Transformation Formula for the Reduced Bergman Kernel and its Application

Analysis Math.

DOI: 10.1007/s10476-022-0136-8

TRANSFORMATION FORMULA FOR THE REDUCED BERGMAN KERNEL AND ITS

APPLICATION

S. GEHLAWAT^1,∗,†, A. JAIN^1,‡and A. D. SARKAR^2,§

1Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India e-mails: sahilg@iisc.ac.in,aakankshaj@iisc.ac.in

2Harish-Chandra Research Institute, Prayagraj (Allahabad) 211019, India e-mail:amardeepsarkar@hri.res.in

(Received July 12, 2021; revised October 24, 2021; accepted December 29, 2021)

Abstract. In this article, we prove the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences between bounded domains in the complex plane. As a corollary, we obtain the transforma- tion formula for the reduced Bergman kernels under proper holomorphic maps.

We also establish the transformation formula for the weighted reduced Bergman kernels under proper holomorphic maps. Finally, we provide an application of this transformation formula.

1. Introduction

Recall that the Bergman space associated with a domain U ⊂C con- sists of square integrable holomorphic functions on it. An important space that has a close relationship with this space is the space of all holomor- phic functions whose derivatives are square-integrable with respect to the area measure. This space can be associated with a closed subspace of the Bergman space, which we call the reduced Bergman space. The reduced Bergman space is a reproducing kernel Hilbert space with its reproducing kernel called the reduced Bergman kernel; see below for the definitions.

It would be interesting to create a dictionary between the Bergman ker- nel and the reduced Bergman kernel, i.e., to know the properties common

∗Corresponding author.

†This author is supported by the CSIR SPM Ph.D. fellowship.

‡This author is supported by PMRF Ph.D. fellowship.

§This author is supported by the postdoctoral fellowship of Harish-Chandra Research Insti- tute, Prayagraj (Allahabad).

Key words and phrases: reduced Bergman space, Bergman space, reduced Bergman kernel, Bergman kernel, transformation formula.

Mathematics Subject Classification: primary 30H20, 46E22, secondary 32H99.

to both the kernels. This note is a small contribution in this direction. The concerned property here is the existence of the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences be- tween bounded planar domains. This is motivated by Bell’s transformation formula for the Bergman kernel under proper mappings and correspondences (see [1], [2], [3], [4] and [5, Theorem 16.5]). The transformation formula un- der proper mappings will follow as a corollary. But we will see that more is true: the transformation formula for the weighted reduced Bergman kernels under proper holomorphic mappings between bounded planar domains. Fi- nally, we will see an application of this transformation formula; any proper holomorphic map from a bounded planar domain to the unit disc is rational if the reduced Bergman kernel of the domain is rational. The techniques of the proofs will be similar to Bell’s [4] work on the transformation formula for the Bergman kernel and its applications.

M. Sakai [8] defined the reduced Bergman kernel in the following way:

Let Ω be a bounded planar domain. Forξ ∈Ω, consider the complex linear space

AD(Ω, ξ) =

f ∈ O(Ω) :f(ξ) = 0,

Ω

|f(z)|²dA <∞ equipped with the inner product f, g=

Ωf(z)g(z)dA, wheredA is the area Lebesgue measure on Ω. Along with this inner product,AD(Ω, ξ) is a complex Hilbert space. It can be proved thatAD(Ω, ξ)f →f(ξ)∈Cis a bounded linear functional. The Riesz representation theorem therefore gives a unique function M(·, ξ)∈AD(Ω, ξ) such that f(ξ) =f, M(·, ξ) for all f ∈AD(Ω, ξ). The function ˜K(z, ξ) := _dz^dM(z, ξ) is defined as the reduced Bergman kernel of Ω.

We will give an another equivalent definition of the reduced Bergman kernel (and the weighted reduced Bergman kernel).

Definition 1.1. Let Ω be a domain inC. The reduced Bergman space of Ω is the space of all the square-integrable holomorphic functions on Ω whose primitive exists on Ω, i.e.,

D(Ω) =

f∈O(Ω) :f=g for someg∈ O(Ω) and

Ω

|f(z)|²dA(z)<∞ . This is a Hilbert space with respect to the inner product

f, g:=

Ω

f(z)g(z)dA(z).

