berezin transform in polynomial bergman spaces

BEREZIN TRANSFORM IN POLYNOMIAL BERGMAN SPACES

YACIN AMEUR, HÅKAN HEDENMALM, AND NIKOLAI MAKAROV

A. We consider smooth weight functions Q : C → R, subject to a growth condition frombelow, and let Km,n denote the reproducing kernel for the space Hm,n of analytic polynomials u ofdegree at most n − 1 of finite norm ‖u‖2mQ =

∫C|u(z)|2 e−mQ(z)dA(z) (dA denotes normalized area

measure in C). For a continuous bounded function f on C, we consider its (polynomial) Berezintransform

Bm,n f (z) =∫C

f (ζ)dB〈z〉m,n(ζ) where dB〈z〉m,n(ζ) =

∣∣∣Km,n(z, ζ)∣∣∣2

Km,n(z, z)e−mQ(ζ)dA(ζ).

We introduce the sets N+ = {z ∈ C; ∆Q(z) > 0} and N+,0 = {z ∈ C; ∆Q(z) ≥ 0}. For a positive realparameter τ, we prove that there exists a compact set Sτ ⊂ N+,0 such that

(1) Bm,n f (z)→ f (z) as m→∞ while n = mτ + o(1),

for all continuous bounded f if z is in the interior of Sτ ∩ N+. Equivalently, the measures B〈z〉m,nconverge to the Dirac measure at z. The set Sτ is defined as the coincidence set for an obstacleproblem.

We also prove that the convergence in (1) is Gaussian when z is in the interior of Sτ ∩N+, in thesense that

dB〈z〉m,n

(z +

ζ√m∆Q(z)

)→ dP(ζ),

where dP(ζ) = e−|ζ|2dA(ζ) is the standard Gaussian measure.

In the model case Q(z) = |z|2, Sτ is the closed disk with center 0 and radius√τ. Let m→∞while

n = mτ + o(1). Then, by the above, the probability measures B〈z〉m,n approach the Dirac point mass atz in a Gaussian fashion, provided that |z| <

√τ. If instead |z| >

√τ, we prove that the measures B〈z〉m,n

converge to harmonic measure for z relative to the domain C∗ \ Sτ (C∗ = C ∪ {∞} is the extendedcomplex plane).

These results have applications to the study of the eigenvalues of random normal matrices.Our auxiliary results include weighted L2 estimates for the equation ∂u = f when f is a suitable

test function and the solution u is restricted by a polynomial growth bound at∞.

1. I

1.1. General introduction to Berezin quantization. In a version of quantum theory, a Bargmann-Fock type space of polynomials plays the role of the quantized system, while the correspondingweighted L2 space is the classical analogue. It is therefore natural to study the asymptotics ofthe quantized system as we approach the semiclassical limit. A particularly useful object is thereproducing kernel of the Bargmann-Fock type space. To make matters more concrete, let µ be afinite positive Borel measure on C, and L2(C;µ) the usual L2 space with inner product

〈 f , g〉L2(C;µ) =

∫C

f (z)g(z)dµ(z).

Research supported by the Göran Gustafsson Foundation. The third author is supported by N.S.F. Grant No. 0201893.1

2 AMEUR, HEDENMALM, AND MAKAROV

The subspace of L2(C;µ) consisting of entire functions is the Bergman space A2(C;µ).We assume that the µ is carried by infinitely many points so that any given polynomial

corresponds to a unique element of L2(C;µ), and write

Jµ = sup{j ∈ Z;

∫C

|z|2( j−1) dµ(z) < ∞}.

As µ is finite, we are ensured that 1 ≤ Jµ ≤ ∞.Let Pn = Pn(C) be the linear space (linear= C-linear) of analytic polynomials of degree at most

n − 1, and writeA2µ,n = L2(C;µ) ∩ Pn(C) ⊂ L2(C;µ).

If we put n′ = min{n, Jµ}, we see that A2µ,n equals Pn′ (C) in the sense of sets, with the norm

inherited from L2(C;µ). Hence A2µ,n = A2

µ,n′ for all n, and there is no loss of generality in assumingthat n = n′ in what follows.

Let e1, . . . , en′ be an orthonormal basis for A2µ,n. The reproducing kernel Kµ,n for the space A2

µ,nis the function

Kµ,n(z,w) =n′∑j=1

e j(z)e j(w).

Then Kµ,n is independent of the choice of an orthonormal basis for A2µ,n, and it is characterized by

the properties that z 7→ Kµ,n(z, z0) is in A2µ,n and

(1.1) u(z0) =⟨u,Kµ,n(·, z0)

⟩L2(C;µ)

=

∫C

u(z)Kµ,n(z, z0)dµ(z), u ∈ A2µ,n, z0 ∈ C.

For a given complex number z0 we now consider the measure

(1.2) dB〈z0〉µ,n (z) =

∣∣∣Kµ,n(z, z0)∣∣∣2

Kµ,n(z0, z0)dµ(z),

which we may call the Berezin measure associated with µ, n and z0. The reproducing property(1.1) applied to u = Kµ,n(·, z0) implies that B〈z0〉

µ,n is a probability measure. In classical physics,B〈z0〉µ,n corresponds to a point mass at z0, so our interest focuses on how closely this measure

approximates the point mass.

1.2. The class of weights. By a weight we mean a (Lebesgue) measurable function φ : C → Rsuch that the measure

dµφ(z) = e−φ(z)dA(z),

is a finite positive measure on C, where dA(z) = dxdy/π with z = x + iy.We write L2

φ for the space L2(C;µφ) and A2φ for A2(C;µφ) and the norm of an element u ∈ L2

φ

will be denoted by

‖u‖2φ =∫C

|u|2 e−φdA.

For a positive integer n, we frequently write

(1.3) L2φ,n =

{u ∈ L2

φ; u(z) = O(|z|n−1

)when z→∞

}and

(1.4) A2φ,n = L2

φ,n ∩ A2φ = L2

φ,n ∩ Pn.

We observe that L2φ,n usually is non-closed in L2

φ, while the finite-dimensional space A2φ,n is always

closed in L2φ.

BEREZIN QUANTIZATION 3

1.3. The weights considered. Let Q : C → R be a given locally bounded Lebesgue measurablefunction, which satisfies a growth condition of the form

(1.5) Q(z) ≥ ρ log |z|2 , |z| ≥ C,

for some positive numbers ρ and C. For a positive number m, we now consider weights φm of theform φm = mQ. By abuse of notation, we will in this context sometimes refer to Q as the weight.To simplify the notation, we set

Hm,n = A2mQ,n = L2

mQ ∩ Pn

and denote by Km,n the reproducing kernel for Hm,n. For a given point z0, the correspondingBerezin measure B〈z0〉

m,n is defined accordingly, cf. (1.2).Adding a real constant to Q means that the inner product in Hm,n is only chanced by a positive

multiplicative constant, and Km,n gets multiplied by the inverse of that number. Hence thecorresponding Berezin measures are unchanged, and we may for instance w.l.o.g. assume thatQ ≥ 1 on Cwhen this is convenient.

1.4. The limits considered; the parameter τ. We think of Q as being fixed while the parametersm and n vary, and also fix a number τ satisfying 0 < τ < ρ. To avoid bulky notation, it is customaryto reduce the number of parameters to one, by regarding n = n(m) as a function of m > 0. We willadopt this convention and study the behavior of the measures B〈z0〉

m,n as m→∞while n = mτ+o(1).

1.5. A word on notation. For real x, we write ]x[ for the largest integer which is strictly smallerthan x. We write frequently

∂z =∂∂z=

12

(∂∂x− i

∂∂y

), ∂z =

∂∂z̄=

12

(∂∂x+ i

∂∂y

), z = x + iy,

and use ∆ = ∆z to denote the normalized Laplacian

∆z = ∂z∂z =14

(∂2

∂x2 +∂2

∂y2

).

We write D(z0, r) for the open disk {z ∈ C; |z − z0| < r}, and we simplify the notation to Dwhen z0 = 0 and r = 1. The boundary of D(z0, r) is denoted T(z0, r). When S is a subset of C,we write S◦ for the interior of S, and S̄ for its closure; the support of a continuous function f isdenoted by supp f . The symbol A ⊂ B means that A is a subset of B, while A b B means that Ais a precompact subset of B. We write dist C(A,B) for the Euclidean distance between A and B.The symbol a.e. is short for "dA-almost everywhere”, and p.m. is short for "probability measure”.Finally, we use the shorthand L2 to denote the unweighted space L2

0 = L2(C; dA).

1.6. Applications to random normal matrices. One of the main motivations for this paper comesfrom the fact that some of our results have applications in random matrix theory. We will returnto this topic in the subsequent article [1]. The relevant results for these applications are primarilyTh. 2.3 and Cor. 8.2.

2. M

2.1. Berezin quantization. Fix a non-negative weight Q and two positive numbers τ and ρ,τ < ρ, such that the growth condition (1.5) is satisfied, and form the measures

dB〈z0〉m,n = B〈z0〉

m,ndA where B〈z0〉m,n(z) =

∣∣∣Km,n(z, z0)∣∣∣2

Km,n(z0, z0)e−mQ(z).

We shall refer to the function B〈z0〉m,n as the Berezin kernel associated with m, n and z0.


Let us now assume that Q ∈ C2(C). In the first instance, we ask for which z0 we have theconvergence

(2.1) B〈z0〉m,n → δz0 as m→∞ while n = mτ + o(1),

in the weak-star sense of measures. Here, we use Cb(C), the Banach space of all boundedcontinuous functions on C, as the predual. In terms of the Berezin transform

Bm,n f (z0) =∫C

f (z)dB〈z0〉m,n(z),

we are asking whether

(2.2) Bm,n f (z0)→ f (z0) as m→∞ while n = mτ + o(1),

for all f ∈ Cb(C). LetN+ andN+,0 be the sets of points defined by

N+ = {z ∈ C; ∆Q(z) > 0}, N+,0 = {z ∈ C; ∆Q(z) ≥ 0}.

We shall find that there exists a compact set Sτ contained inN+,0 such that (2.2) holds for all z0 inthe interior of Sτ ∩N+, while (2.2) fails whenever z0 ∈ C \ Sτ.

2.2. Ingredients from potential theory. To define Sτ, we first need to introduce some notionsfrom weighted potential theory, cf. [19] and [14]. It is here advantageous to slightly relax theregularity assumption on Q and assume that Q is in the class C1,1(C) consisting of all C1-smoothfunctions with Lipschitzian gradients. We will have frequent use of the simple fact that thedistributional Laplacian ∆F of a function F ∈ C1,1(C) makes sense almost everywhere and is ofclass L∞loc(C).

Let Subhτ(C) denote the set of all subharmonic functions f : C → R which satisfy a growthbound of the form

f (z) ≤ τ log |z|2 +O(1) as |z| → ∞.

The equilibrium potential corresponding to the weight Q and the parameter τ is defined by

Q̂τ(z) = sup{

f (z); f ∈ Subhτ(C) and f ≤ Q on C}.

Clearly, Q̂τ is subharmonic on C. What is perhaps not so obvious is that Q̂τ is of class C1,1(C), afact which is well-known in the context of variational inequalities of partial differential equations(associated with the work of Kinderlehrer, Stampacchia, and Caffarelli; see [14] for some details,and Chapter 1 of [17] for the main ingredient). We now define Sτ as the coincidence set

(2.3) Sτ = {z ∈ C; Q(z) = Q̂τ(z)}.

It is easy to see that the set Sτ increases with τ (with respect to containment). Moreover, Q̂τ isharmonic in C \ Sτ.

We collect our observations in a lemma, and add an little concerning the growth of Q̂τ.

Lemma 2.1. The function Q̂τ is of class C1,1(C), and Sτ is compact. Furthermore, Q̂τ is harmonic inC \ Sτ and there is a number C such that Q̂τ(z) ≤ τ log+ |z|

2 + C for all z ∈ C.

Proof. This is [14], Prop. 4.2, p. 10, and [19], Th. I.4.7, p. 54. �

Given the smoothness of Q and Q̂τ, one can show that

∆Q(z) = ∆Q̂τ(z), z ∈ Sτ,

in the almost-everywhere sense. In particular, as Q̂τ is subharmonic, we have

Sτ ⊂ N+,0 = {z ∈ C; ∆Q(z) ≥ 0}.


It is easy to construct subharmonic minorants of critical growth; for C large enough, the functionz 7→ τ log+

(|z|2 /C

)is a minorant of Q of class Subhτ(C). It follows that

(2.4) Q̂τ(z) = τ log |z|2 +O(1) as |z| → ∞.

2.3. Technical tools: the expansion formula for the polynomial Bergman kernel. Many resultsin this paper rely on a suitable expansion formula for Km,n(z,w) when z,w are both close to somepoint z0 ∈ N+, and m and n are large. In the presentation, we shall assume that Q ∈ C2(C) isreal-analytic in S◦τ.

A general expansion formula was recently obtained by R. Berman [2], p. 9, based on themethods of [4]. Here, we will only need a special case of that result. We first need some notation.

If Q is real-analytic in a neighborhood of a point z0, we may introduce the function ψ = ψQ

given by

ψ(z,w) =∞∑

j,k=0

1j!k!

(z − z0) j(w − z̄0)k∂ j∂kQ(z0),

which is holomorphic in (z,w) when z is close to z0 and w is close to z̄0. By Taylor’s formula, wehave

ψ(z, z̄) = Q(z)

for all z close to z0. By varying z0, we find that ψ(z,w) extends to a holomorphic function in aneighborhood of the conjugate diagonal w = z̄, provided that Q is real-analytic throughout C.We shall assume that Q is real-analytic in S◦τ. Then ψ(z,w) is well-defined and holomorphic in aneighborhood of the conjugate diagonal w = z̄, where z ∈ S◦τ. It is clear that ψ(z, z̄) = Q(z) holdsfor all z ∈ S◦τ.

