Ideals and boundaries in Algebras of Holomorphic functions

Ideals and Boundaries in

Algebras of Holomorphic Functions

Linus Carlsson

Ume˚ a University

Department of Mathematics and Mathematical Statistics

Doctoral Thesis No. 33, 2006

To big plans and all the roads home

Linus Carlsson: Ideals and Boundaries in Algebras of Holomorphic Functions

° 2006 Linus Carlsson c

Tryck: Print & Media, Ume˚ a universitet, Ume˚ a isbn 91-7264-011-1

issn 1102-8300, nr. 33 (2006).

2001621

Ideals and Boundaries in

Algebras of Holomorphic Functions

Linus Carlsson

Doctoral Thesis No. 33, Department of Mathematics and Mathematical Statistics, Ume˚a University, 2006

To be publicly discussed in lecture hall MA121, Ume˚a University, on Friday, February 17, 2006 at 10.15 for the degree of Doctor of Philosophy.

Abstract We investigate the spectrum of certain Banach algebras. Properties like generators of maximal ideals and generalized Shilov boundaries are stud- ied. In particular we show that if the ¯ ∂-equation has solutions in the algebra of bounded functions or continuous functions up to the boundary of a domain D ⊂⊂ C

then every maximal ideal over D is generated by the coordinate func- tions. This implies that the fibres over D in the spectrum are trivial and that the projection on C

of the n − 1 order generalized Shilov boundary is contained in the boundary of D.

For a domain D ⊂⊂ C

where the boundary of the Nebenh¨ ulle coincide with the smooth strictly pseudoconvex boundary points of D we show that there always exist points p ∈ D such that D has the Gleason property at p.

If the boundary of an open set U is smooth we show that there exist points in U such that the maximal ideals over those points are generated by the coordinate functions.

An example is given of a Riemann domain, Ω, spread over C

where the fibers over a point p ∈ Ω consist of m > n elements but the maximal ideal over p is generated by n functions.

Mathematics Subject Classification 2000:

Primary 32A17,46J20,32A38; Secondary 46J15,32D15,32D26

Key words and phrases: maximal ideal space, the Gleason problem, generalized

Shilov boundaries, Nebenh¨ ulle, the Koszul complex, Banach algebras of holomor-

phic functions.

Contents

1 Introduction 1

1.1 Summary . . . . 4

1.2 Acknowledgment . . . . 5

2 Preliminaries 7 2.1 Notations and basic definitions . . . . 7

2.2 ∂ problems and the Gleason problem . . . 12 ¯

2.3 Domains and envelopes of holomorphy . . . 13

3 Solving the Gleason problem using ¯ ∂ solutions 15 3.1 Preliminaries . . . 15

3.2 The exterior algebra and interior multiplication . . . 15

3.3 The Koszul complex and solving the Gleason problem . . . 21

3.4 The Koszul complex and a solution to a division problem . . . 25

4 Exhaustion of pseudoconvex domains 29 4.1 Preliminaries . . . 29

4.2 Two ways to exhaust a domain with convergence in C

-norm . . . 29

4.3 Exhaustion of nonsmooth domains . . . 32

5 An equivalence to the Gleason problem 35 5.1 Introduction . . . 35

5.2 Preliminaries . . . 35

5.3 The Gleason Problem . . . 36

5.4 A useful norm . . . 36

5.5 Main result for necessarity . . . 36

5.6 Main result for sufficiency . . . 37

6 A local solution to the Gleason problem 43 7 Nebenh¨ ulle and a local ¯ ∂ solution 49 7.1 Nebenh¨ ulle and pseudoconvexity . . . 49

7.2 Applications . . . 53

8 H

and A as Banach algebras 57 8.1 Domains with trivial fibres in the spectrum . . . 57 8.2 A theorem about the projection of the spectrum . . . 63 8.3 Some properties concerning spectrum-schlichtness . . . 64

9 Generators of function algebras in manifolds 73

9.1 Generalization to Riemann domains spread over C

. . . 73 9.2 Generators on pseudoconvex domains . . . 75

10 Future work 83

10.1 Current research . . . 83

1. Introduction

The theory of complex analysis has its roots in works of Augustin Cauchy, Bernhard Riemann and Karl Weierstrass in the first half of the 19th century. A pioneering Swede was G¨ osta Mittag-Leffler in Stockholm. A student of Weier- strass, Sonya Kovalevsky, brought the theory of complex analysis in higher di- mensions to Sweden and since then Swedish mathematicians have had a strong position in this field of mathematics. A breakthrough in the development was done by Lars H¨ ormander in the beginning of the 1960’s. He saw the subject in the light of partial differential equations. The interplay between several complex variables and partial differential equations has since then been strong. Methods from one of these fields have often been very useful and brought new insights into the other.

Complex analysis in several variables differs substantially from the one di- mensional case. There are several important reasons for this. One is that the fundamental equation, the so called Cauchy-Riemann equation, which character- izes the main objects of the theory - the analytic or holomorphic functions - is replaced by an overdetermined system of equations. There is no Riemann map- ping theorem in higher dimensions and a fundamental problem is to decide where the holomorphic functions can be defined. For general holomorphic functions the so called Levi problem was solved in the 1940’s and the domains of existence for such functions was given a geometric characterization. For subclasses of the holomorphic functions, for example bounded holomorphic functions, the problem is still unsolved and seems quite far from its solution.

