order of pseudo on-vex extendability and the growth exponent of the Bergman metri see [8℄ and [9℄.Using Theorem 1.4 we an showTheorem 1.13 Assume that the pseudo onvex domain D �� Cn has C1-smoothboundary near w0 2 �D. If the type of �D at w0 is �nite, then we have the inequalityhD�w0� � 1N(�D;w0)

Pluri omplex Green fun tion 5The stru ture of this arti le is as follows: se tion 2 ontains the proof of Theorem 1.4,whi h, in fa t, is ompletely parallel to the proof of Theorem 1.1 from [13℄. The shortse tion 3 then ontains what is needed to get Corollary 1.6 and the estimate in thestri tly pseudo onvex ase as stated in Theorem 1.7. Starting with se tion 4 we turnto the boundary behavior of the Bergman metri showing at �rst the basi omparisonlemma (Proposition 4.1) between the Bergman metri s on D and on the sublevel setA(w;D) of the pluri omplex Green fun tion of D. The boundary behavior of su hsublevel sets is studied in se tion 5. The proof of Theorem 1.8 in se tion 6 is thenvery short. Se tion 7, �nally, ontains the proof of Theorem 1.13.2 Proof of Theorem 1.4As said already we an ompletely follow the pattern of the proof of Theorem 1.1from [13℄. For this we always assume in this se tion, that the domain D and the ex-haustion fun tion � with the exponent 0 < � < 1 have been hosen as in Theorem 1.4.We �x a ompa t subset K � D and introdu e two auxiliary fun tions depending onthe hoi e of points z 2 K and w 2 D nK, a number �w > 0, whi h will be hosenlater and an integer k > 0. We putUz;w(�) := maxfGD(�; z);��wg (2.1)and Vw;k(�) := maxfGD(�; w);�kg (2.2)We will study the following three integralsIk(z;w) := RD jVw;kj(dd Uz;w)nI 0k(z;w) := RD jVw;kjn(dd Uz;w)nI(z;w) := RD jGD(�; w)j(dd Uz;w)n (2.3)The main load in proving Theorem 1.4 onsists in showing suitable upper and lowerestimates for the integral I(z;w) as formulated pre isely in the following two lemmasLemma 2.1 There is a onstant C� > 0 (depending on n and �) su h that for allz 2 K and all w 2 D with ÆD(w) < ÆD(K)2 the estimateI(z;w) � C��1� 1nw �Æ��D (K) log 2RDÆD(K)�1=nÆ�=nD (w) (2.4)holds.In formulating the de isive lower estimate for I(z;w) we spe ify the free onstant�w > 0 from (2.1). We get

6 Klas Diederi h and Gregor HerbortLemma 2.2 If, for 0 < t� 1, the number �w is de�ned to be�w := j�(w)j�2t (2.5)then we have for all z 2 K and w 2 D with ÆD(w) < Æ0(t;K) the estimateI(z;w) � jGD(z;w)j � C 0tÆt�D (w)jlog ÆD (w)j (2.6)with some unimportant onstant C 0t > 0. Furthermore, the onstant Æ0(t;K) is givenby Æ0(t;K) = min�Æ1(t; �); C2C1ÆD(K)�with a onstant Æ1(t; �) not depending on K and the onstants C1; C2 from (1.2).Before showing these lemmas we explain at �rst, how Theorem 1.4 follows immediatelyfrom them:Proof of Theorem 1.4: Noti e that one has a ording to (2.5) and (1.2) the inequality�w � C�2t2 Æ�2t�D (w). Hen e we get from (2.4) and (2.6)jGD(z;w)j � C 0tÆt�D (w)jlogÆD (w)j+ I(z;w)� C 0tÆt�D (w)jlogÆD (w)j+ C��Æ��D (K) log 2RDÆD(K)�1=n�1� 1nw Æ�=nD (w)� C 0tÆt�D (w)jlogÆD (w)j+ C��Æ��D (K) log 2RDÆD(K)�1=nÆ(1�2nt)�=nD (w)If we put, for instan e, ~Ct := max�C 0t; C2C1 ; 21 +RDÆ1(t; �)�Theorem 1.4 follows immediately, sin e z 2 K was hosen arbitrarily.We now ome to the proof of the upper bound for I(z;w) from Lemma 2.1. The maintool for it is the following lemma by Z. Blo ki ( [1℄)Lemma 2.3 Assume that u; v1; : : : ; vn : D ! IR� are bounded plurisubharmoni fun tions su h that u(�)! 0 for � ! �D. Then one hasZD jujn(dd v1) ^ : : : ^ (dd vn) � n!kv1k1 � � � kvn�1k1ZD jvnj(dd u)nApplying this to u := Vw;k and v1 = � � � = vn = Uz;w and observing that kvjk1 � �wfor all j = 1; : : : ; n, we get the inequalityI 0k(z;w) � n!�n�1w ZD jUz;wj(dd Vw;k)n (2.7)