Definition 2.2. Let b0 and b1 denote the functions

(2.5) b0(z,w) = ∂z∂wψ(z,w), b1(z,w) =12∂z∂w log[∂z∂wψ(z,w)] =

12∂w∂zb0(z,w)b0(z,w)

.

The function b1(z,w) is well-defined where there is no division by 0, in particular this is so insome neighborhood of the conjugate diagonal w = z̄, where z ∈ S◦τ ∩ N+. Along the conjugate-diagonal w = z̄, we have

b0(z, z̄) = ∆Q(z) and b1(z, z̄) =12∆ log∆Q(z), z ∈ S◦τ ∩N+.

We define the first order approximating Bergman kernel K1m(z,w):

K1m(z,w) = (mb0(z, w̄) + b1(z, w̄)) emψ(z,w̄).

We have the following theorem.

Theorem 2.3. Assume that Q ∈ C2(C) is real-analytic in S◦τ, and suppose K is a compact subset ofS◦τ ∩ N+. Fix a point z0 ∈ K and a real number M > 0. Then there exists a number m0 depending only

on Q, τ,M and positive numbers C and ε depending only on Q, τ,M,K such that∣∣∣Km,n(z,w) − K1m(z,w)

∣∣∣ e− 12 m(Q(z)+Q(w))

≤Cm, z0 ∈ K , z,w ∈ D(z0, ε),

for all m ≥ m0 and n ≥ mτ −M. In particular, by restricting to the diagonal, we get∣∣∣∣Km,n(z, z)e−mQ(z)−

(m∆Q(z) + 1

2∆ log∆Q(z)) ∣∣∣∣ ≤ C

m, z ∈ K ,

when m ≥ m0 and n ≥ mτ −M.

Proof. See Sect. 6. See also Remark 2.5 below. �


Remark 2.4. Since ψ(z,w), b0(z,w), and b1(z,w) are all real-valued along the conjugate-diagonalw = z̄, a suitable version of the reflection principle implies that K1

m is Hermitian, that is,

K1m(z,w) = K1

m(w, z).

Remark 2.5. We should point out that Th. 2.3 has a long history; analogous expansions are well-known for (weighted) Bergman kernels of several complex variables, see, for instance, [11], [7],[16], [10], [4], [2], and the references therein. Moreover, as was mentioned above, Th. 2.3 is aslight modification of a more general result in several complex variables stated by Berman [2],Th. 3.8 (see also [4]). In the proof, we make frequent use of the ideas and techniques developedin [2], [3], [4], and in the book [20].

2.4. Extensions and possible generalizations of the expansion formula. Th. 2.3 is a special caseof a general asymptotic expansion of Bergman kernels, cf. Berman [2], Th. 3.8.

Given a positive integer k, a k-th order approximating Bergman kernel is an expression of the form

Kkm(z,w) =

(mb0(z, w̄) + b1(z, w̄) + . . . +m−k+1bk(z, w̄)

)emψ(z,w̄),

for z and w in a neighborhood of the conjugate diagonal w = z̄, where the coefficient functionsb0, . . . , bk are certain holomorphic functions. It is possible to find b0, . . . , bk such that∣∣∣Km,n(z,w) − Kk

m(z,w)∣∣∣ e− 1

2 m(Q(z)+Q(w))≤ Cm−k, z,w ∈ D(z0, ε),

for m large and n ≥ mτ − 1, provided that z0 ∈ S◦τ ∩ N+ is fixed. In principle, the b0, . . . , bk are

determined from a recursion formula involving partial differential equations of increasing order,cf. Berman et al. [4], equation (2.20), p. 209. However, the analysis required for calculatinghigher order coefficients b j, j ≥ 2, is quite involved, and the first order expansion is sufficient formany practical purposes, cf. [1]. This is why we here prefer a more direct approach.

The real-analyticity assumption on Q in Th. 2.3 is of course excessive. An analogous theoremholds under weaker assumtions, e.g. C∞-smoothness. In this case, the functions b0, b1 and ψare defined using almost-holomorphic extensions (cf. e.g. [7] for a relevant discussion). Themodifications are essentially outlined in [4], Sect. 2.6.

2.5. Asymptotic behavior of the Berezin measure. We begin with the following basic observa-tion.

Proposition 2.6. Let Q ∈ C1,1(C) and z0 ∈ C an arbitrary point. Then, for every open neighborhood Dof Sτ,

B〈z0〉m,n(D)→ 1 as m→∞ while n ≤ mτ + 1.

Proof. See Sect. 3. �

It follows from Prop. 2.6 that for each fixed z0 ∈ C and each open neighborhood D of Sτ,B〈z0〉

m,n(C \D)→ 0 as m→∞while n ≤ mτ+ 1. Hence if z0 ∈ C \Sτ, then (2.1) fails, as the measuresB〈z0〉

m,n remain essentially confined to (neighborhoods of) Sτ. When the point z0 is in Sτ ∩N+, thisconfinement is no longer an obstacle to (2.1). Indeed, for interior points z0 in Sτ ∩ N+, we havethe following.

Theorem 2.7. Assume that Q ∈ C2(C) and let z0 ∈ S◦τ ∩ N+. Then, for any fixed real number M ≥ 0,

the measures B〈z0〉m,n converge to δz0 in the weak-star sense, as m→∞ and n ≥ mτ −M.

Proof. See Sect. 7. See also Rem. 2.12. �


2.6. Gaussian convergence of the Berezin measure. Fix a point z0 ∈ S◦τ ∩ N+. It is convenient

to introduce the rescaled Berezin measure B̂〈z0〉m,n by

(2.6) dB̂〈z0〉m,n = B̂〈z0〉

m,ndA where B̂〈z0〉m,n(z) =

1m∆Q(z0)

B〈z0〉m,n

(z0 +

z√m∆Q(z0)

).

We denote the standard Gaussian p.m. on C by

dP(z) = e−|z|2dA(z).

We have the following CLT, which gives much more precise information than Th. 2.7. Theproof is based on the expansion formula of Th. 2.3.

Theorem 2.8. Assume that Q ∈ C2(C) is real-analytic in S◦τ. Fix a compact subset K b S◦τ ∩ N+, apoint z0 ∈ K , and a number M ≥ 0. We then have

(2.7)∫C

∣∣∣∣B̂〈z0〉m,n(z) − e−|z|

2∣∣∣∣ dA(z)→ 0 as m→∞ while n ≥ mτ −M,

where the convergence is uniform over z0 ∈ K . Equivalently, B̂〈z0〉m,n → P in norm as m → ∞ and

n ≥ mτ −M.


2.7. The Bargmann–Fock case and harmonic measure. When z0 ∈ C\ (S◦τ∩N+), Prop. 2.6 statesthat the measures B〈z0〉

m,n tend to concentrate on Sτ as m → ∞ while n = mτ + o(1). However, ourgeneral results do not suggest what the asymptotic behavior of the p.m. B〈z0〉

m,n might be.We specialize to the Bargmann–Fock weight Q(z) = |z|2, in which case we obviously have

N+ = C. It is convenient to introduce the truncated exponentials

Ek(z) =k∑

j=0

z j

j!.

We check thatKm,n(z,w) = mEn−1(mzw̄),

and that

Q̂τ(z) = |z|2 for |z| ≤√τ, Q̂τ(z) = τ + τ log

|z|2

τfor |z| >

√τ,

so that Sτ = D(0,√τ). We infer that

(2.8) dB〈z0〉m,n(z) = m

|En−1(mzz̄0)|2

En−1(m |z0|2)

e−m|z|2 dA(z).

Let C∗ = C ∪ {∞} denote the extended plane.

Theorem 2.9. Let Q(z) = |z|2 and z0 ∈ C \ Sτ. Then, as m → ∞ and n/m → τ, the measures B〈z0〉m,n

converge to harmonic measure at z0 with respect to C∗ \ Sτ.


Remark 2.10. The above theorem is a special case of Th. 8.7 in [1].

2.8. Technical tools: weighted L2 estimates for the ∂-equation with a polynomial growthconstraint. An essential ingredient in our analysis is an estimate of the norm-minimal solutionu0,n in the space L2

mQ,n to the equation ∂u = f , where f is a suitable bounded measurable functionwith supp f ⊂ Sτ. What makes the problem slightly sophisticated is the growth bound associatedwith L2

mQ,n. In Sect. 4 we show how to modify the standard approach to get the estimates weneed.


2.9. A long-range damping bound on the Berezin density. Th. 2.7 suggests that if z0 is a givenpoint in the interior of Sτ ∩ N+, m is large, and n ≥ mτ −M, then the density B〈z0〉

m,n(z) should bebig near z0 and decay rapidly away from z0. A concrete decay bound is offered by the followingtheorem. The assumption Q ≥ 1 is technical and may be removed at the expense of altering theexpression for the positive constant λ0. The proof is mainly based on the weighted L2 estimatesfor the ∂-equation obtained in Sect. 4.

Theorem 2.11. Assume that Q ∈ C2(C) with Q ≥ 1. Let K be a compact subset of S◦τ ∩N+, and let K̂be the closure of the union⋃

z0∈K

D(z0, r(z0)), where r(z0) =12

dist C(z0,C \ (S◦τ ∩N+)),

which is automatically a compact subset of S◦τ ∩N+. We also put

αK̂= inf{∆Q(z); z ∈ K̂} > 0.

Then, given a positive real M, we have, for all positive integers n confined to mτ −M ≤ n ≤ mτ + 1, theestimate

B〈z0〉m,n(z) ≤ Cme−λ0

√m min{r(z0),|z−z0 |}e−m(Q(z)−Q̂τ(z)), z0 ∈ K , z ∈ C,

where λ0 = 2−3/2√αK̂

e−12 Mτ−1qτ with qτ = sup

SτQ, and the constant C only depends onK , Q, M, and τ.


Remark 2.12. The proof of Th. 2.11 is quite different from that supplied for Th. 2.7. Hence Th.2.11 offers an alternative means to obtain Th. 2.7 in the case of interest when n = mτ + o(1).

3. P

3.1. Preliminaries. In this section, we introduce the cut-off functions needed repeatedly through-out this paper. We also introduce the technique of subharmonicity estimates, which will allowus to turn integral bound into pointwise bounds. An application of the maximum principleallows us to obtain Prop. 2.6. As for the results obtained here, it is enough to suppose that Q isC

1,1-smooth (and satisfies the growth assumption (1.5)). In particular, we have ∆Q ∈ L∞loc(C).

3.2. Cut-off functions. Given reals δ > 0, r > 0 and C > 1, there exists χ ∈ C∞0 (C) such that χ = 1on D(0, δ), χ = 0 outside D(0, δ(1 + r)), χ ≤ 1 and |∂χ|2 ≤ Cr−2δ−2χ on C, where C is an absoluteconstant. In fact, any C > 1 will do. It is then easy to check that ‖∂χ‖2L2 ≤ C(1+2/r). A Lipschitzianχ is given by the radial expression

χ(z) =(1 + r

r−|z|rδ

)2

, δ ≤ |z| ≤ δ(1 + r),

and a smooth χ is obtained by a suitable regularization of this solution. Later on, we will oftenuse the values δ = 3ε/2 and δ(1 + r) = 2ε, where ε > 0 is given. We may then arrange that‖∂χ‖L2 ≤ 3.

3.3. Subharmonicity estimates. We start by giving two simple lemmas which are based on thesub-mean property of subharmonic functions.

Lemma 3.1. Let φ be a weight of class C1,1(C) and put A = ess sup {∆φ(ζ); ζ ∈ D}. Then if u is boundedand holomorphic inD, we have that

|u(0)|2e−φ(0)≤

∫D

|u(ζ)|2e−φ(ζ)eA|ζ|2 dA(ζ) ≤ eA∫D

|u(ζ)|2e−φ(ζ)dA(ζ).


Proof. Consider the function F(ζ) = |u(ζ)|2e−φ(ζ)+A|ζ|2 , which satisfies

∆ log F = ∆ log |u|2 − ∆φ + A ≥ 0 a.e. on D,

making F logarithmically subharmonic. But then F is subharmonic itself, and so

F(0) ≤∫D

F(z)dA(z).

The assertion of the lemma is immediate from this. �

Lemma 3.2. Let Q ∈ C1,1(C) and fix δ > 0 and z ∈ C and put

A = Az,δ,m = ess sup {∆Q(ζ); ζ ∈ D(z,m−1/2δ)}.

Also, let u be holomorphic and bounded inD(z,m−1/2δ). Then

|u(z)|2 e−mQ(z)≤

m eAδ2

δ2

∫D(z,m−1/2δ)

|u(ζ)|2 e−mQ(ζ)dA(ζ).

Proof. The assertion follows if we make the change of variables ζ = z + m−1/2δξ where ζ ∈D(z,m−1/2δ) and ξ ∈ D, and apply Lm. 3.1 with the weight φ(ξ) = mQ(ζ). �

We note the following consequence of the subharmonicity estimates. We will sometimes need itin later sections.

Lemma 3.3. LetK be a compact subset of C and δ a given positive number. We put

Km−1/2δ = {z ∈ C; dist C(z,K ) ≤ m−1/2δ} and A = ess sup {∆Q(z); z ∈ Km−1/2δ}.

Then, for all positive real m and all integers n ≥ 1,∣∣∣Km,n(z,w)∣∣∣2 e−mQ(z)

≤m eAδ2

δ2

∫D(z,m−1/2δ)

∣∣∣Km,n(ζ,w)∣∣∣2 e−mQ(ζ)dA(ζ), z ∈ K , w ∈ C.

Proof. Apply Lm. 3.2 to u(ζ) = Km,n(ζ,w). �

3.4. A weak maximum principle for weighted polynomials. Maximum principles for weightedpolynomials have a long history, see e.g. [19], Chap. III. The following simple lemma will sufficefor our present purposes; it is a consequence of [19], Th. III.2.1. (We recall that Sτ is thecoincidence set given by (2.3).)

Lemma 3.4. Suppose that a polynomial u of degree at most n − 1 satisfies |u(z)|2 e−mQ(z)≤ 1 on Sτ(m,n),

where τ(m,n) = (n − 1)/m < ρ. Then |u(z)|2 e−mQ̂τ(m,n)(z)≤ 1 on C.