Given a domain in the n-dimensional complex Euclidean space, which is not

a domain of existence for, say, bounded holomorphic functions, there might be

no maximal domain to which all these functions can be continued. However, as

was shown by Thullen already in the earl 1930’s, there always exists a maximal

complex manifold, a so called Riemann domain, to which all functions may be

continued. This manifold is called the envelope of the given domain (with respect

to the bounded holomorphic functions). Over each point in the original domain

there might exist several points in the envelope. It is of interest to decide when

this is not the case, i.e. when there are no other points in the envelope than

the trivial ones over the given domain. In this thesis we study some problems in

the interplay between function theory and function algebras, which, among other

things, can be used to draw such conclusions.

In the 1940s, the school of Israil M. Gelfand in Moscow, founded a theory of Banach algebras which turned out to be very fruitful in many different branches of mathematics. In the early 1950’s, some American mathematicians went to work on problems in complex analysis using techniques from the Banach algebra theory. They developed the theory of uniform algebras which formed a new link between classical function theory and functional analysis. This was done by Richard Arens, Charles Rickart, Kenneth Hoffman, Andrew Gleason, Hasley Royden, Errett Bishop, Walter Rudin, John Wermer and others.

A central theorem in the theory of commutative Banach algebras says that the algebra is isomorphic to an algebra of continuous complex-valued functions on a compact Hausdorff space - the so called maximal ideal space or the spectrum of the given algebra. The points in the spectrum can be identified with the maximal ideals in the algebra. A co-worker of Gelfand, Georgi E. Shilov, showed that there always exists a smallest closed subset of the spectrum, on which all the continuous functions mentioned above takes its maximal modulus. This set is called the Shilov boundary for the algebra. How can one explain this phenomenon? The maximum principle is central for holomorphic functions, and one way to explain the Shilov boundary would be to show that the spectrum possesses analytic structure, i.e. show that there are subsets of the spectrum which could be made into complex manifolds such that the functions are holomorphic on these sets.

Andrew Gleason worked on the problem to find analytic structure in the spectrum. Gabriel Stolzenberg had in 1963 given an example of a set in C

such that the maximal ideal space does not contain any analytic variety. In 1964, Gleason was able to prove that if the maximal ideal corresponding to a point in the spectrum is algebraically finitely generated, then some neighborhood of the point can be given the structure of an analytic variety such that every function in the algebra is holomorphic on this variety. The problem was now to be able to decide whether maximal ideals are finitely generated or not. However, in most situations it turns out to be very difficult to determine whether a maximal ideal is finitely generated or not. In connection with the proof of his theorem, Gleason asked whether the maximal ideal consisting of the functions that vanish at the origin in the algebra of functions holomorphic in the unit ball and continuous up to the boundary was finitely generated (by the coordinate functions). This is probably the most simple case and Gleasons question shows how difficult the problem is. The problem (for general domains) is today known as the Gleason problem in the literature. The original question was subsequently solved by the Russian mathematician Z.L. Leibenzon. The method of proof that Leibenzon used yields in fact that all convex domains in C

with C

-boundary have the Gleason property.

A number of authors, using different methods, have contributed to solve the

question for so called strictly pseudoconvex domains. These are domains that

by a holomorphic change of coordinates locally can be made strictly convex. For

such domains there is a strong function theory with good estimates for solu- tions of the Cauchy Riemann equations. The result that has been obtained is that given the algebra of bounded holomorphic functions (or functions holomor- phic in the domain and continuous up to the boundary) on an arbitrary strictly pseudoconvex domain, every maximal ideal consisting of functions vanishing in a point in the domain, is generated by the coordinate functions. This means that the Gleason problem can be solved.

For weakly pseudoconvex domains, partial results have been obtained by Frank Beatrous Jr, John Erik Fornaess and Nils Øvrelid, David Catlin, Ulf Back- lund and Anders F¨ allstr¨ om, Jan Wiegerinck and Oscar Lemmers among others.

The main subject of this thesis is the Gleason problem and consequences thereof, we will now give a mathematical description of the problem. Let D be a domain in C

and denote by B(D) a ring of holomorphic functions which contains the polynomials. If p = (p

, p

, ..., p

) belongs to D we may ask whether or not the maximal ideal J

= {f ∈ B(D) : f(p) = 0} is generated by the functions z

− p

, z

− p

, ..., z

− p

. This is true if and only if for every function f ∈ J

there exist functions f

, f

, ..., f

∈ B(D) such that

f (z) = X

(z

− p

) f

(z)

for all z = (z

, z

, ..., z

) ∈ D. If J

has this property, we say that D has the Gleason B property at p, if this is true for all points p ∈ D then D is said to have the Gleason B property. As mentioned above, Andrew Gleason introduced this subject when studying the Banach algebra A(B(0, 1)). The function space A(B(0, 1)) consist of all holomorphic functions on the unit ball B(0, 1) in C

which can be continuously continued to the boundary. Gleason asked whether the ideal J

was finitely generated, see [Gleason, 1964]. In this thesis we focus on the Banach algebras A(D) or H

(D) denoted by B. Here H

(D) denotes the bounded holomorphic functions on D.