Pluri omplex Green fun tion 7With this we an estimateIk(z;w) = RD jVw;kj(dd Uz;w)n� �RD (dd Uz;w)n�n�1n �RD jVw;kjn(dd Uz;w)n� 1n= (2�)n�1(I 0k(z;w))1=n� Cn�1� 1nw �RD jUz;wj(dd Vw;k)n�1=n(Here we used the fa t that all the positive measures (dd Uz;w)n have total mass (2�)n,see e.g. Lemma 2.1 in [13℄.)So, we have shown, that for all kIk(z;w) � Cn�1� 1nw �ZD jUz;wj(dd Vw;k)n�1=n (2.8)Now noti e, that for k !1 one has jVw;kj % jGD(�; w)j and, hen e, by Beppo Levi,Ik(z;w) !k!1I(z;w) (2.9)Furthermore, sin e all (dd Vw;k)n have the same mass as the measure (2�)nÆw, whereÆw denotes the Dira measure at w, they tend for k !1 in the weak-star topologyto (2�)nÆw. Hen e, we even haveZD f(dd Vw;k)n !k!1(2�)nf(w)for any bounded ontinuous fun tion f on D. Applying this to f := jUz;wj, we get fork !1 ZD jUz;wj(dd Vw;k)n ! (2�)njUz;w(w)j � (2�)njGD(w; z)jCombining this with (2.8) and (2.9), we get the estimateI(z;w) � Cn�1� 1nw jGD(w; z)j1=n (2.10)This inequality implies Lemma 2.1, sin e one also hasLemma 2.4 Let D; �; � be as in Theorem 1.4. Then there exists a onstant C� > 0,su h that for any ompa t set K � D the estimatesupz2KjGD(�; z)j � C��Æ��D (K) log 2RDÆD(K)�Æ�D(�) (2.11)holds for all points � 2 D with ÆD(�) � 12ÆD(K).Proof of Lemma 2.4: We pro eed analogously to [5℄. First we note that we have (1.2).Let, now, x 2 K be arbitrary and Bx;K := B(x; ÆD(K)2 ). We put�x(�) := (maxnC�(�); log j��xjRD o if � 2 D nBx;Klog j��xjRD if � 2 Bx;K

8 Klas Diederi h and Gregor Herbortwhere C = 2�C�12 Æ��D (K) log 2RDÆD(K)Note that inf�Bx;Kj�j � C2 inf�Bx;KÆ�D � C22��Æ�D(K)With this hoi e of C we, therefore, haveC� = �Cj�j � �C inf�Bx;Kj�j � log ÆD(K)2RDon �Bx;K.The above de�nition of �x, hen e, is onsistent, and we obtain a negative plurisub-harmoni fun tion �x on D serving as a andidate for GD(�; x).Let now � 2 D be a point with ÆD(�) � ÆD(K)2 . Then � 62 Bx;K andjGD(�; x)j � j�x(�)j � Cj�(�)j� 2�C�12 �Æ��D (K) log 2RDÆD(K)�j�(�)j� C��Æ��D (K) log 2RDÆD(K)�Æ�D(�)with C� = 2�C1C2 . This proves Lemma 2.4.In order to show the lower bound for I(z;w) as stated in Lemma 2.2, we use thefollowing result shown in [13℄:Lemma 2.5 LetD �� Cn be a hyper onvex domain admitting a ontinuous plurisub-harmoni exhaustion fun tion � : D ! [�1; 0) whi h is uniformly H�older ontinuouson D with exponent � > 0. Then there is a onstant C > 0 (depending on �) andfor any ompa t subset K � D a onstant Æ0 > 0 (only depending on K and �), su hthat one has for any z 2 K and w 2 D nK with ÆD(w) < Æ0 the estimatesup �GD(�; w) : j� � zj < RD exp ����2(w)� � GD(z;w) + Cj�(w)jjlog j�(w)jjAs an be seen from [13℄, the proof of the Lemma yields somemore pre ise informationon the number Æ0(K;�), namely, Æ0 is of the formÆ0(K;�) = minfÆ1(�); Æ2(K)g (2.12)where Æ1(�) is independent of K and Æ2(K) > 0 is a number su h thatj�(x)j < infK j�j 8x 2 D with ÆD(x) < Æ2(K) (2.13)In the situation of Theorem 1.4 we apply Lemma 2.5 for getting Lemma 2.2 with� = �t := �(��)t (2.14)

