The relation between the Bergman kernel and the Szego kernel

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THE RELATION BETWEEN THE BERGMAN KERNEL AND THE SZEGO KERNEL

MOONJA JEONG

We can expect a close relationship between the Bergman kernel and the Szeg8 kernel of a domain because we can change boundary integrals to solid integrals via Green's identity. Indeed, the kernels are very closely related. By inspecting the relation of them we can have the information about the Szeg8 kernel from the information about the Bergman kernel and fruitful applications in mapping problems. For example, by using the relation of the Bergman kernel and the Szeg8 kernel, we get the transformation formula for the Szeg8 kernel under certain proper holomorphic mapping from the transformation formula for the Bergman kernel and we use it to characterize proper rational mapping (see Jeong [8]). The purpose of this paper is to reveal a relationship between the Bergman kernel and the Szeg8 kerneL

Let Q be a bounded n-connected domain in C with C= smooth boundary where n ;::: 2. Let K(z,w) denote the Bergman kernel and S(z,w) denote the Szeg8 kernel associated with Q. The following formula by Bergman [6; p.116]

n - l

K(z, w) =41l"S(z, w)2+L Ai(w)WI(z),

i=l

where WI is the derivative of a multi-valued holomorphic function Wi given by analytically continuing around Qa germ ofWi+wt ^where wt

is a local harmonic conjugate for the harmonic measure function Wi,

represents the relation between the Bergman kernel and the Szeg8 kernel. Bell [4] constructed another formula representing the relationship

Received November 21, 1994.

1991 AMS Subject Classification: 30C40.

Key words: Bergman kernel, Szeg8 kernel.

This work was supported in part by KOSEF, Grant 951-0102-008-2.

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284 MoonjaJeong

between the above two kernels via Garabedian kernel. It is as follows:

whereL(z, a) denotes the Garabedian kernel defined by Garabedian [7]

and for fixeda ES1, the pointsaiare the zeroes of the function S(z, a).

In this paper, we study a proper anti-holomorphic correspondence and make a formula relating the Bergman kernel to the Szego kernel in a different form. Before we state and prove our main theorem, we must explain some notations and basic facts.

For a bounded domain S1 in e ^with ^Coo smooth boundary, let S1*

be the set of points wsuch that w E S1. The Szego kernel S(z,w) associated to S1 defines an analytic subvariety V = {(z,w) E S1 x S1* : S(z,w) = O}. The projection maps 7rl: V --+ S1 and 7r2: V --+ S1*

are proper holomorphic (see Bell [4] and Bell [5; p. 106]). Hence, V is identified with1where the multi-valued map f = 7r2 07r1¹ is a proper anti-holomorphic self-correspondence of S1.

Let S1 be a bounded domain in en. ^Let ^H(S1) denote the space of holomorphic functions in L2(S1) and AH(S1) = {cp E L2(S1) : cp E

H(S1)}. Since AH(S1) is a closed subspace of L2(S1), there exists a unique orthogonal projection Q ofL2(S1) ontoAH(S1) and it satisfies Qcp = Prpfor cp E L2(S1) where P is the Bergman projection of L2(S1) onto H(S1). It is easy to see that Qcp(z) = InK(z,w)cp(w)dVw for everycpE L2(S1)whereK(z,w) denotes the Bergman kernel associated with S1.

The transformation formula for the Bergman kernel under proper holomorphic correspondence is already known (see Bedford and Bell [1]). Now, by a little modification of it, we will make a transformation formula for the Bergman kernel under proper anti-holomorphic correspondence. First we suppose that S1₁ and S1₂ are bounded domains in

en ^{and that} ^f is a bi-anti-holomorphic mapping of S11 onto S12 • Let u(z) =det[if]. For each cp EL²(S12 ), define Acp(z) = u(z)· (cp⁰ f)(z).

We see that A is an isometric anti-isomorphism ofL²(S12 ) onto L²(S1_{1 ),} i.e., it is isometric one-to-one, conjugate-linear map of L²(S1_{2 )} onto L2(S1_{1 ).} It preserves anti-holomorphic functions. Consequently, we get the following proposition.

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PROPOSITION 1. Let n1 and n2 be bounded domains in en and

I: n1 ---+ n2 be a bi-anti-holomorphic mapping. Let Ki(z,w) denote the Bergman kernel associatedtoni fori = 1,2. The Bergman kernels transform according to

K 1(z,w) ⁼ u(w)K2(f(w),f(z))u(z) for every z,wE n1 .

Prool. Let {Vj}~1 denote an orthogonal basis for AH(n2 ). Then

{u'(Vjof)}~1is an orthogonal basis forAH(n1). ^{Note that} K1(',z) E AH(n1 ) when Z E n1 . Itfollows that

00

K 1(w,z) ⁼ I:^u(w)(Vj °f)(w)u(z)(Vj °^f)(z)

j=1

00

= u(w){I:Vj(f(w))Vj(f(z)) }u(z) j=1

= u(w)K2(f(z),f(w))u(z).