As mentioned above, an important concept in Banach space theory, is the maximal ideal space or equivalently the spectrum, denoted by M

. This is the subset of the dual space B

to B consisting of those elements, which respect multiplication, i.e.

m ∈ M

=⇒ m(fg) = m(f)m(g) ∀f, g ∈ B.

If the Banach algebra B(D) is a subalgebra of the holomorphic functions on D ⊂

C

, such that the polynomials belong to B(D), then the projection π : M

→ D

is defined as π(m) = (m(z

), m(z

), ..., m(z

)). Since the point evaluation at

p ∈ D belongs to the spectrum, we get a clear connection between M

and D.

1.1 Summary

In Chapter 3 we study a bounded domain D ⊂ C

. The main results are that;

Assuming that we can solve the ¯ ∂ problem for forms in L

(D) or C

( ¯ D) then D has the Gleason B property, where B ∈ {H

, A}, and if p ∈ C

\ D then we can find functions f

, f

, ..., f

∈ B(D) such that

1 = X

(z

− p

) f

(z).

Both these properties will be used in Chapter 8.

In Chapter 4 different types of exhaustion are investigated, in particular we show that we can exhaust a domain D in C

with domains D

⊂ D

, such that a fixed part of the strictly pseudoconvex points of the boundary of D, will be in the boundary of all D

and

D = [

D

.

Chapter 5 investigates a functional approach to the Gleason problem, a suffi- cient and a necessary condition is given such that the Gleason A property holds true. Also an example of a Reinhardt domain D ⊂⊂ C

is given such that D has the Gleason A property at the origin.

In Chapter 6 we prove that if there exists a peaking function h

for D at ξ and h

∈ H( ˜ D), where D and ˜ D are bounded pseudoconvex domains in C

and D ⊂⊂ ˜ D, then D has the Gleason B property in a neighborhood O

of ξ in D.

In this result we demand that ξ is in the strictly pseudoconvex boundary points of D, the set of strictly pseudoconvex boundary points is denoted by S(bD).

In Chapter 7 the Nebenh¨ ulle, N (D) of a pseudoconvex domain D ⊂⊂ C

is discussed and we show that if K is a nonempty compact set such that K ⊂ S(bD) ∩ b(N(D)) then there exists a strictly pseudoconvex domain ˜ D, such that D ⊂ ˜ D and K ⊂ b ˜ D. This result is used to prove a ¯ ∂ problem for locally supported forms, which in turn is used together with Chapter 6 to prove that all C

smooth domains D ⊂⊂ C

contain points p ∈ D such that D has the Gleason B property at p.

Chapter 8 is devoted to properties of the spectrum, M

, to a Banach algebra B. A correspondence between the Gleason property of a domain D and analytic properties of the spectrum is made clear. Examples of two bounded pseudoconvex domains, D

⊂ C

and D

⊂ C

, with the properties that the projection of the 1-dimensional respectively the 2-dimensional generalized Shilov boundary intersects with D

respectively D

.

In Chapter 9 we generalize various results to Riemann domains spread over

C

, where the Banach algebra B ∈ {H

, A} is studied. If D is such a domain it

is shown that under some hypothesis on D there exist points p ∈ D such that the maximal ideal J

is generated by n function. Also we show that if D ⊂⊂ C

is a domain with smooth boundary then there always exists a point p ∈ D such that D has the Gleason B property at p. Finally we give an example of an m-sheeted Riemann domain R

spread over C

and a point p ∈ R

such that the number of elements in the fibre over p in M

is m but the maximal ideal J

is generated by n functions, R

does not have the Gleason B property at p.

1.2 Acknowledgment

I wish to thank my supervisor Dr. Anders F¨ allstr¨ om, I am grateful for his never ending enthusiasm in life and his ability to make me understand mathematics. He is a generous person who have spent too many hours with me, to share from his broad knowledge. You have been a great tutor for me in the learning process of the world of mathematics. Thanks Anders, I hope you always run in the corridor and that your cup of coffee never dries out.

During my time as a Ph.D. student I have had the opportunity to spend some time at other mathematical departments, I visited UC Berkeley Mathematics and I am proud to thank Professor Michael Christ for being my local supervisor during the spring of 2002. He has given me a broader insight into differential equations and especially the ¯ ∂-Neumann techniques.

Two months of the beginning of 2003, I was a guest Ph.D. student at Institute of Mathematics ”Simion Stoilow” of the Romanian Academy, I would like to thank Professor Mihnea Coltoiu for the time he spent and the great hospitality of the Institute.

I spent autumn 2003 and spring 2004 at the Mathematics Department of the Universitat Aut` onoma de Barcelona. Professor Josep Maria Burgu´ es was my local supervisor, I am very grateful for all the weekly meetings we had, where we discussed several articles. You also taught me about the Nebenh¨ ulle and helped me proof reading some of my results.

A frequent source of happiness has been the pleasure to work with Dr. Oscar Lemmers. Our many and fruitful discussions about the Gleason problem and all the small epsilons and deltas have been a pleasure for me.