Proof. Put q(z) = m−1 log |u(z)|2. The assumptions on u imply that q ∈ Subhτ(m,n)(C) and that q ≤ Qon Sτ(m,n). Hence, by the maximum principle, q ≤ Q̂τ(m,n) on C, as desired. �

Lemma 3.5. Let u ∈ Hm,n, and suppose that m ≥ 1 and n ≤ mτ + 1. Then

|u(z)|2 ≤ m eA‖u‖2mQemQ̂τ(z),

where A denotes the essential supremum of ∆Q over the set {z ∈ C; dist C(z,Sτ) ≤ 1}.

Proof. The assertion that n ≤ mτ + 1 is equivalent to that τ(m,n) ≤ τ where τ(m,n) = (n − 1)/m.Thus Sτ(m,n) ⊂ Sτ and Q̂τ(m,n) ≤ Q̂τ. An application of Lm. 3.2 with δ = 1 now gives

|u(z)|2 e−mQ(z)≤ m eA

∫D(z,m−1/2)

|u(ζ)|2 e−mQ(ζ)dA(ζ) ≤ m eA∫C

|u(ζ)|2 e−mQ(ζ)dA(ζ), z ∈ Sτ.

As a consequence, the same estimate holds on Sτ(m,n). We can thus apply Lm. 3.4, which yieldsthat

|u(z)|2 ≤ m eA‖u‖2mQemQ̂τ(m,n)(z), z ∈ C.

The desired assertion follows, since Q̂τ(m,n) ≤ Q̂τ. �


3.5. The proof of Proposition 2.6. Fix two points z and z0 in C. We apply Lm. 3.5 to thepolynomial

u(ζ) =Km,n(ζ, z0)√Km,n(z0, z0)

,

which is of class Hm,n and satisfies ‖u‖mQ = 1. It follows that |u(z)|2 ≤ m eAeQ̂τ(z), so that

(3.1) B〈z0〉m,n(z) = |u(z)|2 e−mQ(z)

≤ m eAem(Q̂τ(z)−Q(z)), z ∈ C, n ≤ mτ + 1.

Next, let D be an open neighborhood of Sτ. Since Q > Q̂τ on C \ Sτ, the continuity of thefunctions involved coupled with the growth conditions (2.4) and (1.5) shows that Q − Q̂τ isbounded from below by a positive number on C \ D. Moreover, Q − Q̂τ increases sufficientlyrapidly near infinity that it follows that

B〈z0〉m,n(C \ D) =

∫C\D

B〈z0〉m,ndA→ 0

as m→∞ and n ≤ mτ + 1. Since B〈z0〉m,n is a p.m. on C, the assertion of Prop. 2.6 is immediate. �

3.6. An estimate for the one-point function. Our proof of Prop. 2.6 implies a useful estimatefor the one-point function z 7→ Km,n(z, z)e−mQ(z). The following result is implicit in [2].

Proposition 3.6. Suppose that n ≤ mτ + 1. Then there exists a number C depending only on Q, τ suchthat

(i) Km,n(z, z) e−mQ(z)≤ Cm e−m(Q(z)−Q̂τ(z)), z ∈ C,

and

(ii)∣∣∣Km,n(z,w)

∣∣∣2 e−m(Q(z)+Q(w))≤ Cm2e−m(Q(z)−Q̂τ(z))e−m(Q(w)−Q̂τ(w)), z,w ∈ C.

In particular,∣∣∣Km,n(z,w)

∣∣∣2 e−m(Q(z)+Q(w))≤ Cm2 for all z,w ∈ C.

Proof. The function B〈z0〉m,n(z) in the diagonal case z = z0 reduces to

B〈z〉m,n(z) = Km,n(z, z) e−mQ(z).

Thus the estimate (3.1) implies

Km,n(z, z)e−mQ(z)≤ m eAem(Q̂τ(z)−Q(z)),

which proves (i). In order to get (ii) from (i), it is enough to apply the Cauchy–Schwarz inequality.�

4. W ∂-

4.1. General introduction. The Bergman projection and the ∂-equation. Let φ be a weight onC (cf. Subsect. 1.3). We assume throughout that φ is of class C1,1(C) (so that ∆φ ∈ L∞loc(C)), andthat

∫C

e−φdA < ∞, so that L2φ contains the constant functions. We fix a positive integer n and

recall the definition of the "truncated” spaces L2φ,n and A2

φ,n (see (1.3) and (1.4)).Let Kφ,n denote the reproducing kernel for the space A2

φ,n, and let Pφ,n : L2φ → A2

φ,n be theorthogonal projection,

(4.1) Pφ,nu(z) =∫C

u(z)Kφ,n(z,w)e−φ(w)dA(w), u ∈ L2φ.

For f in the class C0(C) (continuous functions with compact support), we consider the Cauchytransform u = C f defined by

C f (z) =∫C

f (w)z − w

dA(w),


which solves the ∂-equation

(4.2) ∂u = f .

Moreover, u = C f is bounded, and is therefore of class L2φ,n for any n ≥ 1. Hence the function

(4.3) u0,n(z) = u(z) − Pφ,nu(z),

solves (4.2) and is of class L2φ,n. It is easy to check that u0,n, as defined by (4.3), is the unique

norm-minimal solution to (4.2) in L2φ,n whenever u ∈ L2

φ,n is a solution to (4.2). In a sense, thestudy of the orthogonal projection Pφ,nu is therefore equivalent to the study of the L2

φ,n-minimalsolution u0,n to (4.2).

It is useful to observe that u0,n is characterized among the solutions of class L2φ,n to (4.2) by the

condition u0,n⊥A2φ,n, that is,

(4.4)∫C

u0,n(z)h(z)e−φ(z)dA(z) = 0, for all h ∈ A2φ,n.

Our principal result in this section, Th. 4.1, states that a variant of the (one-dimensional)weighted L2 estimates of Hörmander apply to u0,n. As for further developments, the importantresults are, however, Cors. 4.5 and 4.4. The reader may consider to glance at those results andskip to the next section, at a first read.

4.2. Hörmander’s weighted L2 estimates. The L2 estimates of Hörmander in the most elemen-tary, one-dimensional form applies only to weights φwhich are strictly subharmonic in the entireplane C. The result states that u0, the L2

φ-minimal solution to (4.2) (where f ∈ C0(C)) satisfies

(4.5)∫C

|u0|2 e−φdA ≤

∫C

∣∣∣ f ∣∣∣2 e−φ

∆φdA,

provided that φ is C2-smooth on C. See [15], eq. (4.2.6), p. 250 (this is essentially just Green’sformula).

It is important to observe that the estimate (4.5) remains valid for strictly subharmonic weightsφ in the larger class C1,1(C) (that φ is strictly subharmonic here means that ∆φ > 0 a.e. on C). Theproof in [15], Subsect. 4.2 goes through without changes in this more general case.

4.3. Weighted L2φ,n estimates. If f ∈ Lp(X) for some Borel setX ⊂ C, it is tacitly extended to all of

C by declaring it to vanish on C \ X.We have the following theorem, where we consider two weights φ and φ̂ with various prop-

erties.

Theorem 4.1. Let T be a compact subset of C, while φ, φ̂, % are three real-valued functions of classC

1,1(C), and n is a positive integer. We assume that:(1) L2

φ̂contains the function z 7→ (1 + |z|)−1, and that there are real constants a, b with 0 ≤ a, b < ∞,

such thatφ̂ ≤ φ + a on C and φ ≤ φ̂ + b on T ,

(2) A2φ̂⊂ A2

φ,n,

(3) ∆(φ̂ + %) > 0 holds a.e. on C,(4) % is locally constant on C \ T , and(5) there exists a number κ, 0 < κ < 1, such that

(5a)|∂%|2

∆(φ̂ + %)≤

κ2

ea+ba.e. on T .


If f ∈ L∞(T ), then u0,n, the L2φ,n-minimal solution to ∂u = f , satisfies∫C

∣∣∣u0,n

∣∣∣2 e%−φdA ≤ea+b

(1 − κ)2

∫T

∣∣∣ f ∣∣∣2 e%−φ

∆(φ̂ + %)dA.

Before we supply the proof, we state a basic lemma.

Lemma 4.2. Suppose that f ∈ L∞(C) has compact support as a distribution. Then (4.2) has a solution uin L2

φ̂. Moreover, v0, the L2

φ̂-minimal solution to (4.2), is of class L2

φ,n.

Proof. As f is bounded with compact support, the Cauchy transform C f is bounded, and decayslike |C f (z)| = O(|z|−1) as |z| → ∞. By assumption (1), the Cauchy transform C f is in L2

φ̂. Thus

the L2φ̂

-minimal solution v0 to (4.2) exists, and it is necessarily of the form v0 = C f + g with some

g ∈ A2φ̂

. In view of property (2), we know that g ∈ A2φ,n. The assertion is now immediate. �

Proof of Theorem 4.1. The assertion is trivial unless∫T

∣∣∣ f ∣∣∣2 e%−φ

∆(φ̂ + %)dA < ∞,

which will be assumed in the rest of the proof. In view of (4.4), the condition that u0,n is L2φ,n-

minimal may be expressed as∫C

u0,ne%h̄e−(φ+%)dA = 0 for all h ∈ A2φ,n.

The latter relation means that the function w0 = u0,ne% minimizes the norm in L2φ+% over elements

of the (non-closed) subspace

e% L2φ,n =

{w; w = e%h, where h ∈ L2

φ,n

}⊂ L2

φ+%

which solve the (modified) ∂-equation

(4.6) ∂w = ∂(u0,ne%

)= f e% + u0,n∂ (e%) .

Since % is locally constant outside the compact set T and bounded on T , it is automaticallybounded throughout C. It follows that we have

L2φ+%,n = e% L2

φ,n = L2φ,n (as sets),

and we conclude that w0 is the norm-minimal solution to (4.6) in L2φ+%,n.

Let v0 denote the L2φ̂

-minimal solution to ∂u = f (see Lm. 4.2). We form the continuous

function g = ∂((u0,n − v0)e%

)= (u0,n − v0) ∂ (e%) , whose support is contained in T , and consider

the ∂-equation

(4.7) ∂ξ = g = (u0,n − v0) ∂ (e%) .

The assertion of Lm. 4.2 remains valid if φ̂ is replaced by φ̂ + %; it follows that (4.7) has asolution ξ ∈ L2

φ̂+%, and moreover, if ξ0 denotes the norm-minimal solution to (4.7) in L2

φ̂+%, we

have ξ0 ∈ L2φ+%,n.

We next continue by forming the function

w1 = v0e% + ξ0.

It is clear that w1 ∈ L2φ+%,n, and a calculation yields that

(4.8) ∂w1 = f e% + v0 ∂ (e%) + (u0,n − v0) ∂ (e%) = ∂(u0,ne%

)= ∂w0.


Since w0 is norm-minimal in L2φ+%,n over functions w such that ∂w = ∂w0, we must have

(4.9)∫C

|w0|2 e−(φ+%)dA ≤

∫C

|w1|2 e−(φ+%)dA ≤ ea

∫C

|w1|2 e−(φ̂+%)dA,

where we have used the condition φ̂ ≤ φ+ a to deduce the second inequality. Moreover, since ξ0

is norm-minimal in L2φ̂+%

amongst solutions to (4.7), we have∫C

w1h̄e−(φ̂+%)dA =∫C

v0h̄e−φ̂dA +∫C

ξ0h̄e−(φ̂+%)dA = 0

for all h ∈ A2φ̂+%

, so the function w1 is in fact the norm-minimal solution to a ∂-equation in L2φ̂+%

.

The ∂-equation satisfied by w1 is (see (4.8))

∂w1 = f e% + u0,n∂ (e%) = f e% + u0,ne%∂%.

By the Hörmander estimate (4.5) applied to the weight φ̂ + %, we obtain that

(4.10)∫C

|w1|2 e−(φ̂+%)dA ≤

∫C

∣∣∣ f e% + u0,ne%∂%∣∣∣2 e−(φ̂+%)

∆(φ̂ + %)dA =

∫C

∣∣∣ f + u0,n∂%∣∣∣2 e%−φ̂

∆(φ̂ + %)dA.

Since f and ∂% are supported in T , and since e−φ̂ ≤ ebe−φ there (see condition (1)), it is seen thatthe right hand side in (4.10) may be estimated by

(4.11) eb∫C

∣∣∣ f + u0,n∂%∣∣∣2 e%−φ

∆(φ̂ + %)dA.

For positive t, we now use the inequality |a + b|2 ≤ (1 + t) |a|2 + (1 + t−1) |b|2 and the condition (5a)to conclude that the integral in (4.11) is dominated by

(1 + t)∫C

| f |2e%−φ

∆(φ̂ + %)dA + (1 + t−1)

∫C

|u0,n|2 |∂%|2

∆(φ̂ + %)e%−φdA

≤ (1 + t)∫C

| f |2e%−φ

∆(φ̂ + %)dA + (1 + t−1)

κ2

ea+b

∫C

|u0,n|2e%−φdA.

(4.12)

Tracing back through (4.9), (4.10), (4.11) and (4.12), we get∫C

|u0,n|2e%−φdA ≤ ea+b(1 + t)

∫C

| f |2e%−φ

∆φ̂ + ∆%dA + (1 + t−1)κ2

∫C

|u0,n|2e%−φdA,

which we write as(1 − (1 + t−1)κ2

) ∫C

|u0,n|2e%−φdA ≤ ea+b(1 + t)

∫C

| f |2e%−φ

∆(φ̂ + %)dA.

The desired estimate now follows if we pick t = κ/(1−κ), and recall that f is supported on T . �

4.4. Implementation scheme. We now fix Q ∈ C1,1(C) satisfying the growth assumption (1.5)with a fixed ρ > 0. Adding a constant to Q does not change the problem and so we may assumethat Q ≥ 1 on C. Let us put

(4.13) qτ = supz∈Sτ{Q(z)}, lτ = sup

z∈Sτ{log(1 + |z|2)}.