For every ζ ∈Ω, the evaluation functional f →f(ζ), f ∈ D(Ω)

is a bounded linear functional, and thereforeD(Ω) is a reproducing kernel

Hilbert space. The reproducing kernel of D(Ω), denoted by ˜K_Ω(·,·), is called the reduced Bergman kernel of Ω. It satisfies the reproducing property:

f(ζ) =

Ω

f(z) ˜K_Ω(z, ζ)dA(z) for all f ∈ D(Ω) and ζ ∈Ω.

It can be seen that the Bergman kernel and the reduced Bergman kernel are same for simply connected domains as every holomorphic function has a primitive on a simply connected domain. As we shall observe in the example below, it need not be the case for non-simply connected domains.

Example 1.2. Consider the annulusA defined by A=

z∈C: 1<|z|<2 .

The Bergman kernel of the annulus is (by the calculations in [7]) K_A(z, w) =

jj∈Z=−1

j+ 1

π(2^2j+2−1)z^jw¯^j+ 1 2πlog 2

1 zζ¯. The functions

ψ_j(z) =z^j, j∈Z\ {−1},

form a complete orthogonal basis of D(A), and therefore the reduced Bergman kernel of A is given by

K˜_A(z, w) =

jj∈Z=−1

j+ 1

π(2^2j+2−1)z^jw¯^j.

Now we are ready to state our main results. First, we recall the def- inition of a proper holomorphic correspondence (see [1]). Let Ω₁ and Ω₂ be domains in C, and π1: Ω1×Ω2 →Ω1, π2: Ω1×Ω2 →Ω2 be the projec- tions. IfV is a complex sub-variety of Ω₁×Ω₂, then consider the associated multi-valued function f: Ω₁Ω₂ given by f(z) =π₂π₁⁻¹(z) ={w: (z, w)

∈V}. The map f is called a holomorphic correspondence and V is called the graph of f. The correspondencef is said to be proper if the projection mapsπ₁:V →Ω₁ andπ₂:V →Ω₂ are proper (being proper means that the inverse images of compact sets are compact).

We remark that there exist sub-varieties V₁ and V₂ of Ω₁ and Ω₂ re- spectively, and positive integers p and q such that π₂π₁⁻¹ is locally given bypholomorphic maps on Ω1\V1which we will denote by{fi}^p_i=1, andπ1π₂⁻¹ is locally given by q holomorphic maps on Ω₂\V₂ which we will denote

by{F_i}^q_i=1. Note that the sub-varietiesV₁ andV₂ are discrete subsets of Ω₁ and Ω2 respectively. Proper holomorphic correspondences arise naturally in the field of complex function theory.

Theorem 1.3. Let Ω1 and Ω2 be bounded domains inC. If f: Ω1 Ω2

is a proper holomorphic correspondence, then the reduced Bergman kernels K˜_l associated with Ω_l, l= 1,2, transform according to

p

i=1

f_i(z) ˜K2(f_i(z), w) = q

j=1

K˜1(z, F_j(w))F_j(w)

for all z∈Ω₁ and w∈Ω₂,where {f_i}^p_i=1,{F_j}^q_j=1,and the positive integers p, q are as above.

Before we state the next result, we recall the definition of a proper holomorphic map. Let Ω₁ and Ω₂ be domains in C. A holomorphic map f: Ω₁ →Ω₂ is called a proper holomorphic map if f⁻¹(K) is compact in Ω₁ wheneverK is compact in Ω2. LetV ⊂Ω2 denote the set of all critical val- ues of f and ˜V =f⁻¹(V)⊂Ω₁. Note that both V and ˜V are discrete sets.