My gratitude goes to Professor Jan Wiegerinck for the time he has spent, reading and making remarks. He has also brought numerous of mistakes in the prior versions of this thesis to my attention, thank you.

From being my playground friend when we were three years old, to proof reading this thesis, Magnus Andersson has been putting up with a lot over the years. I am glad to share your friendship.

I also wish to thank all people at the mathematical department in Ume˚ a, here

I have found a lot of friends.

The several complex variables group, led by Professor Urban Cegrell, has been of incredible value for me. All the seminars, LLS, conferences and social gatherings and meeting all the guests. It has been a privilege to work with you.

I would also like to thank all peoples who have made mathematics interesting for me, to all my teachers; where I would like to mention Sixten Nilsson and Ingemar Strid who first got my mind set on mathematics. I would also like to thank Magnus Skog for long discussions during the nights, about the wonders of mathematics.

To Trina who has been helping me not to play chess nor sleep long in the mornings, but instead made me go to work. I have a wonderful time with you.

Love.

To Joakim Eriksson, my best friend, I hope your flies always will lure the salmon trouts.

To my family and friends who have been encouraging and supportive. I love

you all.

2. Preliminaries

2.1 Notations and basic definitions

The boundary of a set D will be denoted bD, and the interior of a set D is given by D

. An open set which is connected will be called a domain.

We will denote by supp(f ) the support of the function f , and a class of functions with compact support will be given an index 0, e.g. the continuous functions with compact support on D is denoted C

(D).

To say that the boundary of D is C

smooth is equivalent to say that there exists a defining function r ∈ C

such that:

• The domain is given by D = {z : r(z) < 0} and the complement D

= {z : r(z) > 0} .

• Furthermore the gradient of r is non-degenerated for all z ∈ bD = {z : r(z) = 0} .

The notion U ⊂⊂ V will mean that U is a relatively compact set in V , that is, in the given topology ¯ U is a compact set of V .

We will mainly work with domains in C

, where a point z ∈ C

will be given by

z := (z

, z

, ..., z

),

and z

= x

+iy

, j = 1, 2, ..., n. Thus, C

can be identified with R

in a natural manner

z 7−→ (x

, y

, x

, y

, ..., x

, y

).

If U and V are domains in C

such that U ⊂⊂ V then it follows that dist(bU, bV ) > 0. Here dist(S, T ) means the distance between two sets S and T in C

and is defined as

dist(S, T ) := inf

z∈S

⎧ ⎪

⎨

⎪ ⎩ inf

w∈T

⎛

⎝ X

|z

− w

|

⎞

⎠

1/2

⎫

⎪ ⎬

⎪ ⎭ .

It is obvious that the polydisc

∆

(ξ, r) := {z ∈ C

: |z

− ξ

| < r, j = 1, 2, ..., n}

has C

-smooth boundary and the open ball

B(ξ, r) :=

⎧ ⎨

⎩ z ∈ C

: X

|z

− ξ

|

< r

⎫ ⎬

⎭ has C

smooth boundary.

Unless stated otherwise, we will throughout this paper assume that the do- mains are in C

.

The partial differential operators on C

given by

∂

∂z

:= 1 2

µ ∂

∂x

− i ∂

∂y

¶

, j = 1, 2, ..., n,

∂

∂ ¯ z

:= 1 2

µ ∂

∂x

+ i ∂

∂y

¶

, j = 1, 2, ..., n will be very useful. For example, the operator ¯ ∂ act on forms as

∂f := ¯ X

∂f

∂ ¯ z

∧ d¯ z

,

which is central in the definition of a holomorphic function.

There are several equivalent definitions of a function f to be holomorphic.

First we assume that D ⊂ C

is open and that f is complex valued and at least in L

(D). One of the definitions is the following

Definition 2.1. The function f is a holomorphic function on D if f satisfies the homogeneous Cauchy-Riemann equations,

j

f (p) = 0 for all p ∈ D, j = 1, 2, ..., n, which is equivalent to say that ¯ ∂f (p) = 0 for all p ∈ D.

The holomorphic functions on D will be denoted H(D).

Remark 2.2. The property of being holomorphic is a local property.

Note that we only demand the function to be holomorphic in each variable separately.

Remark 2.3. If

j

f (p) = 0 for all p ∈ D, j = 1, 2, ..., n in the weak sense and f ∈ L

(D) then we (by a regularity argument concerning the ¯ ∂ operator, smooth cut off functions, and the Sobolev embedding theorem) in fact have that f ∈ C

(D).

That is: There are no ”fake” holomorphic functions.

For an open set D with defining function r, we define the complex tangent space at p ∈ bD as

T

(bD) :=

⎧ ⎨

⎩ w ∈ C

: X

∂r

∂z

(p)w

= 0

⎫ ⎬

⎭ .

Definition 2.4. Let D ⊂ C

be a domain which has r ∈ C

as its defining function. The Levi form of r at p ∈ bD is defined as

L(r, p, ξ) ≡ X

∂

r

∂z

∂ ¯ z

(p)ξ

ξ ¯

, for all ξ ∈ C

.

