Bergman Kernel

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Riemann Mapping Theorem and the Bergman KernelMth 515 - Fall 1998 - B. E. PetersenOct 21, 1998

These notes summarize the portion of our course that dealt with the Riemann conformal mappingtheorem. I do not provide the detailed proofs that were given in class just an overview but perhaps

I will have included some comments that were omitted or, worse, garbled in class.

1 Riemanns Conformal Mapping Theorem

Let be a nonempty, connected and proper open subset of the complex plane. We say that hasthesquare root property, SRP, if for each function fanalytic on such thatf(z) = 0 for eachz ,we have there exists g analytic on such thatf(z) = g(z)2 for eachz .

A simple calculation shows that iff(z) = 0 for each z and f/fhas a primitive on then thereexistsg analytic on such that f(z) = g(z)

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for each z . It follows that any proper nonemptyconnected simply connected open set has the SRP.

Let D(a, r) ={ z C | | z a | < r } and let D = D(0, 1). Let A() be the space of all functionsanalytic on .

Now Riemanns mapping theorem (1851) may be formulated as:

Theorem 1. Let have the SRP. Ifa then there exists a uniquef A() such that

1. f(a) = 0 andf(a)> 0,

2. f is one-to-one,

3. f() =D.

Recall that a nonconstant analytic function is open. Thus the function f provided by Riemannstheorem is a homeomorphism. Even more, a one-to-one analytic function has non-vanishing derivativeand so has analytic inverse.

Recall also that an analytic function with nonvanishing derivative preserves angles, and their ori-entation, so f is in fact conformal. Conversely a Frechet differentiable conformal map (preservingangles, including orientation) is analytic. So Riemanns mapping theorem is sometimes called thethe conformal mapping theorem. We will refer to the unique function fgiven by theorem (1) as the

Riemann conformal map for the pair (, a).

Note iffis the Riemann conformal map for the pair (, a) and g is be the Riemann conformal mapfor the pair (, b) then

g(z) =| f(b) |

f(b)

f(z) f(b)1 f(b)f(z) .

SinceD is simply connected, and this property is preserved by homeomorphism, we have:


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Corollary 2. A nonempty proper connected open subset ofC has the SRP if and only if it is simply

connected.

Corollary 3. A nonempty proper connected open subset ofC is conformally equivalent to D if andonly if it is simply connected.

Riemanns proof of the conformal mapping theorem required solving the Dirichlet problem. For thishe relied on the Dirichlet principle. His proof therefore technically contained a gap.

The first complete technically correct proof of the conformal mapping theorem is generally acceptedto be the one given by Koebe around 1908, though Osgood may have had one earlier. Other proofswere of course given for example, by Schwarz around 1870. It may prove interesting to study thevarious proofs, their key ideas and their shortcomings!

1.1 The normal families argument

Koebe was the person who introduced the square root argument used in the proof common in moderntextbooks see for example [2], [1] or [3]. It owes much to Koebe, Fejer, F. Riesz, Caratheodory,Bierbach, Gronwall, Lindelof, Montel and others. This proof is a compactness argument based onMontels normal families theorem. Here is how it goes (in outline):

First of all, the lemma of Schwarz, allows us to show that all of the conformal automorphisms of theunit disk are Mobius transformations of the form

w= c z a1 az , | c | = 1.

From this fact the uniqueness is immediate. Next we introduce the family

F= { f A() | f one-to-one, f(a) = 0, f

(a)> 0, f() D } .This family is uniformly bounded, so normal (that is, relatively compact in the compact-open topology,or, if you prefer, in the topology of uniform convergence on compact subsets of ) by Montels theorem.Koebes square root construction is used to show thatFis not empty.