It is well known thatf: Ω₁\V˜ →Ω₂\V is anm-to-1 holomorphic covering map for some m∈N, where m is called the multiplicity of f. We will de- note them local inverses of f on Ω2\V by {F_k}^m_k=1. Now as a special case of Theorem1.3, we have the following corollary:

Corollary 1.4. Let Ω1 and Ω2 be bounded domains in C. If f: Ω1

→Ω₂ is a proper holomorphic map, then the reduced Bergman kernels K˜₁ and K˜2 associated with Ω1 and Ω2 respectively, transform according to

f(z) ˜K₂(f(z), w) = m k=1

K˜₁(z, F_k(w))F_k(w),

for allz∈Ω₁andw∈Ω₂,wheremis the multiplicity of f,and the mapsF_k, for 1≤k≤m,are the local inverses of f.

We would like to take this opportunity to give a transformation for- mula for kernels of a more general class of spaces, namely weighted reduced Bergman spaces. First we give a definition:

Definition 1.5. Let Ω be a domain in C and ν be a positive measur- able function on Ω such that 1/ν ∈L^∞_loc(Ω). The weighted reduced Bergman space of Ω with weightν is the space of all theν-square integrable holomor- phic functions on Ω whose primitive exists on Ω, i.e.,

D^ν(Ω) =

f ∈ O(Ω) :f =g for someg∈ O(Ω) and

Ω

|f(z)|²ν(z)dA(z)<∞ .

This is a Hilbert space with respect to the inner product f, gν :=

Ω

f(z)g(z)ν(z)dA(z).

For everyζ ∈Ω, the evaluation functional

f →f(ζ), f ∈ D^ν(Ω)

is a bounded linear functional, and therefore D^ν(Ω) is a reproducing kernel Hilbert space. The reproducing kernel of D^ν(Ω), denoted by ˜K_Ω^ν(·,·), is called the weighted reduced Bergman kernel of Ω with weight ν. It satisfies the reproducing property:

f(ζ) =

Ω

f(z) ˜K_Ω^ν(z, ζ)ν(z)dA(z) for all f ∈ D^ν(Ω) and ζ ∈Ω.

Remark 1.6. The weighted reduced Bergman kernel ˜K_Ω^ν(z, ζ) of Ω is holomorphic in the first variable and anti-holomorphic in the second variable, and

K˜_Ω^ν(z, ζ) =

Ω

K˜_Ω^ν(ξ, ζ) ˜K_Ω^ν(ξ, z)ν(ξ)dA(ξ).

For a bounded domain Ω⊂C, ˜K_Ω^ν(ζ, ζ)>0 for everyζ ∈Ω.

Theorem 1.7. Let Ω₁ and Ω₂ be bounded domains in C and ν be a positive measurable function on Ω2 such that 1/ν ∈L^∞_loc(Ω2). If f: Ω1→ Ω₂ is a proper holomorphic map,then the weighted reduced Bergman kernels K˜₁^ν◦f and K˜₂^ν associated withΩ₁ and Ω₂ respectively, transform according to

f(z) ˜K₂^ν(f(z), w) = m

k=1

K˜₁^ν◦f(z, F_k(w))F_k(w),

for allz∈Ω₁ andw∈Ω₂,where mis the multiplicity of f,and the mapsF_k are the local inverses of f.

Bell [4] had shown that any proper holomorphic mapf: Ω→ Dfrom a bounded planar domain Ω to the unit disc will be rational if the Bergman kernel of Ω is a rational function. As an application of the above transfor- mation formula, we will prove a similar result replacing the hypothesis of the Bergman kernel of Ω being rational to the reduced Bergman kernel of Ω being rational. To be precise, we prove:

Theorem 1.8. Suppose Ω is a bounded domain in C whose associated reduced Bergman kernel is a rational function,then any proper holomorphic mapping f: Ω→Dmust be rational.

2. Proof of Theorem 1.3

In order to prove the transformation formula for the reduced Bergman kernels under proper holomorphic correspondences, we will need the follow- ing lemmas and propositions. We first state a result which we are going to use.

Lemma 2.1 [5, Theorem 16.3]. Let Ω⊂C be a bounded domain and V ⊂Ω be a discrete subset. Suppose that h is holomorphic on Ω\V and h∈L²(Ω\V), i.e.,

Ω\V|h|²dA <∞,

then h has a removable singularity at each point of V and h∈L²(Ω).