Let D ⊂ C

be an open set, and let r ∈ C

(O

) be a defining function for D, where O

is a neighborhood of p ∈ bD in C

, such that

D ∩ O

= {z ∈ C

: r(z) < 0} ∩ O

,

we say that a domain D is Levi pseudoconvex at p if L(r, p, ξ) ≥ 0 for all ξ ∈ T

(bD).

This definition is independent of the choice of the defining function.

Also we say that p ∈ bD is a strictly Levi pseudoconvex boundary point if L(r, p, ξ) > 0 for all ξ ∈ T

(bD) \ {0}.

When D is (strictly) Levi pseudoconvex at every boundary point, we simply say that D is (strictly) Levi pseudoconvex.

A central subject in this thesis will concern the set of strictly pseudoconvex boundary points of a domain D, denoted by S(bD).

If the boundary of a domain D is C

smooth then S(bD) is open and nonempty (just look at the extreme points of the boundary).

The set SPSH(D) will denote the strictly plurisubharmonic functions on D.

The definition of Levi pseudoconvexity suits our purposes since it is a local property and is easy to check, but sometimes it is convenient or necessary to use other equivalent definitions, some of which are:

Theorem 2.5. Let D ∈ C

be an open set, we say that D is pseudoconvex if one of the following equivalent statements is true; (note, however, that property 2. only makes sense when the boundary of D is C

smooth.)

1. − log µ

is plurisubharmonic on D for any distance function µ

. 2. D is Levi pseudoconvex.

3. D = ∪

D

, where each D

is strictly Levi pseudoconvex with C

- smooth

boundary and D

⊂⊂ D

.

Proof. See for example [Krantz, 2001].

The indices, when doing computation in complex analysis, easily get trouble- some, therefore it is often useful to use multi indices:

If α = (α

, α

, ..., α

) where α

are non negative integers for j = 1, 2, ..., n then we let

|α| = X

α

and

α! = α

! · α

! · ... · α

!

and if v = (v

, v

, ..., v

) is a vector (of functions) we define v

= v

· v

· ... · v

.

If v is a vector of operators, then we will use the same definition.

As an example, assume

= (

1

,

2

) and α = (2, 1) then µ ∂

∂z

¶

= ∂

∂z

∂

∂z

∂

∂z

= ∂

∂z

∂

∂z

= ∂

∂z

∂z

.

Notation 2.6. If the entries of the vector are forms, then the product is changed to the wedge product, e.g.

If d¯ z = (d¯ z

, d¯ z

, ..., d¯ z

) then

d¯ z

= (∧

d¯ z

) ∧ (∧

d¯ z

) ∧ ... ∧ (∧

d¯ z

) .

This definition differs from the usual one. We will use this one because we will only work with increasing order of the indices of the forms.

From now on, we also fix the function for the sign of permutation as ε

which takes the values 1 or −1 so that the equation

d¯ z

= ε

d¯ z

holds true. We define ε

in a similar way.

Example 2.7. Assume we are working in C

. If we have F = dz

∧ dz

∧ dz

then we can write F = −dz

, where α = (0, 1, 1, 0, 1). If G = dz

= dz

, where β = (0, 0, 0, 1, 0). This implies also that

H = F ∧ G = −dz

∧ dz

= −ε

dz

=

= −ε

dz

= −ε

dz

∧ dz

∧ dz

∧ dz

=

= dz

∧ dz

∧ dz

∧ dz

As an example of the use of multiindices we let D be an open set and define the ring of functions which derivatives up to order k, where k is a non negative integer, is continuous in the closure of D, that is

C

( ¯ D) =

½

f ∈ C( ¯ D) : ∂

f

∂z

∂ ¯ z

∈ C( ¯ D), 0 ≤ |α| + |β| ≤ k

¾ .

In the following chapters our main subject to study are the function spaces H

(D) and A(D) .

These are defined as H

(D) = L

(D) ∩ H(D), that are the bounded holo- morphic functions. And A(D) = C( ¯ D) ∩ H(D), the holomorphic functions on D which can be continued continuously to the boundary of D.

Both A(D) and H

(D), equipped with the supremum norm, denoted k.k or k.k

, are Banach algebras.

For an introduction to the theory of Banach algebras we refer to one of the textbooks [Gamelin, 1969], [Stout, 1971], [Alexander and Wermer, 1998], or [Rudin, 1991].

Some of the theory below will as well include the Banach algebras A

(D) = H(D) ∩ C

( ¯ D),

where k is a non negative integer, equipped with the norms

kfk = X

0≤|α|+|β|≤k

° °

° °

° µ ∂

∂z

¶

µ

∂

∂ ¯ z

¶

f

° °

° °

°

,

and sometimes also the Fr´ echet algebra

A

(D) =

\

A

(D),

which can not be equipped with a norm.

The ring of holomorphic functions defined on a neighborhood of D will be denoted H( ¯ D). That is, for every f ∈ H( ¯ D) there exists a domain Ω

such that f ∈ H(Ω

) and ¯ D ⊂⊂ Ω

.