Pluri omplex Green fun tion 9Then we get with � = t� the estimateC t2Æt�D � j�j � C t1Æt�DThis implies that the number Æ2(K) := C2C1 ÆD(K)satis�es (2.13) su h that the number Æ0 in Lemma 2.5 be omes in this aseÆ0 = Æ0(t;K) = min�Æ1(t�); C2C1 ÆD(K)� (2.15)As we will see in the following, this is the number Æ0(t;K) needed in Lemma 2.2, su hthat the number Æ1(t; �) used there is, in fa t, Æ1(t; �) = Æ1(t�). With this we annow give theEnd of the proof of Lemma 2.2: The measure (dd Uz;w)n is supported in the ballBz;w := B(z;RD exp(��w)) = B(z;RD exp(���2t (w))). We, therefore, obtain for allz 2 K and w with ÆD(w) < Æ0(t;K) the inequalityI(z;w) � (2�)n infBz;wjGD(�; w)j= �(2�)nsupBz;wGD(�; w)� jGD(z;w)j � (2�)nCtj�t(w)jjlog j�t(w)jjwhere the inequality and the onstant Ct ome from Lemma 2.5. This �nishes theproof of Lemma 2.2 (and, hen e, also of Theorem 1.4).3 Proofs of Corollary 1.6 and Theorem 1.7We start with the proof of the Corollary. Let for this �; t and ~Ct be as in Theorem 1.4.We take w 2 D su h that ÆD(w) < ~C�1=3nttand put K := �x 2 D : ÆD(x) � Æ1�3ntD (w)Then we have ÆD(w) = Æ3ntD (w)ÆD(K) < ~C�1t ÆD(K)Hen e we getsupK jGD(�; w)j � ~Ct � Æ1�2ntD (w)ÆD(K) ��=n�log 2RDÆ1�3ntD (w)�1=n + Æt�D (w) log 1ÆD(w)!� C 0tÆt�D (w) log 1ÆD(w)< C 0tÆt�t log 1Æt for ÆD(w) < Æt< 1

10 Klas Diederi h and Gregor Herbortprovided Æt � 1. This means that A(w;D) \K = ; as laimed.In showing Theorem 1.7, it suÆ es to prove the �rst laim, sin e the se ond followsfrom it as in the proof of the Corollary. First we hoose a Æ1 > 0 su h that onfw 2 D : ÆD(w) < Æ1g the orthogonal proje tion to �D is well-de�ned. We denotethe image of w by w�. Let now F : D � �D ! �, where � denotes the unit dis inC, be a ontinuous fun tion with the properties:i) F (�; q) is a holomorphi peak fun tion at q 2 �D de�ned on a �xed neighborhoodof D;ii) with some onstant C2 > 0 one has for all w 2 D with ÆD(w) < Æ1j1� F (w;w�)j � C2ÆD(w)(Su h a fun tion is onstru ted for instan e in [11℄.) Furthermore, we noti e that onehas for any two points a; b 2 � the estimate���� a� b1 � ab����2 � 1 � 41 � jbj1� jaj (3.1)This yields for all points z 2 K � D and w 2 D with ÆD(w) < Æ1jGD(z;w)j � jG�(F (z;w�); F (w;w�))j= �12 log ��� F (z;w�)�F (w;w�)1�F (z;w�)F (w;w�)���2� �12 log�1 � 41�jF (w;w�)j1�jF (z;w�)j �� �12 log�1 � 4 C2ÆD(w)1�jF (z;w�)j� (3.2)However, from the Hopf Lemma applied to the plurisubharmoni fun tion jF (�; w�)j�1, we obtain with a onstant C3 > 0 whi h an be hosen to be independent of w�jF (�; w�)j � 1 � �C3ÆD(�)In parti ular, we obtain (sin e z 2 K)�12 log�1� 4 C2ÆD(w)1 � jF (z;w�)j� � � 12 log�1 � 4C2ÆD(w)C3ÆD(K)�If now ÆD(w) < C38C2 ÆD(K)we obtain from (3.2)jGD(z;w)j � �12 log�1� C2ÆD(w)C3ÆD(K)� � 8C2C3 ÆD(w)ÆD(K)This implies the laim.