We next fix a positive number τ < ρ and two positive numbers M0 and M1 such that

(4.14) M1 log(1 + |z|2) ≤M0Q̂τ(z), z ∈ C.


This is possible since Q̂τ ≥ 1 and Q̂τ(z) = τ log |z|2 +O(1) when |z| → ∞ (see (2.4)). In particular,it follows that M1 ≤M0τ. We now put

(4.15) φm = mQ and φ̂m,M0,M1 (z) = (m −M0)Q̂τ(z) +M1 log(1 + |z|2).

Note that φ̂m,M0,M1 is strictly subharmonic on Cwhenever m ≥M0 with

(4.16) ∆φ̂m,M0,M1 (z) = (m −M0)∆Q̂τ(z) +M1

(1 + |z|2

)−2≥M1

(1 + |z|2

)−2,

and that (4.14) implies that (recall that Q̂τ ≤ Q)

(4.17) φ̂m,M0,M1 ≤ φm on C.

In the other direction, we easily see that

(4.18) φm ≤ φ̂m,M0,M1 +M0qτ on Sτ.

Near the point at infinity, we instead have

φm(z) − φ̂m,M0,M1 (z) ≥ m(ρ − τ) log |z|2 +O(1), as |z| → ∞,

given the growth assumption on Q and the asymptotics

(4.19) φ̂m,M0,M1 = ((m −M0)τ +M1) log |z|2 +O(1), as |z| → ∞,

which follows from (2.4). Note also that

A2φm,n = A2

mQ ∩ Pn = Hm,n.

We now check the conditions (1) and (2) of Th. 4.1. We recall that ]x[ denotes the largest integerwhich is strictly smaller than x.

Lemma 4.3. In view of (4.17) and (4.18), condition (1) of Th 4.1 holds for φ = φm, φ̂ = φ̂m,M0,M1 ,provided the compact set T is contained in Sτ, and the constants a, b are given by a = 0, and b = M0qτ,while m is kept m ≥M0. On the other hand, condition (2) of Th 4.1, which requires that A2

φ̂m,M0 ,M1

⊂ Hm,n,

holds if n ≥](m −M0)τ +M1[.

Proof. As for condition (1), it remains to check that the function (1+ |z|)−1 is in L2φ̂m,M0 ,M1

for m ≥M0.

This follows from the asymptotics (4.19), since

(m −M0)τ +M1 > 0

and

(4.20)∫C

(1 + |z|2)−rdA(z) < ∞ ⇐⇒ r > 1.

As for condition (2), we note that since φ̂m,M0,M1 ≤ φm, we have the containment A2φ̂m,M0 ,M1

⊂ A2φm

,

and it remains only to check that A2φ̂m,M0 ,M1

⊂ Pn, provided that n ≥](m −M0)τ +M1[. But this is

immediate from (4.19) and (4.20). �

We first apply Th. 4.1 in a rather simple fashion, cf. [2], p. 10.

Corollary 4.4. Suppose Q ∈ C1,1(C), with Q ≥ 1 on C. If f ∈ L∞(Sτ), then the function u0,n, which isthe L2

mQ,n-minimal solution to ∂u = f , satisfies

‖u0,n‖2mQ ≤

eM0qτ+2lτ

M1‖ f ‖2mQ,

provided that m ≥M0 and n ≥](m −M0)τ +M1[.


Proof. Given Lm. 4.3, the estimate follows if we putT = Sτ and %m = 0 in Th. 4.1, while observingthat κ may be chosen as close to 0 as we like. �

We next apply Th. 4.1 in a more challenging context. The result will be required to obtainoff-diagonal damping estimates for the kernel Km,n(z,w).

Corollary 4.5. Suppose Q ∈ C1,1(C) with Q ≥ 1 on C. Let T be a compact subset of Sτ, such that

∆Q ≥ α > 0 a.e. on T ,

where α is a constant. Also, suppose there is a real-valued function %m ∈ C1,1(C) with

(i) ∂%m = 0 on C \ T ,

such that

(ii) ∆%m ≥ −mα2+M0α a.e. on T ,

while

(iii) |∂%m|2≤

mακ2

2eM0qτa.e. on T ,

for some constant κ, 0 < κ < 1. Then, if m ≥ M0 and n ≥](m −M0)τ +M1[, and if f ∈ L∞(T ), we havethat u0,n, the L2

mQ,n-minimal solution to ∂u = f , satisfies∫C

|u0,n|2e%m−mQdA ≤

2eM0qτ

(1 − κ)2(mα + 2M1e−2lτ )

∫T

| f |2e%m−mQdA.

Proof. The function %m being real-valued, the assumption (i) entails that %m is locally constant inC \ T . Since

(4.21) ∆(φ̂m,M0,M1 + %m) = (m −M0)∆Q̂τ + ∆%m +M1(1 + |z|2)−2,

and since ∆Q̂τ = ∆Q a.e. on Sτ, we get from the assumptions on T and %m that

(4.22) ∆(φ̂m,M0,M1 + %m) ≥mα2+M1e−2lτ a.e. on T .

Moreover, the assumption that ∂%m = 0 on C \ T implies that ∆%m = 0 on C \ T as well, and so,in view of (4.21), we have

∆(φ̂m,M0,M1 + %m) > 0 a.e. on C \ T .

In view of (iii) and (4.22), we have

|∂%m|2

∆(φ̂m,M0,M1 + %m)≤

κ2

eM0qτ,

which means that condition (5) of Th. 4.1 is fulfilled with the parameter choices a = 0 andb = M0qτ of Lm. 4.3. We have already verified that conditions (1)–(4) of Th. 4.1 are satisfied(conditions (1)–(2) are dealt with in Lm. 4.3). The assertion of the corollary is now an immediateconsequence of Th. 4.1. �

5. A B

5.1. Preliminaries. In this section we state and prove a result (Th. 5.2 below), which we lateruse to obtain the asymptotic expansion of Th. 2.3.

5.2. Earlier work. Our approach is based on the paper [4] by Berman, Berndtsson, and Sjöstrand,with some slight modifications. Perhaps it is of value to explain how weighted Bergman kernelexpansions work explicitly in the elementary C1 case. For this reason, our exposition is ratherdetailed.


5.3. A local integral operator. We recall the Ansatz

K1m(z,w) =

(mb0(z, w̄) + b1(z, w̄)

)emψ(z,w̄),

where the functions b0 and b1 are as in Subsect. 2.3. Associated with kernel, we define the localintegral operator

(5.1) I1,χm u(z) =

∫D(z0,2ε)

u(w)χ(w)K1m(z,w) e−mQ(w)dA(w),

where χ is a fixed smooth cut-off function, with the following properties: χ = 1 on D(z0, 32ε),

χ = 0 on C \D(z0, 2ε), 0 ≤ χ ≤ 1 throughout C, and ‖χ‖L2 ≤ 3. The point z0 ∈ N+ is assumed fixed.Moreover, the parameter ε is assumed small but positive, and independent of the parameter m.We specify how small ε should be as we go along. If we put

B1m(z, w̄) = b0(z, w̄) +m−1b1(z, w̄),

we may write

(5.2) I1,χm u(z) = m

∫D(z0,2ε)

u(w)χ(w)B1m(z, w̄) em(ψ(z,w̄)−Q(w))dA(w).

In order for this expression to make sense, we assume that ε is so small that

ψ(z, w̄) =∞∑

k=0

1k!

(w̄ − z̄)k∂kQ(z)

makes sense as a convergent series for all z,w ∈ D(z0, 3ε).

5.4. The phase function. The function

(5.3) φ(z,w) =ψ(z, w̄) −Q(w)

z − w=ψ(z, w̄) − ψ(w, w̄)

z − w,

will be called the phase function. It is clear that φ(z,w) is holomorphic in z and real-analyticallysmooth in w across z = w, and we observe that

(5.4) ∂wem(z−w)φ(z,w) = m(z − w)∂wφ(z,w) em(z−w)φ(z,w) = m(z − w)∂wφ(z,w) em(ψ(z,w̄)−Q(w)).

As a consequence, (5.2) may be written in the form

(5.5) I1,χm u(z) =

∫D(z0,2ε)

u(w)χ(w)z − w

B1m(z, w̄)

∂wφ(z,w)∂wem(z−w)φ(z,w)dA(w).

We note that

(5.6) ∂wφ(z,w)∣∣∣∣w=z= ∆Q(z).

5.5. Diagonal Taylor expansion. By Taylor’s formula,

ψ(z, w̄) −Q(w) = Q(z) + (w̄ − z̄)∂Q(z) +12

(w̄ − z̄)2∂2Q(z)

−

{Q(z) + (w̄ − z̄)∂Q(z) + (w − z)∂Q(z) +

12

(w̄ − z̄)2∂2Q(z) +

12

(w − z)2∂2Q(z) + |w − z|2∆Q(z)}

+O(|w − z|3) = −(w − z)∂Q(z) −12

(w − z)2∂2Q(z) − |w − z|2∆Q(z) +O(|w − z|3),

which in terms of the quadratic expression

Lz(w) = (w − z)∂Q(z) +12

(w − z)2∂2Q(z)

becomesψ(z, w̄) −Q(w) = −Lz(w) − |w − z|2∆Q(z) +O(|w − z|3),


for z,w ∈ D(z0, 2ε). An analogous calculation shows that

2ψ(z, w̄) −Q(z) −Q(w) = −2i Im Lz(w) − |w − z|2∆Q(z) +O(|w − z|3),

and, if we take real parts, we obtain

(5.7) 2 Reψ(z, w̄) −Q(z) −Q(w) = −|w − z|2∆Q(z) +O(|w − z|3).

In several instances we only need a rough estimate of this type:

(5.8) 2 Reψ(z, w̄) −Q(z) −Q(w) ≤ −12|w − z|2∆Q(z0), z,w ∈ D(z0, 3ε).

This entails a restriction on the size of ε > 0. We note that this requirement contains

(5.9) ∆Q(z) ≥12∆Q(z0) > 0, z ∈ D(z0, 3ε).

We recall that ∆Q(z0) > 0 holds as z0 ∈ N+.

5.6. A differential operator and negligible amplitudes. Let Mz−w denote the operator of multi-plication by z − w:

Mz−w f (z,w) = (z − w) f (z,w),

and consider the (partial) differential operator

(5.10) ∇m =1

∂wφ(z,w)∂w +mMz−w.

For smooth functions X(z,w), we see that

(5.11) ∂w

{em(z−w)φ(z,w)X(z,w)

}= ∂wφ(z,w) em(z−w)φ(z,w)

∇mX(z,w).

Functions of the form ∇mX are called negligible amplitudes, because in an integral expression like(5.12) below, the term involving ∇mX decreases exponentially in m.

We turn to the main technical result of this section.

Proposition 5.1. Fix a point z0 ∈ N+. We then have that

B1m(z, w̄)

∂wφ(z,w)=

b0(z, w̄)

∂wφ(z,w)+m−1 b1(z, w̄)

∂wφ(z,w)= 1 +m−1

∇mXm(z,w) +m−2Y(z,w),

where Xm = X0 + m−1X1, and the functions X0, X1, and Y are all holomorphic in z and real-analyticallysmooth in w, for z,w ∈ D(z0, 3ε), provided ε > 0 is small enough. Moreover, if K is a compact subset ofN+, an ε > 0 can be found that works for all z0 ∈ K simultaneously.

We supply the proof later on in the section.

5.7. Application to the local integral operator. We apply Prop. 5.1 to (5.5) (in the second andthird terms on the right hand side, we use relation (5.4) to expand ∂wem(z−w)φ(z,w), and recall that(z − w)φ(z,w) = ψ(z, w̄) −Q(w)) to obtain:

(5.12) I1,χm u(z) =

∫D(z0,2ε)

u(w)χ(w)z − w

∂wem(z−w)φ(z,w)dA(w)

+

∫D(z0,2ε)

u(w)χ(w)∇mXm(z,w) ∂wφ(z,w) em(ψ(z,w̄)−Q(w))dA(w)

+m−1∫D(z0,2ε)

u(w)χ(w) Y(z,w) ∂wφ(z,w) em(ψ(z,w̄)−Q(w))dA(w).

Let C1 be a positive constant such that∣∣∣Y(z,w) ∂wφ(z,w)∣∣∣ ≤ C1, z,w ∈ D(z0, 2ε).


Then, for z ∈ D(z0, 2ε), we have, in view of (5.8) and the Cauchy-Schwarz inequality,∣∣∣∣∣ ∫D(z0,2ε)

u(w)χ(w) Y(z,w) ∂wφ(z,w) em(ψ(z,w̄)−Q(w))dA(w)∣∣∣∣∣

≤ C1

∫D(z0,2ε)

|u(w)| em(Reψ(z,w̄)−Q(w))dA(w) ≤ C1 e12 mQ(z)

∫D(z0,2ε)

|u(w)| e−12 mQ(w)e−

14∆Q(z0)|w−z|2 dA(w)

≤ C1 e12 mQ(z)

∥∥∥ue−12 mQ

∥∥∥L2(D(z0,2ε)

{∫C

e−12∆Q(z0)|w−z|2 dA(w)

}1/2

= C′1 m−1/2 e12 mQ(z)

∥∥∥ue−12 mQ

∥∥∥L2(D(z0,2ε)

,

where C′1 = 21/2[∆Q(z0)]−1/2C1. Next, by (5.11), we have∫D(z0,2ε)


=

∫D(z0,2ε)

u(w)χ(w) ∂w

{em(ψ(z,w̄)−Q(w))Xm(z,w)

}dA(w),

and if u is holomorphic in, sayD(z0, 3ε), we can use Green’s formula to conclude that∫D(z0,2ε)

u(w)χ(w) ∂w

{em(ψ(z,w̄)−Q(w))Xm(z,w)

}dA(w)

= −

∫D(z0,2ε)

u(w)∂χ(w) Xm(z,w) em(ψ(z,w̄)−Q(w))dA(w).