The zero set of a function f is written Z

. When we say a smooth function

we mean that all derivatives of that function are continuous. We will often study

(p, q)-forms with smooth coefficients on a domain, these are written C

(D)

where D is the domain of definition. When working with forms we are going to

adapt this notion, e.g. when we write f ∈ L

we mean that f is a form of

bi-degree p and q and has its coefficients in the function space L.

2.2 ∂ ¯ problems and the Gleason problem

A central problem in the analysis of several complex variables is the lack of the (counterpart of the) Riemann mapping theorem, therefore the study has to be done in types of domains that have some properties in common.

One of the most powerful techniques in function theory of several complex variables is to study the properties of the solutions to the ¯ ∂ equation. These are the solutions u to the system of equations

∂u = λ, ¯ (2.1)

where we want to find a solution u to every (p, q + 1)-form λ in the function space L

(D) ∩ ker ¯ ∂, where ker ¯ ∂ = ©

λ : ¯ ∂λ = 0 ª

and L(D) is a ring of functions on D. Furthermore we want our solutions to have coefficients in the same space of functions, that is u ∈ L

(D).

The ¯ ∂ problem is the question whether or not there exists such a solution.

The reason of ¯ ∂-closed forms in the right hand side of (2.1) is the so-called compatibility condition, which by inspection has to be satisfied if we shall have any success in solving the ¯ ∂-equation.

In some cases, depending both on the ring L and the domain D we can even gain smoothness in the solution compared to L.

Some classical results, which we will use in this thesis, are the following ones:

Theorem 2.8. Let D ⊂ C

be a bounded pseudoconvex domain, and L a space of functions on D, L will be specified below.

For every λ ∈ L

(D) ∩ ker ¯ ∂ there exists a solution u ∈ L

(D) to (2.1), where L is one of the following function spaces:

1. L = C

(D).

2. L = C

( ¯ D) provided that the boundary bD ∈ C

.

3. L = H

(D) provided that D is strictly pseudoconvex and the boundary bD ∈ C

, actually the solutions can be chosen in A(D).

Proof. Part 1. was proved in [H¨ ormander, 1965]

Part 2. was proved in [Kohn, 1973]

Part 3. was proved in [Øvrelid, 1971b].

One can use part 1. in Theorem 2.8 to solve the so-called Levi Problem, that is if D is pseudoconvex then D is a domain of holomorphy. See [H¨ ormander, 1965]

for further details.

Let B(D) be a ring of complex valued functions on a domain D ⊂ C

, usually B(D) will be a Banach algebra of holomorphic functions.

We say that D has the Gleason B property at ξ ∈ D if for all functions

f ∈ B(D) there exists a set of functions {f

}

⊂ B(D) such that

f (z) − f(ξ) = X

f

(z)(z

− ξ

),

when z ∈ D. To determine whether D has the Gleason B property at point ξ ∈ D or not is the so-called Gleason problem, this problem arised in [Gleason, 1964].

For more details on this subject we refer to [Backlund, 1992], [F¨ allstr¨ om, 1994], [Lemmers and Wiegerinck, 2002] or [Lemmers, 2002].

Example 2.9. If we look at the ring of holomorphic functions H(D) on a pseu- doconvex domain D ⊂⊂ C

then the Oka-Hefers Lemma gives an affirmative answer to the Gleason H problem.

Example 2.10. Let D be an open set and let P(D) be the ring of holomorphic polynomials on D. Let ξ ∈ D. For any P ∈ P(D) we can write

P (z) − P (ξ) = X

0<|α|<N

a

(z − ξ)

for some N ∈ N. But since we have a finite sum and every term contains a factor of z

− ξ

for some j = 1, 2..., n we therefore have

P (z) − P (ξ) = X

(z

− ξ

)

⎛

⎝ X

0<|α|<N−1

a

(z − ξ)

⎞

⎠ ,

and defining P

(z) := P

0<|α|<N−1

a

(z − ξ)

, we have P (z) − P (ξ) =

X

(z

− ξ

)P

(z),

where P

∈ P(D). Thus, an open set D ⊂ C

has the Gleason P property.

2.3 Domains and envelopes of holomorphy

Definition 2.11. An open set D ⊂ C

is said to be a domain of holomorphy if the following property holds;

There do not exist nonempty open sets U

, U

with U

connected, U

6⊂ D, U

⊂ U

∩ D, such that for every holomorphic function h on D there exists a holomorphic function H on U

such that h = H on U

.

This definition states that if D is a domain of holomorphy then at each bound- ary point there exists at least one holomorphic function that cannot be extended over that point.

If all holomorphic functions can be extended over a point in the boundary

it may happen that the extended domain is not a domain in C

but rather a

Riemann domain spread over C

so we need the following definition.

Definition 2.12. A pair (Ω, π) is called a Riemann domain spread over C

if:

• Ω is a topological connected Hausdorff space,

• π : Ω → C

is locally homeomorphic, that is, each point in Ω has an open neighborhood U such that π(U ) is open in C

and π|

: U → π(U) is homeomorphic.

Remark 2.13. The map π endows Ω with the structure of a complex manifold.

Remark 2.14. If Ω is a domain in C

then it can be viewed as a Riemann domain in C

with π =identity.