Let f: Ω1Ω2 be a proper holomorphic correspondence between bounded planar domains Ω₁ and Ω₂. For functions u and v on Ω₂ and Ω₁ respectively, define maps Λ₁ and Λ₂ by

Λ1(u) = p

i=1

f_i(u◦f_i) and Λ2(v) = q

j=1

F_j(v◦F_j),

where f_i for 1≤i≤p andF_j for 1≤j≤q are as in the definition of holo- morphic correspondences defined locally on Ω1\V1 and Ω2\V2 respectively.

Lemma 2.2. Given (Ω₁,Ω₂, f) as above, the maps Λ₁, Λ₂ are bounded linear maps that satisfy

(∗) Λ1(A²(Ω2))⊂A²(Ω1) and Λ2(A²(Ω1))⊂A²(Ω2), where A²(Ω_i) denotes the Bergman space of Ω_i for i= 1,2.

Proof. The linearity of maps Λ_i is straightforward. First, we will show that the maps Λ_i preserve square integrable functions, and that Λ1:L²(Ω2)

→L²(Ω₁) and Λ₂: L²(Ω₁)→L²(Ω₂) are bounded operators. Since smooth functions with compact support are dense inL²-space, it is sufficient to work with these functions and the final conclusion will follow by density property and using partition of unity.

Let v∈L²(Ω₁) be a smooth function with very small support U ⊂ Ω₁\V₁ such thatf_i(U)∩f_j(U) =∅ fori=j. Set ˜U_i =f_i(U), and letF_ji be the inverse of f_i on ˜U_i. Observe that

Λ₂(v),Λ₂(v)2 =

Ω₂

|Λ₂(v)(w)|²dA(w)

≤q

Ω₂

q

j=1

|F_j(w)|²|v(F_j(w))|²dA(w)

=q p

i=1

U˜i

q

j=1

|F_j(w)|²|v(Fj(w))|²dA(w)

where ·,·i denotes the inner product on the Hilbert space A²(Ω_i) for i= 1,2. Since F_j( ˜U_i)∩U =∅ if and only if j=j_i,

Λ2(v),Λ2(v)2≤q p

i=1

U˜i

|F_ji(w)|²|v◦F_ji(w)|²dA(w).

Set F_ji(w) =z on ˜U_i, we get w=f_i(z) and Λ₂(v),Λ₂(v)2≤q

p

i=1

U

|v|²dA(z) =pqv, v1 <+∞, that is,

Λ2(v),Λ2(v)2 ≤pqv, v1.

Thus, Λ₂ is a bounded linear operator from L²(Ω₁) into L²(Ω₂). Now, if v∈A²(Ω1), it is clear from the definition of Λ2 that Λ2(v)∈ O(Ω2\V2).

Since V₂ is a discrete subset of Ω₂, it follows by Lemma 2.1 that Λ₂(v)∈ O(Ω2), and therefore Λ2(v)∈A²(Ω2). The analogous statement for Λ1 can be proved similarly.

Lemma 2.3. If f: Ω₁Ω₂ is a proper holomorphic correspondence be- tween bounded planar domains, then

Λ₁(u), v1 =u,Λ₂(v)2

for all u∈L²(Ω₂) andv∈L²(Ω₁).

Proof. We first prove the above result for all smooth u∈L²(Ω₂) sup- ported in small neighbourhoods of points in Ω2\V2 and for all v∈L²(Ω1).

Let W be a neighbourhood of w₀∈Ω₂\V₂ such that F_j for 1≤j≤q are well defined and W_j :=F_j(W) are pairwise disjoint sets. Let f_ij be the inverse of F_j on W_j. Letu∈L²(Ω₂) be such that support of uis contained in W. Therefore, support of Λ1(u) is contained in ^q_j=1W_j where Λ1(u) = _p

i=1f_i(u◦f_i). Now one can see that Λ1(u), v1 =

Ω1

(Λ1(u))¯v dA(z) = q

j=1

Wj

(Λ1(u))¯v dA(z)

= q

j=1

Wj

p

i=1

f_i(u◦f_i)¯v dA(z) = q

j=1

Wj

f_ij(u◦f_ij)¯v dA(z)

= q

j=1

W

uF_j(v◦F_j)dA(w) =

W

u q

j=1

F_j(v◦F_j)dA(w)

=

W

uΛ2(v)dA(w) =

Ω₂

uΛ2(v)dA(w) =u,Λ2(v)2.