We now confine the parameter z to the diskD(z0, ε), and observe that then

|w − z| ≥ε2, w ∈ D(z0, 2ε) \D(z0, 3

2ε),

so that by (5.8),

(5.13) Reψ(z, w̄) ≤12

Q(z) +12

Q(w) −ε2

16∆Q(z0), w ∈ D(z0, 2ε) \D(z0, 3

2ε).

Moreover, we recall that m should be big, at least m ≥ 1, so that there exists a positive constantC2 with

|Xm(z,w)| ≤ C2, z ∈ D(z0, ε), w ∈ D(z0, 2ε),

for small enough ε. As we recall that ∂χ = 0 onD(z0, 32ε), we find that∣∣∣∣∣ ∫

D(z0,2ε)u(w)∂χ(w) Xm(z,w) em(ψ(z,w̄)−Q(w))dA(w)

∣∣∣∣∣≤ C2

∫D(z0,2ε)

∣∣∣u(w)∂χ(w)∣∣∣ em(Reψ(z,w̄)−Q(w))dA(w)

≤ C2 e12 mQ(z)e−

116 mε2∆Q(z0)

∥∥∥ue−12 mQ

∥∥∥L2(D(z0,2ε)

‖∂χ‖L2 .

Since χ was assumed to have ‖∂χ‖L2 ≤ 3, we obtain in summary that∫D(z0,2ε)


≤ 3C2 e12 mQ(z)e−

116 mε2∆Q(z0)

∥∥∥ue−12 mQ

∥∥∥L2(D(z0,2ε)

.

We finally turn to the first term on the right hand side of (5.12), and use Green’s formula to get∫D(z0,2ε)

u(w)χ(w)z − w

∂wem(z−w)φ(z,w)dA(w) = u(z) −∫D(z0,2ε)

u(w)∂χ(w)z − w

em(ψ(z,w̄)−Q(w))dA(w),


where again z ∈ D(z0, ε) and u is assumed holomorphic on D(z0, 3ε). To estimate the rightmostterm, we again recall that ∂χ = 0 onD(z0, 3

2ε), and find that∫D(z0,2ε)

∣∣∣∣∣u(w)∂χ(w)z − w

em(ψ(z,w̄)−Q(w))∣∣∣∣∣dA(w) ≤

2ε

∫D(z0,2ε)

|u(w)∂χ(w)| em(Reψ(z,w̄)−Q(w))dA(w)

≤2ε

e12 mQ(z)e−

116 mε2∆Q(z0)

∫D(z0,2ε)

|u(w)|e−12 mQ(w)

|∂χ(w)|dA(w)

≤2ε

e12 mQ(z)e−

116 mε2∆Q(z0)

∥∥∥ue−12 mQ

∥∥∥L2(D(z0,2ε))

‖∂χ‖L2

≤6ε

e12 mQ(z)e−

116 mε2∆Q(z0)

∥∥∥ue−12 mQ

∥∥∥L2(D(z0,2ε))

, z0 ∈ D(z0, ε),

where we have again used (5.13), the Cauchy-Schwarz inequality, and that ‖∂χ‖L2 ≤ 3. We gatherour observations in a theorem.

Theorem 5.2. Fix a compact subset K of N+. Then there exists an ε > 0 which only depends on Q andK such that the following is true. If z0 ∈ K and if u is holomorphic inD(z0, 3ε), we have∣∣∣I1,χ

m u(z) − u(z)∣∣∣ ≤ Cm−3/2 e

12 mQ(z)

∥∥∥ue−12 mQ

∥∥∥L2(D(z0,2ε)

, z0 ∈ D(z0, ε),

for some positive constant C that only depends on Q,K , and ε.

Proof. The above sequence of estimates shows that for z ∈ D(z0, ε),∣∣∣I1,χm u(z) − u(z)

∣∣∣ ≤ {C′1m−3/2 + (3C2 + 6ε−1) e−

116 mε2∆Q(z0)

}e

12 mQ(z)

∥∥∥ue−12 mQ

∥∥∥L2(D(z0,2ε)

,

whence the assertion is immediate from the observation that the constants C′1,C2 are uniformlybounded for z0 ∈ K , and that ε > 0 can be picked to work simultaneously for all z0 ∈ K . �

Remark 5.3. The above theorem is the reason why we call the local integral operator I1,χm an

approximate Bergman projection.

5.8. The lift of the phase function. As before, we carry out our local analysis near a pointz0 ∈ N+. The points z,w are usually confined to the neighborhoodD(z0, 3ε) of the point z0, whereε > 0 is as small as needed. We will want to replace any occurrence of w̄ previously by anindependent variable ζ. An instance of this is the function

(5.14) θ(z,w, ζ) =ψ(z, ζ) − ψ(w, ζ)

z − w, z,w, ζ ∈ D(z0, 3ε),

which extends the phase function φ(z,w). Indeed, θ(z,wζ) is holomorphic in all three variables

(z,w, ζ) ∈ D(z0, 3ε) ×D(z0, 3ε) ×D(z̄0, 3ε),

and it follows from (5.14) that

(5.15) θ(z,w, w̄) = φ(z,w), z,w ∈ D(z0, 3ε).

By the definition of θ (see (5.14)) and (5.9), we have

∂ζθ(z,w, ζ)∣∣∣∣w=z,ζ=z̄

= ∆Q(z) ≥12∆Q(z0) > 0, z ∈ D(z0, 3ε),

so that if we let ε > 0 be a little smaller, we are guaranteed that

∂ζθ(z,w, ζ) , 0, (z,w, ζ) ∈ D(z0, 3ε) ×D(z0, 3ε) ×D(z̄0, 3ε).

To simplify the notation, we frequently write θ in place of θ(z,w, ζ).


5.9. Some differential operators and a change of variables. We introduce the commuting dif-ferential operators

(5.16) {∂θ} =1∂ζθ

∂ζ, {∂w} = ∂w −∂wθ∂ζθ

∂ζ,

where the curly brackets are used to differentiate these differential operators from the usual partial deriva-tives, e.g. ∂w. We shall adopt the notational convention to write, e.g., {∂θ∂w} for the product{dθ}{∂w}. The differential operators {∂θ}, {∂w} come from the (local) change of variables

(z,w, ζ) 7→ (z,w, θ), where θ = θ(z,w, ζ).

If we put ζ = w̄, we get from (5.16) that

(5.17) {∂θ}∣∣∣ζ=w̄ =

1

∂wφ(z,w)∂w,

and we recognize the expression on the right hand side as a component of the differential operator∇m.

5.10. The lift of the operator ∇m. If we put

(5.18) ∇′

m = {∂θ} +mMz−w,

we have in view of (5.17) defined a lift of ∇m to the context of the three complex variables (z,w, θ).We shall suppress the distinction between ∇m and ∇′m, and write ∇m in place of ∇′m.

5.11. Negligible amplitude equations. We recall that in view of Prop. 5.1, we are to solve theequation

(5.19) Am = 1 +m−1∇mXm +m−2Y,

where

(5.20) Am = A0 +m−1A1, A0(z,w, ζ) =b0(z, ζ)

∂ζθ(z,w, ζ), A1(z,w, ζ) =

b1(z, ζ)∂ζθ(z,w, ζ)

,

More generally, we may consider the “negligible amplitude” equation

(5.21) Am = 1 +m−1∇mXm +m−k−1Ym,

where

Am =

k∑j=0

m− jA j, Xm =

∞∑j=0

m− jX j, Ym =

∞∑j=0

m− jY j.

The key to studying equations of this type is to introduce the generalized differential operator

em−1{∂θ∂w} =

∞∑j=0

1j!m j {∂θ∂w}

j,

because of the identityem−1

{∂θ∂w}∇m = mMz−wem−1{∂θ∂w}.

As we apply this operator em−1{∂θ∂w} to both sides of the equation, we find that

em−1{∂θ∂w}Am = 1 +m−1em−1

{∂θ∂w}∇mXm +m−k−1em−1{∂θ∂w}Ym

= 1 +Mz−wem−1{∂θ∂w}Xm +m−k−1em−1

{∂θ∂w}Ym,

so that by restricting to the diagonal w = z, we get

em−1{∂θ∂w}Am

∣∣∣∣w=z= em−1

{∂θ∂w}

k∑j=0

m− jA j∣∣∣∣w=z= 1 +m−k−1em−1

{∂θ∂w}Ym

∣∣∣∣w=z.


This gives A0|w=z = 1, and also

Al∣∣∣w=z = −

l∑j=1

1j!{∂θ∂w}

jAl− j∣∣∣∣w=z, l = 1, . . . , k.

These equations are compatibility conditions, which must be fulfilled in order for (5.21) to have asolution. In particular, when k = 1, we get

(5.22) A0∣∣∣w=z = 1, A1

∣∣∣w=z = −{∂θ∂w}A0

∣∣∣∣w=z.

To simplify the notation, we put

b0 = b0(z, ζ), b1 = b1(z, ζ),

and see that (5.20) expresses that

A0 =b0

∂ζθ, A1 =

b1

b0A0,

so that the equation (5.22) may be written as

(5.23) A0∣∣∣w=z =

b0

∂ζθ

∣∣∣∣w=z= 1, A1

∣∣∣w=z =

b1

b0= −{∂θ∂w}A0

∣∣∣∣w=z.

We proceed to check that (5.23) holds. It is an easy consequence of (5.14) that ∂ζθ|w=z = ∂z∂ζψ = b0,and so the first part of (5.23) is clearly true. In fact, a Taylor series argument gives more preciseinformation:

(5.24)b0

∂ζθ= 1 −

12

(w − z)∂zb0

b0+O(|w − z|2).

An application of the operator {∂w} to both sides of the above equality gives

(5.25) {∂w}b0

∂ζθ= −

12∂zb0

b0+O(|w − z|),

and if we also apply {∂θ}, the result is

{∂θ∂w}b0

∂ζθ= −

12∂ζθ

∂ζ∂zb0

b0+O(|w − z|) = −

12b0

∂ζ∂zb0

b0+O(|w − z|),

and so

{∂θ∂w}A0∣∣∣w=z = {∂θ∂w}

b0

∂ζθ

∣∣∣∣w=z= −

12b0

∂ζ∂zb0

b0.

A comparison with (2.5) which defines b1 now shows that also the second part of (5.23) is true.

5.12. The solution to the negligible amplitude equation. We put

(5.26) Z0 =1 − A0

z − w, z , w,

which extends holomorphically across w = z:

Z0∣∣∣w=z = {∂w}A0

∣∣∣w=z.

In view of (5.22), the relationA1 + {∂θ∂w}A0 = (z − w)Z1

defines a function Z1 which is holomorphic when z,w are close to z0, and ζ is close to z̄0. Wefinally put

X0 = −Z0, X1 = Z1 + {∂θ∂w}Z0, Y = −{∂θ}Z1− {∂2

θ∂w}Z0.


Then X0,X1,Y are all holomorphic when z,w are close to z0, and ζ is close to z̄0. We rewrite theseexpressions in terms of A0,A1:

X0 =A0− 1

z − w, X1 =

A1

z − w−{∂θ}A0

(z − w)2 , Y = −{∂θ}A1

z − w+{∂2θ}A

0

(z − w)2 .

It follows that

∇mX0 = {∂θ}X0 +m(z − w)X0 ={∂θ}A0

z − w+m(A0

− 1)

and

∇mX1 = {∂θ}X1 +m(z − w)X1 ={∂θ}A1

z − w−{∂2θ}A

0

(z − w)2 +m(A1−{∂θ}A0

z − w

),

so that

∇mXm = ∇m(X0 +m−1X1) = m(A0− 1) + A1 +m−1

({∂θ}A1

z − w−{∂2θ}A

0

(z − w)2

).

By adding up the pieces, we find that

(5.27) 1 +m−1∇mXm +m−2Y = A0 +m−1A1,

which constitutes the solution to the negligible amplitude equation (5.19), for (z,w, ζ) ∈ D(z0, 3ε)×D(z0, 3ε) ×D(z̄0, 3ε).

5.13. The proof of Proposition 5.1. If we put ζ = w̄ in (5.27), and recall the definition (5.20) ofA0,A1, the assertion is immediate. �

6. T T 2.3

6.1. Preliminaries. In this section we prove Th. 2.3, by following the approach outlined in [4].The proof is obtained by combining Lm. 3.2, Th. 5.2, and Cor. 4.4. To facilitate the presentation,we divide the proof into several steps.

6.2. The setting. We first note that adding a constant to Q means that Km,n is changed by amultiplicative constant only, and hence we can (and will) assume that Q ≥ 1 on C.

We fix a point z0 ∈ K , and let ε > 0 be as needed for Th. 5.2. The cut-off function χ is as inSect. 5: it has χ = 1 on D(z0, ε), χ = 0 on C \D(z0, 3

2ε), while 0 ≤ χ ≤ 1 everywhere else. By thedefinition of the approximate Bergman kernel K1

m(z,w) back in Subsect. 2.3, we have

|K1m(z,w)|e−

12 m(Q(z)+Q(w)) = |mb0(z, w̄) + b1(z, w̄)|e

12 m(2 Reψ(z,w̄)−Q(z)−Q(w)),

and if m ≥ 1, we have

|mb0(z, w̄) + b1(z, w̄)|2 ≤ C1m, z,w ∈ D(z0, 2ε),

where C1 is a suitable constant (which only depends on Q,K , ε), and by (5.8), we have

e12 m(2 Reψ(z,w̄)−Q(z)−Q(w))

≤ e−14 m|z−w|2∆Q(z0).

A combination of the above leads to

(6.1) |K1m(z,w)|e−

12 m(Q(z)+Q(w))

≤ C1me−14 m|z−w|2∆Q(z0), z,w ∈ D(z0, 2ε).


6.3. Integral kernel techniques. Let Pm,n : L2mQ → Hm,n denote the orthogonal projection, that is,

Pm,nu(z) =∫C

u(w)Km,n(z,w)e−mQ(w)dA(w).

