The Patterson-Sullivan Interpolation of Pluriharmonic Functions for Determinantal Point Processes on Complex Hyperbolic Spaces
aa r X i v : . [ m a t h . C V ] J a n THE PATTERSON-SULLIVAN INTERPOLATION OFPLURIHARMONIC FUNCTIONS FOR DETERMINANTAL POINTPROCESSES ON COMPLEX HYPERBOLIC SPACES
ALEXANDER I. BUFETOV AND YANQI QIU
To the memory of Alexander Ivanovich Balabanov (1952 – 2018)
Abstract.
The Patterson-Sullivan construction is proved almost surely to recover aBergman function from its values on a random discrete subset sampled with the deter-minantal point process induced by the Bergman kernel on the unit ball D d in C d . Forsuper-critical weighted Bergman spaces, the interpolation is uniform when the functionsrange over the unit ball of the weighted Bergman space. As main results, we obtaina necessary and sufficient condition for interpolation of a fixed pluriharmonic functionin the complex hyperbolic space of arbitrary dimension (cf. Theorem 1.4 and Theo-rem 4.11); optimal simultaneous uniform interpolation for weighted Bergman spaces (cf.Theorem 1.8, Proposition 1.9 and Theorem 4.13); strong simultaneous uniform interpo-lation for weighted harmonic Hardy spaces (cf. Theorem 1.11 and Theorem 4.15); andestablish the impossibility of the uniform simultaneous interpolation for the Bergmanspace A ( D d ) on D d (cf. Theorem 1.12 and Theorem 6.7). Introduction
Consider the unit ball D d in the d -dimensional complex Euclidean space C d and theBergman space A ( D d ) of square Lebesgue integrable holomorphic functions on D d . Itis proved in [8] that almost any realization of the determinantal point process on D d induced by the Bergman kernel is a uniqueness set for A ( D d ), which, however, is far frombeing a sampling set for A ( D d ). This paper is devoted to the explicit interpolation ofholomorphic, pluriharmonic and M -harmonic functions on D d from their restrictions ontoa typical subset of D d sampled with respect to the determinantal point process inducedby the Bergman kernel (see § Patterson-Sullivan construction (cf. Patterson [22] and Sullivan [35]) inour setting of determinantal point processes and establish • a necessary and sufficient condition for interpolation of a fixed pluriharmonic func-tion in the complex hyperbolic space of arbitrary dimension (cf. Theorem 1.4 andTheorem 4.11); • optimal simultaneous uniform interpolation for weighted Bergman spaces (cf. The-orem 1.8, Proposition 1.9 and Theorem 4.13): more precisely, we obtain an ex-plicit critical weight such that the simultaneous uniform interpolation holds forany Bergman space with a super-critical weight but fail for the Bergman spacewith the critical weight; Mathematics Subject Classification.
Primary 60G55; Secondary 37D40, 32A36.
Key words and phrases.
Patterson-Sullivan construction, point processes, interpolation of harmonicfunctions, weighted Bergman spaces, complex hyperbolic spaces. • the impossibility of the uniform simultaneous linear interpolation for the Bergmanspace A ( D d ) (cf. Theorem 1.12 and Theorem 6.7). The proof of the impossibilityrelies on a new identity (1.12) or inequality (6.134), and on universal lower bounds(1.13) and (6.133) on the variance of Bergman kernel-valued linear statistics of ourdeterminantal point processes.Our interpolation formulae can be viewed as discrete version mean-value properties forsquare-integrable M -harmonic or pluriharmonic functions on D d using their “boundaryvalues” which do not exist in the usual non-tangential limit sense (although the “boundaryvalues” in the distributional sense, see (1.4) below, does exist, they however do not seemto be relevant here).The reconstruction of Bergman functions from their restrictions onto a discrete samplingset has been extensively studied in the well-developed theory of interpolation and samplingin Bergman spaces (cf. [28, 29, 30, 4, 6]). Nonetheless, we are not aware of previous workon the reconstruction of Bergman functions from their restrictions onto a uniqueness setthat is not sampling.Generalization of our formalism in metric measure spaces (including in particular realand quaternion hyperbolic spaces and more general hyperbolic spaces) and more generalrandom point processes is given in the sequel to this paper.In the remaining part of the introduction, we illustrate our main results in the case ofdimension d = 1.1.1. The Bergman kernel and Patterson-Sullivan interpolation.
Let dA ( z ) bethe normalized Lebesgue measure on the open unit disk D ⊂ C . Let K D ( z, w ) denote theBergman kernel on the disk D and P K D the determinantal point process induced by K D .In the spirit of the Patterson-Sullivan’s construction, for a locally finite subset X ⊂ D and any s >
1, any z ∈ D , we define the Poincar´e series as (following Sullivan [35], wedenote the Poincar´e series by the letter g ): g X ( s, z ) := X x ∈ X e − sd D ( x,z ) ∈ [0 , ∞ ] , (1.1)where d D ( · , · ) is the Poincar´e metric on the disk D viewed as the Lobachevsky plane, seethe precise formula (1.15) for d D ( · , · ). Given a harmonic function f on D , we shall alsoneed the following definition: g X ( s, z ; f ) := ∞ X k =0 X x ∈ Xk ≤ d D ( x,z ) 1, we have the equality g X ( s, z ) = g X ( s, z ; 1) . Informally, we shall obtain equalities of the following type: f ( z ) = lim s → + g X ( s, z ; f ) g X ( s, z ) , (1.3)for P K D -almost every X ⊂ D and Bergman functions f in A ( D ) and indeed beyond, cf.Theorem 1.4. ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 3 The right hand side of (1.3) does not depend on the values of f on any finite subset of X , it depends only on the boundary behaviour of the function f restricted on X . In thissense, the formula (1.3) should be viewed as a discrete mean-value property of f usingthe “boundary values” of f which may not exist in the usual non-tangential limit sense.If f is a harmonic function on D which is continuous upto the closed unit disk, then itadmits a radial limit on the boundary T = ∂ D and the limit equality (1.3) for P K D -almostevery X ⊂ D can be obtained from a statement of the weak convergence of probabilitymeasures. For a general Bergman function f ∈ A ( D ), the main obstacle for us is, f neednot have a radial limit on the boundary of the unit disk and may have rather complicatedbehaviour near the boundary. Note that, for a general f ∈ A ( D ), we havelim r → − k f r − f k H / ( T ) = 0(1.4)where H / ( T ) is the Sobolev space on T . However, the convergence (1.4) is not strongenough to be applied in our situation.The informal descriptions of our main results are as follows.1. For a harmonic function f : D → C , Theorem 1.4 says thatlim s → + E P K D (cid:16)(cid:12)(cid:12)(cid:12) g X ( s, z ; f ) E P K D ( g X ( s, z )) − f ( z ) (cid:12)(cid:12)(cid:12) (cid:17) = 0(1.5) holds for a fixed point z ∈ D if and only if f satisfies a tempered growth condition :lim α → + α Z D | f ( x ) | (1 − | x | ) α dA ( x ) = 0 . (1.6) Moreover, under the condition (1.6), for any relatively compact subset D ⊂ D , wehave lim s → + sup z ∈ D E P K D (cid:16)(cid:12)(cid:12)(cid:12) g X ( s, z ; f ) E P K D ( g X ( s, z )) − f ( z ) (cid:12)(cid:12)(cid:12) (cid:17) = 0 . Note that while all Bergman functions in A ( D ) satisfy the tempered growth con-dition, the converse is not true. The proof of Theorem 1.4 relies on upper andlower estimates of the variance of g X ( s, z ; f ) (cf. Proposition 1.5). The lowerestimate of the variance of g X ( s, z ; f ) (cf. Proposition 4.10) will also play a rolein the following results.2. Theorem 1.8 gives optimal simultaneous uniform interpolation for weighted Bergmanspaces: for any relatively compact subset D ⊂ D ,lim s → + sup z ∈ D E P K D (cid:16) sup f ∈B ( W ) (cid:12)(cid:12)(cid:12) g X ( s, z ; f ) E P K D ( g X ( s, z )) − f ( z ) (cid:12)(cid:12)(cid:12) (cid:17) = 0 , provided that W ∈ L ( D , dA ) is any non-negative function withlim | z |→ − W ( z )(1 − | z | ) − [log( −| z | )] − = ∞ (1.7) and B ( W ) is the unit ball of the weighted Bergman space with weight W : B ( W ) := n f : D → C (cid:12)(cid:12)(cid:12) f is holomorphic and Z D | f ( x ) | W ( x ) dA ( x ) < o . ALEXANDER I. BUFETOV AND YANQI QIU A weight W satisfying (1.7) will be called supercritical . Our result in theweighted Bergman spaces is optimal: while the simultaneous uniform interpola-tion holds for all super-critical weights, it fails (see Proposition 1.9) for the criticalweight itself: W cr ( z ) = 1(1 − | z | ) log (cid:0) −| z | (cid:1) for all z ∈ D . The proof of the optimality relies on a lower estimate of the variance of vector-valued linear statistics: there exists a numerical constant c > 0, such that for anyHilbert space vector-valued harmonic function F : D → H with sub-exponentialmean-growth (see (1.22) for the precise meaning), for all s ∈ (1 , 2] and all z ∈ D ,we have Var P K D ( g X ( s, z ; F )) ≥ c Z D e − sd D ( x,z ) k F ( x ) k dA ( x )(1 − | x | ) . The following estimate (cf. Lemma 6.5) is also used: there exists a constant c > K W cr ( x, y ) of the weighted Bergman space A ( D , W cr ) satisfies Z π K W cr ( xe iθ , y ) | K D ( xe iθ , y ) | dθ π ≥ c (1 − | xy | ) log (cid:16) − | xy | (cid:17) (1.8) for all x, y ∈ D . We note that the precise formula for K W cr ( x, y ) is not known andthe proof of the lower estimate (1.8) does not follow from the diagonal-asymptotics(cf. Lemma 5.1) of the weighted Bergman kernel K W cr .3. Theorem 1.11 gives strong simultaneous uniform interpolation for weighted har-monic Hardy spaces: Let µ be any Borel probability measure on T and set H ( D ; µ ) := n f : D → C (cid:12)(cid:12)(cid:12) f ( z ) = P [ hµ ] := Z T − | z | | − ¯ ζz | h ( ζ ) dµ ( ζ ) , h ∈ L ( T , µ ) o . Then for P K D -almost every X , simultaneously for all f ∈ H ( D ; µ ), all s > z ∈ D , the series (1.2) converges and we have f ( z ) = lim s → + g X ( s, z ; f ) g X ( s, z ) , (1.9) where the convergence is uniform for z ranges over any compact subset of D andfor f ranges over the unit ball of H ( D ; µ ): H ( D ; µ ) := { f = P [ hµ ] : k h k L ( µ ) ≤ } . Note that if µ is not absolutely continuous with respect to the Lebesgue measureon T , then a function in H ( D , µ ) may have no radial limit on the boundary T = ∂ D . The proof of Theorem 1.11 relies on (i) a concept of sharply temperedgrowth condition (cf. Definition 4.2 and Lemma 4.19) involving the followingestimate: there exists a constant C > ε > Z D Z T (1 − | w | ) ε P ( w, ζ ) dµ ( ζ ) dA ( w ) ≤ Cε ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 5 and on (ii) the proof of the following inequality (cf. Lemma 3.2 and the inequalities(4.94) and (4.106)): for any relatively compact subset D ⊂ D , there exists aconstant C D > ε ∈ (0 , 1) and any s ∈ (1 , P K D (cid:16) sup f ∈H ( D ; µ ) sup z ∈ D (cid:12)(cid:12)(cid:12) g X ( s, z ; f ) E P K D ( g X ( s, z )) − f ( z ) (cid:12)(cid:12)(cid:12) > ε (cid:17) ≤ C D · s − ε . (1.10)4. In Theorem 1.12, we consider interpolation with general radial weights. Namely,instead of considering the weights e − sd D (0 ,z ) used in the classical Patterson-Sullivantheory, we consider all compactly supported radial weights R : D → R + and set g R X ( z ; f ) := X x ∈ X R ( ϕ z ( x )) f ( x ) , g R X ( z ) := X x ∈ X R ( ϕ z ( x )) where ϕ z ( x ) = x − z − ¯ zx . (1.11) Here we require the compact support assumption on the weights for avoiding theconvergences issue, see Remark 1.2 below. We study the uniform simultaneousinterpolation of all functions in A ( D ) and obtain a universal positive lower boundfor the variance of the associated Bergman kernel-valued linear statistics. There-fore, we show that, in this very general setting, for the determinantal point process P K D , any uniform linear interpolation of A ( D ) is not possible in L -sense. Theproof of the impossibility of the uniform linear interpolation of A ( D ) relies on aprecise formula (cf. Proposition 6.6) for the variance of the following linear statis-tics: for any z ∈ D , any bounded compactly supported radial weight R : D → R + , E P K D " sup f ∈ A ( D ): k f k≤ (cid:12)(cid:12)(cid:12) g R X ( z, f ) − f ( z ) E P K D ( g R X ( z )) (cid:12)(cid:12)(cid:12) = 12(1 − | z | ) Z D Z D |R ( x ) − R ( y ) | (1 − | xy | ) · I z ( x, y ) dA ( x ) dA ( y ) , (1.12) where I z ( x, y ) is given by the formula: I z ( x, y ) = 1 + (3 + 8 | z | ) | xy | + (3 | z | + 8 | z | ) | xy | + | z | | xy | . The proof of the identity (1.12) relies on a reduction formula (see Lemma 6.2)for the variance of the Bergman kernel-valued linear statistics of P K D : for anybounded compactly supported radial function R : D → R + , we haveVar P K D ( g R X ( z ; F D )) = 12 Z D Z D |R ( ϕ z ( x )) − R ( ϕ z ( y )) | K D ( x, y ) | K D ( x, y ) | dA ( x ) dA ( y ) , where F D : D → A ( D ) is the Bergman kernel-valued function defined by F D ( w ) := K D ( · , w ) ∈ A ( D ) . In particular, for obtaining the above reduction formula, we will use the followinggeneneral identity (see the proof of Lemma 6.3) for any weight W on D inducinga weighted Bergman space A ( D , W ): Z D K W ( w, z ) | K D ( z, w ) | dA ( w ) = K W ( z, z ) K D ( z, z ) , where K W ( · , · ) is the reproducing kernel of A ( D , W ). ALEXANDER I. BUFETOV AND YANQI QIU The right hand side double integral in (1.12) seems to related to certain Sobolev-type norm of R . From (1.12), we derive the following universal lower boundshowing the impossibility of the uniform linear interpolation of A ( D ): for any z ∈ D , we haveinf R E P K D " sup f ∈ A ( D ): k f k≤ (cid:12)(cid:12)(cid:12) g R X ( z, f ) E P K D ( g R X ( z )) − f ( z ) (cid:12)(cid:12)(cid:12) ≥ , (1.13) where R ranges over all compactly supported radial weights on D . Remark . The reader may notice that sometimes it is more convenient for us to workwith the normalization g X ( s, z ; f ) / E P K D ( g X ( s, z )) than to work with the normalization g X ( s, z ; f ) /g X ( s, z ). We will show (cf. Theorem 1.11 applied to the constant function f ≡ 1) that, for P K D -almost every configuration X ∈ Conf( D ), the following limit equalitylim s → + sup z ∈ D (cid:12)(cid:12)(cid:12) g X ( s, z ) E P K D ( g X ( s, z )) − (cid:12)(cid:12)(cid:12) = 0holds for all relatively compact subsets D ⊂ D . Therefore, for dealing with the P K D -almostsure convergence, we can use equally both normalizations g X ( s, z ; f ) / E P K D ( g X ( s, z )) and g X ( s, z ; f ) /g X ( s, z ). For instance, the following almost sure limit equalities are equivalent: f ( z ) = lim s → + g X ( s, z ; f ) g X ( s, z ) and f ( z ) = lim s → + g X ( s, z ; f ) E P K D ( g X ( s, z )) . However, for dealing with the L -convergence, the normalization g X ( s, z ; f ) / E P K D ( g X ( s, z ))has its advantage since the following equality E P K D (cid:16)(cid:12)(cid:12)(cid:12) g X ( s, z ; f ) E P K D ( g X ( s, z )) − f ( z ) (cid:12)(cid:12)(cid:12) (cid:17) = Var P K D ( g X ( s, z ; f ))[ E P K D ( g X ( s, z ))] does not have a counterpart for the normalization g X ( s, z ; f ) /g X ( s, z ). In fact, it is notclear to us whether the limit equality (1.5) has the following analogue:lim s → + E P K D (cid:16)(cid:12)(cid:12)(cid:12) g X ( s, z ; f ) g X ( s, z ) − f ( z ) (cid:12)(cid:12)(cid:12) (cid:17) = 0 . Remark . For the classical Bergman space A ( D ), the simultaneous uniform interpo-lation of all f ∈ A ( D ) is much more harder, in fact, for P K D -almost every X , we have the series g X ( s, z ; f ) does not converge simulataneous for all f ∈ A ( D ) . See the statements in (1.31) below and Proposition 1.13 for more details. Remark . In the case of weighted Bergman space, it is not clear whether there is asimilar inequality as the inequality (1.10) for a general fixed f ∈ A ( D , W ). Remark . The choice of the weight e − sd D ( x,z ) (as in the classical Patterson-Sullivantheory) has its own advantage, see the proofs of Theorem 4.15 and Theorem 4.18.Note that the standard procedure in sampling theory of the reconstruction of Bergmanfunctions from their restrictions on a sampling set (see Seip [29, p. 53] or Duren [11, p.165]) is not applicable in our setting since it is proved in [8] that the determinantal pointprocess P K D almost surely gives rise to a uniqueness but non-sampling set for A ( D ). ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 7 All the previous reconstruction results for functions on D have their higher dimen-sional counterparts. Note that in particular, in higher dimension, we shall distinguish the M -harmonicity and pluriharmonicity: pluriharmonicity implies M -harmonicity but notconversely. Main results: dimension d = 1 Main results: dimension d ≥ harmonic func-tion on D , the necessary and sufficientcondition for the Patterson-Sullivan in-terpolation is the tempered growth con-dition, cf. (1.21). Theorem 4.11: for a fixed pluriharmonic function on D d , the necessary and suffi-cient condition for the Patterson-Sullivaninterpolation is the tempered growthcondition, cf. Definition 4.1.Theorem 1.6: a fixed harmonic functionon D with tempered growth satisfies thePatterson-Sullivan interpolation. Theorem 4.8: a fixed M -harmonic func-tion on D d with tempered growth satisfiesthe Patterson-Sullivan interpolation.Theorem 1.8: optimal simultaneous uni-form Patterson-Sullivan interpolation forweighted Bergman spaces on D withsuper-critical weights. Theorem 4.13: optimal simultaneousuniform Patterson-Sullivan interpolationfor weighted Bergman spaces on D d withsuper-critical weights.Theorem 1.11: strong simultaneous uni-form Patterson-Sullivan interpolation forweighted harmonic Hardy spaces on D . Theorem 4.15: strong simultaneous uni-form Patterson-Sullivan interpolation forweighted harmonic Hardy spaces on D d .Theorem 1.12: impossibility of the uni-form linear interpolation of A ( D ). Theorem 6.7: impossibility of the uni-form linear interpolation of A ( D d ).1.2. The reconstruction problem. Let dA ( z ) = i π d ¯ z ∧ dz be the normalized Lebesguemeasure on the open unit disk D ⊂ C and consider the Bergman space A ( D ) defined by A ( D ) := n f : D → C (cid:12)(cid:12)(cid:12) f is holomorphic and Z D | f ( z ) | dA ( z ) < ∞ o . The Hilbert space A ( D ) admits a reproducing kernel given by K D ( z, w ) = 1(1 − z ¯ w ) , z, w ∈ D . Let N be the set of non-negative integers and let ( a n ) n ∈ N be the sequence of independentcomplex Gaussian random variables with expectation 0 and variance 1. The random series S D ( z ) = ∞ X n =0 a n z n almost surely has radius of convergence 1 and thus defines a holomorphic function on D .Peres and Vir´ag [23] proved that the zero set Z ( S D ) ⊂ D of S D is the determinantal pointprocess on D induced by the Bergman kernel K D . More precisely, let Conf( D ) denote theset of (locally finite) configurations on D and let P K D be the determinantal probabilitymeasure on Conf( D ) induced by the kernel K D . The Peres-Vir´ag Theorem states that thedistribution of Z ( S D ) is given by P K D . The precise definitions of configurations, pointprocesses and determinantal point processes are recalled in § ALEXANDER I. BUFETOV AND YANQI QIU For P K D -almost every X ∈ Conf( D ), it is proved in [8] that any function f ∈ A ( D ),equal to 0 in restriction to X , must be the zero function; in other words, P K D -almostevery X is a uniqueness set for A ( D ) and consequently all functions f ∈ A ( D ) are simultaneously uniquely determined by their restrictions f | X . It is thus natural to ask Question A. How to recover a Bergman function f ∈ A ( D ) from its restriction to a P K D -typical configuration X ∈ Conf( D ) ? And how to recover simultaneously all Bergmanfunctions in A ( D ) from their restrictions to a P K D -typical configuration X ∈ Conf( D ) ?Remark . For a general uniquenss set of any reproducing kernel Hilbert space, theinterpolation using the Gram-Schmidt procedure is possible. More precisely, let H ( K )be a reproducing kernel Hilbert space with reproducing kernel K : S × S → C and let { x k } ∞ k =1 ⊂ S be a uniqueness set for H ( K ). Then the Gram-Schmidt procedure appliedto the sequence { K ( · , x k ) } ∞ k =1 yields an orthonormal basis { ϕ n } ∞ n =1 of H ( K ). Write ϕ n = n X k =1 c n,k K ( · , x k ) , where c n,k ∈ C , then for any f ∈ H ( K ), we have f = ∞ X n =1 ϕ n · h f, ϕ n i H ( K ) = ∞ X n =1 ϕ n n X k =1 c n,k f ( x k ) , (1.14)where the convergence is in norm. See [15, p. 135] for more details.Note however that ϕ n and c n,k are not explicit and moreover depend on the orderingof points in the uniqueness set, while the order of points is irrelevant to the uniquenessproperty. In the case of Bergman space A ( D ), the sequence { ϕ n } ∞ n =1 changes drasticallyif we remove or add a finite number of points from a uniqueness set for A ( D ), whileremoving or adding a finite number of points from a uniqueness set for A ( D ) does notviolate its uniqueness property.1.3. Patterson-Sullivan construction for point processes. We view D as the Poincar´emodel of the Lobachevsky plane. Recall that the Poincar´e metric on the Lobachevskyplane D is given by the following explicit formula: d D ( x, z ) := log (cid:16) | ϕ x ( z ) | − | ϕ x ( z ) | (cid:17) for x, z ∈ D , where ϕ x ( z ) = z − x − ¯ xz .(1.15)In the spirit of the Patterson-Sullivan construction (cf. Patterson [22] and Sullivan [35]),we consider, for P K D -almost every X ∈ Conf( D ), for any z ∈ D and s > 1, the probabilitymeasure (viewed as measure on the closed unit disk D ): ν X ( s, z ) := 1 g X ( s, z ) X x ∈ X e − sd D ( x,z ) δ x with g X ( s, z ) := X x ∈ X e − sd D ( x,z ) . (1.16)We can easily show (which is also an immediate consequence of Theorem 1.11 below) thatthe following weak convergence holds for P K D -almost every configuration X ∈ Conf( D ):lim s → + ν X ( s, z ) = P z for all z ∈ D ,(1.17)where P z is the harmonic measure on the unit circle T = ∂ D associated to z , that is, P z ( dζ ) = P ( z, ζ ) dm ( ζ ) with dm the normalized Lebesgue measure on T and P ( z, ζ ) the ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 9 Poisson kernel: P ( z, ζ ) = 1 − | z | | − ¯ ζz | = 1 − | z | | z − ζ | , z ∈ D and ζ ∈ T . (1.18) Remark . The exponent s = 1 is critical since for P K D -almost every X ∈ Conf( D ), wehave g X ( s, z ) < ∞ if and only if s > Remark . The Lobachevsky metric on D coincides with the Bergman metric on D defined using the Bergman kernel, cf. Krantz [17, Chapter 1].As an immediate consequence of (1.17), for P K D -almost every X ∈ Conf( D ), simulta-neously for all z ∈ D and all harmonic functions u : D → C that is continuous upto theclosed disk D , we can recover u ( z ) from the values of u | X as follows: u ( z ) = lim s → + Z D udν X ( s, z ) = lim s → + g X ( s, z ) X x ∈ X e − sd D ( x,z ) u ( z ) . (1.19)However, (1.19) does not hold for a general f ∈ A ( D ). The main reasons are:(1): A general f ∈ A ( D ) is not necessarily uniformly bounded on D and thus for P K D -almost every X ∈ Conf( D ), it is unclear whether the series X x ∈ X e − sd D ( x,z ) | f ( x ) | (1.20) is convergent or not when s > D need not be contin-uous upto the closed disk D , the formula (1.19) for a general uniformly boundedharmonic function u on D does not follow from the convergence (1.17).1.4. Patterson-Sullivan interpolation of a fixed harmonic function. For a fixedfunction, we will obtain interpolation formula for a harmonic function f : D → C belong-ing to the following class (which we will call the class of tempered harmonic functions ): T ( D ) := n f : D → C (cid:12)(cid:12)(cid:12) f is harmonic, lim α → + α Z D | f ( z ) | (1 − | z | ) α dA ( z ) = 0 o . (1.21)Note that while all harmonic functions with R D | f | dA < ∞ are tempered, a temperedharmonic function may satisfy R D | f | dA = ∞ . For the purpose of simultaneous interpo-lation in § T ( D ) of tempered harmonicfunctions (see Lemma 4.12 and Remark 4.2 below).Recall that a non-negative function Λ on R + (or on N ) is called sub-exponential iflim t → + ∞ Λ( t ) e − αt = 0 for any α > k ∈ N and z ∈ D , set A k ( z ) = { x ∈ D : k ≤ d D ( z, x ) < k + 1 } . We say that a harmonic function f : D → C has sub-exponential mean-growth if thereexists a sub-exponential function Λ : N → R + such that Z A k (0) | f ( w ) | dA ( w )(1 − | w | ) ≤ Λ( k ) e k for all k ∈ N .(1.22) Set S exp ( D ) := n f : D → C (cid:12)(cid:12)(cid:12) f is harmonic with sub-exponential mean-growth o . (1.23) Remark . For a harmonic function f : D → C , one can show that the condition (1.22)is equivalent to the following condition:12 π Z π | f ( re iθ ) | dθ ≤ − r · e Λ (cid:16) log 11 − r (cid:17) , for all r ∈ [0 , e Λ : R → R + is a sub-exponential function depends on f . Nonetheless, the condition(1.22) is more convenient for us and has a simpler counterpart in higher dimension case. Lemma 1.1. We have the inclusion: T ( D ) ⊂ S exp ( D ) . Since the series (1.20) need not be convergent for a general f ∈ A ( D ) or f ∈ T ( D ),we shall use summation over annuli to ensure convergence of our series: for any harmonicfunction f on D and any X ∈ Conf( D ), define g ( k ) X ( s, z ; f ) := X x ∈ X ∩ A k ( z ) e − sd D ( z,x ) f ( x ) . (1.24) Lemma 1.2. Assume that f ∈ S exp ( D ) . Then for any s > and any relatively compactsubset D ⊂ D , we have ∞ X k =0 sup z ∈ D n E P K D h | g ( k ) X ( s, z ; f ) | io / < ∞ . In particular, for P K D -almost every X ∈ Conf( D ) , we have ∞ X k =0 | g ( k ) X ( s, z ; f ) | < ∞ for Lebesgue almost every z ∈ D . (1.25) Remark . For a general f ∈ S exp ( D ), it is not clear whether (1.25) holds for all z ∈ D .This should be compared to the stronger result obtained in Lemma 1.10.From Lemma 1.2, fixing any f ∈ S exp ( D ), any z ∈ D and any s > 1, for P K D -almostevery X ∈ Conf( D ), we can define g X ( s, z ; f ) := ∞ X k =0 g ( k ) X ( s, z ; f ) = ∞ X k =0 X x ∈ X ∩ A k ( z ) e − sd D ( z,x ) f ( x ) . (1.26)The series g X ( s, z ; f ) should be viewed as a certain principal value of the linear statisticsfor the observable function e − sd D ( z,x ) f ( x ). Lemma 1.3. Assume that f ∈ S exp ( D ) . Then for any s > and any z ∈ D , we have E P K D ( g X ( s, z ; f )) = f ( z ) · E P K D ( g X ( s, z )) , where g X ( s, z ) is defined in (1.16) . Theorem 1.4. Assume that f ∈ S exp ( D ) . Then the limit equality lim s → + E P K D (cid:16)(cid:12)(cid:12)(cid:12) g X ( s, z ; f ) E P K D ( g X ( s, z )) − f ( z ) (cid:12)(cid:12)(cid:12) (cid:17) = 0(1.27) ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 11 holds for a fixed point z ∈ D if and only if f ∈ T ( D ) . Moreover, for any f ∈ T ( D ) , the convergence (1.27) holds locally uniformly on z ∈ D . The key ingredient in the proof of Theorem 1.4 is given in the following Proposition 1.5. For any z ∈ D , there exist two constants c ( z ) , c ( z ) > such that forany f ∈ S exp ( D ) and any s ∈ (1 , , we have c ( z ) ≤ Var P K D ( g X ( s, z ; f )) Z D | f ( w ) | (1 − | w | ) s − dA ( w ) ≤ c ( z ) . Remark . The idea behind Lemma 1.2 is the following: although a general f ∈ A ( D )or more generally a general f ∈ S exp ( D ) is not necessarily uniformly bounded on D , itsaverage over all circles C ( z, r ) = { w ∈ D : d D ( w, z ) = r } depends only on the centre z ∈ D and thus is bounded in r ∈ (0 , ∞ ): f ( z ) = 1 | C ( z, r ) | B Z C ( z,r ) f ( w ) ds B ( w ) , (1.28)where ds B is the length element of the Bergman metric and | C ( z, r ) | B is the length of thecircle C ( z, r ) under the Bergman metric. Lemma 1.2 implies that that for any f ∈ S exp ( D )and P K D -almost every X ∈ Conf( D ), there is enough cancellation inside the partial sum(1.24) in such a way that for all s > 1, we have ∞ X k =0 | g ( k ) X ( s, z ; f ) | < ∞ . Theorem 1.6. Fix f ∈ T ( D ) . Let ( s n ) n ≥ be a fixed sequence in (1 , ∞ ) converging to and satisfying ∞ X n =1 ( s n − Z D | f ( w ) | (1 − | w | ) s n − dA ( w ) < ∞ . Then for P K D -almost every X ∈ Conf( D ) , the equality Z D lim sup n →∞ (cid:12)(cid:12)(cid:12) g X ( s n , z ; f ) E P K D ( g X ( s n , z )) − f ( z ) (cid:12)(cid:12)(cid:12) dA ( z ) = 0 holds for any relatively compact subset D ⊂ D and moreover f ( z ) = lim n →∞ g X ( s n , z ; f ) g X ( s n , z ) for all Lebesgue almost every z ∈ D . (1.29) Remark . For a general f ∈ T ( D ), it is not clear whether the convergence (1.29) holdsfor all z ∈ D . This should be compared to Theorem 1.11 below where the convergence(1.29) for all z ∈ D is established for weighted harmonic Hary functions. Remark . In Lemma 1.2 and in Theorem 1.4, the class S exp ( D ) can be replaced bythe larger class b S ( D ) consisting of all harmonic functions such that ∞ X k =1 e − ( α +1) k (cid:16) Z π (cid:12)(cid:12) f ((1 − e − k ) e iθ ) (cid:12)(cid:12) dθ π (cid:17) / < ∞ for all α > , and one can show (the proof involves similar estimates as those in Proposition 1.5) thatfor a harmonic function f , the statement in Lemma 1.2 holds if and only if f belongs tothis larger class b S ( D ). Nonetheless, it is sufficient for us to work with this smaller andsimpler class S exp ( D ).1.5. Simultaneous Patterson-Sullivan interpolation for families of functions. Now we consider the simultaneous Patterson-Sullivan interpolation for families of holo-morphic or harmonic functions on D .1.5.1. Informal description of the simultaneous interpolation. Clearly the almost everystatement in Theorem 1.6 can be extended to any fixed countable dense family F ⊂ A ( D ). At the same time, for any 1 < s ≤ and any z ∈ D , we have (cf. Proposition 1.13)sup N ∈ N E P K D (cid:16) sup f ∈ A ( D ): k f k≤ (cid:12)(cid:12)(cid:12) N X k =0 g ( k ) X ( s, z ; f ) (cid:12)(cid:12)(cid:12) (cid:17) = ∞ . (1.30) Remark . In the sequel to this paper, the almost sure version of (1.30) is proved: P K D (cid:16)n X ∈ Conf( D ) (cid:12)(cid:12)(cid:12)X k g ( k ) X ( s, z ; f ) converges for all f ∈ A ( D ) o(cid:17) = 0 . (1.31)Hence for a fixed z ∈ D , once s > P K D -almostevery X ∈ Conf( D ), the normalization g X ( s, z ; f ) /g X ( s, z ) can not be simultaneouslydefined for all f ∈ A ( D ) and it is impossible to extend the our Patterson-Sullivan inter-polation (1.29) to the whole space A ( D ).The above discussions lead to the following considerations.I): Instead of considering the whole space A ( D ), we consider smaller families of func-tions inside A ( D ). The main results along this line are: 1) an optimal simultaneousand uniform interpolation for weighted Bergman spaces is obtained in Theorem 1.8; 2)a strong simultaneous and uniform interpolation for weighted harmonic Hardy spaces, isobtained in Theorem 1.11.II): By (1.31), there is an issue of defining g X ( s, z ; f ) simultaneously for all f ∈ A ( D ).For bypassing this issue, we replace e − sd D ( z,x ) = e − sd D (0 ,ϕ z ( x )) in the definition (1.26) of g X ( s, z ; f ) by R ( ϕ z ( x )) with R ∈ L ( D , dA ) ranges over the set of all non-negativebounded compactly supported and radial functions on D and define, for all X ∈ Conf( D ), g R X ( z ; f ) := X x ∈ X R ( ϕ z ( x )) f ( x ) and g R X ( z ) := X x ∈ X R ( ϕ z ( x )) . (1.32)Here the radial assumption is related to (1.28) and the compact support assumption isimposed on R to ensure the convergence of g R X ( z ; f ) for all configurations X ∈ Conf( D ).However, we obtain in Theorem 1.12 below that for any z ∈ D ,inf R E P K D h sup f ∈ A ( D ): k f k≤ (cid:12)(cid:12)(cid:12) g R X ( z, f ) g R P K D − f ( z ) (cid:12)(cid:12)(cid:12) i ≥ g R P K D := E P K D [ g R X ( z )] , where the infimum runs over all non-negative bounded compactly supported radial func-tions R on D (the fact that E P K D [ g R X ( z )] is independent of z follows from the conformalinvariance of P K D ). This demonstrates the impossiblity of the simultaneous Patterson-Sullivan interpolation for all functions in A ( D ). ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 13 Remark . One can show the following analogue of (1.29): for any fixed f ∈ A ( D ),there exists a sequence ( R n ) n ≥ of non-negative bounded compactly supported radialfunctions on D such that for P K D -almost every X , f ( z ) = lim n →∞ g R n X ( z ; f ) g R n X ( z ) for Lebesgue almost every z ∈ D . Indeed, we may take R n ( x ) = ( d D ( x, < r n ) for a certain sequence ( r n ) n ≥ of positivenumbers converging to infinity fast enough. Question: Is it possible to obtain an optimal result, analogous to Theorem 1.8, for thesimultaneous uniform interpolation of weighted Bergman spaces by using g R X ( z ; f ) ?1.5.2. Weighted Bergman spaces. The following weight is essential for us: W cr ( z ) = 1(1 − | z | ) log (cid:0) −| z | (cid:1) for all z ∈ D . (1.33)Here the subcript “cr” comes from the word “critical”. A function W ∈ L ( D , dA ) with W ( z ) ≥ super-critical weight if it satisfieslim | z |→ − W ( z ) W cr ( z ) = ∞ . Note that, although W cr is radial, super-critical weights need not be.The weighted Bergman space associated to a weight W is given by A ( D , W ) := n f : D → C (cid:12)(cid:12)(cid:12) f is holomorphic and k f k W := Z D | f ( z ) | W ( z ) dA ( z ) < ∞ o . Let B ( W ) denote the unit ball of A ( D , W ): B ( W ) := n f ∈ A ( D , W ) (cid:12)(cid:12)(cid:12) k f k W < o . Lemma 1.7. Let W be a weight on D , either equal to W cr or super-critical. Then for anyrelatively compact subset D ⊂ D and any s > , ∞ X k =0 sup z ∈ D n E P K D h sup f ∈B ( W ) | g ( k ) X ( s, z ; f ) | io / < ∞ . Fix s > , z ∈ D and a weight W which is equal to W cr or super-critical. Then for P K D -almost every X ∈ Conf( D ), Lemma 1.7 resolves the problem (discussed in § g X ( s, z ; f ) for all f ∈ A ( D , W ): g X ( s, z ; f ) = ∞ X k =0 g ( k ) X ( s, z ; f ) . (1.34) Theorem 1.8. Let W be a super-critical weight on D . Then for any relatively compactsubset D ⊂ D , lim s → + sup z ∈ D E P K D (cid:16) sup f ∈B ( W ) (cid:12)(cid:12)(cid:12) g X ( s, z ; f ) g P K D ( s ) − f ( z ) (cid:12)(cid:12)(cid:12) (cid:17) = 0 , (1.35) where g P K D ( s ) := E P K D [ g X ( s, z )] . In particular, there exists a sequence ( s n ) n ≥ in (1 , ∞ ) converging to such that for P K D -almost every X ∈ Conf( D ) , we have that for Lebesgue almost every z ∈ D , f ( z ) = lim n →∞ g X ( s n , z ; f ) g X ( s n , z ) simultaneously for all f ∈ A ( D , W ) ,where the convergence is uniform for f in the unit ball B ( W ) of A ( D , W ) . We shall see later in § s n ) n ≥ in Theorem 1.8 can be taken to beany sequence ( s n ) n ≥ in (1 , ∞ ) satisfying ∞ X n =1 ( s n − Z D (1 − | x | ) s n − K W ( x, x ) dA ( x ) < ∞ . Note that our condition depends only on the given super-critical weight W .For the critical weight W cr , we have the following Proposition 1.9. Take z = 0 , then we have lim inf s → + E P K D (cid:16) sup f ∈B ( W cr ) (cid:12)(cid:12)(cid:12) g X ( s, z ; f ) g P K D ( s ) − f ( z ) (cid:12)(cid:12)(cid:12) (cid:17) > . (1.36)As we mentioned before, in the proof of Proposition 1.9, we use the following estimate(cf. Lemma 6.5): there exists a constant c > K W cr ( x, y )of the weighted Bergman space A ( D , W cr ) satisfies Z π K W cr ( xe iθ , y ) | K D ( xe iθ , y ) | dθ π ≥ c (1 − | xy | ) log (cid:16) − | xy | (cid:17) for all x, y ∈ D .This estimate does not follow from the asymptotics (cf. Lemma 5.1) of the diagonal values K W cr ( x, x ) when x is close to the boundary of D .It would be interesting to prove a pointwise version of (1.36) in the same spirit of (1.31):lim inf s → + sup f ∈B ( W cr ) (cid:12)(cid:12)(cid:12) g X ( s, z ; f ) g P K D ( s ) − f ( z ) (cid:12)(cid:12)(cid:12) > P K D -almost every X ∈ Conf( D ) . Weighted harmonic Hardy spaces. Recall that the Poisson transformation a signedRadon measure ν on T is a harmonic function on D given by P [ ν ]( z ) := Z T P ( z, ζ ) dν ( ζ ) = Z T − | z | | − ¯ ζz | dν ( ζ ) . Given any Borel probability measure µ on T , we define the associated weighted harmonicHardy space by H ( D ; µ ) := n f : D → C (cid:12)(cid:12)(cid:12) f ( z ) = P [ hµ ]( z ) = Z T P ( z, ζ ) h ( ζ ) dµ ( ζ ) , h ∈ L ( T , µ ) o . Lemma 1.10. Let µ be any Borel probability measure on T . Then for P K D -almost every X ∈ Conf( D ) , simultaneously for all f ∈ H ( D ; µ ) , all z ∈ D and all s > , we have X x ∈ X e − sd D ( x,z ) | f ( x ) | < ∞ . ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 15 For a fixed Borel probability measure µ on T , Lemma 1.10 implies that for P K D -almostevery X ∈ Conf( D ), we can define g X ( s, z ; f ) simultaneous for all f ∈ H ( D , µ ), all s > z ∈ D . And, in this situation, g X ( s, z ; f ) defined in (1.26) can be simplified as g X ( s, z ; f ) = X x ∈ X e − sd D ( x,z ) f ( x ) . Theorem 1.11. Let µ be any Borel probability measure on T . Then for P K D -almost every X ∈ Conf( D ) , simultaneously for all f ∈ H ( D ; µ ) and all z ∈ D , we have f ( z ) = lim s → + g X ( s, z ; f ) g X ( s, z ) , (1.37) where the convergence is uniform for f ∈ { P [ hµ ] : k h k L ( µ ) ≤ } and locally uniformly on z ∈ D , that is, for any relatively compact D ⊂ D , lim s → + sup f = P [ hµ ] k h k L µ ) ≤ sup z ∈ D (cid:12)(cid:12)(cid:12) g X ( s, z ; f ) g X ( s, z ) − f ( z ) (cid:12)(cid:12)(cid:12) = 0 . Remark . Note that in Theorem 1.11, the measure µ is not involved explicitly in (1.37).The implied subset of full P K D -measure in Conf( D ) in our statement might however dependon this measure µ and it is not clear whether the limit equality (1.37) can be extended tothe following family: n f : D → C (cid:12)(cid:12)(cid:12) f = P [ ν ] for a signed Radon measure ν on T o . Remark . We note that, although a function f ∈ H ( D ; µ ) is harmonic, neither thefunction g X ( s, z ; f ) nor the normalization g X ( s, z ; f ) /g X ( s, z ) is harmonic in z . On theother hand, for fixed x ∈ D , one can show that the function z e − sd D ( x,z ) is subharmonicin the region O s,x := n z ∈ D : | ϕ x ( z ) | > s − √ s − o . Note that for a fixed s > 1, any fixed compact subset D eventually is contained in theregion O s,x when x is close enough to the boundary.1.5.4. Comments. Let us describe a simple reconstruction algorithm for all harmonicHardy functions and explain why it is not applicable to weighted Bergman spaces andmore general spaces of harmonic functions.Let dm denote the normalized Lebesgue measure on T . If h ∈ L ( T ) = L ( T , m ), thenwe write P [ h ] := P [ hdm ]. The space of harmonic Hardy functions on D is defined by H ( D ) := n u : D → C (cid:12)(cid:12)(cid:12) u = P [ h ] , h ∈ L ( T ) o . We can show that P K D -almost every X satisfies that for Lebesgue almost every ζ ∈ T ,the Stolz angle S ζ , the closed convex hull of { ζ }∪{ z ∈ D : | z | ≤ / √ } , contains infinitelymany points. Then for any u = P [ h ] ∈ H ( D ) with h ∈ L ( T ), for Lebesgue almost everypoint ζ ∈ T , the non-tangential limit of u exists and is equal to h : h ( ζ ) = u ∗ ( ζ ) := lim S ζ ∋ z → ζ u ( z ) = lim X ∩ S ζ ∋ z → ζ u ( z ) for Lebesgue almost every ζ ∈ T . Therefore, for all z ∈ D , we have u ( z ) = P [ h ]( z ) = P [ u ∗ ]( z ) . However, the above reconstruction algorithm is not applicable to any weighted Bergmanspace, since it always contains functions without non-tangential limit (cf. MacLane [20] or Duren [10, Thm. 5.10]). Also, it is not applicable to harmonic functions of the form u = P [ ν ] with a singular measure ν on T . For instance, if ν is singular to dm , then u ∗ = ( P [ ν ]) ∗ vanishes Lebesgue almost everywhere (cf. e.g. Duren [10, Thm 1.2] and[3, 24, 25]), therefore, we obtain u = P [ u ∗ ] = 0.1.5.5. Outline of the proofs of Theorems 1.8 and 1.11. Let H be a Hilbert space. Theproofs of Theorems 1.8 and 1.11 rely on an extension of Theorem 1.4 to a fixed vector-valued harmonic function F : D → H belonging to the following class T ( D ; H ) := n F : D → H (cid:12)(cid:12)(cid:12) f is harmonic and lim α → + α · Z D k F ( w ) k (1 − | w | ) α dA ( w ) = 0 o . Let us illustrate the main steps for the proof of Theorem 1.8 for a fixed z ∈ D as follows.Similarly to the definition (1.24), for a fixed vector-valued function F : D → H , we define g ( k ) X ( s, z ; F ) := X x ∈ X ∩ A k ( z ) e − sd D ( z,x ) F ( x ) . (1.38) Step 1. The definition (1.22) of harmonic function with sub-exponential mean-growthnaturally extends to H -valued harmonic function F : D → H . We denote S exp ( D , H ) := n F : D → H (cid:12)(cid:12)(cid:12) F is harmonic with sub-exponential mean-growth o . It turns out (cf. Lemma 4.6) that if F ∈ S exp ( D , H ), then for any s > z ∈ D , ∞ X k =0 (cid:16) E P K D h k g ( k ) X ( s, z ; F ) k i(cid:17) / < ∞ and thus for P K D -almost every X ∈ Conf( D ), the following series converges in H : g X ( s, z ; F ) := ∞ X k =0 g ( k ) X ( s, z ; F ) . Then we prove (cf. Lemma 4.7) that for any F ∈ S exp ( D , H ), any s > z ∈ D , E P K D ( g X ( s, z ; F )) = F ( z ) · E P K D ( g X ( s, z ))and Var P K D ( g X ( s, z ; F )) ≤ Z D e − sd D ( z,w ) k F ( w ) k dA ( w )(1 − | w | ) . (1.39) Step 2. We then show (cf. Lemma 4.1) that T ( D , H ) ⊂ S exp ( D , H ). By the definition of T ( D , H ), we derive from (1.39) that there exists a sequence ( s n ) n ≥ in (1 , ∞ ) convergingto 1 such that for P K D -almost every X ∈ Conf( D ) and our fixed point z ∈ D ,lim n →∞ (cid:13)(cid:13)(cid:13) g X ( s n , z ; F ) g X ( s n , z ) − F ( z ) (cid:13)(cid:13)(cid:13) = 0 . (1.40) Step 3. Let W be a super-critical weight on D . Then A ( D , W ) is a reproducing kernelHilbert space with a reproducing kernel denoted by K W ( · , · ). Set F W : D → A ( D , W ) by F W ( w ) = K W ( · , w ) . We show (cf. Lemma 4.12) that once W is super-critical, then F W ∈ T ( D , A ( D , W )) andTheorem 1.8 follows from the equality (1.40) and the reproducing property of A ( D , W ): f ( z ) = h f, F W ( z ) i A ( D ,W ) for all f ∈ A ( D , W ) and all z ∈ D . ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 17 The proof of Theorem 1.11 requires some further steps involving the concept of sharplytempered growth condition (cf. Definition 4.2 and Theorem 4.18) and the proof of factthat the Poisson kernel satisfies this condition (cf. Lemma 4.19).1.6. Impossibility of simultaneous uniform interpolation of A ( D ) . Let R : D → R + be a non-negative bounded compactly supported radial function. For any z ∈ D , any f ∈ A ( D ) and any X ∈ Conf( D ), we define g R X ( z, f ) and g R X ( z ) as in (1.32). Recall that E P K D [ g R X ( z )] is independent of z ∈ D . Theorem 1.12. For any z ∈ D , we have inf R E P K D h sup f ∈ A ( D ): k f k≤ (cid:12)(cid:12)(cid:12) g R X ( z, f ) g R P K D − f ( z ) (cid:12)(cid:12)(cid:12) i ≥ with g R P K D := E P K D [ g R X ( z )] ,where the infimum is taken over all non-negative bounded compactly supported radial func-tions R on D .Remark . Recall that the natural radial assumption on R is related to (1.28).The following proposition is an average version of (1.31), which shows the difficulty ofdefining simultaneously g X ( s, z ; f ) for all f ∈ A ( D ). Proposition 1.13. For any < s ≤ , we have sup N ∈ N E P K D (cid:16) sup f ∈ A ( D ): k f k≤ (cid:12)(cid:12)(cid:12) N X k =0 g ( k ) X ( s, z ; f ) (cid:12)(cid:12)(cid:12) (cid:17) = ∞ . Questions and Conjectures.Question. Can we extend the equality (1.37) to all functions of the form f = P [ ν ] with ν signed Radon measures on T ? Do we have lim s → + E P K D h sup ζ ∈ T (cid:12)(cid:12)(cid:12) g X ( s, z ; P ( · , ζ )) g P K D ( s ) − P ( z, ζ ) (cid:12)(cid:12)(cid:12)i = 0 ? Here g P K D ( s ) is defined as in (1.35) and P ( z, ζ ) is the Poisson kernel given in (1.18) . Question. Can we construct in deterministic ways explicit configurations X ∈ Conf( D ) with critical upper density (namely, the number of points of X inside any compact subsetof D is controlled by the hyperbolic area of that compact subset) such that it satisfies thesimultaneous interpolation property as follows: f ( z ) = lim s → + g X ( s, z ; f ) g X ( s, z ) for all f ∈ H ∞ ( D ) and all z ∈ D ?(1.41) Here H ∞ ( D ) is the space of all bounded holomorphic functions on D . Conjecture. In the spirit of Theorem 1.12, we conjecture that inf R sup f ∈ A ( D ): k f k≤ (cid:12)(cid:12)(cid:12) g R X ( z, f ) g R P K D − f ( z ) (cid:12)(cid:12)(cid:12) > for P K D -almost every X ∈ Conf( D ) , where the infimum is taken over all non-negative bounded compactly supported radial func-tions R on D . Acknowledgements. Mikhael Gromov taught the Patterson-Sullivan theory to the olderof us in 1999; we are greatly indebted to him. We are deeply grateful to AlexanderBorichev, S´ebastien Gou¨ezel, Pascal Hubert, Alexey Klimenko and Andrea Sambusettifor useful discussions. Part of this work was done during a visit to the Centro De Giorgidella Scuola Normale Superiore di Pisa. We are deeply grateful to the Centre for its warmhospitality. AB’s research has received funding from the European Research Council(ERC) under the European Union’s Horizon 2020 research and innovation programmeunder grant agreement No 647133 (ICHAOS). YQ’s research is supported by the NationalNatural Science Foundation of China, grants NSFC Y7116335K1, 11801547 and 11688101.2. Preliminaries on point processes Configurations and point processes. Let E be a metric complete separablespace, equipped with a σ -finite positive Radon measure µ . A configuration X on E is a collection of points of E , possibly with finite multiplicities and considered withoutregard to order, such that any relatively compact subset B ⊂ E contains only finitelymany points. Let Conf( E ) denote the space of all configurations on E . A configuration X ∈ Conf( E ) may be identified with a purely atomic Radon measure P x ∈ X δ x , where δ x is the Dirac mass at the point x , and the space Conf( E ) is a complete separable metricspace with respect to the vague topology on the space of Radon measures on E . A Borelprobability measure P on Conf( E ) is called a point process on E . For further backgroundon point processes, see Daley and Vere-Jones [9], Kallenberg [16].A configuration is called simple if all its points have multiplicity one. A point process P is called simple, if P -almost every configuration is simple. For a simple point process P on E and an integer n ≥ 1, we say that a σ -finite measure ξ ( n ) P on E n is the n -th correlationmeasure of P if for any bounded compactly supported function φ : E n → C , we have E P (cid:16) X x ,...,xn ∈ Xxi = xj φ ( x , . . . , x n ) (cid:17) = Z E n φ ( y , . . . , y n ) dξ ( n ) P ( y , · · · , y n ) . If ξ ( n ) P is absolutely continuous to the measure µ ⊗ n , then the Radon-Nikodym derivative ρ ( P ) n ( x , · · · , x n ) := dξ ( n ) P dµ ⊗ n ( x , · · · , x n ) where ( x , · · · , x n ) ∈ E n ,is called the n -th correlation function of P with respect to the reference measure µ .2.2. Determinantal point processes. Let K be a locally trace class positive contractive integral operator on the complex Hilbert space L ( E, µ ). By slightly abusing the notation,we denote by K ( x, y ) the kernel of the integral operator K . By a theorem obtained byMacchi [19] and Soshnikov [32], as well as by Shirai and Takahashi [31], there exists aunique simple point process P K on E such that for any positive integer n ∈ N , its n -thcorrelation function, with respect to the reference measure µ , exists and is given by ρ ( P K ) n ( x , · · · , x n ) = det( K ( x i , x j )) ≤ i,j ≤ n . The point process P K is called the determinantal point process induced by the kernel K .Let us recall the standard expression for the variance of linear statistics under deter-minantal point processes induced by orthogonal projections. ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 19 Lemma 2.1. Let ( E, µ ) be a locally compact metric complete separable space equippedwith a Radon measure µ . Let P K be the determinantal point process on E induced by alocally trace class orthogonal projection K : L ( E, µ ) → L ( E, µ ) . Let H be a Hilbertspace and F : E → H be a function with E P K [ P x ∈ X k F ( x ) k + k F ( x ) k ] < ∞ . Then Var P K h X x ∈ X F ( x ) i = 12 Z E Z E k F ( x ) − F ( y ) k · | K ( x, y ) | dµ ( x ) dµ ( y ) and thus Var P K h X x ∈ X F ( x ) i ≤ Z E k F ( x ) k · K ( x, x ) dµ ( x ) . Preliminaries on the complex hyperbolic spaces Let d ≥ z, w ∈ C d , write z · w = P dk =1 z k w k and ¯ z = (¯ z , · · · , ¯ z d ).The Euclidean norm is denoted by | z | = √ z · ¯ z . Let D d = { z ∈ C d : | z | < } be the openunit ball in C d . The origin of C d will be denoted by o , that is, o = (0 , , · · · , ∈ C d .3.1. The Bergman metric on D d . Recall that any bounded complex domain carries anatural Riemannian metric, the Bergman metric (cf. Krantz [17, Chapter 1]), defined interms of the reproducing kernel of the space of square-integrable holomorphic functionson the bounded domain. For the open unit ball D d , the Bergman metric takes the form ds B := 4 | dz | + · · · + | dz d | − | z | + 4 | z dz + · · · + z d dz d | (1 − | z | ) . Let d B ( · , · ) denote the distance under the Bergman metric on D d . Let Aut( D d ) denote thegroup of all biholomorphic functions ψ : D d → D d . Note that d B is invariant under theaction of Aut( D d ), that is, d B ( ϕ ( z ) , ϕ ( w )) = d B ( z, w ) for all z, w ∈ D d and ϕ ∈ Aut( D d ).The ball D d endowed with d B is a model for the complex hyperbolic space.For w = o , set ϕ o ( z ) = − z . For w ∈ D d \ { o } , set ϕ w ( z ) := w − z · ¯ w | w | w − p − | w | (cid:0) z − z · ¯ w | w | w (cid:1) − z · ¯ w , z ∈ D d . (3.42)By Rudin [27, Thm. 2.2.2], the map ϕ w defines a biholomorphic involution of D d inter-changing w and o . For any z, w ∈ D d , we have d B ( z, w ) = log (cid:18) | ϕ w ( z ) | − | ϕ w ( z ) | (cid:19) . (3.43)3.2. The conformal invariant measure on D d . Let dv d ( z ) denote the normalizedLebesgue measure on D d such that v d ( D d ) = 1. The volume measure µ D d associated tothe Bergman metric and invariant under the group action of Aut( D d ) is given by dµ D d ( z ) = dv d ( z )(1 − | z | ) d +1 . (3.44)Let B ( z, r ) := { w ∈ D d : d B ( w, z ) < r } denote the ball in D d with respect to d B . Theelementary asymptotics in Lemmas 3.1 and 3.2 will be useful for us. Lemma 3.1. For any z ∈ D d , we have lim r →∞ µ D d ( B ( z, r )) e dr = 14 d . (3.45) Proof. Note that µ D d ( B ( z, r )) = µ D d ( B ( o, r )) for any z ∈ D d and any r > 0. Now by theformula of integration in polar coordinates and change of variables, we have µ D d ( B ( o, r )) = Z B ( o,r ) dv d ( z )(1 − | z | ) d +1 = d d Z r e − dx ( e x − d − ( e x + 1) dx, By l’Hˆopital’s rule, we obtain the desired limit equality (3.45). (cid:3) Lemma 3.2. For any z ∈ D d , we have lim s → d + ( s − d ) Z D d e − sd B ( x,z ) dµ D d ( x ) = d d . (3.46) Proof. For any s, t > 0, we can write e − st = Z R + se − sr · ( t < r ) dr. (3.47)Thus for any z ∈ D d , we have Z D d e − sd B ( x,z ) dµ D d ( x ) = Z D d e − sd B ( x,o ) dµ D d ( x )= Z D d dµ D d ( x ) Z R + se − sr ( d B ( x, o ) < r ) dr = s Z R + e − sr µ D d ( B ( o, r )) dr. Set κ ( r ) := µ D d ( B ( o,r )) e rd . Then for any s > d , Z D d e − sd B ( x,z ) dµ D d ( x ) = s Z R + κ ( r ) e − ( s − d ) r dr = ss − d Z R + κ (cid:16) ts − d (cid:17) e − t dt. The equality (3.46) follows from (3.45) and the Dominated Convergence Theorem. (cid:3) M -harmonic and pluriharmonic functions. The invariant Laplacian ∆ B on D d is given by ∆ B = (1 − | z | ) d X i,j =1 ( δ ij − z i ¯ z j ) ∂ ∂z i ∂ ¯ z j . A function f ∈ C ( D d ) is called M -harmonic if ∆ B f ≡ D d . Note that whileholomorphic functions on D d are M -harmonic, an Euclidean harmonic function on D d need not be. Set S d = { z ∈ C d : | z | = 1 } and let σ S d be the normalized surface measureon S d such that σ S d ( S d ) = 1. By Rudin [27, Cor. 2 of Thm. 4.2.4], a continuous function u ∈ C ( D d ) is M -harmonic if and only if it satisfies the invariant mean-value property : u ( ψ ( o )) = Z S d u ( ψ ( tζ )) dσ S d ( ζ ) for any ψ ∈ Aut( D d ) and any 0 < t < . (3.48)The Poisson-Szeg˝o kernel P b : D d × S d → R + is defined by the formula P b ( w, ζ ) = (1 − | w | ) d | − ζ · ¯ w | d , w ∈ D d and ζ ∈ S d . (3.49)The Poisson transformation of a signed Borel measure ν on S d of finite total variation isan M -harmonic function on D d and is defined by P b [ ν ]( z ) := Z S d P b ( z, ζ ) dν ( ζ ) , z ∈ D d . (3.50) ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 21 The definition of M -harmonicity naturally extends to vector-valued functions. Let H be a Hilbert space and F : D d → H be an M -harmonic function, then (3.48) implies F ( z ) = 1 µ D d ( B ( z, r )) Z B ( z,r ) F ( x ) dµ D d ( x ) for any z ∈ D d and r > F : D d → H is called pluriharmonic if ∂ ∂z j ∂ ¯ z k F = 0 for all 1 ≤ j, k ≤ d .Note that a function on D d is pluriharmonic if and only if it is both M -harmonic andEuclidean harmonic, see [27, Section 4.4, p. 59].3.4. Weighted Bergman spaces. A function W ∈ L ( D d , dv d ) with W ( z ) ≥ R D d W dv d > weight on D d . Given a weight W on D d , set A ( D d , W ) := n f : D d → C (cid:12)(cid:12)(cid:12) f is holomorphic and k f k W := Z D d | f ( z ) | W ( z ) dv d ( z ) < ∞ o . We call W Bergman-admissible if for any compact subset S ⊂ D d ,sup z ∈ S sup f ∈B ( W ) | f ( z ) | < ∞ where B ( W ) is the unit ball of A ( D d , W ) . Let W be a Bergman-admissible weight on D d . Then A ( D d , W ) is closed in L ( D d , W )and is called the weighted Bergman space associated with the weight W . It is a reproducingkernel Hilbert space whose reproducing kernel will be denoted by K W ( · , · ). We use theconvention that the function K W ( z, w ) is holomorphic in z and anti-holomorphic in w .The following equality will be useful for us: for any z ∈ D d , K W ( z, z ) = sup n | f ( z ) | (cid:12)(cid:12)(cid:12) f ∈ A ( D d , W ) and Z D d | f | W dv d ≤ o . (3.52)The Bergman kernel K D d (corresponding to W ≡ 1) is given by (cf. Rudin [27, § K D d ( z, w ) = 1(1 − z · ¯ w ) d +1 , z, w ∈ D d . (3.53)The reader is referred to Hedenmalm-Korenblum-Zhu [15, Chapters 1 and 9] for moredetails on weighted Bergman spaces and to Hedenmalm-Jakobsson-Shimorin [14] forweighted Bergman spaces associated with logarithmically subharmonic weights.One can show (cf. Duren [11, Thm. 1 in Chapter 1]) that if Φ : [0 , → [0 , ∞ )is an integrable function and R r Φ( t ) dt > r ∈ (0 , W Φ ( z ) := Φ( | z | ) on D d is Bergman-admissible. Note also that if a weight W is Bergman-admissible, then so is the weight ρ ( z ) W ( z ) for any ρ ∈ L ( D d , W ) with inf z ρ ( z ) > Patterson-Sullivan interpolations In this section, the determinantal point process P K D d will be denoted simply by P d := P K D d . Outline of this section. The main results of this section are described as follows.1). In Theorem 4.8 we obtain the Patterson-Sullivan interpolation of a fixed M -harmonic function on D d satisfying the tempered growth condition (the precise meaning isgiven in Definition 4.1).2). We show in Theorem 4.11 that in the case of pluriharmonic functions, the temperedgrowth condition is necessary and sufficient such that the Patterson-Sullivan interpolationholds in the sense of L -mean convergence.3). In Theorem 4.13, we obtain the simultaneous uniform interpolation of all functionsin the weighted Bergman space A ( D d , W ) for a super-critical weight W on D d (see (4.72)for the precise meaning of super-critical weight).4). In Theorem 4.15, we obtain the simultaneous uniform interpolation of all functionsin a weighted harmonic Hardy space H ( D d , µ ) (see (4.74) for the precise definition)associated to any Borel probability measure µ on the sphere S d . The proof of Theorem 4.15relies on a sharply tempered growth condition (see Definition 4.2 and Lemma 4.19).4.2. Preliminary properties of tempered M -harmonic functions. Let H be aHilbert space over R or C . For any integer k ≥ z ∈ D d , set A k ( z ) = { x ∈ D d | k ≤ d B ( x, z ) < k + 1 } . Definition 4.1. An M -harmonic function F : D d → H is called tempered iflim α → + α Z D d k F ( x ) k (1 − | x | ) α + d − dv d ( x ) = 0 . (4.54) Remark . By (3.46), it is easy to show that the condition (4.54) is equivalent tolim s → d + Z D d e − sd B ( x,o ) k F ( x ) k dµ D d ( x ) h Z D d e − sd B ( x,z ) dµ D d ( x ) i = 0 . (4.55) Definition 4.2. A tempered M -harmonic function F : D d → H is called sharply tempered if there exists a strictly decreasing sequence ( ε n ) n ≥ in R + satisfyinglim n →∞ ε n = 0 and lim n →∞ ε n +1 ε n = 1 , (4.56)such that ∞ X n =1 ε n Z D d e − ε n d B ( x,o ) k F ( x ) k e − d · d B ( x,o ) dµ D d ( x ) < ∞ . (4.57) Definition 4.3. An M -harmonic function F : D d → H is said to have sub-exponentialmean-growth if there is a sub-exponential function Λ : N → R + such that Z A k ( o ) k F ( x ) k dµ D d ( x ) ≤ Λ( k ) e kd for all k ∈ N .(4.58) ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 23 In what follows, we denote T ( D d , H ) := n F : D d → H (cid:12)(cid:12)(cid:12) F is tempered M -harmonic function o ; T ph ( D d , H ) := n F ∈ T ( D d , H ) (cid:12)(cid:12)(cid:12) F is pluriharmonic o ; S exp ( D d , H ) := n F : D → H (cid:12)(cid:12)(cid:12) F is M -harmonic with sub-exponential mean-growth o . Lemma 4.1. We have the inclusion: T ( D d , H ) ⊂ S exp ( D d , H ) . Proof. Note that there exist constants c , c > c e − k ≤ − | x | ≤ c e − k for all x ∈ A k ( o ).Then by taking α k = 1 / √ k and noting (3.44), we have e − kd Z A k ( o ) k F ( x ) k dµ D d ( x ) = e − kd Z A k ( o ) k F ( x ) k (1 − | x | ) α k + d − dv d ( x )(1 − | x | ) α k +2 d ≤ e − kd c α k +2 d e − k ( α k +2 d ) Z A k ( o ) k F ( x ) k (1 − | x | ) α k + d − dv d ( x ) ≤ ke √ k c α k +2 d · α k Z D d k F ( x ) k (1 − | x | ) α k + d − dv d ( x ) . Since lim k c α k = 1, the assumption (4.54) implies that there exists C > e − kd Z A k ( o ) k F ( x ) k dµ D d ( x ) ≤ C · ke √ k c d =: Λ( k ) . This completes the proof of the lemma. (cid:3) Lemma 4.2. An M -harmonic function F : D d → H is tempered provided that Z D d k F ( w ) k (1 − | w | ) d − dv d ( w ) < ∞ . (4.59) Proof. The condition (4.59) clearly implies the condition (4.54). (cid:3) Lemma 4.3. Let Θ : N → R + be any function such that lim k →∞ Θ( k ) = 0 . Then an M -harmonic function F : D d → H is tempered provided that Z A k ( o ) k F ( x ) k dµ D d ( x ) ≤ Θ( k ) · ( k + 1) · e kd for all k ∈ N . (4.60) Proof. By Remark 4.1, it suffices to show the limit equality (4.55). By (4.60), we have Z D d e − sd B ( o,x ) k F ( x ) k dµ D d ( x ) ≤ ∞ X k =0 e − sk Z A k ( o ) k F ( x ) k dµ D d ( x ) ≤ ∞ X k =0 e − s − d ) k Θ( k )( k + 1) . It is easy to see that the assumption lim k →∞ Θ( k ) = 0 implieslim s → d + P ∞ k =0 e − s − d ) k · Θ( k ) · ( k + 1) P ∞ k =0 e − s − d ) k · ( k + 1) = 0 . By direct computation, we have ∞ X k =0 e − s − d ) k · ( k + 1) = 1(1 − e − s − d ) ) and lim s → d + ( s − d ) ∞ X k =0 e − s − d ) k · ( k + 1) = 14 . It follows that lim s → d + ( s − d ) Z D d e − sd B ( o,x ) k F ( x ) k dµ D d ( x ) = 0 . Combining the above equality with (3.46), we obtain the desired limit equality (4.55). (cid:3) Corollary 4.4. Let Θ : R + → R + be any function with lim t →∞ Θ( t ) = 0 . Then an M -harmonic function F : D d → H is tempered provided that k F ( z ) k ≤ Θ (cid:16) − | z | (cid:17) · − | z | ) d log (cid:16) − | z | (cid:17) for all z ∈ D d . Proof. For any k ∈ N and for any z ∈ A k ( o ), we have e k / ≤ (1 − | z | ) − ≤ e k . Note alsothat µ D d ( A k ( o )) ≤ C d · e dk , where C d > d . Thus Corollary 4.4 followsfrom Lemma 4.3. (cid:3) Lemma 4.5. Let F : D d → H be a tempered M -harmonic function. Assume that thereexist constants C > , α > such that Z D d e − εd B ( x,o ) k F ( x ) k e − d · d B ( x,o ) dµ D d ( x ) ≤ Cε − · ε α for any ε ∈ (0 , . (4.61) Then F is sharply tempered. More generally, if in the upper estimate (4.61) , the term ε α is replaced by log − − α (1 /ε ) , then F is also sharply tempered.Proof. Assume (4.61), then take ε n = n − /α , both conditions (4.56) and (4.57) are sat-isfied. Now assume that in (4.61), the term ε α is replaced by log − − α (1 /ε ), then take ε n = e − n γ with (1 + α ) − < γ < 1, both conditions (4.56) and (4.57) are satisfied. (cid:3) Interpolation of a fixed tempered M -harmonic function. We adapt the def-inition (1.38) to a function F : D d → H : g ( k ) X ( s, z ; F ) := X x ∈ X ∩ A k ( z ) e − sd B ( z,x ) F ( x ) . Lemma 4.6. Assume that F ∈ S exp ( D d , H ) . Then for any s > d and any relativelycompact subset D ⊂ D d , we have ∞ X k =0 sup z ∈ D n E P d (cid:16) k g ( k ) X ( s, z ; F ) k (cid:17)o / < ∞ . (4.62) In particular, for P d -almost every X ∈ Conf( D d ) , we have ∞ X k =0 k g ( k ) X ( s, z ; F ) k < ∞ for Lebesgue almost every z ∈ D d . (4.63)Therefore, by Lemmas 4.1 and 4.6, fixing any F ∈ T ( D d , H ) and any s > d, z ∈ D d , for P d -almost every X ∈ Conf( D d ), we may define g X ( s, z ; F ) := ∞ X k =0 g ( k ) X ( s, z ; F ) , (4.64) ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 25 where the series converges absolutely in H . Introduce also the following notation: g ( k ) X ( s, z ) : = X x ∈ X ∩ A k ( z ) e − sd B ( z,x ) and g X ( s, z ) := X x ∈ X e − sd B ( z,x ) ,g P d ( s ) := E P d [ g X ( s, z )] = Z D d e − sd B ( x,z ) dµ D d ( x ) = Z D d e − sd B ( x,o ) dµ D d ( x ) . Lemma 4.7. Assume that F ∈ S exp ( D d , H ) . Then for any s > d, z ∈ D d , we have E P d ( g X ( s, z ; F )) = F ( z ) · g P d ( s )(4.65) and (4.66) Var P d ( g X ( s, z ; F )) ≤ Z D d e − sd B ( x,z ) k F ( x ) k dµ D d ( x ) . Theorem 4.8. Assume that F ∈ T ( D d , H ) . Then for any relatively compact subset D ⊂ D d , we have lim s → d + sup z ∈ D E P d (cid:16)(cid:13)(cid:13)(cid:13) g X ( s, z ; F ) g P d ( s ) − F ( z ) (cid:13)(cid:13)(cid:13) (cid:17) = 0 . (4.67) Proposition 4.9. Assume that F ∈ T ( D d , H ) . Let ( s n ) n ≥ be a sequence in ( d, ∞ ) converging to d and satisfying ∞ X n =1 g P d ( s n ) Z D d e − s n d B ( x,o ) k F ( x ) k dµ D d ( x ) < ∞ . (4.68) Then for P d -almost every X ∈ Conf( D d ) , the equality Z D lim sup n →∞ (cid:13)(cid:13)(cid:13) g X ( s n , z ; F ) g P d ( s n ) − F ( z ) (cid:13)(cid:13)(cid:13) dv d ( z ) = 0(4.69) holds for any relatively compact subset D ⊂ D d and moreover, lim n →∞ (cid:13)(cid:13)(cid:13) g X ( s n , z ; F ) g X ( s n , z ) − F ( z ) (cid:13)(cid:13)(cid:13) = 0 for Lebesgue almost every z ∈ D d . (4.70)For pluriharmonic functions, we obtain a necessary and sufficient condition such thatthe Patterson-Sullivan interpolation formula holds in the sense of L -mean convergence. Proposition 4.10. There exists a constant c > depending only on d such that for anypluriharmonic function F : D d → H with sub-exponential mean-growth, for all s ∈ ( d, d ] and all z ∈ D d , we have Var P d ( g X ( s, z ; F )) ≥ c Z D d e − sd B ( x,z ) k F ( x ) k dµ D d ( x ) . Theorem 4.11 (Necessary and sufficient condition for the interpolation of pluriharmonicfunctions) . Assume that F : D d → H is a pluriharmonic function with sub-exponentialmean-growth. Then the following limit equality lim s → d + E P d (cid:16)(cid:13)(cid:13)(cid:13) g X ( s, z ; F ) g P d ( s ) − F ( z ) (cid:13)(cid:13)(cid:13) (cid:17) = 0(4.71) holds for a fixed point z ∈ D d if and only if F ∈ T ph ( D d , H ) . Moreover, for any F ∈ T ph ( D d , H ) , the convergence (4.71) holds locally uniformly on z ∈ D d . Simultaneous uniform interpolation for weighted Bergman spaces. The fol-lowing radial weight on D d is Bergman-admissible and is essential for us. W cr ( z ) = 1(1 − | z | ) log (cid:0) −| z | (cid:1) for all z ∈ D d . By the discussion in § W ∈ L ( D , dA ) with W ( z ) ≥ | z |→ − W ( z ) W cr ( z ) = ∞ (4.72)is a Bergman-admissible weight on D d . Weights satisfying (4.72) will be called super-critical . Note that super-critical weights need not be radial .For a Bergman-admissible weight W on D d , we denote by K W the reproducing kernelof A ( D d , W ). Define an M -harmonic function F W : D d → A ( D d , W ) by F W ( w ) := K W ( · , w ) . (4.73) Lemma 4.12. Let W be a super-critical weight on D d . Then F W is tempered.Remark . For a super-critical weight W on D d , the function F W need not satisfy thecondition (4.59).Therefore, if W is a super-critical weight on D d , then by Lemmas 4.6 and 4.12, for anyfixed s > d, z ∈ D d , for P d -almost every X ∈ Conf( D d ), the series g X ( s, z ; F W ) = ∞ X k =0 g ( k ) X ( s, z ; F W )is absolutely convergent in A ( D d , W ) and consequently, we may define g X ( s, z ; f ), simul-taneously for all f ∈ A ( D d , W ), by g X ( s, z ; f ) := h f, g X ( s, z ; F W ) i A ( D d ,W ) = ∞ X k =0 g ( k ) X ( s, z ; f ) . Theorem 4.13. Let W be a super-critical weight on D d . Then there exists a sequence ( s n ) n ≥ in ( d, ∞ ) converging to d such that for P d -almost every X ∈ Conf( D d ) , we havethat for Lebesgue almost every z ∈ D d , f ( z ) = lim n →∞ g X ( s n , z ; f ) g X ( s n , z ) simultaneously for all f ∈ A ( D d , W ) ,where the convergence is uniform for f in the unit ball of A ( D d , W ) . Simultaneous uniform interpolation for weighted harmonic Hardy spaces. Recall the Poisson transformation (3.50). For any Borel probability measure µ on S d , set H ( D d ; µ ) : = n f : D d → C (cid:12)(cid:12)(cid:12) f = P b [ hµ ] , h ∈ L ( S d , µ ) o . (4.74) Lemma 4.14. Let µ be any Borel probability measure on S d . Then for P d -almost every X ∈ Conf( D d ) , simultaneously for all f ∈ H ( D d ; µ ) , all z ∈ D d and all s > d , we have X x ∈ X e − sd B ( x,z ) | f ( x ) | < ∞ . ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 27 Therefore, fixing a Borel probability measure µ on S d , for P d -almost every X ∈ Conf( D d ),we can define g X ( s, z ; f ) simultaneously for all f ∈ H ( D d , µ ), all s > d and all z ∈ D d by g X ( s, z ; f ) = X x ∈ X e − sd B ( x,z ) f ( x ) . Theorem 4.15. Let µ be any Borel probability measure on S d . Then for P d -almost every X ∈ Conf( D d ) , simultaneously for all f ∈ H ( D d ; µ ) and all z ∈ D d , we have f ( z ) = lim s → d + g X ( s, z ; f ) g X ( s, z ) , where the convergence is uniform for f ∈ { P b [ hµ ] : k h k L ( µ ) ≤ } and locally uniformlyon z ∈ D d . Proofs of Lemma 4.6 and Lemma 4.7.Lemma 4.16. Let F : D d → H be M -harmonic. Then for any s > , R > and z ∈ D d , E P d (cid:16) X x ∈ X ∩ B ( z,R ) e − sd B ( z,x ) F ( x ) (cid:17) = F ( z ) · E P d (cid:16) X x ∈ X ∩ B ( z,R ) e − sd B ( z,x ) (cid:17) . Proof. The identity (3.47) implies that for any s > , t > , R > 0, we have e − st ( t < R ) = Z R se − sr ( t < r ) dr + Z ∞ R se − sr ( t < R ) dr. This identity and the mean value property equality (3.51) together imply E P d (cid:16) X x ∈ X ∩ B ( z,R ) e − sd B ( z,x ) F ( x ) (cid:17) = Z D d e − sd B ( z,x ) ( d B ( z, x ) < R ) F ( x ) dµ D d ( x )= Z D d F ( x ) dµ D d ( x ) h Z R se − sr ( d B ( z, x ) < r ) dr + Z ∞ R se − sr ( d B ( z, x ) < R ) dr i = Z R se − sr dr Z B ( z,r ) F ( x ) dµ D d ( x ) + Z ∞ R se − sr dr Z B ( z,R ) F ( x ) dµ D d ( x )= F ( z ) · (cid:16) Z R se − sr µ D d ( B ( z, r )) dr + Z ∞ R se − sr µ D d ( B ( z, R )) dr (cid:17) . The same compution applies to F ≡ F and the constant function 1. (cid:3) Proof of Lemma 4.6. Fix s > d . Let D ⊂ D d be a relatively compact subset. Let N D ∈ N be the smallest integer with N ≥ sup z ∈ D d B ( z, o ) . Then for any integer k ≥ N D and z ∈ D , we have A k ( z ) ⊂ N D [ ℓ =0 A k − N D + ℓ ( o ) . Replacing the sub-exponential function Λ( k ) by e Λ( k ) := sup ≤ n ≤ k Λ( n ) if necessary, we may assume that Λ is non-decreasing. Therefore, by Lemma 2.1 and(4.58), for any integer k ≥ N D and z ∈ D , we haveVar P d ( g ( k ) X ( s, z ; F )) ≤ Z A k ( z ) e − sd B ( x,z ) k F ( x ) k dµ D d ( x ) ≤ e − sk Z A k ( z ) k F ( x ) k dµ D d ( x ) ≤ e − sk N D X ℓ =0 Z A k − ND + ℓ ( o ) k F ( x ) k dµ D d ( x ) ≤ e − sk N D X ℓ =0 Λ( k − N D + ℓ ) e k − N D + ℓ ) d ≤≤ N D + 1) e dN D Λ( k + N D ) e − s − d ) k ≤ Ce − (2 s − d ) k Λ( k + N D ) , where C > d and D . Then applying Lemma 4.16, we obtain { E P d ( k g ( k ) X ( s, z ; F ) k ) } / = n Var P d ( g ( k ) X ( s, z ; F )) + k E P d ( g ( k ) X ( s, z ; F )) k o / ≤≤ √ Ce − ( s − d ) k p Λ( k + N D ) + k F ( z ) k · E P d ( g ( k ) X ( s, z )) . Now since s > d and Λ is sub-exponential, we have ∞ X k =0 e − ( s − d ) k p Λ( k + N D ) < ∞ and ∞ X k =0 E P d ( g ( k ) X ( s, z )) = E P d ( g X ( s, z )) = Z D d e − sd B ( z,x ) dµ D d ( x ) = Z D d e − sd B ( o,x ) dµ D d ( x ) < ∞ . The desired convergence (4.62) follows immediately.Finally, the convergence (4.62) implies ∞ X k =0 Z D E P d (cid:16) k g ( k ) X ( s, z ; F ) k (cid:17) dv d ( z ) < ∞ . It follows that the convergence (4.63) holds for P d -almost every X ∈ Conf( D d ) andLebesgue almost every z ∈ D . We then complete the proof of the lemma by takingan exhausting sequence ( D k ) k ≥ of relatively compact subsets of D d . (cid:3) Proof of Lemma 4.7. Clearly, Lemmas 4.6 and 4.16 imply the desired equality (4.65).Now for any positive integer N , by Lemmas 2.1 and 4.16, we have E P d (cid:16)(cid:13)(cid:13)(cid:13) N − X k =0 g ( k ) X ( s, z ; F ) (cid:13)(cid:13)(cid:13) (cid:17) = E P d (cid:16)(cid:13)(cid:13)(cid:13) X x ∈ X ∩ B ( z,N ) e − sd B ( x,z ) F ( x ) (cid:13)(cid:13)(cid:13) (cid:17) == Var P d (cid:16) X x ∈ X ∩ B ( z,N ) e − sd B ( x,z ) F ( x ) (cid:17) + (cid:13)(cid:13)(cid:13) E P d (cid:16) X x ∈ X ∩ B ( z,N ) e − sd B ( x,z ) F ( x ) (cid:17)(cid:13)(cid:13)(cid:13) ≤≤ Z B ( z,N ) e − sd B ( x,z ) k F ( x ) k dµ D d ( x ) + k F ( z ) k (cid:16) Z B ( z,N ) e − sd B ( x,z ) dµ D d ( x ) (cid:17) ≤≤ Z D d e − sd B ( x,z ) k F ( x ) k dµ D d ( x ) + k F ( z ) k g P d ( s ) . ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 29 Therefore, by applying (4.62) and (4.65), we obtainVar P d ( g X ( s, z ; F )) = E P d ( k g X ( s, z ; F ) k ) − k E P d ( g X ( s, z ; F )) k == lim N →∞ E P d (cid:16)(cid:13)(cid:13)(cid:13) N − X k =0 g ( k ) X ( s, z ; F ) (cid:13)(cid:13)(cid:13) (cid:17) − k F ( z ) k g P d ( s ) ≤ Z D d e − sd B ( x,z ) k F ( x ) k dµ D d ( x ) . This is the desired inequality (4.66). (cid:3) Proof of Theorem 4.8 and Proposition 4.9. In what follows, set R X ( s, z ; F ) := g X ( s, z ; F ) g X ( s, z ) and R X ( s, z ; F ) := g X ( s, z ; F ) E P d [ g X ( s, z )] = g X ( s, z ; F ) g P d ( s ) . (4.75) Proof of Theorem 4.8. Let D ⊂ D d be a relatively compact subset. Then C D = sup z ∈ D d B ( z, o ) < ∞ . Let F : D d → H be a tempered M -harmonic function. For any s > d , by (4.65), we have E P d ( R X ( s, z ; F )) = F ( z ). Whence by Lemma 4.7,sup z ∈ D E P d ( k R X ( s, z ; F ) − F ( z ) k ) ≤ g P d ( s ) sup z ∈ D Z D d e − sd B ( x,z ) k F ( x ) k dµ D d ( x ) ≤ e sC D g P d ( s ) Z D d e − sd B ( x,o ) k F ( x ) k dµ D d ( x ) . (4.76)The desired relation (4.67) now follows from (4.55). (cid:3) Proof of Proposition 4.9. For proving (4.70), we may assume that F is not identicallyzero. Fix any sequence ( s n ) n ≥ in ( d, ∞ ) converging to d and satisfying the condition(4.68). By (4.68) and (4.76), we obtain ∞ X n =1 E P d (cid:16) Z D k R X ( s n , z ; F ) − F ( z ) k dv d ( z ) (cid:17) < ∞ . It follows that for P d -almost every X ∈ Conf( D d ), we have the limit equality Z D lim sup n →∞ k R X ( s n , z ; F ) − F ( z ) k dv d ( z ) = 0 . (4.77)This is the desired relation (4.69).Now since F is not identically zero, we havesup d 1. Therefore, we may apply (4.77) to the scalar function F ≡ Z D lim sup n →∞ (cid:12)(cid:12)(cid:12)(cid:12) g X ( s n , z ) g P d ( s n ) − (cid:12)(cid:12)(cid:12)(cid:12) dv d ( z ) = 0 for P d -almost every X ∈ Conf( D d ) . (4.78) By (4.77) and (4.78), for P d -almost every X ∈ Conf( D d ) and Lebesgue almost every z ∈ D , we havelim n →∞ k R X ( s n , z ; F ) − F ( z ) k = lim n →∞ (cid:13)(cid:13)(cid:13)(cid:13) g P d ( s n ) g X ( s n , z ) · R X ( s n , z ; F ) − F ( z ) (cid:13)(cid:13)(cid:13)(cid:13) = 0 . Finally, we complete the proof of Theorem 4.8 by taking an exhausting sequence ( D k ) k ≥ of relative compact subsets D k ⊂ D d . (cid:3) Proofs of Proposition 4.10 and Theorem 4.11. Proof of Proposition 4.10. Fix z ∈ D d . Recall the definition (3.42) of the M¨obius transfor-mation ϕ z on D d . Assume that F : D d → H is pluriharmonic such that F ∈ S exp ( D d , H )then so is F ◦ ϕ z . By [27, Thm. 4.4.9], it is easy to see that there exist two sequences( ξ n ) n ∈ N d and ( η n ) n ∈ N d \{ } of vectors in H such that F ( ϕ z ( x )) = X n ∈ N d ξ n x n + X n ∈ N d \{ } η n x n , where x ∈ D d and x n = x n · · · x n d d .Then by the conformal invariance of the measure µ D d , we have(4.79) Z D d e − sd B ( x,z ) k F ( x ) k dµ D d ( x ) = Z D d e − sd B ( x,o ) k F ( ϕ z ( x )) k dµ D d ( x )= X n ∈ N d k ξ n k Z D d | x n | e − sd B ( x,o ) dµ D d ( x ) + X n ∈ N d \{ } k η n k Z D d | x n | e − sd B ( x,o ) dµ D d ( x ) . Define a measure on D d × D d by dM ( x, y ) := | K D d ( x, y ) | dv d ( x ) dv d ( y ) . Then dM is invariant under the transformation ( x, y ) ( ϕ z ( x ) , ϕ z ( y )). Thus by Lemma 2.1,we haveVar P d ( g X ( s, z ; F )) = 12 Z D d Z D d k F ( x ) e − sd B ( x,z ) − F ( y ) e − sd B ( y,z ) k dM ( x, y )= 12 Z D d Z D d (cid:13)(cid:13)(cid:13) F ( ϕ z ( x )) e − sd B ( x,o ) − F ( ϕ z ( y )) e − sd B ( y,o ) | {z } denoted δ ( x, y ) (cid:13)(cid:13)(cid:13) dM ( x, y ) . Set F := { ξ n x n e − sd B ( x,o ) : n ∈ N d } ∪ { η n x n e − sd B ( x,o ) : n ∈ N d \ { }} . (4.80)Denote a general element in F by E s , we have δ ( x, y ) = X E s ∈F ( E s ( x ) − E s ( y )) . Now take any θ = ( θ , · · · , θ d ) ∈ [0 , π ) d , write D θ ( x ) := ( x e iθ , · · · , x d e iθ d ) . Since K D d ( x, y ) = K D d ( D θ ( x ) , D θ ( y )), we haveVar P d ( g X ( s, z ; F )) = 12 Z D d Z D d h π ) d Z [0 , π ) d k δ ( D θ ( x ) , D θ ( y )) k dθ i dM ( x, y ) . ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 31 It is easy to see that for any two distinct elements E s , f E s ∈ F , we have1(2 π ) d Z [0 , π ) D E s ( D θ ( x )) − E s ( D θ ( y )) , f E s ( D θ ( x )) − f E s ( D θ ( y )) E H dθ = 0 . Therefore, by using the equality k E s ( D θ ( x )) − E s ( D θ ( y )) k = k E s ( x ) − E s ( y ) k , we have1(2 π ) d Z [0 , π ) d k δ ( D θ ( x ) , D θ ( y )) k dθ = X E s ∈F k E s ( x ) − E s ( y ) k . Hence Var P d ( g X ( s, z ; F )) = 12 X E s ∈F Z D d Z D d k E s ( x ) − E s ( y ) k dM ( x, y ) | {z } denoted I ( E s ) . (4.81)Let us now estimate the double integral I ( E s ) for E s ∈ F . Claim I: there is a constant c > s ∈ [ d, d ] and any E s ∈ F , we have I ( E s ) ≥ c Z D d k E s ( x ) k dµ D d ( x ) = c Z D d k E s ( x ) k K D d ( x, x ) dv d ( x ) . (4.82)Assuming Claim I, then by (4.79), (4.81) and (4.82), we obtain the desired inequalityVar P d ( g X ( s, z ; F )) ≥ c X E s ∈F Z D d k E s ( x ) k dµ D d ( x ) = c Z D d e − sd B ( x,z ) k F ( x ) k dµ D d ( x ) . It remains to prove Claim I. Note that I ( E s ) = 2 Z D d k E s ( x ) k K D d ( x, x ) dv d ( x ) − ℜ Z D d Z D d h E s ( x ) , E s ( y ) i dM ( x, y ) . Thus it suffices to show that there exists a constant γ ∈ (0 , 1) such that for any s ∈ [ d, d ],we have ℜ Z D d Z D d h E s ( x ) , E s ( y ) i dM ( x, y ) ≤ γ · Z D d k E s ( x ) k K D d ( x, x ) dv d ( x ) . (4.83)Now for any n ∈ N d , set N ( n, s ) : = Z D d Z D d x n y n e − sd B ( x,o ) e − sd B ( y,o ) dM ( x, y ) ,D ( n, s ) : = Z D d | x n | e − sd B ( x,o ) K D d ( x, x ) dv d ( x ) . By the definition (4.80) of the family F , for proving (4.83), it suffices to showsup n ∈ N d sup s ∈ [ d, d ] ℜ ( N ( n, s )) D ( n, s ) < . (4.84)Note that by Cauchy-Bunyakovsky-Schwarz inequality, it is easy to see that for any fixed n ∈ N d and any s ∈ [ d, d ], we have the strict inequality ℜ ( N ( n, s )) < D ( n, s ). Therefore,by using the continuity on s , to prove the inequality (4.84), it suffices to showlim | n |→∞ sup s ∈ [ d, d ] ℜ ( N ( n, s )) D ( n, s ) = 0 where | n | = n + · · · + n d . (4.85) By expanding (1 − x · ¯ y ) − d − , we may write N ( n, s ) as follows: N ( n, s ) = Z D d Z D d x n y n (cid:12)(cid:12)(cid:12) ∞ X k =0 Γ( k + d + 1) k !Γ( d + 1) ( x · ¯ y ) k (cid:12)(cid:12)(cid:12) dv d ( x ) dv d ( y ) e sd B ( x,o ) e sd B ( y,o ) = Z D d Z D d h Z π e i | n | θ x n y n (cid:12)(cid:12)(cid:12) ∞ X k =0 Γ( k + d + 1) k !Γ( d + 1) ( e inθ x · ¯ y ) k (cid:12)(cid:12)(cid:12) dθ π | {z } denoted by A ( x, y ) i dv d ( x ) dv d ( y ) e sd B ( x,o ) e sd B ( y,o ) . Clearly, we have A ( x, y ) = x n y n ∞ X k =0 Γ( k + d + 1) k !Γ( d + 1) Γ( k + | n | + d + 1)( k + | n | )!Γ( d + 1) ( x · ¯ y ) k · (¯ x · y ) k + | n | . By Cauchy-Bunyakovsky-Schwarz inequality, for any n ∈ N d and k ∈ N , we have (cid:12)(cid:12)(cid:12) Z D d Z D d x n y n ( x · ¯ y ) k · (¯ x · y ) k + | n | dv d ( x ) dv d ( y ) e sd B ( x,o ) e sd B ( y,o ) (cid:12)(cid:12)(cid:12) ≤ (cid:16) Z D d Z D d | x n | | x · ¯ y | k + | n | dv d ( x ) dv d ( y ) e sd B ( x,o ) e sd B ( y,o ) (cid:17) / (cid:16) Z D d Z D d | y n | | x · ¯ y | k + | n | dv d ( x ) dv d ( y ) e sd B ( x,o ) e sd B ( y,o ) (cid:17) / = Z D d Z D d | x n | | x · ¯ y | k + | n | dv d ( x ) dv d ( y ) e sd B ( x,o ) e sd B ( y,o ) ≤ Z D d Z D d | x n | | x · ¯ y | k + | n | (1 − | x | ) s (1 − | y | ) s dv d ( x ) dv d ( y ) | {z } denoted by B s ( k, n ) . It is clear that B s ( k, n ) = a k,n · Z D d Z D d | x | | n | · | x | k + | n | | y | k + | n | (1 − | x | ) s (1 − | y | ) s dv d ( x ) dv d ( y )= a k,n (2 d ) Z Z r d − r d − r k +3 | n | r k + | n | (1 − r ) s (1 − r ) s dr dr ≤ d · a k,n Γ(2 k + 3 | n | + 1)Γ( s + 1)Γ(2 k + 3 | n | + s + 2) Γ(2 k + | n | + 1)Γ( s + 1)Γ(2 k + | n | + s + 2) , where a k,n = Z S d Z S d | ζ n | | ζ · ¯ ξ | k + | n | dσ S d ( ζ ) dσ S d ( ξ ) = Γ( d )Γ( k + | n | + 1)Γ( d + k + | n | ) · Γ( d ) n ! · · · n d !Γ( d + | n | ) . The computation of a k,n relies on [27, section 1.4.5] and [27, Prop. 1.4.9]. Therefore, ℜ ( N ( n, s )) ≤ | N ( n, s ) | ≤ ∞ X k =0 Γ( k + d + 1) k !Γ( d + 1) Γ( k + | n | + d + 1)( k + | n | )!Γ( d + 1) B s ( k, n ) . ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 33 By the classical asymptotics for Gamma function, there exists a constant C d > d such that for all s ∈ [ d, d ] and all n ∈ N d with | n | ≥ 1, we have ℜ ( N ( n, s )) ≤ C d · Γ( d ) n ! · · · n d !Γ( d + | n | ) · ∞ X k =0 k d ( k + | n | ) s +1 . (4.86)On the other hand, for D ( n, s ), it is easy to see that D ( n, s ) = Z S d | ζ n | dσ S d ( ζ ) · Z D d | x | | n | (cid:16) − | x | | x | (cid:17) s dv d ( x )(1 − | x | ) d +1 ≥ s + d +1 Z S d | ζ n | dσ S d ( ζ ) · Z D d | x | | n | (1 − | x | ) s − d − dv d ( x )= 2 d s + d +1 Z S d | ζ n | dσ S d ( ζ ) · Z r d − | n | (1 − r ) s − d − dr = 2 d s + d +1 Γ( d ) n ! · · · n d !Γ( d + | n | ) Γ(2 d + 2 | n | )Γ(2 s − d )Γ(2 | n | + 2 s + d ) . Hence there exists a constant C ′ d > s ∈ [ d, d ] and any n ∈ N d with | n | ≥ 1, we have D ( n, s ) ≥ C ′ d · Γ( d ) n ! · · · n d !