Pluri omplex Green fun tion 114 A omparison lemma for the Bergman metri The basi tool allowing to dedu e estimates on the Bergman metri of a boundeddomain D �� Cn from estimates on the Green fun tion GD(�; �) of D is the followingfundamental omparison lemma:Proposition 4.1 For any bounded pseudo onvex domain D �� Cn there is a on-stant C1 > 0, su h that for all (w;X) 2 D � (Cn n f0g) the following estimate holds1C1BA(w;D)(w;X) � BD(w;X) � C1BA(w;D)(w;X) (4.1)The proof of this Proposition is based on the well-known fa t, that for arbitrarybounded domains �� Cn the Bergman kernelK(z) := K(z; z) an be hara ter-ized by the extremal propertyK(z) = maxn jf(z)j2 : f 2 H2(); kfk := kfkL2() � 1o (4.2)and the quantity � Cn 3 (z;X) 7�! b2(z;X) := K(z)B2(z;X) is given by thevariational problemb2(z;X) = max� jh�f(z);Xij2 : f 2 H2(); f(z) = 0; kfk � 1 (4.3)where H2() := L2() \ O(). Therefore, the following interpolation result an beused for proving Proposition 4.1.Lemma 4.2 Let D �� Cn be a bounded pseudo onvex domain and �x a pointw 2 D and a number S > 0. Denote by AS(w;D) the sublevel setAS(w;D) := fGD(�;w) < �SgThen we an �nd for every fun tion f 2 H2(AS(w;D)) a fun tion ~f 2 H2(D) su hthati) ~f(w) = f(w)ii) � ~f(w) = �f(w)iii) the estimate ~f 2D � 1 + e(4n+6)S+1+R2S2 !ZAS(w;D) jf j2 d2nzholds with R :=maxD jzj.We will show at �rst, how this lemma implies Proposition 4.1:

12 Klas Diederi h and Gregor HerbortProof of Proposition 4.1: Choosing S = 1 in Lemma 4.2 we get for any nonzerof 2 H2(A(w;D)) be ause of (4.2) resp. (4.3)KD(w) � ��� ~f (w)���2 ~f 2 � 1CD(1 + 4e4n+6) jf(w)j2kfk2A(w;D)and b2D(w;X) � ���h� ~f(w);Xi���2 ~f 2 � 1CD(1 + 4e4n+6) jh�f(w);Xij2kfk2A(w;D)for all X 2 Cn. Here the onstant CD an be de�ned asCD = 1 + 4e1+R2Taking the supremum over all f 2 H2(A(w;D)) with kfkA(w;D) = 1 we get at �rstKD(w) � 1CD(1 + 4e4n+6)KA(w;D)(w) (4.4)and taking the supremum over all f 2 H2(A(w;D)) with kfkA(w;D) = 1 and f(w) = 0we also get b2D(w;X) � 1CD(1 + 4e4n+6)b2A(w;D)(w;X) (4.5)for any X 2 Cn.Putting together (4.3) and (4.5) yieldsB2D(w;X) � 1CD(1 + 4e4n+6)B2A(w;D)(w;X)the lower estimate of Proposition 4.1. The upper estimate follows similarly using (4.4).It remains to show Lemma 4.2:Proof of Lemma 4.2: The proof is a re�nement of the argument showing Proposition3.6 of [12℄. We denote for small � > 0 by D� the set D� := fz 2 D : dist(z; �D) > �g.Furthermore, let 1 2 C10 (Cn) be a non-negative polyradially symmetri fun tionwith RCn 1 d2nz = 1. We put �(z) := ��2n 1���1z� for z 2 CnWith it we de�ne on D� the weight fun tions��(z) := (2n + 2)V� (z) + eV� (z) + � jzj2with V� := GD(�; w) � � and� := (2n + 2)GD(�; w) + eGD(�;w)