We introduce the functions

(6.2) Km,n;z(ζ) = Km,n(ζ, z) and K1,χm;w(ζ) = χ(ζ)K1

m(ζ,w),

with the tacit understanding that K1,χw (ζ) = 0 whenever ζ ∈ C \ D(z0, 2ε). We also need the

difference

(6.3) Rχw(ζ) = K1,χm;w(ζ) − Pm,nK1,χ

m;w(ζ), ζ ∈ C.

From the definition (5.1) of the local integral operator I1,χm , we have

I1,χm Km,n;z(ζ) =

∫D(z0,2ε)

Km,n;z(ξ)χ(ξ)K1m(ζ, ξ) e−mQ(ξ)dA(ξ)

=

∫D(z0,2ε)

Km,n(ξ, z)χ(ξ)K1m(ζ, ξ) e−mQ(ξ)dA(ξ),

so that as we take complex conjugates, we find that

(6.4) I1,χm Km,n;z(w) =

∫C

K1m(ξ,w)χ(ξ)Km,n(z, ξ)e−mQ(ξ)dA(ξ) = Pm,nK1,χ

m;w(z),

Next, by Th. 5.2, there is a constant C2 (which only depends on Q,K , ε), such that∣∣∣I1,χm Km,n;z(w) − Km,n;z(w)

∣∣∣ ≤ C2m−3/2e12 mQ(w)

∥∥∥Km,n;ze−12 mQ

∥∥∥L2(D(z0,2ε))

,

and in view of (6.4) we may rewrite this as

(6.5)∣∣∣Pm,nK1,χ

m;w(z) − Km,n(z,w)∣∣∣ ≤ C2m−3/2e

12 mQ(w)


∥∥∥L2(D(z0,2ε))

.

We note that∥∥∥Km,n;ze−12 mQ

∥∥∥L2(D(z0,2ε))

≤


∥∥∥L2(C)= ‖Km,n;z‖mQ = Km,n(z, z)1/2

≤ C3m1/2e12 mQ(z),

with a number C3 that only depends on Q, τ (see Prop. 3.6), so that by (6.5),

(6.6)∣∣∣Pm,nK1,χ

m;w(z) − Km,n(z,w)∣∣∣ ≤ C4m−1e

12 m(Q(z)+Q(w)),

where C4 = C2C3.

6.4. Application of weighted ∂ methods. The function Rχw given by (6.3) is the L2mQ,n-minimal

norm solution to the ∂-problem

∂ζRχw(ζ) = ∂ζK

1,χm,w(ζ) = K1

m;w(ζ)∂χ(ζ),

with the understanding that the rightmost expression vanishes on C \D(z0, 2ε). Here, we use thenotation K1

m;w(ζ) = K1m(ζ,w). We now add the requirement on ε that

D(z0, 2ε) ⊂ Sτ,

so that ∂χ vanishes off Sτ. By Cor. 4.4, then,∥∥∥Rχw∥∥∥

mQ ≤e

12 M0qτ+lτ

M11/2

∥∥∥K1m;w∂χ

∥∥∥mQ,


provided that m ≥ M0 and n ≥](m −M0)τ +M1[. We recall that the positive reals M0,M1 areconstrained by (4.14), and that qτ, lτ are given by (4.13). Next, we recall that ‖∂χ‖L2 ≤ 3, so that∥∥∥K1

m;w∂χ∥∥∥

mQ =∥∥∥K1

m;w∂χe−12 mQ

∥∥∥L2

≤ ‖∂χ‖L2

∥∥∥K1m;we−

12 mQ

∥∥∥L∞(D(z0,2ε)\D(z0, 3

2 ε))≤ 3

∥∥∥K1m;we−

12 mQ

∥∥∥L∞(D(z0,2ε)\D(z0, 3

2 ε)).

A combination of the above gives

(6.7)∥∥∥Rχw

∥∥∥mQ ≤

3e12 M0qτ+lτ

M11/2

∥∥∥K1m;we−

12 mQ

∥∥∥L∞(D(z0,2ε)\D(z0, 3

2 ε)).

In view of (6.1), we have∥∥∥K1m;we−

12 mQ

∥∥∥L∞(D(z0,2ε)\D(z0, 3

2 ε))≤ C1m e−

116 mε2∆Q(z0)e

12 mQ(w), w ∈ D(z0, ε),

so that (6.7) entails that

(6.8)∥∥∥Rχw

∥∥∥mQ ≤

3C1e12 M0qτ+lτ

M11/2

m e−1

16 mε2∆Q(z0)e12 mQ(w), w ∈ D(z0, ε).

As Rχw is holomorphic in the disk D(z0, 32ε), we may use Lm. 3.2 to get a pointwise estimate in

place of an integral one. If we assume ε is so small that ∆Q ≤ 2∆Q(z0) onD(z0, ε), we find that

(6.9) |Rχw(z)|e−12 mQ(z)

≤2√

me14 ε

2∆Q(z0)

ε

∥∥∥Rχwe−12 mQ

∥∥∥L2(D(z, 1

2 m−1/2ε))

≤2√

me14 ε

2∆Q(z0)

ε

∥∥∥Rχw∥∥∥

mQ, z,w ∈ D(z0, ε),

where m ≥ 1 is assumed. We get from a combination of (6.8) and (6.9) that

(6.10) |Rχw(z)| ≤ C5 m3/2 e−1

16 mε2∆Q(z0)e12 m(Q(z)+Q(w)), z,w ∈ D(z0, ε),

where C5 is the constant

C5 =6C1e

12 M0qτ+lτ+ 1

4 ε2∆Q(z0)

M11/2ε

.

6.5. Conclusion of the proof of Theorem 2.3. We have assumed that m ≥ m0 and n ≥ mτ −M,for some large positive real m0. We need to find M0,M1 such that (4.14) holds, while

n ≥ mτ −M =⇒ n ≥](m −M0)τ +M1[,

for all m ≥ m0. We should also make sure m0 ≥ max{1,M0}. For the implication to hold, it isenough to require that

mτ −M ≥ (m −M0)τ +M1,

which amounts toM0τ −M1 ≥M.

By choosing, e.g., M1 = 1 and M0 appropriately big, we can make sure the above inequalityis fulfilled, while (4.14) holds. In conclusion, if we recall the definition (6.3) of Rχw, we get bycombining (6.6) with (6.10) the estimate∣∣∣Km,n(z,w) − K1

m(z,w)∣∣∣ = ∣∣∣Km,n(z,w) − Pm,nK1,χ

m;w(z) − Rχw(z)∣∣∣ ≤ ∣∣∣Km,n(z,w) − Pm,nK1,χ

m;w(z)∣∣∣ + |Rχw(z)|

≤

(C4m−1 + C5m3/2e−

116 mε2∆Q(z0)

)e

12 m(Q(z)+Q(w)), z,w ∈ D(z0, ε).

Finally, we note thatinf

z0∈K∆Q(z0) > 0,


and that ε is to be chosen independently of z0 ∈ K , so the term m3/2e−116 mε2∆Q(z0) is dominated by

m−1 for large m (we assume that m ≥ m0). The constants C4,C5 may also be chosen independentlyof z0 ∈ K , This completes the proof. �

7. B G

7.1. Preliminaries. In this section we use the expansion formula for Km,n (Th. 2.3) to prove Ths.2.7 and 2.8. We fix a compact subset K b S◦1 ∩ N+, a positive real ε with the properties listed inTh. 2.3, and write, for m ≥ 3,

δm = εm−1/2 log m.

7.2. Application of the expansion formula for the polynomial Bergman kernel. By Th. 2.3, wehave for each z0 ∈ K b S◦τ ∩N+,

(7.1) Km,n(z, z0) e−12 m(Q(z)+Q(z0))

= m(b0(z, z̄0) +m−1b1(z, z̄0)) e12 m(2 Reψ(z,z̄0)−Q(z)−Q(z0)) +O(m−1), z ∈ D(z0, ε),

when m→ ∞ while n ≥ mτ −M, for some fixed positive real M. The O expression is uniform inz0 ∈ K . In particular, with z = z0, we have

(7.2) Km,n(z0, z0) e−mQ(z0) = m∆Q(z0) + 12∆ log∆Q(z0) +O(m−1), z ∈ D(z0, ε).

Moreover, by (5.7) with z in place of w, and z0 in place of z, we get

2 Reψ(z, z̄0) −Q(z) −Q(z0) = −|z − z0|2∆Q(z0) + Rz0 (z),

where Rz0 is a function with

|Rz0 (z)| ≤ C1|z − z0|3, z ∈ D(z0, 2ε),

for some constant C1 independent of z0 ∈ K . By Taylor’s formula,

b0(z, z̄0) = b0(z0, z̄0) +O(|z − z0|) = ∆Q(z0) +O(|z − z0|), z ∈ D(z0, ε),

so that

(7.3) b0(z, z̄0) = ∆Q(z0) +O(m−1/2 log m), z ∈ D(z0, δm),

since – by the way we defined δm – we have δm ≤ ε for m ≥ 3. As b1(z, z̄0) is bounded forz ∈ D(z0, 2ε), we conclude from (7.1) that From the bound on Rz0 we get that

m|Rz0 (z)| ≤ C1mδ3m ≤ C1ε

3m−1/2[log m]3, z ∈ D(z0, δn), z0 ∈ K ,

which gives

(7.4) emRz0 (z) = 1 +O(m−1/2[log m]3), z ∈ D(z0, δn), z0 ∈ K .

As we implement this into (7.1), while using (7.3) and that b1(z, z̄0) is bounded for z ∈ D(z0, ε), wefind that

(7.5) Km,n(z, z0) e−12 m(Q(z)+Q(z0)) = m

(∆Q(z0) +O(m−1/2[log m]3)

)e−

12 m|z−z0 |

2∆Q(z0) +O(m−1),

for z ∈ D(z0, δn) as m → ∞ while n ≥ mτ −M. Forming the square of the modulus of each sidegives us∣∣∣Km,n(z, z0)

∣∣∣2 e−m(Q(z)+Q(z0)) = m2((∆Q(z0))2 +O(m−1/2[log m]3)

)e−m∆Q(z0)|z−z0 |

2

+O(e−12 m∆Q(z0)|z−z0 |

2) +O(m−2),

so that in view of (7.2), we obtain

(7.6) B〈z0〉m,n(z) =

∣∣∣Km,n(z, z0)∣∣∣2

Km,n(z0, z0)e−mQ(z) = m

(∆Q(z0) +O(m−1/2[log m]3)

)e−m∆Q(z0)|z−z0 |

2+O(m−1),


for z ∈ D(z0, δn) as m→∞while n ≥ mτ −M.

7.3. Proof of Theorem 2.7. Our next step is to calculate the integral

(7.7)∫D(z0,δm)

e−m∆Q(z0)|z−z0 |2dA(z) =

1m∆Q(z0)

(1 − e−ε

2∆Q(z0)[log m]2)=

1m∆Q(z0)

(1 +O(m−1)),

which allows us to conclude from (7.6) that

B〈z0〉(D(z0, δm)) =∫D(z0,δm)

B〈z0〉m,n(z) dA(z) = 1 +O(m−1/2[log m]3).

Since the Berezin measure B〈z0〉 is a p.m., we get

(7.8) B〈z0〉(C \D(z0, δm)) =∫C\D(z0,δm)

B〈z0〉m,n(z) dA(z) = O(m−1/2[log m]3),

which tends to 0 as m→ ∞. In particular, since δm → 0 as m→ ∞, the mass of B〈z0〉 concentratesto the point z0, which concludes the proof. �

7.4. Proof of Theorem 2.8. By the linear change of variables

z = z0 +ζ√

m∆Q(z0),

we have

(7.9)∫C

∣∣∣∣B̂〈z0〉m,n(ζ) − e−|ζ|

2∣∣∣∣ dA(ζ) =

∫C

∣∣∣∣B〈z0〉m,n(z) −m∆Q(z0)e−m∆Q(z0)|z−z0 |

2∣∣∣∣ dA(z).

If we use (7.7), we get from (7.6) that

(7.10)∫D(z0,δm)

∣∣∣∣B〈z0〉m,n(z) −m∆Q(z0)e−m∆Q(z0)|z−z0 |

2∣∣∣∣ dA(z) = O(m−1/2[log m]3),

while (7.8) and

m∆Q(z0)∫C\D(z0,δm)

e−m∆Q(z0)|z−z0 |2dA(z) = e−ε

2∆Q(z0)[log m]2= O(m−1)

show that

(7.11)∫C\D(z0,δm)

∣∣∣∣B〈z0〉m,n(z) −m∆Q(z0)e−m∆Q(z0)|z−z0 |

2∣∣∣∣ dA(z) = O(m−1/2[log m]3).

We add up (7.10) and (7.11):∫C

∣∣∣∣B〈z0〉m,n(z) −m∆Q(z0)e−m∆Q(z0)|z−z0 |

2∣∣∣∣ dA(z) = O(m−1/2[log m]3).

The assertion is immediate, since the right hand side approaches 0 uniformly in z0 ∈ K whenm→∞. �

8. O-

Background. Th. 8.1 is the main result of this section. It supplies an off-diagonal damping boundof the polynomial Bergman kernel Km,n. Off-diagonal damping techniques are well-known for theBergman kernel, under the assumption that the weight Q is globally strictly subharmonic. See,e.g., Lindholm’s paper [18], Prop. 9, pp. 404–407, for background on the damping technique. Inour situation, two non-standard elements arise: (1) polynomial Bergman kernels are considered,and (2) the weight Q need not be globally strictly subharmonic. Our proof is based on he weightedL2 estimate provided by Cor. 4.5.


8.1. Some notation. In this section we prove Th. 2.11. We shall obtain that theorem from Cor.8.2 below, which is derived from Th. 8.1. The estimate of Th. 8.1 and Cor. 8.2 is of independentinterest, and has applications to random matrix theory, see [1]. It will be convenient to define theset

Sτ,1 = {ζ ∈ C; dist C(ζ,Sτ) ≤ 1}.We begin by picking a point z0 ∈ S

◦τ ∩N+, and a radius r0 > 0 such that

(8.1) D(z0, r0) ⊂ Sτ ∩N+.