The next step is to show that the closure ofF is obtained by simply adjoining{0}. This part ofthe argument makes use of Hurwitz theorem: a sequence of one-to-one analytic functions converginguniformly on compact sets has limit that is one-to-one or constant Rouches theorem, which countssolutions of an equation, could be used instead. It follows thatF {0} is compact and thereforethe continuous map gg (a) has a maximum at some point f F. The continuity of the map isestablished by an argument that may be based on the Cauchy formula. By the maximum principlef()

D. Finally Koebes square root trick and some facts about Mobius transforms are used to

show f() =D.

1.2 The potential theory argument

Riemanns original potential theory approach based on the Dirichlet problem has been perfected overthe years and can now also be used to give a complete proof of the conformal mapping theorem

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see [4]. Some of the contributors to this line of development were Perron, C. Neumann, Schwarz,Poincare, Hilbert, Osgood, E.H. Taylor, and others.

A physical argument for using the Dirichlet principle to solve the Dirichlet problem was given as earlyas 1840 by Gauss. It was popularized in the late 1840s by Dirichlet, Kelvin, and others, and proofs

were attempted. In 1870 Weierstrass gave an example to show that the Dirichlet principle is false, atleast in the form usually given, and it fell into disrepute. In 1898 Hilbert resurrected it by rigorouslyproving it under suitable, but useful, hypotheses. Schwarz, C. Neumann and Poincare developed newmethods for solving the Dirichlet problem and considerable generality was achieved in the work ofPerron.

The basic idea in this approach is that by solving the Dirichlet problem one can prove the existenceof Greens function but there is a simple relation between the Riemann map and Greens function(see below).

1.3 Another version of Riemanns theorem

Once we have a version of the Riemann mapping theorem it is not difficult to produce some usefulvariations. Here is an example:

Theorem 4. Let be a connected simply connected nonempty open subset ofC. Leta and let

F= { h A() | h one-to-one, h(a) = 0, h(a) = 1, | h | bounded} .

Then there exists a uniqueg Fsuch that

supz

| g(z) |= infhF

supz

| h(z) | .

Moreoverg () is a disk,

g() =D(0, r(a)).

The radius r(a) is called the interior mapping radiusof at a. It is not difficult to show that ifgis any one-to-one analytic map of onto D(0, 1) then

r(z) =1

|h(z)

|2

| h(z) | .

Thus for the unit disk D we have rD(z) = 1 | z |2 and for the upper half plane Uwe have rU(z) =2e(z).

Note ifh A() is one-to-one, h(a) = 0, h(a) = 1 and h() =D(0, R) thenR = r(a) andh is theunique map given by the above theorem.

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2 Remarks on the Dirichlet Problem

2.1 The Dirichlet problem

In the historical remarks above I probably lied to you through oversight or outright ignorance butit was accidental. Now I am going to lie to you on purpose by over-simplifying.

Let be an open subset ofC and let q be a real function defined on the boundary of . TheDirichlet problem is

u= 0 in

u= qon .(1)

Here is the Laplace operator sometimes denoted2.

Harmonic functions, that is, solutions of Laplaces equation, u = 0, are invariant under conformalmapping. Thus in a sense Riemanns conformal mapping theorem reduces the Dirichlet problem for

any connected simply connected proper open subset ofC to the Dirichlet problem for the unit disk,where we have Poissons formula available! Of course there is a catch the conformal map can bevery badly behaved at the boundary in general we cannot obtain a solution in this manner. Forsets with sufficiently nice boundaries though we can extend the conformal map to a homeomorphismof the closure of and the closed unit disk and proceed along the lines indicated.

2.2 The Dirichlet principle

TheDirichlet principlestates that the solution to the Dirichlet problem is the functionuwhich amongall functions v with v = qon minimizes the Dirichlet integral

v

x

2

+

v

y

2

dxdy.