Therefore, by the conjugate-symmetry properties of the Bergman kernels,

K 1(z,w) ⁼ u(w)K2(f(w),/(z))u(z) for z, w E n1 . 0

We want to move our concern to the case when I : f!1 ~ f!2 is a proper anti-holomorphic correspondence between two bounded domains in en. Let f!i denote the set of points w such that ill E f!2.

Then I is given by I(z) = {ill E f!2 : (z,w) EVe f!1 x f!n where V is an analytic subvariety of f!1 x f!i. The projections 1Tl :V ---+ f!1 and 1T2 : V ---+ f!i are proper maps. There are subvarieties VI and V₂ ofn1 and f!i, respectively and positive integers p and q satisfying the following conditions:

(1) Near a point z E f!1-VI, there are exactlypanti-holomorphic mappings {/df=1 defined near z that represent the multi-valued mapping

- -1

1T201Tl

(2) Near a point w Ef!2 - V₂* whereV₂*= {( Ef!2 :(EV2},there are q anti-holomorphic mappings {FjH=1 that represent 1Tl °^7f2-1.

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286 Moonja Jeong

Let Ui(Z) - det[W] and Uj(w) = det[~~]. ^Let Qi denote the orthogonal projection ofL2(Qi)^onto AH(Qi), i = 1,2. By modifying Theorem 2 in Bedford and Bell [2], we can obtain the following lemma.

LEMMA 2. Suppose f : Q₁ ---() Q2^is ^a proper anti-holomorpmc correspondence between two bounded domains inen. If'P EL2(Q2)' then

2::p ^Ui('P⁰ ^Ji) ^E ^L²^(Q^{1 )}

i=l

p p

and Ql(2::^Ui('P⁰ ^Ji)) ⁼ 2::^{Ui( Q2'P} ⁰ ^fi).

i=l i=l

Proof. We already know that L:f=lUi('P0fi) is well defined on Q -

VI. So we get

p ² [ p

II~Ui('POJi)II£2(Ol_Vd~p^lo₁^-v_l ~IUiI21'POJiI2

= pq [ 1'P1² ⁼ pqll'PII},2(02)'

lo²-v2*

Since VI is a set of measure 0 in nI, we see that L:f=lUi('P0Ji) extends to be in L2(Q1 ) and that

p

II.?=^Ui('P ⁰ Ji)II£2(Od ::; ~11'PIIL2(o2)'

z=l

If'P E AH(Q2), then L:f=lUi('P 0Ji) E AH(Q1 - VI) nL2(Q1 ) implies that L:f=lUi('P°fi) E AH(Q1 ) by the L2Removable Singularity Theorem (see Bell [3]). So Ql(L:f=l Ui('P 0Ji)) = L:f=lUi('P °Ji) =

L:f=lUi( Qz'P0Ji).

Therefore, we will be finished if we show that the transformation formula holds for 'P E L^Z(n2) that is orthogonal to AH(Q2). ^If 'P is orthogonal to AH(Qz) ^and 9 E AH(Ql), then L:j=lUj(g 0 Fj ) E AH(Qz)implies that

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by Lemma 2

So2:f=1^Ui(cp0^Ji)is orthogonal toAH(fh)and hence Ql(2:;=1 Ui(cp0

Ji)) = O. This completes the proof of Lemma 2. 0

By using the above lemma, we get the following transformation formula for the Bergman kernel under the proper anti-holomorphic correspondence.

PROPOSITION 3. Assume the same hypothesisasin Lemma 2. Let Ki(z,w) denote theBergman kernel associated to Oi fori =1,2. The Bergman kernels transform accordingto

p q

L ui(z)K2(Ji(z),w) ⁼ L ^U^j^(w)K¹^(F^j^(w),z)

i=1 j=1

for all z E 01 andw E 02⁰

Proof. For w E O_{2 -} V₂*, there exist an arbitrarily small neigh- borhood W of w in O_{2 -} V₂* and Fj : W -+ Dj which are bi-anti- holomorphic mappings for 1 :::; j :::; q such that Fj ⁰ ^fi(j) = ^idDj for some 1 :::; i(j) :::;^p where {Dj13=1 C 01 - VI are disjoint and for each 1:::;j :::; q, {fi(Dj)}f=1 are disjoint. Take8w E Co(W) which is radi- ally symmetric about w with frh 8w = 1. For any h E AH(02 - V2*), we haveh(w) = f0

2h8w^o As a result, we get Q28w(-) =K2(·,w). Now, for z E 01 - VI,

p p

L ui(z)K2(fi(z), w) =L ^Ui(Z)(Q2⁸^w ⁰ ^Ji)(z)

i=1 i=1

P

=Ql~=^ui(9^w ⁰ ^fi))(z) i=1

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288 Moonja Jeong

p q

L ^U^i(Z)K²^(Ji(z),w) ⁼I,:Uj(w)KI(Fj(w),z)

i=I j=1

for ^W EQ_{2 -} V₂* and z E Q_{1 -} VI' By the £2 Removable Singularity theorem, it holds for all^W E_{Q 2}and z E Ql. 0

Nowweare ready to prove the following main theorem relating the Bergman kernel to the Szego kernel.