Definition 2.15. Let B be a nonempty subset of H(Ω). The Riemann domain (E

(Ω), π

) is said to be an B-envelope of holomorphy of the Riemann domain (Ω, π) if the following properties hold true:

1. There exists a locally holomorphic map φ : Ω → E

(Ω) such that

• π

◦ φ = π,

• for every f ∈ B there exists a function F ∈ H(E

(Ω)) such that F ◦ φ = f.

2. For every Riemann domain (Ω

, π

) which satisfies 1. with φ

: Ω → Ω

there exists a locally holomorphic map ϕ : Ω

→ E

(Ω) such that

• π

◦ ϕ = π

,

• ϕ ◦ φ

= φ,

• if F and F

are the continuations to E

(Ω) and Ω

of f ∈ B then F ◦ ϕ = F

.

Part 1. says that the functions of B can be holomorphically continued to (E

, π

) and that the extensions are unique while part 2. states that if there is another extension (Ω

, π

) then that one can be holomorphically extended to (E

, π

). As a consequence we get that (E

, π

) is unique up to analytic isomor- phisms.

A standard theorem, whose proof can be found in, e.g. [Jarnicki and Pflug, 2000], is the celebrated

Theorem 2.16. (Thullen Theorem) Let (Ω, π) be a Riemann domain over C

and let ∅ 6= B ⊂ H(Ω). Then (Ω, π) has a B-envelope of holomorphy.

It is a well known fact that if we have two Riemann domains (Ω

, π

) and (Ω

, π

) over C

and B a nonempty subset of H(Ω

) such that (Ω

, π

) are B- envelope of holomorphy then the intersection (Ω

∩ Ω

, π

) is a B-envelope of holomorphy.

Also the following is true:

If (Ω

, π

) is an B-envelope of holomorphy then it is also an H-envelope of

holomorphy.

3. Solving the Gleason problem using ¯ ∂ solutions

3.1 Preliminaries

The main result of this chapter is that we can show the Gleason B property if we can solve the equation ¯ ∂u = w for all ¯ ∂-exact (p, q)-forms w, where the coefficients belong to an algebra L implies that the coefficients of u can be chosen to belong to the same algebra L. To prove this we will use the ideas of the Koszul complex, see [H¨ ormander, 1967]. We will also solve a division problem.

In this chapter L will be either L

(D) or C( ¯ D) and B = (L ∩ H)(D).

We will denote L

the algebra of (p, q)-forms with coefficients in L.

3.2 The exterior algebra and interior multiplication

Let D be an analytic manifold of complex dimension n, that is, locally D can be biholomorphically mapped onto an open set in C

which in turn can be identified with R

. On D we let the dual tangent space T

have the 2n dimensional basis denoted by (dz

, d¯ z

, dz

, d¯ z

, ..., dz

, d¯ z

) and as usual we let the (p, q)-forms be the space

T

=

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩ X

|α|=p

|β|=q

f

(z)dz

∧ d¯ z

⎫ ⎪

⎪ ⎬

⎪ ⎪

⎭ ,

here f

are the coefficient functions or simply the coefficients. We will also let

T

(D) =

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩ X

|α|=p

|β|=q

f

(z)dz

∧ d¯ z

: f

∈ L(D)

⎫ ⎪

⎪ ⎬

⎪ ⎪

⎭ .

We will soon define an operator δ

, which works on a space that we are about

to define, this will require some technicalities.

The operator δ

is defined via a mapping G := (g

, g

, ..., g

) such that G : D → C

,

where each g

∈ B(D), i = 1, 2, ..., d. Once the dimension d is fixed we let E denote the trivial vector bundle of complex dimension d and let (e

, e

, ..., e

) be the canonical basis.

We let E

be the dual space to E with the dual basis {e

, ..., e

} such that he

, e

i = δ

(the Kronecker delta function) for 1 ≤ i, j ≤ d. Finally we let E

= ∧

E be the product bundle of dimension p. Also E

will denote the dual product vector bundle.

For 0 ≤ s ≤ n, we let L

(D) =

½

f ∈ L : ∂

f

∂ ¯ z

∈ L(D), 0 ≤ |α| ≤ s

¾ .

Remark 3.1. In the proofs of this chapter, all the derivatives of the functions f ∈ L

(D), will only be of the form

, hence the definition.

Now we define the space on which the operator δ

will be defined.

Let f L

(D) be the space of forms of degree (0, q) on D and degree p on E

with coefficients in L(D), that is

L f

(D) = T

(D) ⊗

E

or equivalently, with e = (e

, e

, ..., e

) L f

(D) = nX

f

d¯ z

⊗ e

: f

∈ L(D), |α| = q, |β| = p o . And define L

(D) as

L

(D) = nX

f

d¯ z

⊗ e

: f

∈ L

(D), |α| = q, |β| = p o . For simplicity we will sometimes write L

instead of L

(D) and f L

for f L

(D).