Pluri omplex Green fun tion 13Finally, we hoose a ut-o� fun tion � 2 C1(IR) with j�0j � 2S�1 and su h that�(x) = � 1 for x � �2S0 for x � �SLet us now take an arbitrary f 2 H2(AS(w;D)) and de�ne the (0; 1)-form�� = nXj=1 ��;j dzj := �(� Æ V� � f) = �0 Æ V� � f�V�It is smooth and its support is ontained in the set f�2S � V� � �Sg � AS(w;D).We also observe that the weight fun tion �� satis�es on D� the estimate�� � 1 +R2 (4.6)and on sup p�� we have �� � �(4n+ 4)S (4.7)Next we want to onvin e ourselves thatZD� j�� j2e��� d2nz <1 (4.8)Namely, kGD(�;w)kL1(D) � C for all w 2 D, C > 0 a onstant, hen e j�V� j � C��2n�1on D� . In onjun tion with (4.6) and (4.7) the inequality (4.8) therefore follows.As a next step the Levi form of �� an be estimated from below byL�� � eV��V� �V� � e�2S�V� �V� on supp�� (4.9)If we write Q = (Qjk) for the inverse of the oeÆ ient matrix of L�� , we obtain usingat �rst (4.9) and then (4.7) the ru ial estimateRDPnj;k=1Qjk(z)��;j(z)��;k(z)e��� (z) d2nz � e2SRsupp�� j�0 Æ V� j2jf(z)j2e��� (z) d2nz� 4S�2e(4n+6)Skfk2AS(w;D)This shows, that the methods of the proof of Lemma 4.4.1 from [14℄ apply to thepseudo onvex open set D� for small � > 0. We obtain a solution u� 2 C1(D� ) of theequation �u� = �� satisfying the estimateZD� ju� j2e��� d2nz � 4S�2e(4n+6)SZsupp�� jf j2 d2nz � 4S�2e(4n+6)Skfk2AS(w;D) (4.10)This shows, in parti ular, that the Alaoglu-Bourbaki theorem applies to the familyof L2-fun tions de�ned by v� := � e���=2u� on D�0 on D nD�

14 Klas Diederi h and Gregor HerbortHen e there is a sequen e (�k)k of parameters onverging to 0 and a fun tion v 2L2(D), su h that ZD jvj2 d2nz � 4S�2e(4n+6)Skfk2AS(w;D)and vk ! v in the weak-�-topology on L2(D) as k !1.We, now, laim, that the fun tion u := e�=2v has the following properties:i) The L2-estimateZD juj2 d2nz � e1+R2ZD juj2e�� d2nz � 4S�2e(4n+6)S+R2kfk2AS(w;D)holds.ii) There is a holomorphi fun tion ~f on D with ~f = � Æ GD(�; w) � f � u almosteverywhere.iii) One has ~f (w) = f(w), � ~f(w) = �f(w).Namely, the �rst laim follows from the de�nition of v together with (4.6). In orderto prove the se ond assertion we put ~f1 := � ÆGD(�; w) � f �u. Sin e u 2 L2(D), so is~f1. It, therefore, suÆ es to verify � ~f1 = 0 on Dnfwg in the distributional sense. This an be done exa tly as in the proof of Proposition 3.6 in [12℄. For the onvenien eof the reader we mention the essential steps also here. For this we denote by h�; �ithe inner produ t on L2(D) resp. on (p; q)-forms with oeÆ ients in this spa e. We hoose a smooth (0; 1)-form � with ompa t support in D n fwg and al ulate forlarge k with # meaning the formal adjoint of �he�=2v�k; #�i = he12��k v�k; e12(����k )#�i= he12��k v�k; #�i+ he12��kv�k ;�e12(����k ) � 1�#�i= h��k ; �i+ hu�k ;�e12(����k ) � 1�#�i= h� Æ V�k � f; #�i+ hu�k ;�e12 (����k ) � 1�#�i (4.11)From (4.10) we get ku�kk2D�k � 4S�2e(4n+6)Skfk2uniformly in k and Beppo-Levi gives k(exp(12(� � ��k)) � 1)#�kD ! 0 as k ! 1.Altogether the Cau hy-S hwarz inequality, therefore, gives as estimate for the se ondterm of the right-hand side of (4.11)hu�k ;�e12(����k ) � 1�#�i ! 0 for k !1This yields h ~f1; #�i = h� ÆGD(�; w)f; #�i � hu; #�i= limk!1h� Æ V�k � f; #�i � limk!1he�=2v�k ; #�i= 0