Hence the result follows by density argument, as the linear span of all suchu is dense inL²(Ω₂).

We need the following lemma about the extension of holomorphic func- tions.

Lemma 2.4. Let Ω⊂C be a domain and V ⊂Ωbe a discrete set. Sup- pose that h is holomorphic on Ω\V and h has removable singularities at each point in V. Thenh also has removable singularities at each point inV. Proof. Since V is discrete set, for p∈V, choose U ⊂Da simply con- nected neighbourhood of p with

U \ {p}

∩V =∅. By assumption, the holomorphic functionhhas removable singularity atpwhen restricted onU. LetH be a primitive of h onU, then (h−H)(z) =h(z)−h(z) = 0 for all z∈U \ {p}. Hence h(z) =H(z) +c for all z∈U\ {p}, and for some con- stant c∈C. Now, using the Riemann removable singularity theorem, h is holomorphic onU. Hence all points of V are removable singularities for h.

This completes the proof.

Now we have the following proposition about reduced Bergman spaces.

Proposition 2.5. If f: Ω₁ Ω₂ is a proper holomorphic correspon- dence between bounded planar domains, then

Λ˜₁(u) := Λ₁|_D(Ω2)(u)∈ D(Ω₁) and Λ˜₂(v) := Λ₂|_D(Ω1)(v)∈ D(Ω₂) for allu∈ D(Ω₂) and v∈ D(Ω₁).

Proof. Let u∈ D(Ω₂). Thereforeu= ˜u for some ˜u∈ O(Ω₂). Observe Λ˜₁(u) = Λ₁(u) =

p

i=1

f_i(u◦f_i) = p

i=1

(˜u◦f_i) = p

i=1

˜ u◦f_i

. Consider the functionh₁=_p

i=1u˜◦f_i. It is easy to see thath₁∈ O(Ω₁\V₁), and sinceV₁ is a discrete set and ˜Λ₁(u)∈L²(Ω₁) i.e.h₁∈L²(Ω₁), it follows from Lemma 2.1 that h₁ has removable singularities at points of V₁. Now, by Lemma2.4,h₁ also has removable singularities at points of V₁. Therefore h₁ ∈ O(Ω₁) and hence for all u∈ D(Ω₂), we have

Λ˜₁(u)∈ D(Ω₁).

The other part of the proposition can be proved in a similar manner.

Proposition 2.6. If f: Ω₁Ω₂ is a proper holomorphic correspon- dence between bounded planar domains,then

Λ˜1(u), v1 =u,Λ˜2(v)2

for all u∈ D(Ω2) and v∈ D(Ω1).

Proof. Since ˜Λ_i is just the restriction of Λ_i onD(Ω_j) wherej=i, this result holds trivially from Lemma 2.3and Proposition 2.5.

Proof of Theorem1.3. For a fixedw∈Ω₂\V₂, consider the function G(z) =

q

j=1

K˜₁(z, F_j(w))F_j(w), z∈Ω₁.

Since ˜K₁(·, ζ)∈ D(Ω₁) for allζ ∈Ω₁, we observeG∈ D(Ω₁). Now using the reproducing property of the reduced Bergman kernel, we get for an arbitrar- ily chosenv∈ D(Ω₁)

v, G1 = q

j=1

F_j(w)v,K˜₁(·, F_j(w))1= q

j=1

F_j(w)v(F_j(w))

= ( ˜Λ₂(v))(w) =Λ˜₂(v),K˜₂(·, w)2=v,Λ˜₁(u)1,

where u= ˜K₂(·, w). So, v, G1 =v,Λ˜₁(u)1 for all v∈ D(Ω₁). Therefore, Λ˜₁(u) =G. Thus, we have proved that

p

i=1

f_i(z) ˜K₂(f_i(z), w) = q

j=1

K˜₁(z, F_j(w))F_j(w) forz∈Ω₁, w∈Ω₂\V₂. Since the expression on the left side of the equation is anti-holomorphic inw andV₂ is discrete, the points inV₂ are removable singularities of the expres- sion on the right side of the equation. Hence, the formula holds everywhere by continuity.