THEOREM 4. Suppose that Q is a bounded n-connected domain in C with C<XJ smooth boundary where n _~ 2. The Bergman kernel K( z,w) associated toQ isrelated to the Szego kernel via the identity

n-I n-I

~K(' )Sa(ai,a) _ '"'K( . )Sw(Wj,w) L.-t. anW Szai,a.( ) - L.-t w;,a Szw;',w( )

1=1 ;=1

where for fixed a E Q, w E Q, the points {ai}?::::-l and {wj}j,;I are the zeroes of the functions S(z, a) andS(z, w),respectively. Note that Sz(ai,a) = :zS(z,a)IZ=4' andSz(Wj,w) = :zS(z,w)lz=wj'

Proof. For a EQ, S(z,a) has n -1 zeroes {ai}?:::iÎ in nwith mul- tiplicity. The multi-valued map a 1--+ a b " " an-I is a proper anti- holomorphic self-correspondence ofQ. Let f denote this multi-valued map. There exists a subvariety VI ofQ such that {Ji}?::::-11 denote the mappings that locally define f and fie a) = âi ^for â Ê ^{Q -} ^VI. Ît îs

obvious that the inverse correspondencef-¹is equal to f.

For a E Q - VI, let ;4, in Q be a sufficiently small simple closed curve surroundingai not to include other zeroes ofS( z, a) inside or on

; 4 ; ' By using the fact that

Sz(z, a) ~ z F'(z)

zS(z,a) =~z-aj+zF(z)

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by the Residue Theorem.

where F(z) is holomorphic with no zeroes inside or on fan we can represent h by

( ) 1 1 ^SzCz,^a)

h a = _~ z S( ) dz.

7rZ "'I,,; Z,a

Hence

ah(a) = ^_11 ^z~ (Sz(z, a)) dz

aa 27ri ^"'1,,- aa S(z,a)

•

= _11 ^{z (Saz(z,}^{a) _} Sz(z, a)Sa(Z, a)) dz

27ri ^"'I,,; S(z,a) S(z, a)2

= _11 ^z~ (Sa(z,a)) dz

27ri "'I". az S(z, a)

= _11 ^~ (zSa(z,a)) dz - _11 ^{Sa(z,a) dz}

27ri "'I,,; az S(z, a) 27ri "'I,,; S(z,a)

= --1-1 ^Sa(Z,^a) ^dz

27ri "'1,,- S(z,a)

• Sa( ai, a) SzCai,a)

Sincef is a proper anti-holomorphic correspondence and{/i}i::::-l1define

f locally, it follows that by Proposition 3,

n-l ar ^n-l af-

?:K(/i(a),w) aa&(a) =?:K(!i(w),a)a~(w).

&=1 )=1

Therefore,

n - l ( ) n-l S ( )

' " K( _ )Sa ai, a _ ' "K( _ ) w ^Wj,w L..J_ a&,w Sz ai,a( ) - L..J_ w),a SZ(W)-,W)

&=1 )=1

for a, w E 0 - VI. By the £2 Removable Singularity theorem, it holds for all a, w E O. 0

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290 Moonja Jeong

References

1. E. Bedford and S. Bell, H olomorphic correspondences of bounded domains in

en, Springer Lee. Notes in Math. 1094(1984), 1-14.

2. , Extension of proper holomorphic mappings past the boundary, Manuscripta math. 50 (1985), 1-10.

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

Amer. Math. Soc. 270 (1982), 685-691.

4. , The Szego projection and the classical objects of potential theory in the plane,Duke Math. J. 64(1991),1-26.

5. , The Cauchy transform, Potential Theory, and Conformal mapping, CRC Press, Boca Raton, Florida, 1992.

6. S. Bergman, The Kernel function and Conformal mapping, Math Surveys, Amer. Math. Soc., Providence, 1970.

7. P. Garabedian, Schwartz's lemma and the Szego kernel function, Trans. Amer.

Math. Soc. 67 (1949), 1-35.

8. M. Jeong, The SzegfJ kernel and the rational proper mappings between planar domains, Complex Variables Theory and Appl. 23 (1993), 157-162.

Department of Mathematics The University of Suwon Suwon 445-743, Korea