Pluri omplex Green fun tion 15This proves ii).The equalities iii) follow immediately from i). Namely, on an open neighborhood Uof w one has GD(�; w) � �3S, hen e, f � ~f = u almost everywhere on U . From i) weget ZU ���f � ~f���2e�� d2nz <1Therefore and be ause of the fa tor (2n + 2) in the de�nition of �, the holomorphi fun tion f � ~f must vanish to at least se ond order at w. This �nishes the proof ofLemma 4.2.5 The boundary behavior of the sublevel sets ofthe Green fun tionThe next essential tool needed for the proof of Theorem 1.8 is the onvergen e prop-erty limw!w0 diam(A(w;D)) = 0 (5.1)for the sublevel sets of the pluri omplex Green fun tion for any point w0 2 �D. Inorder to formulate the exa t hypothesis whi h allows us to show (5.1), we remind thereader of the following de�nition from [13℄De�nition 5.1 We say that a bounded domain D �� Cn satis�es property (P�) ata point w0 2 �D if for any ompa t set K � Dlimw!w0supz2KjGD(z;w)j = 0Remark 5.2 Theorem 1.2 as stated in the Introdu tion says, that any domainD ��Cn admitting a bounded uniformly H�older ontinuous plurisubharmoni exhaustionfun tion satis�es property (P�) at any of its boundary points.We will show in this se tionProposition 5.3 LetD �� Cn be a domain satisfying property (P�) at every bound-ary point. Assume, furthermore, that any point w0 2 �D is a peak point for thefamily P(D) := C0(D)\PSH(D) of ontinuous fun tions on D plurisubharmoni onD. Then one has for all w0 2 �Dlimw!w0 diam(A(w;D)) = 0As a �rst step in the proof of this proposition we will show, that the sublevel setsA(w;D) annot ome arbitrarily lose to any point of �D n fw0g as w! w0. For thiswe will need the hypothesis on erning plurisubharmoni peak fun tions, but not yetproperty (P�). We laim:

16 Klas Diederi h and Gregor HerbortLemma 5.4 Let D �� Cn be a pseudo onvex domain, su h that any point w0 2 �Dis a peak point for the family P(D) := C0(D) \ PSH(D). If then K1 � �D is ompa t and L � D is a set with L \K1 = ;, then, for every s > 0, there is an openneighborhood Us � K1 with U s \ L = ; and su h thatGD(z;w) > �sfor all (z;w) 2 (Us \D) � L.Proof of Lemma 5.4: We put Æ := 14 dist(K1; L). Let q 2 K1 be arbitrary and 2 P(D) a plurisubharmoni fun tion on D, ontinuous on D, with (q) = 0 and (z) < 0 for z 2 D n fqg. We then de�ne for any w 2 L with the onstantCq := 2 log 2RD=Æminjz�qj�3Æj (z)jthe fun tion vw(z) :=8<:maxnCq (z); log jz�wj2RD o for jz � wj � Ælog jz�wj2RD for jz � wj < ÆNoti e, that vw is well-de�ned, sin e for z 2 D with jz � wj = Æ it follows Cq (z) �log jz�wj2RD . Namely, for su h z we get jz � qj � jw � qj � jz � wj � 4Æ � Æ = 3Æ and,hen e, Cq (z) = �Cqj (z)j � �Cq min�B(q;3Æ)\Dj j = 2 log Æ2RD < log jz � wj2RDThe fun tion vw, therefore, is for ea h w 2 L plurisubharmoni on D and, even, a andidate in the de�nition of GD(�; w). Hen e,GD(z;w) � Cq (z)if jz � wj > Æ and w 2 L. The fun tion being ontinuous, we an, after �xinga number s > 0, �nd an open neighborhood U sq of q, su h that U qs \ L = ; andCq (z) > �s on U qs \D. The laim of the lemma now follows from the ompa tnessof K1, namely, we an over K1 by �nitely many su h neighborhoods U qs for a �nitenumber of points q 2 K1. The union Us of these neighborhoods has the desiredproperty.We are now ready for theProof of Proposition 5.3: Let w0 2 �D. It suÆ es to prove:Claim: For any numberm > 0 there is a number Æ > 0, su h that A(w;D) � B(w0; 1m)for all w 2 D with jw �w0j < Æ.