2.3 The Dirichlet integral

We can give a geometric interpretation of the Dirichlet integral (defined above). Let g = u+ iv bea one-to-one analytic function on . Then in view of the CauchyRiemann equations the integrandin the Dirichlet integral for u may be written

u

x

2

+

u

y

2

=

u

x

v

y u

y

v

x =J(u, v)

=

u

x

2

+

v

x

2

= | g |2

whereJ(u, v) is the Jacobian of the transformation (x, y) (u, v). It follows that

v

x

2

+

v

y

2

dxdy=

| g(z) |2 dxdy= area ofg ().

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2.4 Greens function

The Greens functionof is defined by

g(z, w) = log | z w | +uw(z)

whereuw is the solution to the Dirichlet problem

uw = 0 in

uw(z) = log | z w | forz .

The logarithm occurs here because it is the fundamental solution of the Laplacian in dimension 2. Indimension 3 or higher the appropriate term is| z w |2n.

Greens formulais that the solution to the Dirichlet problem 1 is given by

u(w) = 1

2

q(z)

g

nz (z, w)dsz. (2)

Here derivative is the normal derivative at ofg(z, w) with respect toz and dsz denotes the elementof arc length along.

As an example, Greens function for the disk D(0, R) is given by

g(z, w) =1

2log

R2 2e(zw) + | z |2 | w |2 R2| z |2 2e(zw) + | w |2

.

In this case Greens formula just reduces to Poissons formula for the disk.

2.5 Relation between Greens function and Riemanns conformal mapping

If is a simply connected, connected proper nonempty open subset ofC and for each w we letfw : D be the unique analytic bijection withfw(w) = 0 and fw(w)> 0 thne the Greens functionis given by

g(z, w) = log | fw(z) | . (3)

Conversely if we know Greens function then equation (3) allows us to find the Riemann conformalmap. If we locally choose a harmonic conjugatev for z g(z, w) thenfw(z) = exp(g(z, w) iv(z))is actually well-defined and has a removeable singularity at w. Thenfw is the Riemann map for thepair (, w) (see [4]).

In summary we see if we can find the Riemann map then we can find Greens function and so solvethe Dirichlet problem. Conversely if we can solve the Dirichlet problem then we can find Greensfunction and therefore we can find the Riemann conformal map.

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2.6 Hilbert transform

Lets make a few more informal connections. Let qbe a function on the circle D. Define

h(z) = 1

2

D

w+z

w

z

q(w)

w dw

where D is parametrized in the usual way. If we let u = eh then a quick calculation shows

u(rei) = 1

2

1 r21 2r cos( ) +r2 q(e

i) d.

This is the Poisson formula for the unit disk, and as mentioned above, is a special case of Greensformula. Now letv = mhso v is the harmonic conjugate ofu which satisfiesv(0) = 0. A calculationshows

v(rei) = 1

2

2r sin( )1 2r cos( ) +r2 q(e

i)d.

If we formally letr tend to 1 we obtain a divergent integral which does however exist in the principalvalue sense:

v(ei

) =

1

2

cot

phi

2

q(ei

)d.

This is the Hilbert transform we view a function qas the boundary value of a harmonic function,then the bounday value of the harmonic conjugate is the Hilbert transform ofq. The Hilbert transformis the prototype for the CalderonZygmund singular integral operators, which in turn led to manyimportant applications in partial differential equations.

3 The Bergman Kernel

Letf

A(), z

and 0< r < R whereR = dist(z, ) is the distance from z to the boundary of. Then by Cauchys formula

f(n)(z) = n!

2

f(z+rei) rnein d.

If we take the absolute value, multiply this equation by rn+1, and then integrate r from 0 to R weobtain

Rn+2

n+ 2

f(n)(z)

n!2

D(z,R)

| f(z) | dxdy.

If we estimate the right side by the Cauchy-Schwarz inequality we obtain

f(n)(z)

n+ 2

2

n!

1/2Rn+1

D(z,R) |f(z)

|

2 dxdy

1/2

.

If we define theL2 norm off by

f =

| f(z) |2 dxdy 1/2

then we have shown

f(n)(z)

n! (n+ 2)2

f1/2 (dist(z, ))n+1

.