For p > 0 and for a holomorphic mapping G = (g

, g

, ..., g

) where g

∈ B(D), i = 1, 2, ..., d, we define the interior product δ

from f L

−→ ] L

as follows:

For u = P

|α|=p

u

⊗ e

∈ f L

, where each u

is a f L

-form we let δ

act on u as;

δ

(u

⊗ e

) = X

g

u

⊗ (e

ye

) , (3.1)

where

e

ye

=

½ (−1)

e

if α

= 1, k

= P

α

,

0 if α

6= 1, (3.2)

here ε

= (0, 0, ..., 0, 1, 0, ..., 0) is the trivial vector except in position j where the entry is 1. Finally on u we let δ

act linearly as

δ

(u) = δ

⎛

⎝ X

|α|=p

u

⊗ e

⎞

⎠ = X

|α|=p

δ

(u

⊗ e

) . (3.3)

Remark 3.2. To make this definition consistent we let the operator vanish except when 0 ≤ p ≤ d and 0 ≤ q ≤ n.

Example 3.3. For G(z) = z = (z

, z

, ..., z

) and for u ∈ f L

, that is u = P

i=1

f

(z)e

, we have that

δ

u = X

z

f

(z),

which has an obvious connection to the Gleason problem at the origin.

Notation 3.4. When it is clear from the context we will often omit the dimension p and write δ

for δ

. If G = z as in the example above we will also omit G, that is δ = δ

.

Remark 3.5. The operator image clearly satisfies δ

(L

) ⊂ L

. On the space f L

we define the differential operators ∂ as

∂

⎛

⎝ X

|α|=p

u

⊗ e

⎞

⎠ ≡ X

|α|=p

¡ ∂u

¢

⊗ e

.

This gives that ∂(L

) ⊂ L

.

Some properties of the δ

-operator We will now look at some properties of the δ

-operator just defined. We show that δ

and ¯ ∂ commutes and that (δ

)

vanishes. Also a type of Leibnitz rule will follow.

Remark 3.6. From equation (3.2) we get δ

f = 0 for any f ∈ L

, q ∈ Z

. Proposition 3.7. For any u = P

|α|=p

u

⊗e

∈ L

and v = P

|β|=r

v

⊗e

∈ L

we have;

1. δ

(δ

u) = 0, 2. δ

( ¯ ∂(u)) = ¯ ∂(δ

(u)),

3. δ

(u ∧ v) = δ

(u) ∧ v + (−1)

u ∧ δ

(v).

Proof. Since it is obvious which space dimension we are working on, we will assume that G = (g

, g

, ..., g

) and work with δ

.

From equation (3.1) and (3.3) we get δ

(u) = X

|α|=p

u

⊗ δ

(e

).

Proof of 1. Letting δ

act on δ

(u) we get δ

(δ

(u)) = X

|α|=p

u

⊗ δ

(δ

(e

)).

If |α| < 2 then δ

δ

(u) vanishes by Remark 3.6. When |α| ≥ 2 we get by a straight forward calculation that

δ

(e

) = X

g

(e

ye

) which gives

δ

(δ

(e

)) = X

X

g

g

¡

e

y (e

ye

) ¢

. (3.4)

If i = j we get from equation (3.2) that e

y (e

ye

) = 0 and if i 6= j are fixed integers then we get the two terms in the sum

g

g

¡

e

y (e

ye

) + e

y ¡

e

ye

¢¢

. (3.5)

If we assume that i > j then from equation (3.2) we get e

y (e

ye

) = e

y ³

(−1)

e

´

=

= (−1)

e

= e

y ³

(−1)

e

´

=

= −e

y ¡

e

ye

¢

and the same is true when i < j by symmetry, so (3.5) vanishes for all i 6= j. Therefore all terms in equation (3.4) cancel out. So 1. is proved.

Proof of 2. Since

δ

(u) = X

|α|=p

u

⊗ δ

(e

)

and ¯ ∂(δ

(e

)) = 0 (observe that this holds for all holomorphic mappings

G in δ

), we get

∂δ ¯

(u) = X

|α|=p

∂(u ¯

) ⊗ δ

(e

)

and since

∂(u) = ¯ X

|α|=p

∂(u ¯

) ⊗ e

the desired result follows, that is δ

∂ = ¯ ¯ ∂δ

.

Proof of 3. In the definition of δ

we see that it is defined on e

, which means that the wedge product is taken over an increasing index set e

∧ e

∧ ... ∧ e

, therefore we rewrite u ∧ v in such a way;

δ

(u ∧ v) = δ

⎛

⎝ X

|α|=p

u

⊗ e

∧ X

|β|=r

v

⊗ e

⎞

⎠ =

= δ

⎛

⎝ X

|α|=p

X

|β|=r

(u

∧ v

) ⊗ ¡

e

∧ e

¢ ⎞

⎠ =

= X

|α|=p

X

|β|=r

(u

∧ v

) ⊗ δ

¡

e

∧ e

¢ ,

So we have to show that

δ

(e

∧ e

) = δ

(e

) ∧ e

+ (−1)

e

∧ δ

(e

), (3.6)

which we do by induction on p from p = 1, 2, ..., d, where d is the dimension of the mapping G = (g

, g

, ..., g

).

1. For p = 0 we have e

= 1 which is a function so from Remark 3.6 we have δ

(1) = 0 so

δ

(e

∧ e

) = δ

(e

) ∧ e

+ (−1)

e

∧ δ

(e

)

trivially.

2. For p = 1 we have for e

= e

for some 1 ≤ m ≤ d that