Γ( d + | n | ) · | n | s − d . (4.87)Combining (4.86) and (4.87), for all s ∈ [ d, d ] and all n ∈ N d with | n | ≥ 1, we obtain ℜ ( N ( n, s )) D ( n, s ) ≤ C d C ′ d ∞ X k =0 k d · | n | s − d ( k + | n | ) s +1 . Finally, it suffices to show lim m →∞ sup s ∈ [ d, d ] ∞ X k =1 k d m s − d ( k + m ) s +1 = 0 . (4.88)Indeed, write ∞ X k =1 k d m s − d ( k + m ) s +1 = 1 m ∞ X k =1 ( k/m ) d (1 + k/m ) s +1 ≤ m ∞ X k =1 ( k/m ) d (1 + k/m ) d +1 . (4.89)Set H ( t ) = t d / (1 + t ) d +1 . Note that H is increasing on a finite interval [0 , t ] and thendecreasing on the interval [ t , ∞ ). It follows that ∞ X k =1 ( k/m ) d (1 + k/m ) d +1 ≤ max t ∈ R + H ( t ) + 2 Z ∞ H ( t ) dt. (4.90)The desired limit equation (4.88) follows immediately from (4.89) and (4.90). (cid:3) Proof of Theorem 4.11. Theorem 4.11 is an immediate consequence of Lemma 4.7 andProposition 4.10. (cid:3) Proofs of Lemma 4.12 and Theorem 4.13.Proposition 4.17. Let W be a super-critical weight on D d . Then there exists a function Θ : R + → R + with lim t →∞ Θ( t ) = 0 such that K W ( z, z ) ≤ Θ (cid:0) − | z | (cid:1) · − | z | ) d log (cid:0) − | z | (cid:1) for all z ∈ D d . The proof of Proposition 4.17 is postponed to § Proof of Lemma 4.12. By the reproducing property of the kernel K W , we have k F W ( z ) k A ( D d ,W ) = h K W ( · , z ) , K W ( · , z ) i A ( D d ,W ) = K W ( z, z ) for all z ∈ D d .Thus Lemma 4.12 follows from Corollary 4.4 and Proposition 4.17. (cid:3) Proof of Theorem 4.13. Theorem 4.8 and Lemma 4.12 together imply that there exists asequence ( s n ) n ≥ in ( d, ∞ ) converging to d such that if we fix a countable dense subset D ⊂ D d , then for P d -almost every X ∈ Conf( D d ), we havelim n →∞ (cid:13)(cid:13)(cid:13) g X ( s n , z ; F W ) g X ( s n , z ) − F W ( z ) (cid:13)(cid:13)(cid:13) A ( D d ,W ) = 0 for all z ∈ D .We complete the proof of Theorem 4.13 by noting that for all f ∈ A ( D d , W ), z ∈ D d , g X ( s, z ; f ) = h f, g X ( s, z ; F W ) i A ( D d ,W ) and f ( z ) = h f, F W ( z ) i A ( D d ,W ) and (cid:13)(cid:13)(cid:13) g X ( s n , z ; F W ) g X ( s n , z ) − F W ( z ) (cid:13)(cid:13)(cid:13) A ( D d ,W ) = sup f ∈B ( W ) (cid:12)(cid:12)(cid:12)D f, g X ( s n , z ; F W ) g X ( s n , z ) − F W ( z ) E A ( D d ,W ) (cid:12)(cid:12)(cid:12) , where B ( W ) is the unit ball of A ( D d , W ). (cid:3) Proofs of Lemma 4.14 and Theorem 4.15. In this subsection, assume that theHilbert space H is over the field R and is of the form H = L R ( ν ) = L R (Σ , ν ) , where ν is a Borel probability measure on a metric complete seperable space Σ and L R ( ν ) isthe space of real-valued square-integrable functions on (Σ , ν ). The subset of non-negativefunctions in L R ( ν ) will be denoted by L R ( ν ) + .Recall that a convergent series P ∞ n =1 v n in L R ( ν ) is said to converge unconditionally ifits sum does not change under any reordering of the terms. Theorem 4.18. Let F : D d → L R ( ν ) + be a sharply tempered M -harmonic function. Let D ⊂ D d be a relatively compact subset. Then P d -almost every X ∈ Conf( D d ) satisfies: (i) The series g X ( s, z ) = P x ∈ X e − sd B ( x,z ) converges for all s > d and all z ∈ D d . (ii) The series g X ( s, z ; F ) = P x ∈ X e − sd B ( x,z ) F ( x ) converges unconditionally in L R ( ν ) for all s > d and all z ∈ D d . (iii) For all z ∈ D d , we have lim s → d + (cid:13)(cid:13)(cid:13) g X ( s, z ; F ) g X ( s, z ) − F ( z ) (cid:13)(cid:13)(cid:13) L R ( ν ) = 0 . (4.91)(iv) The following local uniform convergence holds: lim s → d + sup z ∈ D (cid:13)(cid:13)(cid:13) g X ( s, z ; F ) g X ( s, z ) − F ( z ) (cid:13)(cid:13)(cid:13) L R ( ν ) = 0 . ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 35 The proof of Theorem 4.18 is postponed to the end of this subsection. We now showhow to derive Lemma 4.14 and Theorem 4.15 from Theorem 4.18. Recall the Poisson-Szeg˝o kernel P b ( w, ζ ) defined in (3.49). For any Borel probability measure µ on S d , wedefine a function F µ : D d → L R ( S d , µ ) by F µ ( w ) := P b ( w, · ) ∈ L R ( S d , µ ) . (4.92)Note for any ζ ∈ S d , the function w P b ( w, ζ ) is M -harmonic. Therefore, F µ definedin (4.92) is a vector-valued M -harmonic function. The following Lemma 4.19 shows that F µ is sharply tempered in the sense of Definition 4.2. Lemma 4.19. Let µ be a Borel probability measure on S d . Then there exists a constant C > , such that for any k ∈ N , we have Z A k ( o ) k P b ( w, · ) k L ( µ ) dµ D d ( w ) ≤ Ce kd . (4.93) Moreover, there exists a constant C ′ > such that for any ε ∈ (0 , , we have Z D d e − εd B ( w,o ) k P b ( w, · ) k L ( µ ) e − d · d B ( w,o ) dµ D d ( w ) ≤ C ′ ε . (4.94) In particular, the M -harmonic function F µ defined by (4.92) is sharply tempered.Proof. Let U d be the compact group of d × d unitary matrices equipped with the normalizedHaar measure dU . For any k ∈ N and any U ∈ U d , since µ D d is radial, we have Z A k ( o ) k P b ( w, · ) k L ( µ ) dµ D d ( w ) = Z A k ( o ) k P b ( U w, · ) k L ( µ ) dµ D d ( w ) . Therefore, Z A k ( o ) k P b ( w, · ) k L ( µ ) dµ D d ( w ) = Z A k ( o ) Z U d k P b ( U w, · ) k L ( µ ) dU dµ D d ( w ) == Z A k ( o ) Z S d h Z U d (cid:18) (1 − | U w | ) d | − ζ · U w | d (cid:19) dU i dµ ( ζ ) dµ D ( w ) . Clearly, for any ζ ∈ S d , since | U w | = | w | and ζ · U w = ( U − ζ ) · w , we have Z U d (cid:18) (1 − | U w | ) d | − ζ · U w | d (cid:19) dU = Z U d (cid:18) (1 − | w | ) d | − ( U − ζ ) · ¯ w | d (cid:19) dU = Z S d (cid:18) (1 − | w | ) d | − ξ · ¯ w | d (cid:19) dσ S d ( ξ ) . By [27, Prop. 1.4.10], there exists a constant c > w ∈ D d , Z S d (cid:18) (1 − | w | ) d | − ξ · ¯ w | d (cid:19) dσ S d ( ξ ) ≤ ce d · d B ( w,o ) . Thus we obtain Z A k ( o ) k P b ( w, · ) k L ( µ ) dµ D d ( w ) ≤ Z A k ( o ) Z S d ce d · d B ( w,o ) dµ ( ζ ) dµ D d ( w ) ≤ ce ( k +1) d µ D d ( A k ( o )) . By Lemma 3.1, there exists a constant c ′ > µ D d ( A k ( o )) ≤ c ′ e kd and we obtainthe desired inequality (4.93). By Lemma 4.3, the mean-growth estimate (4.93) impliesthat F µ : D d → L R ( µ ) defined in (4.92) is tempered in the sense of Definition 4.1. Since 1 − e − x ≥ e − x for any x ∈ (0 , ε ∈ (0 , Z D d e − εd B ( w,o ) k P b ( w, · ) k L ( µ ) e − d · d B ( w,o ) dµ D d ( w ) ≤ ∞ X k =0 e − (2 d +2 ε ) k Z A k ( o ) k P b ( w, · ) k L ( µ ) dµ D d ( w ) ≤ C ∞ X k =0 e − εk = C − e − ε ≤ e C ε . This implies the desired inequality (4.94). By Lemma 4.5, F µ is sharply tempered. (cid:3) Proof of Lemma 4.14. By Thereom 4.18, Lemma 4.19, for P d -almost every X ∈ Conf( D d ),simultaneously for all s > d and all z ∈ D d , the series g X ( s, z ; F µ ) = X x ∈ X e − sd B ( x,z ) F µ ( x ) = X x ∈ X e − sd B ( x,z ) P b ( x, · )converges unconditionally in L R ( µ ) and thus for all f = P b [ hµ ] ∈ H ( D d ; µ ), the series X x ∈ X e − sd B ( x,z ) f ( x ) = X x ∈ X e − sd B ( x,z ) Z S d h ( ζ ) P b ( x, ζ ) dµ ( ζ ) = X x ∈ X e − sd B ( x,z ) h h, P b ( x, · ) i L ( µ ) converges unconditionally. We complete the proof of the lemma by using the equivalencebetween the unconditional convergence and the absolute convergence for scalar series. (cid:3) Proof of Theorem 4.15. Let µ be a Borel probability measure on S d . Thereom 4.18 andLemma 4.19 together imply that if D ⊂ D d is a relatively compact subset, then for P d -almost every X ∈ Conf( D d ), we havelim s → d + sup z ∈ D (cid:13)(cid:13)(cid:13) g X ( s, z ; P b ( x, · )) g X ( s, z ) − P b ( z, · ) (cid:13)(cid:13)(cid:13) L ( µ ) = 0 . By observing that for any f = P b [ hµ ] ∈ H ( D d ; µ ), g X ( s, z ; f ) g X ( s, z ) = D h, g X ( s, z ; P b ( x, · )) g X ( s, z ) E L ( µ ) and f ( z ) = h h, P b ( z, · ) i L ( µ ) , we complete the whole proof of Theorem 4.15. (cid:3) Let us now proceed to the proof of Theorem 4.18. Proof of Theorem 4.18. Fix a non-identically zero sharply tempered M -harmonic func-tion F : D d → L R ( ν ) + . Then we can fix a strictly decreasing sequence ( ε n ) n ≥ convergingto 0 and satisfying (4.56) and (4.57). Let ( s n ) n ≥ be defined by s n = d + ε n . (4.95)Then the sequence ( s n ) n ≥ converges to d and satisfies the condition (4.68).Fix a countable dense subset D ⊂ D d . By Lemma 4.6 and Theorem 4.8, there exists asubset Ω ⊂ Conf( D d ) with P d (Ω) = 1 such that for any X ∈ Ω and any z ∈ D , we have • < g X ( s n , z ) < ∞ for all n ∈ N ; • the following limit holds (cf. (4.78) in the proof of Theorem 4.8):lim n →∞ g X ( s n , z ) g P d ( s n ) = 1;(4.96) ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 37 • the following series converges in L R ( ν ) for all n ∈ N : g X ( s n , z ; F ) = ∞ X k =0 g ( k ) X ( s n , z ; F ) = ∞ X k =0 X x ∈ X ∩ A k ( z ) e − s n d B ( x,z ) F ( x );(4.97) • the following limit equality holds:lim n →∞ (cid:13)(cid:13)(cid:13) g X ( s n , z ; F ) g X ( s n , z ) − F ( z ) (cid:13)(cid:13)(cid:13) L ( ν ) = 0 . (4.98)We now complete the proof of Theorem 4.15 by proving the following claims. Claim I. For any X ∈ Ω, the limit equality (4.96) holds for all z ∈ D d .For an arbitrary z ′ ∈ D d , there exists a sequence ( z k ) k ≥ in D converging to z ′ . For any X ∈ Ω, any k, n ∈ N , we have e − s n d B ( z k ,z ′ ) g X ( s n , z k ) ≤ g X ( s n , z ′ ) ≤ e s n d B ( z k ,z ′ ) g X ( s n , z k ) . Therefore, using the limit equality (4.96) for z k ∈ D , we obtain e − d · d B ( z k ,z ′ ) ≤ lim inf n →∞ g X ( s n , z ′ ) g P d ( s n ) ≤ lim sup n →∞ g X ( s n , z ′ ) g P d ( s n ) ≤ e d · d B ( z k ,z ′ ) . The desired limit equality (4.96) for the point z ′ then follows since lim k d B ( z k , z ′ ) = 0. Claim II. For any X ∈ Ω, the following series converges unconditionally in L R ( ν ) for all s > d and all z ∈ D d : g X ( s, z ; F ) = X x ∈ X e − sd B ( x,z ) F ( x ) . (4.99)We will use an elementary fact: Let ( u n ) n ≥ be a sequence in L R ( ν ) + and ( b n ) n ≥ bea sequence of positive numbers with sup n ∈ N b n < ∞ . If the series P ∞ n =0 u n converges in L ( ν ), then it converges unconditionally and so is the series P ∞ n =0 b n u n .Fix any X ∈ Ω. Let s > d and z ′ ∈ D d . Fix any point z ∈ D and any integer n , largeenough such that s n ≤ s . Since F takes values in L R ( ν ) + , the convergence of the series(4.97) for s n and z implies the unconditional convergence of the series (4.99) for s n and z . But then, by using e − sd B ( x,z ′ ) F ( x ) = e − sd B ( x,z ′ )+ s n d B ( x,z ) · e − s n d B ( x,z ) F ( x )and sup x ∈ X e − sd B ( x,z ′ )+ s n d B ( x,z ) ≤ sup x ∈ X e − sd B ( x,z ′ )+ sd B ( x,z ) ≤ e sd B ( z ′ ,z ) < ∞ , we immediately obtain the unconditional convergence of the series (4.99) for s and z ′ . Claim III. For any X ∈ Ω, the series g X ( s, z ) = X x ∈ X e − sd B ( x,z ) converges and g X ( s, z ) > s > d , all z ∈ D d . The proof of Claim III is similar tothat of Claim II. Claim IV. For any X ∈ Ω, the limit equality (4.98) holds for all z ∈ D d .Fix any X ∈ Ω. For an arbitrary z ′ ∈ D d , there exists a sequence ( z k ) k ≥ in D convergingto z ′ . Clearly, for any n, k ∈ N , we have e − s n d B ( z k ,z ′ ) g X ( s n , z k ; F ) g X ( s n , z k ) ≤ g X ( s n , z ′ ; F ) g X ( s n , z ′ ) ≤ e s n d B ( z k ,z ′ ) g X ( s n , z k ; F ) g X ( s n , z k )and thus, by using the simplified notation k · k = k · k L ( ν ) , we have (cid:13)(cid:13)(cid:13) g X ( s n , z ′ ; F ) g X ( s n , z ′ ) − F ( z ′ ) (cid:13)(cid:13)(cid:13) ≤ max ± (cid:13)(cid:13)(cid:13) e ± s n d B ( z k ,z ′ ) g X ( s n , z k ; F ) g X ( s n , z k ) − F ( z ′ ) (cid:13)(cid:13)(cid:13) ≤ max ± n(cid:13)(cid:13)(cid:13) e ± s n d B ( z k ,z ′ ) (cid:16) g X ( s n , z k ; F ) g X ( s n , z k ) − F ( z k ) (cid:17)(cid:13)(cid:13)(cid:13) + k e ± s n d B ( z k ,z ′ ) F ( z k ) − F ( z ′ ) k o ≤ e s n d B ( z k ,z ′ ) n(cid:13)(cid:13)(cid:13) g X ( s n , z k ; F ) g X ( s n , z k ) − F ( z k ) (cid:13)(cid:13)(cid:13) + k F ( z k ) − F ( z ′ ) k o + | e s n d B ( z k ,z ′ ) − |k F ( z ′ ) k . Therefore, for any k ∈ N , by using (4.98) for z k and lim n s n = d , we obtainlim sup n →∞ (cid:13)(cid:13)(cid:13) g X ( s n , z ′ ; F ) g X ( s n , z ′ ) − F ( z ′ ) (cid:13)(cid:13)(cid:13) ≤ e dd B ( z k ,z ′ ) k F ( z k ) − F ( z ′ ) k + | e dd B ( z k ,z ′ ) − |k F ( z ′ ) k . Finally, by using the assumption lim k d B ( z k , z ′ ) = 0 and the continuity of F ( M -harmonicfunctions are continuous), we obtain the desired limit equality (4.98) for z ′ . Claim V. For any X ∈ Ω, the limit equality (4.91) holds for all z ∈ D d .Fix any X ∈ Ω and any z ∈ D d . Recall the notation R X ( s, z ; F ) and R X ( s, z ; F )introduced in (4.75). By Claim I and Claim IV, writing k · k = k · k L ( ν ) , we havelim n →∞ k R X ( s n , z ; F ) − F ( z ) k = 0 . (4.100)Recall the definition (4.95) of s n . Let s ∈ ( d, s ). Since ( s n ) n ∈ N is strictly decreasing andconverges to d , there is a unique n s ∈ N such that s n s +1 ≤ s < s n s and we may define R + X ( s, z ; F ) := g X ( s n s +1 , z ; F ) g P d ( s n s +1 ) , R − X ( s, z ; F ) := g X ( s n s , z ; F ) g P d ( s n s ) , β ( s ) := g P d ( s n s +1 ) g P d ( s n s ) . (4.101)The limit equality (4.100) implies thatlim s → d + k R ± X ( s, z ; F ) − F ( z ) k = lim n →∞ k R X ( s n , z ; F ) − F ( z ) k = 0 . (4.102)By Lemma 3.2, we have lim s → d + ( s − d ) g P d ( s ) = d d . Thus by (4.56) and the definition (4.95) of s n , we havelim s → d + β ( s ) = lim n →∞ g P d ( s n +1 ) g P d ( s n ) = 1 . (4.103)The monotonicity of s e − sd B ( x,z ) and the assumption F ( x ) ∈ L R ( ν ) + together imply β ( s ) − R − X ( s, z ; F ) ≤ R X ( s, z ; F ) ≤ R + X ( s, z ; F ) β ( s ) . ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 39 Therefore, by noting that β ( s ) ≥ 1, we have(4.104) k R X ( s, z ; F ) − F ( z ) k ≤ ± k β ( s ) ± R ± X ( s, z ; F ) − F ( z ) k≤ ± n k β ( s ) ± ( R ± X ( s, z ; F ) − F ( z )) k + k β ( s ) ± F ( z ) − F ( z ) k o ≤ β ( s ) max ± k R ± X ( s, z ; F ) − F ( z ) k + 2 | β ( s ) − |k F ( z ) k . By (4.102), (4.103) and (4.104), we have lim s → d + k R X ( s, z ; F ) − F ( z ) k = 0 , which, com-bined with Claim I, implies the desired equalitylim s → d + k R X ( s, z ; F ) − F ( z ) k = 0 . (4.105) Claim VI. For any relatively compact subset D ⊂ D d , there exists C D > ε ∈ (0 , 1) and any s ∈ ( d, d ), we have P d (cid:16) sup z ∈ D k R X ( s, z ; F ) − F ( z ) k > ε (cid:17) ≤ C D ε d +2 g P d ( s )] Z D d e − sd B ( x,o ) k F ( x ) k dµ D d ( x ) . (4.106)Indeed, since D is relatively compact and F is M -harmonic, there exists C > k F ( x ) − F ( y ) k ≤ C | x − y | for all x, y ∈ D and C := sup x,y ∈ D e dd B ( x,y ) < ∞ , C := sup x ∈ D k F ( x ) k < ∞ . (4.107)Note that for any x, y ∈ D , we have e − sd B ( x,y ) R X ( s, y ; F ) ≤ R X ( s, x ; F ) ≤ e sd B ( x,y ) R X ( s, y ; F ) . Therefore, for any x, y ∈ D and any s ∈ ( d, d ), by using the elementary inequality | e ± sd B ( x,y ) − | ≤ e sd B ( x,y ) − ≤ e dd B ( x,y ) − C and C , we obtain k R X ( s, x ; F ) − F ( x ) k ≤ max ± k e ± sd B ( x,y ) R X ( s, y ; F ) − F ( x ) k≤ max ± n e ± sd B ( x,y ) (cid:16) k R X ( s, y ; F ) − F ( y ) k + k F ( y ) − F ( x ) k (cid:17) + | e ± sd B ( x,y ) − |k F ( x ) k o ≤ e sd B ( x,y ) (cid:16) k R X ( s, y ; F ) − F ( y ) k + k F ( y ) − F ( x ) k (cid:17) + | e sd B ( x,y ) − |k F ( x ) k≤ C k R X ( s, y ; F ) − F ( y ) k + C C | x − y | + C ( e dd B ( x,y ) − . (4.108)For any ε ∈ (0 , c > D such thatsup x,y ∈ D : | x − y |≤ cε n C C | x − y | + C ( e dd B ( x,y ) − o < ε . (4.109)Let D ε ⊂ D be any fixed finite cε -net of D with respect to the Euclidean metric, that is,for any x ∈ D , we have inf z ∈ D ε | z − x | ≤ cε. By a classical volume argument, there exists a constant c ′ > D ε ⊂ D can be chosen with cardinality D ε ≤ c ′ ε d . (4.110)Combining (4.108) with (4.109), by our choice of D ε , we obtain(4.111) P d (cid:16) sup z ∈ D k R X ( s, z ; F ) − F ( z ) k > ε (cid:17) ≤ P d (cid:16) sup z ∈ D ε k R X ( s, z ; F ) − F ( z ) k > ε C (cid:17) ≤ X z ∈ D ε P d (cid:16) k R X ( s, z ; F ) − F ( z ) k > ε C (cid:17) . The Chebychev inequality and Lemma 4.7 imply that, for any s ∈ ( d, d ) and z ∈ D ε ,(4.112) P d (cid:16) k R X ( s, z ; F ) − F ( z ) k > ε C (cid:17) ≤ C ε g P d ( s )] Z D d e − sd B ( x,z ) k F ( x ) k dµ D d ( x ) ≤ C ε C [ g P d ( s )] Z D d e − sd B ( x,o ) k F ( x ) k dµ D d ( x ) , where C := sup z ∈ D e dd B ( z,o ) < ∞ . Combining (4.110), (4.111) and (4.112), we obtain the desired inequality: P d (cid:16) sup z ∈ D k R X ( s, z ; F ) − F ( z ) k > ε (cid:17) ≤ C C c ′ ε d +2 g P d ( s )] Z D d e − sd B ( x,o ) k F ( x ) k dµ D d ( x ) . Claim VII. Let D ⊂ D d be any relatively compact subset. Then for P d -almost every X ∈ Conf( D d ), we have lim n →∞ sup z ∈ D k R X ( s n , z ; F ) − F ( z ) k = 0 . (4.113)where the sequence ( s n ) n ≥ is defined in (4.95).Indeed, recall that the sequence ( s n ) n ≥ defined in (4.95) satisfies the condition (4.68).Thus the inequality (4.106) implies that for any ε ∈ (0 , ∞ X n =1 P d (cid:16) sup z ∈ D k R X ( s n , z ; F ) − F ( z ) k > ε (cid:17) < ∞ . Since ε ∈ (0 , 1) is arbitrary, we obtain that for P d -almost every X ∈ Conf( D d ), the desiredlimit equality (4.113) holds. Claim VIII. Let D ⊂ D d be any relatively compact subset. Then for P d -almost every X ∈ Conf( D d ), we have lim s → d + sup z ∈ D k R X ( s, z ; F ) − F ( z ) k = 0 . (4.114)Indeed, repeating exactly the same derivation of the P d -almost sure limit equality(4.105) from (4.100), we obtain the proof of the P d -almost sure limit equality (4.114)from (4.113). (cid:3) ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 41 The reproducing kernels with super-critical weights This section is devoted to the proof of Proposition 4.17. Lemma 5.1. There exist two constants c, C > depending only on d , such that c (1 − | z | ) d log (cid:16) − | z | (cid:17) ≤ K W cr ( z, z ) ≤ C (1 − | z | ) d log (cid:16) − | z | (cid:17) for all z ∈ D d . The proof of the following elementary lemma is routine and will be omitted. Lemma 5.2. Let ( a k ) k ∈ N , ( b k ) k ∈ N be two sequences in R + with lim k →∞ a k /b k = 0 . Assumethat P k ∈ N a k t k and P k ∈ N b k t k converge for all t ∈ (0 , and P k ∈ N b k = ∞ . Then lim t → − P k ∈ N a k t k P k ∈ N b k t k = 0 . Lemma 5.3. For any integer d ≥ , there exist constants c d , C d > such that c d (1 − t ) d log (cid:16) − t (cid:17) ≤ ∞ X k =0 ( k +1) d − log( k +2) t k ≤ C d (1 − t ) d log (cid:16) − t (cid:17) for all t ∈ (0 , . Proof. For any integer m ≥ t ∈ (0 , − t ) ∞ X k =0 ( k + 1) m log( k + 2) t k = ∞ X k =0 h ( k + 1) m log( k + 2) − k m log( k + 1) | {z } denoted a k,m i t k . Clearly, there exist contants c m , C m > m such that c m · ( k + 1) m − log( k + 2) ≤ a k,m ≤ C m · ( k + 1) m − log( k + 2) for all k ≥ . Therefore, we have c m − t ≤ P ∞ k =0 ( k + 1) m log( k + 2) t k P ∞ k =0 ( k + 1) m − log( k + 2) t k ≤ C m − t for all t ∈ (0 , . It follows that, there exist constants c ′ d , C ′ d > c ′ d (1 − t ) d − ≤ P ∞ k =0 ( k + 1) d − log( k + 2) t k P ∞ k =0 log( k + 2) t k ≤ C ′ d (1 − t ) d − for all t ∈ (0 , . Finally, note that for any t ∈ (0 , − t ) ∞ X k =0 log( k + 2) t k = ∞ X k =0 log (cid:16) k + 1 (cid:17) t k ≤ log 2 + ∞ X k =1 t k k = log (cid:16) − t (cid:17) and there exists c ′′ > t ∈ (0 , − t ) ∞ X k =0 log( k + 2) t k = ∞ X k =0 log (cid:16) k + 1 (cid:17) t k ≥ c ′′ (cid:16) log 2 + ∞ X k =1 t k k (cid:17) = c ′′ log (cid:16) − t (cid:17) . Combining the above inequalities, we complete the proof of the lemma. (cid:3) Proof of Lemma 5.1. Since W cr is radial, the polynomials ( z n := z n · · · z n d d ) n ∈ N d are or-thogonal and complete in A ( D d , W cr ). Thus K W cr ( z, w ) = X n ∈ N d a n ( W cr ) z n ¯ w n with a n ( W cr ) = k z n k − A ( D d ,W cr ) . (5.115) For any n ∈ N d , we have the identity (see e.g. Zhu [36, Lem. 1.11]), Z S d | ζ n | dσ S d ( ζ ) = ( d − n ! · · · n d !( d − | n | )! . Therefore, by the formula of integration in polar coordinates, a n ( W cr ) − = Z D d | z n | W cr ( z ) dv d ( z ) = 2 d Z r d +2 | n |− W cr ( r ) dr Z S d | ζ n | dσ S d ( ζ )= d · n ! · · · n d !( d − | n | )! Z t | n | + d − (1 − t ) − log − (cid:16) − t (cid:17) dt = d · n ! · · · n d !( d − | n | )! · Z ∞ log 4 (1 − e − x ) | n | + d − x dx. Claim: There exist two constants c , c > c log(4 k + 4) ≤ Z ∞ log 4 (1 − e − x ) k x dx ≤ c log(4 k + 4) for all integer k ≥ . Indeed, for any integer k ≥ 0, we have the lower-estimate of the integral: Z ∞ log 4 (1 − e − x ) k dxx ≥ Z ∞ log(4 k +4) (1 − e − x ) k dxx ≥ Z ∞ log(4 k +4) (cid:16) − k + 1 (cid:17) k dxx ≥ c log(4 k + 4) . Now set H k ( x ) := (1 − e − x ) k x for x ∈ [log 4 , ∞ ) . For k ≥ 1, we can show, by studying the derivative H ′ k ( x ), that H k is increasing on theinterval [log 4 , log(4 k + 4)] and hence H k ( x ) ≤ H k (log(4 k + 4)) ≤ (4 k + 4) for all x ∈ [log 4 , log(4 k + 4)] . Therefore, for any k ≥ 1, we have Z ∞ log 4 (1 − e − x ) k x dx ≤ Z log(4 k +4)log 4 (1 − e − x ) k x dx + Z ∞ log(4 k +4) (1 − e − x ) k x dx ≤≤ sup log 4 ≤ x ≤ log(4 k +4) H k ( x ) · Z log(4 k +4)log 4 dx + Z ∞ log(4 k +4) x dx ≤ k + 4) . Consequently, there exist constants c , c > n ∈ N d , we have c ≤ a n ( W cr ) − d · n ! · · · n d !( d − | n | )! 1log(4( | n | + d )) ≤ c . (5.116)By using the elementary identity (see e.g. Zhu [36, formula (1.1)]) X n ∈ N d : | n | = k r n · · · r n d d n ! · · · n d ! = ( r + · · · + r d ) k k ! , ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 43 we obtain, for any z ∈ D d , thatΦ( z ) =: X n ∈ N d ( d − | n | )! d log(4( | n | + d )) n ! · · · n d ! | z n | = ∞ X k =0 ( d − k )! d · k ! log(4( k + d )) | z | k . Therefore, by the limit equalitieslim k →∞ ( d − k )!( k + 1) d − · k ! = 1 and lim k →∞ log(4( k + d ))log( k + 2) = 1and Lemma 5.3, there exist constants c , c > 0, such that c (1 − | z | ) d log (cid:16) − | z | (cid:17) ≤ Φ( z ) ≤ c (1 − | z | ) d log (cid:16) − | z | (cid:17) . Finally, comparing (5.115) with the definition of Φ( z ) and using (5.116), we obtain c Φ( z ) ≤ K W cr ( z, z ) ≤ c Φ( z ) for all z ∈ D d . This completes the whole proof. (cid:3) Let Φ : [0 , → R + be a function in L ([0 , R δ Φ( r ) dr > δ ∈ (0 , B : [0 , → R + be a function such that B Φ ∈ L ([0 , r → − B ( r ) = ∞ . Define two radial Bergman-admissible weights on D d by W Φ ( z ) = Φ( | z | ) and W B Φ ( z ) = B ( | z | )Φ( | z | ) . We shall compare the reproducing kernels K W Φ and K W B Φ . Note thatlim k →∞ R r k Φ( r ) dr R r k B ( r )Φ( r ) dr = 0 . (5.117)Indeed, for any ε > 0, take r ε ∈ (0 , 1) such that ε · B ( r ) ≥ r ∈ [ r ε , δ ε with r ε < δ ε < 1, we havelim sup k →∞ R r ε r k Φ( r ) dr R r k B ( r )Φ( r ) dr ≤ lim sup k →∞ R r ε r k Φ( r ) dr R δ ε r k B ( r )Φ( r ) dr ≤ lim sup k →∞ r kε · R r ε Φ( r ) drδ kε · R δ ε B ( r )Φ( r ) dr = 0 . Therefore, we havelim sup k →∞ R r k Φ( r ) dr R r k B ( r )Φ( r ) dr ≤ lim sup k →∞ R r ε r k Φ( r ) dr + ε · R r ε r k B ( r )Φ( r ) dr R r k B ( r )Φ( r ) dr ≤ ε. Since ε > Lemma 5.4. Under the above assumptions on Φ and B , we have lim | z |→ − K W B Φ ( z, z ) K W Φ ( z, z ) = 0 . Proof. Since W Φ : D d → R + is radial, we have K W Φ ( z, w ) = X n ∈ N d a n (Φ) z n ¯ w n with a n (Φ) = k z n k − A ( D d ,W Φ ) = k z n · · · z n d d k − A ( D d ,W Φ ) . For any n ∈ N d , by the formula of integration in polar coordinates, a n (Φ) − = Z D d | z n | Φ( | z | ) dv d ( z ) = 2 d Z r d +2 | n |− Φ( r ) dr Z S d | ζ n | dσ S d ( ζ ) . Replacing Φ by B Φ, we obtain the corresponding formulas for K W B Φ and for a n ( B Φ). Inparticular, we see that the ratio a n ( B Φ) a n (Φ) depends only on | n | : R | n | := a n ( B Φ) a n (Φ) = R r d +2 | n |− W ( r ) dr R r d +2 | n |− B ( r ) W ( r ) dr . Then for any z ∈ D d , by writing z = rζ with r = | z | and ζ = z/r ∈ S d , we have K W Φ ( z, z ) = ∞ X k =0 (cid:16) X n ∈ N d : | n | = k a n (Φ) | z n | (cid:17) = ∞ X k =0 (cid:16) X n ∈ N d : | n | = k a n (Φ) | ζ n | (cid:17) r k ,K W B Φ ( z, z ) = ∞ X k =0 (cid:16) X n ∈ N d : | n | = k a n ( B Φ) | z n | (cid:17) = ∞ X k =0 R k · (cid:16) X n ∈ N d : | n | = k a n (Φ) | ζ n | (cid:17) r k . Since W Φ is radial, the transformation f ( · ) f ( U · ) is unitary on A ( D d , W Φ ) for any d × d unitary matrix U . Hence by (3.52), the function z K W Φ ( z, z ) is radial and thus A k (Φ) := X n ∈ N d : | n | = k a n (Φ) | ζ n | does not depend on ζ ∈ S d . Now by (5.117), we have lim k →∞ R k = 0. Finally, by Lemma 5.2, we obtainlim sup | z |→ − K W B Φ ( z, z ) K W Φ ( z, z ) = lim sup r → − P ∞ k =0 R k A k (Φ) r k P ∞ k =0 A k (Φ) r k = 0 . This completes the proof of Lemma 5.4. (cid:3) Proof of Proposition 4.17. We can write W cr = W Φ for a function Φ : [0 , → R + . Let W be a super-critical weight on D d . Set B ( r ) := inf ζ ∈ S d W ( rζ )Φ( r )for any r ∈ [0 , r → − B ( r ) = ∞ . Hence by Lemma 5.4, we havelim | z |→ − K W B Φ ( z, z ) K W cr ( z, z ) = lim | z |→ − K W B Φ ( z, z ) K W Φ ( z, z ) = 0 . Since B ( | z | )Φ( | z | ) ≤ W ( z ), by (3.52), we have K W ( z, z ) ≤ K W B Φ ( z, z ) . The desiredestimate of K W ( z, z ) now follows from Lemma 5.1. (cid:3) Impossibility of simultaneous uniform interpolations In this section, we are going to prove Proposition 1.9, Theorem 1.12 and Proposi-tion 1.13 and their higher dimensional counterparts. As in § 4, the determinantal pointprocess P K D d will be denoted simply by P d := P K D d . In particular, for d = 1, we use the simplified notation P := P = P K D . ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 45 The variances of some linear statistics. Recall the definition (1.33) of W cr .Define a harmonic function F cr : D → A ( D , W cr ) by F cr ( w ) := K W cr ( · , w ) . By Lemma 4.6 and Lemma 5.1, for any s > , z ∈ D , for P -almost every X ∈ Conf( D ),we may define g X ( s, z ; F cr ) as in (4.64). In particular, we will denote M X ( s ) := g X ( s, F cr ) = ∞ X k =0 X x ∈ X ∩ A k (0) e − sd D ( x, F cr ( x ) . Define a positive symmetric measure dM on D = D × D by dM ( x, y ) := | K D ( x, y ) | dA ( x ) dA ( y ) . (6.118) Lemma 6.1. For any s > , the variance of M X ( s ) is given by Var P ( M X ( s )) = 12 Z D | e − sd D ( x, − e − sd D ( y, | K W cr ( x, y ) dM ( x, y ) . (6.119) Remark . Although K W cr ( x, y ) is complex-valued, the integral in (6.119) is positive.Let F D : D → A ( D ) be the function defined by F D ( w ) := K D ( · , w ) ∈ A ( D ) . For any bounded compactly supported radial function R : D → R + and any X ∈ Conf( D ),we define g R X ( z ; F D ) := X x ∈ X R ( ϕ z ( x )) F D ( x ) where ϕ z ( x ) = z − x − ¯ zx . (6.120) Lemma 6.2. For any bounded compactly supported radial function R : D → R + , we have Var P ( g R X ( z ; F D )) = 12 Z D |R ( ϕ z ( x )) − R ( ϕ z ( y )) | K D ( x, y ) dM ( x, y ) . Clearly, both Lemma 6.1 and Lemma 6.2 will follow from Lemma 2.1 and the following Lemma 6.3. Let W be a Bergman-admissible weight on D such that inf w ∈ D W ( w ) > .Let P : D → R be a real-valued function such that Z D P ( x ) | K W ( x, y ) | dM ( x, y ) < ∞ . Then, by writing k · k = k · k A ( D ,W ) and recalling the definition (4.73) of F W , we have Z D kP ( x ) F W ( x ) − P ( y ) F W ( y ) k dM ( x, y ) = Z D |P ( x ) − P ( y ) | K W ( x, y ) dM ( x, y ) . Proof. We first show that for any z ∈ D , Z D K W ( w, z ) | K D ( z, w ) | dA ( w ) = K W ( z, z ) K D ( z, z );(6.121) Z D K W ( z, w ) | K D ( z, w ) | dA ( w ) = K W ( z, z ) K D ( z, z ) . (6.122) Fix any z ∈ D , define f z ( w ) := K W ( w, z ) K D ( w, z ) for w ∈ D . Then f z ∈ A ( D ) since it isholomorphic and Z D | f z ( w ) | dA ( w ) ≤ sup w ∈ D | K D ( w, z ) | W ( w ) · Z D | K W ( w, z ) | W ( w ) dA ( w )= sup w ∈ D | K D ( w, z ) | W ( w ) · K W ( z, z ) < ∞ . Therefore, we have f z ( z ) = R D K D ( z, w ) f z ( w ) dA ( w ) . Using f z ( z ) = K W ( z, z ) K D ( z, z ), K W ( z, w ) f z ( w ) = K W ( w, z ) | K D ( z, w ) | , we obtain the equality (6.121). Taking complexcongugate on both sides of (6.121), we obtain (6.122).Now define H ( x, y ) := kP ( x ) F W ( x ) − P ( y ) F W ( y ) k . Note that we have H ( x, y ) = P ( x ) K W ( x, x ) + P ( y ) K W ( y, y ) − P ( x ) P ( y ) K W ( x, y ) − P ( x ) P ( y ) K W ( y, x ) . By exchanging the integration variables x, y , we have Z D P ( x ) P ( y ) K W ( x, y ) dM ( x, y ) = Z D P ( x ) P ( y ) K W ( y, x ) dM ( x, y ); Z D P ( x ) K W ( x, x ) dM ( x, y ) = Z D P ( y ) K W ( y, y ) dM ( x, y ) . Using first the equality (6.122) and then R D | K D ( x, y ) | dA ( y ) = K D ( x, x ), we obtain I ( P ) := Z D P ( x ) K W ( x, y ) dM ( x, y ) = Z D P ( x ) K W ( x, x ) K D ( x, x ) dA ( x )= Z D P ( x ) K W ( x, x ) dM ( x, y ) . Hence I ( P ) ∈ R . Thus, by taking complex conjugate (using the equality K W ( x, y ) = K W ( y, x )) and then exchanging the integration variables x, y , we obtain Z D P ( x ) K W ( x, y ) dM ( x, y ) = Z D P ( y ) K W ( x, y ) dM ( x, y ) . Combining all the above equalities, we obtain Z D HdM = 2 Z D P ( x ) K W ( x, x ) dM ( x, y ) − Z D P ( x ) P ( y ) K W ( x, y ) dM ( x, y )= Z D P ( x ) K W ( x, y ) dM ( x, y ) + Z D P ( y ) K W ( x, y ) dM ( x, y ) − Z D P ( x ) P ( y ) K W ( x, y ) dM ( x, y )= Z D |P ( x ) − P ( y ) | K W ( x, y ) dM ( x, y ) . This completes the proof of Lemma 6.3. (cid:3) ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 47 Proof of Proposition 1.9. The proof of Proposition 1.9 relies on the following Lemma 6.4. There exists C > such that Var P ( M X ( s )) ≥ C · ( s − − for s ∈ (1 , .Proof of Proposition 1.9. Writing k · k = k · k A ( D ,W cr ) . By (4.65), we have E P (cid:16) sup f ∈B ( W cr ) (cid:12)(cid:12)(cid:12) g X ( s, f ) g P ( s ) − f (0) (cid:12)(cid:12)(cid:12) (cid:17) = E P (cid:16) k M X ( s ) − g P ( s ) F cr (0) k g P ( s ) (cid:17) = Var P ( M X ( s )) g P ( s ) . Therefore, Proposition 1.9 follows from Lemma 3.2 and Lemma 6.4. (cid:3) It remains to prove Lemma 6.4. Set E ( t ) := 1(1 − t ) log (cid:16) − t (cid:17) , t ∈ (0 , . (6.123) Lemma 6.5. There exists a constant c > such that for any s > , we have Var P ( M X ( s )) ≥ c Z D Z D | e − sd D (0 ,x ) − e − sd D (0 ,y ) | E ( | xy | ) dA ( x ) dA ( y ) . Proof. Set D ( x, y ) := K W cr ( x, y ) | K D ( x, y ) | . Since the function x e − sd D (0 ,x ) is radial,by Lemma 6.1, we haveVar P ( M X ( s )) = 12 Z D Z D | e − sd D (0 ,x ) − e − sd D (0 ,y ) | D ( x, y ) dA ( x ) dA ( y )= 12 Z D Z D | e − sd D (0 ,x ) − e − sd D (0 ,y ) | h Z π D ( xe iθ , y ) dθ π | {z } denoted by D ♯ ( x, y ) i dA ( x ) dA ( y ) . By (5.115) and (5.116), we have K W cr ( x, y ) = ∞ X n =0 a n ( W cr ) x n ¯ y n with a n ( W cr ) ≥ c ′ · log( n + 2) for any n ∈ N ,where c ′ > D ♯ ( x, y ) = ∞ X n =0 a n ( W cr ) Z π ( x ¯ ye iθ ) n · (cid:12)(cid:12)(cid:12) − x ¯ ye iθ ) (cid:12)(cid:12)(cid:12) dθ π = ∞ X n =0 a n ( W cr ) | xy | n h ( n + 1) | xy | n (1 − | xy | ) + 2 | xy | n +2 (1 − | xy | ) i ≥ c ′ ∞ X n =0 ( n + 1) log( n + 2) | xy | n (1 − | xy | ) + 2 c ′ ∞ X n =0 log( n + 2) | xy | n +2 (1 − | xy | ) . Then by Lemma 5.3, there exists c > D ♯ ( x, y ) ≥ c (1 − | xy | ) log (cid:16) − | xy | (cid:17) = c · E ( | xy | ) . This completes the proof of the lemma. (cid:3) Proof of Lemma 6.4. By Lemma 5.3, there exists c > E ( t ) = 1(1 − t ) log (cid:16) − t (cid:17) ≥ c ∞ X n =0 n log( n + 2) t n for all t ∈ (0 , G s ( x, y ) := | e − sd D (0 ,x ) − e − sd D (0 ,y ) | . By Lemma 6.5, there exists c ′ > s > 1, we have(6.125) Var P ( M X ( s )) ≥ c Z D Z D G s ( x, y ) E ( | xy | ) dA ( x ) dA ( y ) ≥≥ c ′ ∞ X n =0 n log( n + 2) Z D Z D G s ( x, y ) | xy | n dA ( x ) dA ( y ) | {z } denoted by I n ( s ) . Note that for any n ∈ N , the integral I n ( s ) is finite for any s ≥ I n ( s ) = 2 V ( n, s ) − U ( n, s ) , where U ( n, s ) and V ( n, s ) are defined for all n ∈ N and all s ≥ U ( n, s ) : = Z D Z D e − sd D (0 ,x ) − sd D (0 ,y ) | xy | n dA ( x ) dA ( y ) = (cid:16) Z (cid:16) − r r (cid:17) s r n +1 dr (cid:17) ; V ( n, s ) : = Z D Z D e − sd D (0 ,x ) | xy | n dA ( x ) dA ( y ) = 12 n + 2 Z (cid:16) − r r (cid:17) s r n +1 dr. (6.126) Claim A: we have γ := sup n ∈ N sup s ∈ [1 , U ( n, s ) V ( n, s ) < . (6.127)Let us complete the proof of the lemma by using Claim A. By (6.127), we have I n ( s ) = 2 V ( n, s ) − U ( n, s ) ≥ − γ ) V ( n, s ) for any n ∈ N and any s ∈ (1 , P ( M X ( s )) ≥ c ′ (1 − γ ) ∞ X n =0 n log( n + 2) V ( n, s ) =: 2 c ′ (1 − γ )Σ V ( s ) . Now by Lemma 5.3, there exists a constant c ′′ > V ( s ) = ∞ X n =0 n log( n + 2)2 n + 2 Z (cid:16) − r r (cid:17) s r n +1 dr ≥ c ′′ Z (cid:16) − r r (cid:17) s · r log( − r )(1 − r ) dr == c ′′ Z (cid:16) − √ t √ t (cid:17) s · log( − t )(1 − t ) dt ≥ c ′′ s Z (1 − t ) s − log (cid:16) − t (cid:17) dt. By