Let α and A be two positive constants such that

(8.2) ∆Q ≥ α on D(z0, r0), ∆Q ≤ A on Sτ,1.

8.2. Application of weighted estimates for ∂-equation with a growth constraint. Suppose wehave found a real-valued function %m, of class C1,1(C), such that

(i) %m is constant on C \D(z0, r0),(ii) ∆%m ≥ −

12 mα +M0α a.e. onD(z0, r0),

(iii) |∂%m|2≤

18 mαe−M0qτ a.e. onD(z0, r0),

where qτ and lτ are given by (4.13), and the general setting with M0,M1 is as in the implementationscheme of Sect. 4 (so we of course assume Q ≥ 1). This means that the conditions on %m of Cor.4.5 are fulfilled with κ = 1

2 and T = D(z0, r0). Cor. 4.5 then asserts that if f ∈ L∞(D(z0, r0)), and ifm ≥M0 and n ≥](m −M0)τ +M1[, then u0,n, the L2

mQ,n-minimal solution to ∂u = f , satisfies

(8.3)∫C

|u0,n|2e%m−mQdA ≤

8eM0qτ

mα + 2M1e−2lτ

∫D(z0,r0)

| f |2e%m−mQdA.

We make one more assumption on %m, namely

(iv) %m = 0 onD(z0,m−1/2).

Then we get immediately from (8.3) that

(8.4)∫D(z0,m−1/2)

|u0,n|2e−mQdA ≤

8eM0qτ

mα + 2M1e−2lτ

∫D(z0,r0)

| f |2e%m−mQdA.

An application of Lm. 3.2 with δ = 1 shows that for m ≥ 1,

|u0,n(z0)|2e−mQ(z0)≤ meA

∫D(z0,m−1/2)

|u0,n|2e−mQdA,

and if we combine this with (8.4), the result is

(8.5) |u0,n(z0)|2e−mQ(z0)≤

8meA+M0qτ

mα + 2M1e−2lτ

∫D(z0,r0)

| f |2e%m−mQdA.

8.3. Juggling with test functions. We now require of f that it should be of the form f = g∂χ0,where g is a polynomial in Hm,n, and χ0 is a smooth real-valued function with 0 ≤ χ0 ≤ 1everywhere, and, more specifically,

(8.6) χ0 = 0 on D(z0, r1), and χ0 = 1 on C \D(z0, 2r1),

where 0 < r1 ≤12 r0. It is possible to require in addition that

(8.7) |∂χ0|2≤ 2r−2

1 χ0,

cf. the subsection on cut-off functions in Sect. 3. One solution of the equation ∂u = g∂χ0 is givenby u = gχ0, so the difference gχ0 − u0,n is entire. In fact, from Subsect. 4.1, we have that

gχ0 − u0,n = Pm,n[gχ0],


where Pm,n : L2mQ → Hm,n is the orthogonal projection. As χ0 vanishes at z0, we see that

u0,n(z0) = −Pm,n[gχ0](z0),

so that (8.5) with f = gχ0 expresses that

(8.8) |Pm,n[gχ0](z0)|2e−mQ(z0)≤

8meA+M0qτ

mα + 2M1e−2lτ

∫D(z0,r0)

|g|2|∂χ0|2e%m−mQdA.

If we also use the estimate (8.7), we get in place of (8.8) the estimate

(8.9) |Pm,n[gχ0](z0)|2e−mQ(z0)≤

16mr−21 eA+M0qτ

mα + 2M1e−2lτ

∫D(z0,r0)

|g|2χ0e%m−mQdA

≤16mr−2

1 eA+M0qτ

mα + 2M1e−2lτ

∫C

|g|2χ0e%m−mQdA.

This suggests that we should introduce the norm

‖g‖2χ0,mQ =

∫C

|g|2χ0e−mQdA,

with associated inner product. The Hilbert space of polynomials of degree ≤ n− 1 supplied withthis new norm will be denoted by Hχ0,m,n. We write ball(Hχ0,m,n) for the closed unit ball of Hχ0,m,n.Next, we recall that Pm,n is expressed in terms of the reproducing kernel Km,n,

Pm,n[F](ζ) =∫C

Km,n(ζ, ξ) F(ξ)e−mQ(ξ)dA(ξ) =⟨F,Km,n(·, ζ)

⟩mQ.

This allows us to make the following calculation:

(8.10) sup{∣∣∣Pm,n[gχ0](z0)

∣∣∣2; g ∈ ball(Hχ0,m,n)}= sup

{∣∣∣⟨gχ0,Km,n(·, z0)⟩

mQ

∣∣∣2; g ∈ ball(Hχ0,m,n)}

= sup{∣∣∣⟨g,Km,n(·, z0)

⟩χ0,mQ

∣∣∣2; g ∈ ball(Hχ0,m,n)}=

∥∥∥Km,n(·, z0)∥∥∥2

χ0,m,n

=

∫C

|Km,n(ζ, z0)|2χ0(ζ)e−mQ(ζ)dA(ζ).

We realize from (8.10) that our estimate (8.9) entails that

(8.11) e−mQ(z0)∫C

|Km,n(ζ, z0)|2χ0(ζ)e−mQ(ζ)dA(ζ) ≤16mr−2

1 eA+M0qτ

mα + 2M1e−2lτsup

{e%m(ζ); ζ ∈ C \D(z0, r1)

},

as χ0 vanishes onD(z0, r1). We need to turn this integral estimate into a pointwise one. We picka point z1 ∈ Sτ \D(z0, 2r1 +m−1/2), and note thatD(z1,m−1/2) ⊂ Sτ,1 for m ≥ 1. In addition, χ0 = 1onD(z1,m−1/2), and so, by Lm. 3.2, with δ = 1,

(8.12) |Km,n(z1, z0)|2e−mQ(z1)≤ meA

∫D(z1,m−1/2)

|Km,n(ζ, z0)|2e−mQ(ζ)dA(ζ)

≤ meA∫C

|Km,n(ζ, z0)|2χ0(ζ)e−mQ(ζ)dA(ζ),

where it is tacitly assumed that m ≥ 1. A combination of (8.11) and (8.12) leads to

(8.13) |Km,n(z1, z0)|2e−m(Q(z0)+Q(z1))≤

16m2r−21 e2A+M0qτ

mα + 2M1e−2lτsup

{e%m(ζ); ζ ∈ C \D(z0, r1)

},

If we suppose r1 ≥12 m−1/2 and recall that M1 > 0, we may relax (8.13) to

(8.14) |Km,n(z1, z0)|2e−m(Q(z0)+Q(z1))≤ 64m2α−1e2A+M0qτ sup

{e%m(ζ); ζ ∈ C \D(z0, r1)

}.

Our hope is that %m can be made rather big negative onC\D(z0, r1), for then the estimate (8.14)supplies strong decay off the diagonal.


8.4. Construction of %m. We now construct a function %m which fulfills the conditions (i)–(iv)above, under the assumption

(8.15) m ≥ max{4M0,

(√

2 +√α)2

αr20

}.

We look for a radial function of the form

%m(ζ) = −√

mσm(|ζ − z0|),

where the function σm : [0,∞)→ [0,∞) is of class C1,1. By condition (i), we should have

(8.16) σ′m(t) = 0, r0 < t < ∞.

Also, condition (iv) requires that

(8.17) σm(t) = 0, 0 ≤ t ≤ m−1/2.

Condition (iii) is fulfilled if

(8.18) 0 ≤ σ′m(t) ≤ 2−1/2√α e−12 M0qτ , 0 < t < r0.

As for condition (ii), we first check that

∆%m(ζ) = −√

m4

{σ′′m(|ζ − z0|) +

σ′m(|ζ − z0|)|ζ − z0|

},

so that condition (ii) is equivalent to having

(8.19) σ′′m(t) +σ′m(t)

t≤ 2αm1/2

− 4αM0m−1/2, 0 < t < r0.

By (8.15), m ≥ 4M0, and we see that (8.19) is implied by the slightly simpler-looking condition

(8.20) σ′′m(t) +σ′m(t)

t≤ αm1/2, 0 < t < r0.

We define σm(t) in the following manner. We introduce points t1, t2, t3, t4, with

m−1/2 = t1 < t2 < t3 < t4 = r0,

and we letσ′′m(t) = am1[t1,t2](t) − bm1[t3,t4](t),

where the positive real constants am, bm remain to be determined. Here, 1X denotes the character-istic function of the set X. The condition (8.16) then requires that

am(t2 − t1) − bm(t4 − t3) = 0.

In this problem the interval [t3, t4] should be really short, and the number bm correspondinglylarge. As for am, we pick

am =12αm1/2,

because then

σ′′m(t) +σ′m(t)

t= am + am

t − t1

t≤ 2am = αm1/2, t1 < t < t2,

and (8.20) holds at least on the interval [t1, t2]. In the remaining interval [t2, t4], we have

σ′′m(t) +σ′m(t)

t≤σ′m(t)

t≤ am

t2 − t1

t≤ am =

12αm1/2, t2 < t < t4,

and so (8.20) holds throughout. It remains to arrange that (8.18) holds as well. The maximumvalue of σ′m(t) is attained at t = t2, so that (8.18) holds once

(8.21) σ′m(t2) = am(t2 − t1) =12αm1/2(t2 − t1) ≤ 2−1/2√α e−

12 M0qτ .


If we putt2 = (1 + 21/2α−1/2e−

12 M0qτ )m−1/2,

then (8.21) is valid with equality. The condition (8.15) guarantees that t2 < t4 = r0. The choice oft3 is essentially trivial; we should just pick it very close to t4 = r0. We consider the function

σ̃m(t) = 2−1/2√α e−12 M0qτ max

{0,min{t − t2, t4 − t2}

}= max

{0, 2−1/2√αe−

12 M0qτ min{t, r0} −m−1/2

(2−1/2√α e−

12 M0qτ + e−M0qτ

)}= max

{0, 2λ0 min{t, r0} −m−1/2(2λ0 + λ1)

},

whereλ0 = 2−3/2√αe−

12 M0qτ , λ1 = e−M0qτ ,

and note that the above construction produces a function σm(t) with

(8.22) σ̃m(t) ≤ σm(t), 0 ≤ t < ∞.

8.5. Implementation of the estimates. We return to (8.14), with the given choice of %m:

(8.23) |Km,n(z1, z0)|2e−m(Q(z0)+Q(z1))≤ 64m2α−1e2A+M0qτ e−

√mσm(r1)

≤ 64m2α−1e2A+M0qτ e−√

m σ̃m(r1),

where we use (8.22) and recall that it is assumed that r1 ≥12 m−1/2. We now turn things around,

and try to pick first z1 and only later r1. If we begin with

|z1 − z0| ≥ 2m−1/2,

we may put

r1 =12

min{r0, |z1 − z0| −m−1/2

},

and it is an easy matter to obtain the estimate

e−√

m σ̃m(r1)≤ e3λ0+λ1 e−λ0

√m min{r0,|z0−z1 |}.

If we suppose thatr0 ≥ m−1/2,

which is a consequence of (8.15), thenr1 ≥

12 m−1/2

as needed, and we may derive from (8.23) that

(8.24) |Km,n(z1, z0)|2e−m(Q(z0)+Q(z1))≤ 64m2α−1e2A+M0qτ+3λ0+λ1 e−λ0

√m min{r0,|z1−z0 |}.

It remains to pick r0 and α so that both (8.1) and (8.2) are fulfilled. One way – which we shallfollow – is to assume Q is C2-smooth, and choose

r0 =12

dist C(z0,C \ (S◦τ ∩N+)),

and let α be the biggest possible such that (8.2) is met. We recall that the assumptions leadingup to (8.24) are as follows: the parameters M0,M1 are as in Subsection 4.4, the parameter m isassumed so big that (8.15) holds, that n ≥](m −M0)τ +M1[, and |z1 − z0| ≥ 2m−1/2.

Theorem 8.1. Assume that Q ∈ C2(C) and that Q ≥ 1. Fix a point z0 ∈ S◦τ ∩ N+ and a real parameter

M > 0. Finally, put

r0 =12

dist C(z0,C \ (S◦τ ∩N+)), α = infD(z0,r0)

∆Q, A = supSτ,1

∆Q.

Then there exists a positive constant C such that for all m ≥ 1 and all positive integers n with n ≥ mτ−M,

(8.25) |Km,n(z1, z0)|2e−m(Q(z0)+Q(z1))≤ Cm2e−λ0

√m min{r0,|z1−z0 |}, z1 ∈ Sτ,


where λ0 = 2−3/2√αe−12 Mτ−1qτ and qτ is given by (4.13). The constant C only depends on the parameters

α,A, qτ,Mτ−1.

Proof. We may consider in the context of Subsect. 4.4 that M1 → 0 while M0 → Mτ−1. In otherwords, whenever M1 appears, we replace it with 0, and when M0 appears, we replace it by Mτ−1.It is important that (4.14) no longer restrains M0. By (8.24), the asserted estimate (8.25) holds forany C with

C ≥ 64α−1e2A+Mτ−1qτ+3λ0+λ1 ,

provided that (8.15) holds and |z1 − z0| ≥ 2m−1/2. We note that by Lm. 3.2, one can show that (cf.Prop. 3.6)

|Km,n(z1, z0)|2e−m(Q(z0)+Q(z1))≤ e2Am2, z1 ∈ Sτ.

In case |z1 − z0| < 2m−1/2, we have

e−λ0√

m min{r0,|z1−z0 |} ≥ e−2λ0 ,

so that (8.25) holds wheneverC ≥ e2A+2λ0 .

Next, we observe that the opposite to (8.15) splits in two possibilities:

m < 4Mτ−1 or m1/2 <

√2 +√α

√αr0

.

If m < 4Mτ−1, then

e−λ0√

m min{r0,|z1−z0 |} ≥ e−2λ0r0√

Mτ−1,

so that (8.25) holds whenever

C ≥ e2A+2λ0r0√

Mτ−1.