3. Proof of Theorem 1.7

We start this Section by proving a weighted version of Lemma 2.1.

Lemma 3.1. Let Ω⊂C be a domain and V ⊂Ω be a discrete subset.

Suppose h is holomorphic on Ω\V and h∈L^2,μ(Ω\V), i.e.,

Ω\V |h|²μ dA <∞

where μ is a positive measurable function on Ω such that 1/μ∈L^∞_loc(Ω).

Thenh has a removable singularity at each point in V and h∈L^2,μ(Ω).

Proof. Let p∈V and U be a relatively compact neighourhood of p in Ω such that U∩V ={p}. By definition, μ≥c a.e. onU for somec >0.

Observe that

U\{p}|h|²dA= 1 c

U\{p}|h|²c dA

≤ 1 c

U\{p}|h|²μ dA≤ 1 c

Ω\V|h|²μ dA <∞.

Therefore, by Lemma 2.1, h has a removable singularity at p. Since p∈V is arbitrary,hhas a removable singularity at each point in V, i.e., h∈ O(Ω)

∩L^2,μ(Ω).

Letf: Ω₁→Ω₂ be a proper holomorphic map between two bounded pla- nar domains Ω1 and Ω2 andν be a weight function on Ω2 as in Theorem1.7.

As we defined in the previous section, the corresponding maps Λ1 and Λ2

are given by:

Λ1(u) =f(u◦f) and Λ2(v) = m

k=1

F_k(v◦F_k),

where u is function on Ω₂ and v is a function on Ω₁, and {F_k}^m_k=1 are the local inverses of f defined outside the set of critical values V ⊂Ω₂ with m being the multiplicity of f.

Using similar arguments as in the proofs of Lemmas2.2and2.3, Propo- sitions 2.5 and 2.6, we have the following analogous statements for the weighted case.

Lemma 3.2. Given (Ω1,Ω2, f, ν) as above, the maps Λ_i, for i∈ {1,2}, are bounded linear maps that satisfy

Λ₁(A^2,ν(Ω₂))⊂A^2,ν◦f(Ω₁) and Λ₂(A^2,ν◦f(Ω₁))⊂A^2,ν(Ω₂), whereA^2,μ(Ω) =

f ∈ O(Ω) :

Ω|f|²μ dA <∞

denotes the weighted Bergman space of Ω with weight μ.

Lemma 3.3. If f: Ω₁ →Ω₂ is a proper holomorphic mapping between bounded planar domains and ν is a weight on Ω2 as given in Theorem 1.7, then

Λ₁(u), v_ν◦f =u,Λ₂(v)ν

for allu∈L^2,ν(Ω₂) andv ∈L^2,ν◦f(Ω₁).

Proposition 3.4. If f: Ω₁ →Ω₂ is a proper holomorphic mapping be- tween bounded planar domains and ν is a weight on Ω2 as in Theorem 1.7, then

Λ˜1(u) := Λ1|_D^ν(Ω₂)(u)∈ D^ν◦f(Ω1) and Λ˜2(v) := Λ2|_D^ν◦f(Ω₁)(v)∈ D^ν(Ω2) for all u∈ D^ν(Ω₂) and v∈ D^ν◦f(Ω₁).

Proposition 3.5. If f: Ω1 →Ω2 is a proper holomorphic mapping be- tween bounded planar domains and ν is a weight on Ω₂ as in Theorem 1.7, then

Λ˜₁(u), v_ν◦f =u,Λ˜₂(v)ν

for all u∈ D^ν(Ω₂) and v∈ D^ν◦f(Ω₁).