Pluri omplex Green fun tion 17We apply Lemma 5.4 to the ompa t setK1 := �D \ �z 2 D : jz �w0j � 1m�and the set L := D \ B�w0; 12m�with s = 14, obtaining an open neighborhood Um of K1 su h thatA(w;D) \ Um = ; for w 2 LThe set K2 := �z 2 D : jz � w0j � 1m; z 62 Um�is a ompa t subset of D. By property (P�) one haslimw!w0 supz2K2jGD(z;w)j = 0meaning, that for small enough Æ0 > 0A(w;D) \K2 = ; for w 2 D \B(w0; Æ0)Hen e, we obtain for all w 2 D with jw � w0j < Æ := minfÆ0; 12mgA(w;D) \ �D nB�w0; 1m�� = (A(w;D) \ Um) [ (A(w;D) \K2) = ;Sin e this implies A(w;D) � D \B(w0; 1m ) for su h points, Proposition 5.3 has beenproved.6 Proof of Theorem 1.8This is now very brief. Namely, take (w;X) 2 D �Cn. Proposition 4.1 says thatBD(w;X) � C1BA(w;D)(w;X)with some onstant C1 > 0 independent of w. Furthermore, a trivial argument yieldsBA(w;D)(w;X) � jXjdiam(A(w;D)) (6.1)Sin e, however, Proposition 5.3 applies in the given situation, the right-hand sidein (6.1) goes to in�nity as w goes to w0. This implies the Theorem.

18 Klas Diederi h and Gregor Herbort7 Proof of Theorem 1.13Again, this proof is, of ourse, based on the �-te hnique for the onstru tion ofgood andidates of holomorphi L2-fun tions ontributing to the variational problemswhi h give KD(w) and bD(w;X) (see (4.2) and (4.3)). The main tool obtained bythis method is ontained in the following lemma whi h is a suitable modi� ation ofLemma 4.2.Lemma 7.1 Let ~D �� Cn be a pseudo onvex domain. Fix a point w 2 ~D and aunit ve torX 2 Cn. Assume, we are given a negative plurisubharmoni fun tion V (w)on ~D, su h that on an open neighborhood of w the fun tion z ! V (w)(z)� log jz�wjis bounded from above, and denote for any number S > 0AS�w; V (w)� := �z : V (w)(z) < �S gThen we an �nd for every fun tion f 2 H2(AS(w; V (w))) a fun tion ~f 2 H2( ~D) withthe following properties:i) ~f(w) = f(w)ii) � ~f(w) = �f(w)iii) k ~fk2~D � (1 + 4S�2e(4n+6)S+1+R2~D)kfk2AS(w;V (w)), where eRD := max eD jzj.Be ause of the omplete analogy to the proof of Lemma 4.2 we do not prove thislemma here.We now ome to the proof of Theorem 1.13.Let 0 < t� 1 be arbitrarily small and hoose � � 2 su h that D is pseudo onvexlyextendable of order � at a given boundary point w0. Let � be a orresponding ex-tending fun tion de�ned on an open neighborhood U of w0. More pre isely, supposethat � satis�es �C�ÆD(z) + ��z � w0��2� � �(z) � ���z � w0��� (7.1)on D \ U .It suÆ es to show:Claim: One has hD�w0� � 1� 2nt� (7.2)Let us next hoose a pseudo onvex domain D1 � U \ f� < 0g with C3-smoothboundary and the following properties:1) There exists an open neighborhood U1 �� U of w0 su h that�D1 \ U1 = f� = 0g \ U12) D \ U1 � D1 \ U13) On U1 the inequality j�j � C 0ÆD1 holds with some onstant C 0.