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One can obtain the same estimate, but with (n + 2)/2 replaced by

n+ 1 by expanding fin a powerseries in D(z, R) and then computing theL2 norm overD(z, R).

The estimate above certainly implies that for each compact set K there is a constant CK > 0such that

sup

zK|f(z)

| CK

f

.

It follows that the spaceA2() =A() L2()

is a closed subspace ofL2(). Thus A2() is a separable Hilbert space.

The estimate also shows that for each z the linear functional f f(z) is continuous on A2().By the Riesz representation theorem there exists a unique function Bz A2() such that

f(z) = ( f | Bz) , for eachf A2().We define

B(z, w) =Bz(w).

The function B is called the Bergman kernel of . It was introduced by Bergman in 1921. Thedefinition shows

f(z) =

B(z, w)f(w) dxdy , f A2(),

wherew = x+iy. This property is known as the reproducing property.

Let (n)n0 be an orthonormal Hilbert basis ofA2(). Then we have the FourierBessel series

Bz =

n=0

bn(z)n

with convergence in A2() for each z

. Since

n(z) = ( n| Bz) =bn(z)we see that

n=0

| n(z) |2 = Bz

and

B(z, w) =

n=0

n(z)n(w),

with, a priori, convergence inL2(), for each z .

For anyf A2() we have the FourierBessel series

f=

n=0

( f | n) n

which converges inA2(). Actually more is true:

Lemma 5. Iff A2() then the FourierBessel series off converges to f uniformly on compactsubsets of.

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Proof. Indeed

f(z) m

n=0

( f | n) n(z)

f m

n=0( f | n) n1/2dist(z, )

.

In particular the series

Bz =

n=0

bn(z)n

converges uniformly on compact subsets of for eachz . Thus the series

B(z, w) =

n=0

n(z)n(w)

converges at least pointwise on . It follows that

B(z, z) =

n=0| n(z) |

2

=

n=0| bn(z) |

2

= Bz2

.

NowBz A2() implies| Bz(w) | Bz

1/2dist(w, ).

Thus

Bz2 =B(z, z) = | Bz(z) | Bz1/2dist(w, )

which implies;

Theorem 6.

n=0

| n(z) |2 1dist(z, )2

.

By the CauchySchwarz inequality we now deduce that the series

B(z, w) =

n=0

n(z)n(w)

converges absolutely, and converges uniformly on compact subsets of .

For anyf A2() by CauchySchwarz we have

n=0

| ( f | n) | | n(z) | 2

n=0

| ( f | n) |2

n=0

| n(z) |2

f2 1dist(z, )

which implies:

Corollary 7. Iff A2() then the FourierBessel series offconverges absolutely.

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References

[1] Lars. V. Ahlfors, Complex analysis, 2nd ed., McGrawHill, New York, St. Louis, San Francisco,Toronto, London, Sydney, 1966.

[2] John B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathemat-

ics, vol. 11, SpringerVerlag, New York, Berlin, Heidelberg, London, Paris, Tokyo, Hong Kong,Barcelona, 1978.

[3] Einar Hille,Analytic function theory, vol. 2, Ginn and Co., Boston, New York, Chicago, Atlanta,Dallas, Palo Alto, Toronto, 1962.

[4] Oliver Dimon Kellogg,Foundations of potential theory, Springer-Verlag, Berlin, Heidelberg, NewYork, 1929 (reprinted 1967).

Copyright c 1998 Bent E. Petersen. Permission is granted to duplicate thisdocument for nonprofit educational purposes provided that no alterations are

made and provided that this copyright notice is preserved on all copies.

Bent E. Petersen phone numbersDepartment of Mathematics office (541) 737-5163Oregon State University home (541) 753-1829Corvallis, OR 97331-4605 fax (541) 737-0517

[email protected]://osu.orst.edu/peterseb

http://www.peak.org/petersen

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