Proof of Theorem1.7. LetV ={f(z) :f(z) = 0}. Since f is proper, V is a discrete subset of Ω₂. For a fixed w∈Ω₂\V, consider the function

G(z) = m k=1

K˜₁^ν◦f(z, F_k(w))F_k(w), z∈Ω₁.

Since ˜K₁^ν◦f(·, ζ)∈ D^ν◦f(Ω₁) for all ζ ∈Ω₁, we have G∈ D^ν◦f(Ω₁). Follow- ing the similar argument as in the proof of Theorem 1.2, we have for an arbitrarily chosen v∈ D^ν◦f(Ω1)

v, Gν◦f = v,Λ˜1

K˜₂^ν(·, w)

ν◦f. Therefore, ˜Λ₁K˜₂^ν(·, w)

=G. That is, f(z) ˜K₂^ν(f(z), w) =

m k=1

K˜₁^ν◦f(z, F_k(w))F_k(w) forz∈Ω₁, w∈Ω₂\V.

As before, the formula holds everywhere by continuity.

4. Proof of Theorem 1.8

Let ˜K(z, w) and K(z, w) denote the reduced Bergman kernel functions associated to Ω and D respectively, where Ω is a bounded planar domain and D is the unit disc in C. For a proper holomorphic map f: Ω→D, let {F_k}^m_k=1 denote the local inverses of f defined on D\V where m is the multiplicity of f and V is the set of all critical values of f. According to Corollary 1.4, the kernel functions transform according to

f(z)K(f(z), w) = m k=1

K(z, F˜ _k(w))F_k(w), z∈Ω, w∈D.

For fixedw∈Dandα∈N∪ {0}, define a linear functional Λ on the Bergman space of Ω, i.e., A²(Ω) by

Λ(h) =∂^α m

k=1

F_k(h◦F_k)

(w),

where ∂^α = _∂z^∂αα denotes the standard holomorphic differential operator of orderα. A lemma by Bell about Λ reads as follows.

Lemma 4.1 (Bell, [4]). Let ξ1, ξ2, . . . , ξ_q denote the points in f⁻¹(w).

There exist a positive integer s and constants c_l,β such that for h∈A²(Ω), Λ(h) =

q

l=1

β≤s

c_l,β∂^βh(ξ_l).

proof of Theorem1.8. Letf⁻¹(0) ={ζ₁, . . . , ζ_q}. For the linear func- tional Λ on A²(Ω) defined above, corresponding to α= 1∈N∪ {0} and 0∈D, i.e.

Λ(h) = ∂

∂w m

k=1

F_k(h◦F_k)

(0),

the above lemma gives a positive integers >0 and constantsc_l,β such that

∂

∂w m

k=1

F_k(h◦F_k)

(0) = q

l=1

β≤s

c_l,β∂^βh(ζ_l) for all h∈A²(Ω).

Therefore, by differentiating the transformation formula for the reduced Bergman kernels underf with respect to ¯w and settingw= 0, we get

f(z) ∂

∂w¯K(f(z),0) = ∂

∂w¯ m

k=1

K(z, F˜ _k(·))F_k(·)

(0)

= ∂

∂w m

k=1

K(z, F˜ _k(·))F_k(·)

(0) = q

l=1

β≤s

¯

c_l,β∂^βK(z, ζ˜ _l)

= q

l=1

β≤s

¯

c_l,β∂^βK˜(z, ζ_l),

where ¯∂^β = _∂^∂_w¯^ββ. Since ˜K is a rational function,f(z)_∂^∂_w¯K(f(z),0) is a ra- tional function inz. Similarly, takingα= 0 proves thatf(z)K(f(z),0) is a

rational function in z. Note that for disc D, the reduced Bergman kernel is equal to the Bergman kernel as D(D) =A²(D). Therefore,

K(z, w) = 1 π

1 (1−zw)¯ ². So, _∂^∂_w¯K(z,0) = ^2z_π andK(z,0) = _π¹. Thus,

f(z)_∂^∂_w¯K(f(z),0)

f(z)K(f(z),0) = 2f(z).

Hence, f is a rational function.