change of variable t = 1 − e − x , we obtain, for any s ∈ (1 , Z (1 − t ) s − log (cid:16) − t (cid:17) dt = 2 s − Z ∞ log 2 e − s − x xdx ≥≥ s − Z ∞ ( s − 1) log 2 e − x xdx ≥ s − Z ∞ log 2 e − x xdx. Thus, there exists C > P ( M X ( s )) ≥ C · ( s − − for any s ∈ (1 , ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 49 It remains to prove Claim A. By Cauchy-Bunyakovsky-Schwarz inequality, U ( n, s ) 1. For any n ∈ N , since U ( n,s ) V ( n,s ) is continuous on s , we have γ n := sup s ∈ [1 , U ( n, s ) V ( n, s ) < . (6.128)Write r n := 1 − √ n +1 . There exists c > s ∈ [1 , 2] and any n ∈ N , Z r n (cid:16) − r r (cid:17) s r n +1 dr ≤ sup ≤ r ≤ r n (cid:16) − r r (cid:17) · Z r n r n +1 dr ≤≤ √ n + 1 · (1 − √ n +1 ) n +2 n + 2 ≤ e − c √ n ( n + 1) / . and Z (cid:16) − r r (cid:17) s r n +1 dr ≥ Z (1 − r ) r n +1 dr = Γ(2 n + 2)Γ(2 n + 5) ≥ n + 1) . It follows that for any s ∈ [1 , R r n ( − r r ) s · r n +1 dr R ( − r r ) s · r n +1 dr ≤ n + 1) / e − c √ n | {z } denoted by α n and hence Z (cid:16) − r r (cid:17) s r n +1 dr ≤ − α n Z r n (cid:16) − r r (cid:17) s r n +1 dr. Thus for any s ∈ [1 , 2] and any n ∈ N , we have U ( n, s ) V ( n, s ) ≤ [ − α n R r n ( − r r ) s · r n +1 dr ] n +2 R ( − r r ) s · r n +1 dr ≤ − α n ) r n ) s [ R (1 − r ) s r n +1 dr ] n +2 12 s R (1 − r ) s r n +1 dr == 2 n + 2(1 − α n ) (cid:16) 21 + r n (cid:17) s · Γ( s + 1) Γ(2 n + 2) Γ(2 n + 3 + s ) Γ(2 n + 3 + 2 s )Γ(2 s + 1)Γ(2 n + 2) == (cid:16) 21 + r n (cid:17) s − α n ) Γ( s + 1) Γ(2 s + 1) Γ(2 n + 3)Γ(2 n + 3 + 2 s )Γ(2 n + 3 + s ) . By Cauchy-Bunyakovsky-Schwarz inequality, we haveΓ( s + 1) = (cid:16) Z ∞ t s e − t dt (cid:17) < Z ∞ t s e − t dt · Z ∞ e − t dt = Γ(2 s + 1) . Since the function [1 , ∋ s Γ( s +1) Γ(2 s +1) is continuous, we have sup s ∈ [1 , 2] Γ( s +1) Γ(2 s +1) < . Note thatthe limit equality lim n →∞ Γ( n + t ) n t Γ( n ) = 1 holds uniformly for t in a compact subset of R , we havelim n →∞ sup s ∈ [1 , (cid:12)(cid:12)(cid:12) Γ(2 n + 3)Γ(2 n + 3 + 2 s )Γ(2 n + 3 + s ) − (cid:12)(cid:12)(cid:12) = 0and thus lim sup n →∞ γ n = lim sup n →∞ sup s ∈ [1 , U ( n, s ) V ( n, s ) ≤ sup s ∈ [1 , Γ( s + 1) Γ(2 s + 1) < . (6.129)Combining (6.129) and (6.128), we obtain the desired inequality (6.127). (cid:3) Proof of Theorem 1.12. Recall the definition (6.120) of g R X ( z ; F D ). Proposition 6.6. Let R : D → R + be bounded compactly supported and radial. Then Var P ( g R X ( z ; F D )) = 12(1 − | z | ) Z D Z D |R ( x ) − R ( y ) | (1 − | xy | ) · I z ( x, y ) dA ( x ) dA ( y ) for any z ∈ D , where I z ( x, y ) is given by the formula: I z ( x, y ) = 1 + (3 + 8 | z | ) | xy | + (3 | z | + 8 | z | ) | xy | + | z | | xy | . Proof. Recall the definition (6.118) of the measure dM ( x, y ) on D . We have dM ( x, y ) = Φ( x, y ) dµ D ( x ) dµ D ( y ) with Φ( x, y ) = | K D ( x, y ) | K D ( x, x ) K D ( y, y ) . Note that Φ( x, y ) = Φ( ϕ z ( x ) , ϕ z ( y )). Indeed, this can be derived from the identity (cf.Rudin [27, Thm. 2.2.2]):1 − ϕ z ( x ) · ϕ z ( y ) = (1 − | z | )(1 − x · ¯ y )(1 − x · ¯ z )(1 − z · ¯ y ) x, y, z ∈ D . (6.130)Therefore, by Lemma 6.2 and the conformal invariance of the measure µ D , we haveVar P ( g R X ( z ; F D )) = 12 Z D |R ( x ) − R ( y ) | K D ( ϕ z ( x ) , ϕ z ( y )) | K D ( x, y ) | dA ( x ) dA ( y ) . Now since R is radial, we haveVar P ( g R X ( z ; F D )) = 12 Z D |R ( x ) − R ( y ) | J z ( x, y ) dA ( x ) dA ( y ) , where J z ( x, y ) is defined by J z ( x, y ) := Z π Z π K D ( ϕ z ( xe iθ ) , ϕ z ( ye − iθ )) | K D ( xe iθ , ye − iθ ) | dθ π dθ π . Using (6.130), we have J z ( x, y ) = 1(1 − | z | ) Z π Z π (1 − xe iθ · ¯ z ) (1 − z · ¯ ye iθ ) (1 − x ¯ ye iθ e iθ ) (1 − ¯ xye − iθ e − iθ ) dθ dθ π = 1(1 − | z | ) · π I C dζ iζ I C dζ iζ (1 − x ¯ zζ ) (1 − z ¯ yζ ) (1 − x ¯ yζ ζ ) (1 − ¯ xyζ − ζ − ) , where C the unit circle oriented counterclockwise. Using the residue method, we obtain J z ( x, y ) = 1(1 − | z | ) h | zxy | + 3 | zxy | (1 − | xy | ) + 4 | xy | (1 + 4 | zxy | + | zxy | )(1 − | xy | ) i = 1 + (3 + 8 | z | ) | xy | + (3 | z | + 8 | z | ) | xy | + | z | | xy | (1 − | z | ) · (1 − | xy | ) . This completes the proof of the proposition. (cid:3) Proof of Theorem 1.12. Fix any z ∈ D . Note that E P h sup f ∈ A ( D ): k f k≤ (cid:12)(cid:12)(cid:12) g R X ( z, f ) g R P − f ( z ) (cid:12)(cid:12)(cid:12) i = Var P ( g R X ( z ; F D ))( g R P ) . ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 51 Since R : D → R + is radial and compactly supported, there exists Φ : [0 , → R + withsupp(Φ) ⊂ [0 , − ε ] for some ε > R ( z ) = Φ( | z | ). Set V ( t ) := Φ( √ − t ) , G ( t ) := V ( t ) t = Φ( √ − t ) t , where t ∈ (0 , . (6.131)Note that supp( V ) = supp( G ) ⊂ [ ε ′ , 1] for some ε ′ ∈ (0 , g R P = E P ( g R X ( z )) = E P ( g R X (0)) = Z D Φ( | x | )(1 − | x | ) dA ( x ) = Z Φ( √ − t ) t dt = Z G ( t ) dt. By Proposition 6.6, we haveVar P ( g R X ( z ; F D )) ≥ Z D Z D |R ( x ) − R ( y ) | (1 − | xy | ) dA ( x ) dA ( y )= 2 Z Z | Φ( r ) − Φ( r ) | (1 − r r ) r r dr dr = 12 Z Z | Φ( √ t ) − Φ( √ t ) | (1 − t t ) dt dt . Then by using the change of variables t = 1 − s , t = 1 − s , we obtainVar P ( g R X ( z ; F D )) ≥ Z Z | V ( s ) − V ( s ) | ( s + s − s s ) ds ds = Z ≤ s ≤ s ≤ | V ( s ) − V ( s ) | ( s + s − s s ) ds ds . Using change of variables s = λt, s = t , we obtainVar P ( g R X ( z ; F D )) ≥ Z ≤ s ≤ s ≤ | V ( s ) − V ( s ) | ( s + s − s s ) ds ds = Z Z | V ( λt ) − V ( t ) | ( λt + t − λt ) tdλdt ≥≥ Z Z | V ( λt ) − V ( t ) | ( λt + t ) tdλdt = Z Z | λ G ( λt ) − G ( t ) | ( λ + 1) dλdt ≥≥ Z Z | λ G ( λt ) − G ( t ) | dλdt. Now note that Z Z (cid:12)(cid:12) λ G ( λt ) (cid:12)(cid:12) dλdt = Z λ dλ Z λ G ( t ′ ) dt ′ ≤ Z G ( t ) dt. Hence, using the triangle inequality on the space L ( dλdt ) = L ([0 , ), we obtain(6.132) q · Var P ( g R X ( z ; F D )) ≥ k λ G ( λt ) − G ( t ) k L ( dλdt ) ≥≥ k G ( t ) k L ( dλdt ) − k λ G ( λt ) k L ( dλdt ) ≥≥ k G ( t ) k L ( dλdt ) − k G ( t ) k L ( dλdt ) k G ( t ) k L ( dλdt ) k G ( t ) k L ( dt ) . Therefore, Var P ( g R X ( z ; F D ))( g R P ) ≥ k G ( t ) k L ( dt ) [ R G ( t ) dt ] ≥ . This completes the proof of Theorem 1.12. (cid:3) Proof of Proposition 1.13. Fix an exponent s ∈ (1 , ]. For any N ∈ N , setΦ( r ) = (cid:16) − r r (cid:17) s , Φ N ( r ) = Φ( r ) (cid:16) log (cid:16) − r r (cid:17) ≤ N + 1 (cid:17) , R N ( x ) = Φ N ( | x | ) . Writing k · k = k · k A ( D ) , by (6.132), for any N ∈ N and any z ∈ D , we have E P (cid:16) sup f ∈ A ( D ): k f k≤ (cid:12)(cid:12)(cid:12) N X k =0 g ( k ) X ( s, z ; f ) (cid:12)(cid:12)(cid:12) (cid:17) = E P ( k g R N X ( z ; F D ) k ) ≥≥ Var P ( g R N X ( z ; F D )) ≥ (cid:16) Z Φ N ( √ − t ) t dt (cid:17) . Therefore, by using the assumption 1 < s ≤ , we obtain the claimed result:sup N ∈ N E P (cid:16) sup f ∈ A ( D ): k f k≤ (cid:12)(cid:12)(cid:12) N X k =0 g ( k ) X ( s, z ; f ) (cid:12)(cid:12)(cid:12) (cid:17) ≥ (cid:16) Z Φ( √ − t ) t dt (cid:17) = ∞ . The case of dimension d ≥ . In this subsection, we always assume that thedimension d ≥ 2. Let F d : D d → A ( D d ) be the function defined by F d ( w ) := K D d ( · , w ) ∈ A ( D d ) . For any bounded compactly supported radial function R : D d → R + and any configuration X ∈ Conf( D d ), we define g R X ( z ; F d ) := X x ∈ X R ( ϕ z ( x )) F d ( x ) , where ϕ z ( x ) is defined as in (3.42). Theorem 6.7. For any integer d ≥ , there exists a constant c d > , such that for anycompactly supported radial weight R : D d → R + and any z o ∈ D d , we have E P d h sup f ∈ A ( D d ): k f k≤ (cid:12)(cid:12)(cid:12) g R X ( z o , f ) g R P d − f ( z o ) (cid:12)(cid:12)(cid:12) i ≥ c d d (1 − | z o | ) d +1 d d +2 (cid:16) − √ d + 2 (cid:17) , (6.133) where g R P d = E P d (cid:16) X x ∈ X R ( ϕ z o ( x )) (cid:17) = E P d (cid:16) X x ∈ X R ( x ) (cid:17) . Proposition 6.8 below will play the same role as that that of Proposition 6.6 for d = 1. Proposition 6.8. For any integer d ≥ , there exists a constant c d > , such that forany compactly supported radial weight R : D d → R + and any z o ∈ D d , Var P d ( g R X ( z o ; F d )) ≥ c d (1 − | z o | ) d +1 Z D d Z D d |R ( z ) − R ( w ) | (1 − | z | | w | ) d +1 dv d ( z ) dv d ( w ) . (6.134) Remark . By taking d = 1 on the right hand side of the inequality (6.134), we obtaina term (1 − | z | | w | ) − , which is different to the correct order (1 − | z | | w | ) − obtained inProposition 6.6 for the case of dimension d = 1. ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 53 Lemma 6.9. If P : D d → R is a compact supported real-valued measurable function, then Var P d h X x ∈ X P ( x ) F d ( x ) i = 12 Z D d Z D d |P ( z ) − P ( w ) | · K D d ( z, w ) | K D d ( z, w ) | dv d ( z ) dv d ( w ) . Proof. Lemma 6.9 is a consequence of Lemma 2.1 and the higher dimensional counterpartof Lemma 6.3 (whose proof is exactly the same as Lemma 6.3). (cid:3) Proof of Proposition 6.8. Fix z o ∈ D d . Using Lemma 6.9, the invariance of the measure | K D d ( z, w ) | dv d ( z ) dv d ( w )under the diagonal action of Aut( D d ) and the identity (cf. Rudin [27, Theorem 2.2.2]):1 − ϕ z o ( z ) · ϕ z o ( w ) = (1 − | z o | )(1 − z · ¯ w )(1 − z · ¯ z o )(1 − z o · ¯ w ) z, w ∈ D d , we haveVar P d ( g R X ( z o ; F d )) = 12 Z D d Z D d |R ( z ) −R ( w ) | K D d ( ϕ z o ( z ) , ϕ z o ( w )) | K D d ( z, w ) | dv d ( z ) dv d ( w )= 12 Z D d Z D d |R ( z ) − R ( w ) | (cid:20) (1 − z · ¯ z o )(1 − z o · ¯ w )(1 − | z o | )(1 − z · ¯ w ) (cid:21) d +1 | − z · ¯ w | d +2 | {z } denoted T ( z, w ; z o ) dv d ( z ) dv d ( w ) . Denote e = (1 , , · · · , ∈ C d . By the rotational invariance of R and dv d , we haveVar P d ( g R X ( z o ; F d )) = 12 Z D d Z D d |R ( z ) − R ( w ) | T ( z, w ; | z o | e ) dv d ( z ) dv d ( w )= 12 Z D d Z D d |R ( z ) − R ( w ) | b T ( z, w ; | z o | ) dv d ( z ) dv d ( w )= 12 Z D d Z D d |R ( z ) − R ( w ) | e T ( z, w ; | z o | ) dv d ( z ) dv d ( w ) , (6.135)where b T ( z, w ; | z o | ) and e T ( z, w ; | z o | ) are given by b T ( z, w ; | z o | ) : = Z π T ( e iθ z, e iθ w ; | z o | e ) dθ π ; e T ( z, w ; | z o | ) : = Z S d Z S d b T ( | z | · ζ , | w | · ξ ; | z o | ) dσ S d ( ζ ) dσ S d ( ξ ) . Direct computation yields b T ( z, w ; | z o | ) = K D d ( z, w ) | K D d ( z, w ) | (1 − | z o | ) d +1 d +1 X k =0 (cid:18) d + 1 k (cid:19) | z o | k h z, e i k h w, e i k . Note that by expanding the equality (3.53), we have K D d ( z, w ) = ∞ X k =0 a k ( z · ¯ w ) k , a k ≥ . Since the function ( ζ , ξ ) ζ · ¯ ξ is non-negative definite, by a classical result due toSchur (the pointwise products of non-negative definite functions are still non-negative),the functions ( ζ , ξ ) ( ζ · ¯ ξ ) k for all k ∈ N are all non-negative definite. Thus the function( ζ , ξ ) K D d ( | z | ζ , | w | ξ ) = ∞ X k =0 a k | z || w | ( ζ · ¯ ξ ) k is non-negative definite and so is the function ( ζ , ξ ) K D d ( | w | ξ, | z | ζ ). Again by Schur’sresult on pointwise product of non-negative definite functions, the function( ζ , ξ ) F z,w ( ζ , ξ ) := K D d ( | z | ζ , | w | ξ ) · | K D d ( | z | ζ , | w | ξ ) | (1 − | z o | ) d +1 is non-negative definite. Therefore, we have e T ( z, w ; | z o | ) = Z S d Z S d F z,w ( ζ , ξ ) d +1 X k =0 (cid:18) d + 1 k (cid:19) ( | z o | | z || w | ) k h ζ , e i k h ξ, e i k dσ S d ( ζ ) dσ S d ( ξ ) ≥ ( d + 1) Z S d Z S d F z,w ( ζ , ξ ) dσ S d ( ζ ) dσ S d ( ξ )= ( d + 1) (1 − | z o | ) d +1 Z S d Z S d − | z || w | ζ · ¯ ξ ) d +1 dσ S d ( ζ ) dσ S d ( ξ ) | − | z || w | ζ · ¯ ξ | d +2 . By using [36, Lemma 1.9 & formula (1.13)], we have e T ( z, w ; | z o | ) ≥ ( d + 1) ( d − − | z o | ) d +1 Z D − | z || w | z ′ ) d +1 | (1 − | z || w | z ′ ) | d +2 dV ( z ′ )2 π == ( d + 1) ( d − − | z o | ) d +1 Z π Z − | z || w | re iθ ) d +2 − | z || w | re − iθ ) d +1 rdrdθ π == ( d + 1) ( d − − | z o | ) d +1 Z rdr πi I C η d (1 − | z || w | rη ) d +2 η − | z || w | r ) d +1 dη == ( d + 1) ( d − − | z o | ) d +1 Z (cid:20) d ! ∂ d ∂η d (cid:12)(cid:12)(cid:12) η = | z || w | r (cid:18) η d (1 − | z || w | rη ) d +2 (cid:19)(cid:21) rdr. Since for any integer 0 ≤ k ≤ d , ∂ k ∂η k (cid:12)(cid:12)(cid:12) η = | z || w | r ( η d ) ≥ , ∂ k ∂η k (cid:12)(cid:12)(cid:12) η = | z || w | r (cid:18) − | z || w | rη ) d +2 (cid:19) ≥ , we have ∂ d ∂η d (cid:12)(cid:12)(cid:12) η = | z || w | r (cid:18) η d (1 − | z || w | rη ) d +2 (cid:19) == d X k =0 (cid:18) dk (cid:19) ∂ d − k ∂η d − k ( η d ) ∂ k ∂η k (cid:18) − | z || w | rη ) d +2 (cid:19) (cid:12)(cid:12)(cid:12) η = | z || w | r ≥≥ (cid:20) ∂ d ∂η d ( η d ) 1(1 − | z || w | r ) d +2 + η d ∂ d ∂η d (cid:18) − | z || w | rη ) d +2 (cid:19)(cid:21) (cid:12)(cid:12)(cid:12) η = | z || w | r == d !(1 − | z | | w | r ) d + (2 d +1+ d )!(2 d +1)! ( | z | | w | r ) d (1 − | z | | w | r ) d +2 ≥ c ′ d (1 − | z | | w | r ) d +2 , ATTERSON-SULLIVAN INTERPOLATION OF PLURIHARMONIC FUNCTIONS 55 where c ′ d := min x ∈ [0 , (cid:20) d !(1 − x ) d + (2 d + 1 + d )!(2 d + 1)! x d (cid:21) > . Therefore, there exist constants c ′′ d > , c ′′′ d > d , such that e T ( z, w ; | z o | ) ≥ c ′′ d (1 − | z o | ) d +1 Z − | z | | w | r ) d +2 rdr ≥≥ c ′′′ d (1 − | z o | ) d +1 − | z | | w | ) d +1 (cid:20) − (1 − | z | | w | ) d +1 | z | | w | (cid:21) . By taking c d := c ′′′ d · min x ∈ (0 , (cid:20) − (1 − x ) d +1 x (cid:21) > , we obtain e T ( z, w ; | z o | ) ≥ c d (1 − | z o | ) d +1 − | z | | w | ) d +1 . Finally, by substituting the lower bound obtained above for e T ( z, w ; | z o | ) into the equality(6.135), we complete the proof of Proposition 6.8. (cid:3) Proof of Theorem 6.7. Fix z o ∈ D d . Since R : D d → R + is radial and compactly sup-ported, there exists a function Φ : [0 , → R + whose support is contained in [0 , − ε ] forsome ε > R ( z ) = Φ( | z | ). Set V ( t ) := Φ((1 − t ) / d ) , G ( t ) := V ( t ) t d +1 , where t ∈ (0 , . Then there exists ε ′ > G ) ⊂ [ ε ′ , P d ( g R X ( z o ; F d )) ≥ c d (1 − | z o | ) d +1 Z D d Z D d |R ( z ) − R ( w ) | (1 − | z | | w | ) d +1 dv d ( z ) dv d ( w ) == c d · (2 d ) (1 − | z o | ) d +1 Z Z | Φ( r ) − Φ( r ) | (1 − r r ) d +1 ( r r ) d − dr dr == c d (1 − | z o | ) d +1 Z Z | Φ( t / d ) − Φ( t / d ) | (1 − ( t t ) /d ) d +1 dt dt . Since ( t t ) /d ≥ t t whenever 0 ≤ t t ≤ 1, we haveVar P d ( g R X ( z o ; F d )) ≥ c d (1 − | z o | ) d +1 Z Z | Φ( t / d ) − Φ( t / d ) | (1 − t t ) d +1 dt dt == c d (1 − | z o | ) d +1 Z Z | V ( t ) − V ( t ) | ( t + t − t t ) d +1 dt dt ≥≥ c d (1 − | z o | ) d +1 Z Z | V ( t ) − V ( t ) | ( t + t ) d +1 dt dt == 2 c d (1 − | z o | ) d +1 Z ≤ t ≤ t ≤ | V ( t ) − V ( t ) | ( t + t ) d +1 dt dt . Using change of variables t = λt, t = t in the last integral, we obtainVar P d ( g R X ( z o ; F d )) ≥ c d (1 − | z o | ) d +1 Z Z | V ( λt ) − V ( t ) | ( λ + 1) d +1 t d dλdt ≥≥ c d d (1 − | z o | ) d +1 Z Z | V ( λt ) − V ( t ) | t d dλdt == c d d (1 − | z o | ) d +1 Z Z | λ d +1 G ( λt ) − G ( t ) | t d − dλdt ≥ c d d (1 − | z o | ) d +1 Z Z | λ d +1 G ( λt ) − G ( t ) | dλdt. Now note that Z Z (cid:12)(cid:12) λ d +1 G ( λt ) (cid:12)(cid:12) dλdt = Z λ d +1 dλ Z λ G ( t ′ ) dt ′ ≤ d + 2 Z G ( t ) dt. The above inequality combined with the triangle inequality and then Cauchy-Bunyakovsky-Schwarz inequality yields (cid:18)Z Z (cid:12)(cid:12) λ d +1 G ( λt ) − G ( t ) (cid:12)(cid:12) dλdt (cid:19) / ≥ (cid:18)Z Z | G ( t ) | dλdt (cid:19) / − (cid:18)Z Z (cid:12)(cid:12) λ d +1 G ( λt ) (cid:12)(cid:12) dλdt (cid:19) / ≥ (cid:16) − √ d + 2 (cid:17) (cid:18)Z G ( t ) dt (cid:19) / ≥ (cid:16) − √ d + 2 (cid:17) Z G ( t ) dt. 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Joliot Curie 13453, Marseille,France; Steklov Mathematical Institute of RAS, Moscow, Russia Email address : [email protected], [email protected] Yanqi QIU: Institute of Mathematics and Hua Loo-Keng Key Laboratory of Mathe-matics, AMSS, Chinese Academy of Sciences, Beijing 100190, China; CNRS, Institut deMath´ematiques de Toulouse, Universit´e Paul Sabatier Email address ::