Finally, if

m1/2 <

√2 +√α

√αr0

,

thene−λ0

√m min{r0,|z1−z0 |} ≥ e−λ0(1+21/2α−1/2

,

so that (8.25) holds whenever

C ≥ e2A+λ0(1+21/2α−1/2).

In conclusion, it follows that (8.25) holds generally, with

C = e2A max{64α−1eMτ−1qτ+3λ0+λ1 , e2λ0 , eλ0r0

√

Mτ−1, eλ0(1+21/2α−1/2)

}.

The proof is complete. �

8.6. Application of the maximum principle. The maximum principle allows us to extend theestimate of Th. 8.1 to general z1 ∈ C.

Corollary 8.2. Assume Q ∈ C2(C) and that Q ≥ 1. Fix a point z0 ∈ S◦τ ∩ N+ and a real parameter

M > 0. Let r0, α,A be as in Th. 8.1. Then, if the positive integer n is confined to mτ −M ≤ n ≤ mτ + 1,we have

|Km,n(z0, z1)|2e−m(Q(z0)+Q(z1))≤ Cm2e−λ0

√m min{r0,|z0−z1 |}e−m(Q(z1)−Q̂τ(z1)), z1 ∈ C,

where λ0 and the constant C are as in Th. 8.1.


Proof. By Th. 8.1, it is enough to obtain the claimed estimate when z1 ∈ C \ Sτ, as Q̂τ = Q holdson Sτ. To this end, we consider the function

f (ζ) = log∣∣∣Km,n(ζ, z0)

∣∣∣2 −mQ̂τ(ζ).

Since Q̂τ is harmonic on C \ Sτ, f is subharmonic there. Moreover, since n− 1 ≤ mτ and since thepolynomial Km,n(·, z0) is of degree ≤ n − 1, we have that

log∣∣∣Km,n(ζ, z0)

∣∣∣2 ≤ (n − 1) log |ζ|2 +O(1) ≤ mτ log |ζ|2 +O(1) when |ζ| → ∞,

while the relation (2.4) says that Q̂τ(ζ) = τ log |ζ|2+O(1) when |ζ| → ∞. Hence f is bounded abovein the plane C (the bound might depend on m, though). Moreover, it is clear that f is harmonicin a punctured neighborhood of infinity, so that f has a representation

f (ζ) = h(ζ) − c log |ζ|

for all large enough |ζ|, where c is a non-negative number and h is harmonic at infinity (see e.g.[19], Cor. 0.3.7, p. 12). In particular, f extends to a subharmonic function on C∗ \ Sτ. By Th. 8.1,we have

f (ζ) ≤ log(Cm2) +mQ(z0) − λ0√

mr0, when ζ ∈ ∂Sτ,

since it is clear that r0 < |ζ − z0| for ζ ∈ ∂Sτ. The maximum principle extends this estimate to allζ ∈ C∗ \ Sτ. The conclusion is that∣∣∣Km,n(z0, z1)

∣∣∣2 e−mQ̂τ(z1)−mQ(z0)≤ Cm2e−λ0r0

√m, when z1 ∈ C \ Sτ,

and the claimed estimate follows. �

8.7. The proof of Theorem 2.11. Let K be a compact subset of S◦τ ∩N+, and pick M > 0. By Th.2.3 we have that Km,n(z0, z0)e−mQ(z0) = m∆Q(z0) +O(1) as m→∞ and n ≥ mτ −M, where the O(1)is uniform in z0 ∈ K . It follows that

(8.26) B〈z0〉m,n(z) =

∣∣∣Km,n(z, z0)∣∣∣2

Km,n(z0, z0)e−mQ(z) =

∣∣∣Km,n(z, z0)∣∣∣2

m∆Q(z0) +O(1)e−m(Q(z)+Q(z0)), z ∈ C,

as m → ∞ and n ≥ mτ −M. Since ∆Q is bounded from below by a positive number on K , theright hand side in (8.26) can be estimated by

Cm−1∣∣∣Km,n(z, z0)

∣∣∣2 e−m(Q(z)+Q(z0)), z ∈ C,

where C depends on the lower bound of ∆Q onK . The assertion now follows from Cor. 8.2. �

9. T B–F

9.1. Preliminaries. In this section we prove Th. 2.9. We therefore put Q(z) = |z|2. Recall that inthis case Sτ = D(0,

√τ), and (see (2.8))

(9.1) dB〈z0〉m,n(z) = m

|En−1(mzz̄0)|2

En−1(m |z0|2)

e−m|z|2 dA(z) where Ek(z) =k∑

j=0

z j

j!.

9.2. The action on polynomials. We first consider the Berezin transform on polynomials.

Proposition 9.1. Fix a complex number z0 , 0, a positive integer d and let n be an integer, n ≥ d + 1.Then, for all analytic polynomials u of degree at most d, we have

(9.2) pv∫C

u(z−1)dB〈z0〉m,n → u(z−1

0 ), as m→∞,

uniformly in n, n ≥ d + 1.


Proof. It is sufficient to prove the statement for u(z) = z j with j ≤ d. The left hand side in (9.2) canthen be written

pv∫C

z− jdB〈z0〉m,n =

mb jm,n(z0)

En−1(m |z0|2),

where we have put

b jm,n(z0) = pv

∫C

z− j∣∣∣∣∣ n−1∑

k=0

(mz0z̄)k

k!

∣∣∣∣∣2e−m|z|2 dA(z).

Expanding the square yields

b jm,n(z0) =

n−1∑k,l=0

mk+lz0kz̄l

0

k!l!pv

∫C

z̄kzl− je−m|z|2 dA(z).

Clearly only those k, l for which k = l − j give a non-zero contribution to the sum, and therefore,

b jm,n(z0) = z0

− jn−1∑l= j

m2l− j|z0|

2l

(l − j)!l!

∫C

|z|2(l− j) e−m|z|2 dA(z) =1

mz0j

n−1∑l= j

ml|z0|

2l

l!.

It follows that

b jm,n(z0) =

1mz0

j

n−1∑l= j

(m |z0|2)l

l!=

1mz0

j

(En−1(m |z0|

2) − E j−1(m |z0|2)),

and somb j

m,n(z0)

En−1(m |z0|2)=

1z0

j

1 −E j−1(m |z0|

2)

En−1(m |z0|2)

.Finally, since j ≤ d < n,

E j−1(m |z0|2)

En−1(m |z0|2)≤

Ed−1(m |z0|2)

Ed(m |z0|2)→ 0 as m→∞.

�

Proposition 9.2. Fix z0 ∈ C \D(0,√τ) and let u be an analytic polynomial. Then, for any fixed r with

0 < r <√τ, we have

pv∫D(0,r)

u(z−1)dB〈z0〉m,n(z)→ 0 as m→∞ while n = mτ + o(m).

Proof. Put, for ν = 0, 1, 2, . . .,

bνm,n(z0, r) = pv∫D(0,r)

z−νdB〈z0〉m,n .

A straightforward calculation based on (9.1) leads to

(9.3) bνm,n(z0, r) =1

zν0En−1(m |z0|2)

n−1∑j=ν

(m |z0|2) j

j!( j − ν)!

∫ mr2

0s j−νe−sds.

We suppose n is greater than ν by at least two units, so that we may pick an integer k withν < k < n, and split the sum (9.3) accordingly:

(9.4) bνm,n(z0, r) =1

zν0En−1(m |z0|2)

{ k−1∑j=ν

(m |z0|2) j

j!( j − ν)!

∫ mr2

0s j−νe−sds +

n−1∑j=k

(m |z0|2) j

j!( j − ν)!

∫ mr2

0s j−νe−sds

}.


We estimate the first term trivially as follows:

(9.5)k−1∑j=ν

(m |z0|2) j

j!( j − ν)!

∫ mr2

0s j−νe−sds ≤

k−1∑j=ν

(m |z0|2) j

j!( j − ν)!

∫∞

0s j−νe−sds =

k−1∑j=0

(m |z0|2) j

j!= Ek−1(m|z0|

2).

As for the second term, we use the fact that the function s 7→ s j−νe−s is increasing on the interval[0, j − ν], to say that ∫ mr2

0s j−νe−sds ≤ (mr2) j−ν+1e−mr2

,

provided that j ≥ mr2 + ν. It follows that if k ≥ mr2 + ν, thenn−1∑j=k

(m |z0|2) j

j!( j − ν)!

∫ mr2

0s j−νe−sds ≤ (mr2)1−νe−mr2

n−1∑j=k

(mr |z0|)2 j

j!( j − ν)!.

By Stirling’s formula, j! ≥√

2π j j+1/2e− j, so that

(mr |z0|)2 j

j!e−mr2

≤1√2π j

m j|z0|

2 j(

mr2

je1− mr2

j

) j

.

Since the function x 7→ xe1−x is increasing on the interval [0, 1], it yields that

(mr |z0|)2 j

j!e−mr2

≤1√2π j

m j|z0|

2 j(

mr2

ke1− mr2

k

) j

, mr2 + ν ≤ k ≤ j.

We write

(9.6) ck,m =mr2

ke1− mr2

k ≤ 1, mr2 + ν ≤ k,

and conclude that

(9.7)n−1∑j=k

(m |z0|2) j

j!( j − ν)!

∫ mr2

0s j−νe−sds ≤ (mr2)1−ν

n−1∑j=k

(m|z0|2ck,m) j

( j − ν)!√

2π j

≤(mr2)1−ν

√2π

(ck,m)kn−1∑j=k

(m|z0|2) j

( j − ν)!≤

mr2

√2π

( |z0|

r

)2ν(ck,m)kEn−ν−1(m|z0|

2).

Now, a combination (9.5) and (9.7) applied to (9.4) yields

(9.8)∣∣∣zν0bνm,n(z0, r)

∣∣∣ ≤ Ek−1(m |z0|2)

En−1(m |z0|2)+

mr2

√2π

( |z0|

r

)2ν(ck,m)k En−ν−1(m|z0|

2)En−1(m|z0|

2)

≤Ek−1(m |z0|

2)

En−1(m |z0|2)+

mr2

√2π

( |z0|

r

)2ν(ck,m)k.

We would like to show that each of the terms on the right hand side of (9.8) can be made smallby choosing k cleverly. As for the first term, we appeal to a theorem of Szegö, [21], Hilfssatz 1, p.54, which states that

El(lx) =1√

2πl(ex)l x

x − 1(1 + εl(x)) x > 1,

where εl(x)→ 0 uniformly on compact subintervals of (1,∞) as l→∞. It follows that

Ek−1(m |z0|2)

En−1(m |z0|2)=

√n − 1k − 1

m|z0|2− n + 1

m|z0|2 − k + 1

(em|z0|2

k − 1

)k−1(em|z0|2

n − 1

)1−n 1 + εk−1( m|z0 |2

k−1 )

1 + εn−1( m|z0 |2

n−1 ).

Finally, we decide to pick k such thatk/m→ β


as k,m → ∞, where r2 < β < τ. We observe that with this choice of k, the above epsilons tend tozero as k,m,n→∞. The function

y 7→ (e/y)y, 0 < y ≤ 1

is is strictly increasing, so that with

y1 =k − 1m|z0|

2 ≈β

|z0|2 , y2 =

n − 1m|z0|

2 ≈τ

|z0|2 ,

we have(e/y1)y1

(e/y2)y2≤ θ < 1,

where at least for large k,m,n, the number θmay be taken to be independent of k,m,n. It followsthat (em|z0|

2

k − 1

)k−1(em|z0|2

n − 1

)1−n≤ θ−m|z0 |

2,

so thatEk−1(m |z0|

2)

En−1(m |z0|2)≤ (1 + o(1))

√τβ|z0|

2− τ

|z0|2 − β

θ−m|z0 |2→ 0

exponentially quickly as k,m,n→∞.Finally, as for the second term, we observe that the numbers ck,m defined by (9.6) have the

property that

ck,m →r2

βe1− r2

β < 1,

as k,m,n → ∞ in the given fashion. In particular, the second term converges exponentiallyquickly to 0. The proof is complete. �

Corollary 9.3. Let z0 ∈ C \D(0,√τ), and letD be an open set in C which contains the circle T(0,

√τ).

Further, let u be an analytic polynomial. Then∫D

u(z−1)dB〈z0〉m,n(z)→ u(z−1

0 ) as m→∞ while n = mτ + o(m).

Proof. This follows from Props. 9.1, 9.2, and 2.6. �

9.3. The proof of Theorem 2.9. LetCh(C;C\Sτ) denote the class of bounded continuous functionsC→ Cwhich are harmonic on C \Sτ. For a function f ∈ Cb(C) we write f̃ for the unique functionof class Ch(C;Sτ) which coincides with f onD(0,

√τ). We must show that∫

C

f (z)dB〈z0〉m,n(z)→ f̃ (z0)

as m→∞ and n/m→ τ. See, e.g., [12], p. 90. By convolving with the Féjer kernel, we find that fmay be uniformly approximated on a neighborhoodD of T(0,

√τ) by functions which are of the

form u(z−1), with u a harmonic polynomial. We may therefore w.l.o.g. suppose that f itself is ofthis form, i.e. f (z) = u(z−1) when z ∈ D. Thus f (z) = f̃ (z) = u(z−1) onD. By Prop. 2.6,∫

C

( f (z) − f̃ (z))dB〈z0〉m,n(z)→ 0,

as m→∞while n = mτ + o(m). Moreover, Cor. 9.3 gives that∫C

f̃ (z)dB〈z0〉m,n(z) =

∫C

u(z−1)dB〈z0〉m,n(z)→ u(z−1

0 ) = f̃ (z0),

as m→∞while n = mτ + o(m). �


R

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Y A, D M, T R I T, S – 100 44 S, SWEDENE-mail address: [email protected]

H̊ H, D M, T R I T, S – 100 44 S,SWEDEN

E-mail address: [email protected]

NM, D M, C I T, P, CA 91125, USAE-mail address: [email protected]

berezin transform in polynomial bergman spaces

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