5. Questions and comments

(1) The transformation formula for the Bergman kernels was originally conceived not just for planar domains but for domains inCⁿ,n≥1. We tried to do the same with the reduced Bergman kernel, but there were some ob- structions. To begin with, since the definition of the reduced Bergman space involves primitives of holomorphic functions; it is not clear what should be the right analogue of the reduced Bergman space in higher dimensions. We tried with a few possible candidates, that is for a domainU ⊂Cⁿ, consider the followings spaces

D¹(U) =

f∈O(U)∩L²(U) :f= n

i=1

∂g_i

∂z_i for some g1, g2, . . . , g_n∈O(U)

,

D²(U) =

f ∈ O(U)∩L²(U) :f = ∂g

∂z_i for some g∈ O(U)

.

As one can observe, two important properties of the reduced Bergman space for planar domains in the proof of transformation formula are (i) being a closed subspace of the Bergman space, (ii) being invariant under the map Λ_i (defined in (∗)). In the case of higher dimensions, both the candidate spaces D¹(U),D²(U) either don’t have both or one of these two properties. So, the problem of finding the correct analogue of the reduced Bergman space in higher dimensions is still open.

(2) Theorem1.8is analogous to a result due to Bell for Bergman kernels.

It is therefore natural to ask if there is any bounded planar domain whose reduced Bergman kernel is rational, but the Bergman kernel is not rational.

Bell proved that if Ω is an n-connected domain, n >1, with C^∞ smooth boundary, then the Bergman kernel of Ω can not be a rational function. We know that the Bergman kernel and the reduced Bergman kernel for a simply

connected domain are same. We tried to construct the desired example using the following relation between the Bergman kernel and the reduced Bergman kernel for ann-connected planar domain Ω (due to Schiffer and Bergman [6])

K_Ω(z, ζ) =

n−1

i,j=1

p_ijω_i(z)ω_j(ζ) + ˜K_Ω(z, ζ), z, ζ∈Ω,

where the coefficientsp_ij are real numbers andω_i, for 1≤i≤n−1, are the harmonic measures on Ω corresponding to then−1 inner boundary compo- nents. But we cannot conclude anything from this formula unless we have explicit information about the first term on the right hand side in the rela- tion.

(3) In the weighted case, the transformation formula for the weighted reduced Bergman kernels under a proper holomorphic map f: Ω1→Ω2 is obtained with respect to the weights ν on Ω₂ and weight ν◦f on Ω₁. In general, iff is not a function, for instance in the case of proper holomorphic correspondence, it is not clear how to define a weight on Ω1 such that the transformation formula remains true.

Acknowledgements. The authors would like to thank Kaushal Verma for all the helpful discussions and suggestions. The authors would also like to thank the referee for carefully reading the article and giving suggestions to improve the previous version of this manuscript.

References

[1] E. Bedford and S. Bell, Holomorphic correspondences of bounded domains inCⁿ, in:

Complex Analysis(Toulouse, 1983), Lecture Notes in Math., vol. 1094, Springer (Berlin, 1984), pp. 1–14.

[2] S. R. Bell, Proper holomorphic mappings and the Bergman projection,Duke Math. J., 48(1981), 167–175.

[3] S. R. Bell, The Bergman kernel function and proper holomorphic mappings, Trans.

Amer. Math. Soc.,270(1982), 685–691.

[4] S. R. Bell, Proper holomorphic mappings that must be rational,Trans. Amer. Math.

Soc.,284(1984), 425–429.

[5] S. R. Bell,The Cauchy Transform, Potential Theory, and Conformal Mapping, Studies in Advanced Mathematics, CRC Press (Boca Raton, FL, 1992).

[6] S. Bergman and M. Schiffer, Kernel functions and conformal mapping, Compositio Math.,8(1951), 205–249.

[7] S. G. Krantz, A tale of three kernels,Complex Var. Elliptic Equ.,53(2008), 1059–1082.

[8] M. Sakai, The sub-mean-value property of subharmonic functions and its application to the estimation of the Gaussian curvature of the span metric,Hiroshima Math. J., 9(1979), 555–593.