Pluri omplex Green fun tion 19Be ause of the well-known lo alization property of the Bergman metri it suÆ es todo all work on the domain ~D := D \ U1instead of D.Let, now, U2 �� U1 be another open neighborhood of w0 and hoose w 2 D \ U2arbitrarily. We de�ne for any unit ve tor X 2 Cn the fun tionf(z) := hz � w;XiK ~D(z;w)pK ~D(w)Lemma 7.1 with this domain ~D, the fun tion V (w) := GD1(�; w) and S = 1 applied tothis fun tion f yields a fun tion ~f 2 H2(D) su h thatb ~D(w;X) � ���h� ~f(w);Xi��� ~f ~D � C� pK ~D(w)kfkA(w;D1)\ ~Dand, hen e, B ~D(w;X) � C� 1kfkA(w;D1)\ ~D (7.3)with an unimportant onstant C�.It, therefore, remains to estimate the norm kfkA(w;D1)\ ~D.For this we noti e, that on ~D � U \D��z � w0�� � j�(z)j1=� � C 0Æ1=�D1 (z)and, hen e, jf(z)j � jz � wj jK ~D(z;w)jpK ~D(w)� (jz � w0j+ jw � w0j) jK ~D(z;w)jpK ~D(w)� �C 0Æ1=�D1 (z) + jw � w0j� jK ~D(z;w)jpK ~D(w) (7.4)By Lemma 1.6 we have with some number Æt � 1 for all w 2 D1 with ÆD1(w) < Ætthe estimate A(w;D1) � �ÆD1 < Æ1�2ntD1 (w)Inserting this into (7.4) we see that, on ~D \A(w;D1) one even hasjf(z)j � C 00�Æ1�2nt=�D1 (w) + ��w � w0���jK ~D(z;w)jpK ~D(w)if ÆD1(w) < Æt. By integration we obtain for su h points wkfkA(w;D1)\ ~D � ~C�Æ1�2nt=�D1 (w) + ��w � w0���

20 Klas Diederi h and Gregor HerbortFinally, we observe that, on a one within ~D with vertex at w0, we an estimateÆ1�2nt=�D1 (w) + ��w � w0�� � CÆ1�2nt=�~D (w)This, together with (7.3) provesB ~D(w;X) � C�� 1Æ1�2nt=�~D (w)for all w suÆ iently lose to w0 and inside a one within ~D with vertex at w0. Thisproves the Claim (7.2) and, therefore, Theorem 1.13.Referen es1. Blo ki, Z.: Estimates for the omplex Monge-Amp�ere operator, Bull. Pol. A ad.S i. 41 (1993), 151{157.2. Blo ki, Z., P ug, P.: Hyper onvexity and Bergman ompleteness, Nagoya Math.J. 151 (1998), 221{225.3. Carlehed, M.: Comparison of the pluri omplex and the lassi al Green fun tions,Mi higan Math. J. 45 (1998), 399{407.4. Carlehed, M., Cegrell, U., Wikstr�om, F.: Jensen measures, hyper onvexity, andboundary behavior of the pluri omplex Green's fun tion, Ann. Pol. Math. 71(1999), 87{103.5. Demailly, J. P.: Mesures de Monge-Amp�ere et mesures pluriharmoniques, Math.Z. 194 (1987), 519{564.6. Diederi h, K., Forn�ss, J. E.: Pseudo onvex domains: Bounded stri tly plurisub-harmoni exhaustion fun tions, Invent. Math. 39 (1977), 129{141.7. Diederi h, K., Forn�ss, J. E.: Pseudo onvex domains with real analyti boundary,Ann. Math. 107 (1978), 371{384.8. Diederi h, K., Herbort, G.: Geometri and analyti boundary invariants on pseu-do onvex domains . Comparison results, J. Geom. Analysis 3 (1993), 237{267.9. Diederi h, K., Herbort, G.: Pseudo onvex domains of semiregular type, Contri-butions to Complex Analysis (H. Skoda, J. M. Tr�epreau, ed.), Aspe ts of Math-emati s, vol. E 26, Vieweg- Verlag, 1994, pp. 127{162.10. Diederi h,K. , Ohsawa, T.: An estimate on the Bergman distan e on pseudo onvexdomains, Ann. of Math. 141 (1995), 181{190

Pluri omplex Green fun tion 2111. Graham, I.: Boundary behavior of the Caratheodory and Kobayashi metri s onstrongly pseudo onvex domains in Cn with smooth boundary, Trans. Amer. Math.So . 207 (1975), 219{240.12. Herbort, G.: The Bergman metri on hyper onvex domains, Math. Zeit. 232(1999), 183{196.13. Herbort, G.: On the pluri omplex Green fun tion on pseudo onvex domains witha smooth boundary, Interntl. J. of Math. (To appear).14. H�ormander, L.: An introdu tion to omplex analysis in several variables, NorthHolland, Amsterdam, 2nd ed. ed., 1973.15. Klimek, M.: Extremal plurisubharmoni fun tions and invariant pseudodistan es,Bull. So . Math. Fran e 113 (1985), 231{240.Klas Diederi h and Gregor HerbortMathematikUniversit�at WuppertalGausstr. 20D-42095 Wuppertal GERMANYklas�math.uni-wuppertal.de and gregor�math.uni-wuppertal.de