Pluripolar Sets and Pluripolar Hulls

UPPSALA DISSERTATIONS IN MATHEMATICS 41

Pluripolar sets and pluripolar hulls

Tomas Edlund

Department of Mathematics

Uppsala University

Dissertation at Uppsala University to be publicly examined in Room 2347, Pollacksbacken, Thursday, September 22, 2005 at 13:15 for the Degree of Doctor of Philosophy. The examina- tion will be conducted in English

Abstract

Edlund, T. 2005. Pluripolar sets and pluripolar hulls. Acta Universitatis Upsaliensis. Uppsala Dissertations in Mathematics 41. vii, 23 pp. Uppsala. ISBN 91-506-1812-1

For many questions of complex analysis of several variables classical potential theory does not provide suitable tools and is replaced by pluripotential theory. The latter got many important applications within complex analysis and related fields.

Pluripolar sets play a special role in pluripotential theory. These are the exceptional sets in this theory. Complete pluripolar sets (i.e. subsets of a domain Ω which are the exact −∞-locus of a plurisubharmonic function in Ω) are especially important.

In the thesis we study complete pluripolar sets and pluripolar hulls. (In "good" cases the latter are the smallest complete pluripolar sets containing a given pluripolar set).

We show that in some sense there are many complete pluripolar sets. E.g., smooth closed complete pluripolar curves are dense in the space of closed C

curves in C

(k is any natural number). We also show that on each closed subset of the complex plane there is a continuous function whose graph is complete pluripolar in C

.

On the other hand we study the propagation of pluripolar sets, equivalently we study pluripo- lar hulls. We relate the pluripolar hull of a graph to fine analytic continuation of the function.

Fine analytic continuation of an analytic function over the unit disk is related to the fine topol- ogy introduced by Cartan and to the previously known notion of finely analytic functions. We show that fine analytic continuation implies non-triviality of the pluripolar hull. Concerning the inverse direction, we show that the projection of the pluripolar hull is finely open. The difficulty to judge from non-triviality of the pluripolar hull about fine analytic continuation lies in possi- ble multi-sheetedness. If however the pluripolar hull contains the graph of a smooth extension of the function over a fine neighborhood of a boundary point we indeed obtain fine analytic continuation.

Keywords: plurisubharmonic functions, pluripolar set, complete pluripolar set, pluripolar hull, fine topology, finely analytic function, fine analytic continuation

Tomas Edlund, Department of mathematics. Uppsala University. Box 480, SE-751 06 Uppsala, Sweden

Tomas Edlund 2005 c ISSN 1401-2049 ISBN 91-506-1812-1

urn:nbn:se:uu:diva-5872 (http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-5872)

List of Papers

This thesis is based on the following papers, which are referred to in the text by their Roman numerals.

I Edlund, T. (2004) Complete pluripolar curves and graphs. Ann.

Polon. Mat. 84(1):75-86.

II Edlund, T., Jöricke, B. The pluripolar hull of a graph and fine analytic continuation. Arkiv för matematik (to appear).

III Edlund, T. Pluripolar hulls of graphs and extension to fine neigh- borhoods. Manuscript.

IV Edlund, T. A note on the pluripolar hull of a graph and analytic structure. Manuscript.

Reprints were made with permission from the publishers.

Contents

1 Introduction . . . . 1

1.1 Properties of plurisubharmonic functions . . . . 2

1.2 Pluripolar sets . . . . 3

1.3 Complete pluripolar sets . . . . 4

2 Description of the results of the thesis . . . . 7

2.1 Pluripolar completeness of graphs and of closed curves . . . . 7

2.2 Propagation of pluripolar sets . . . . 9

Acknowledgments . . . . 15

Swedish summary . . . . 17

References . . . . 21

1. Introduction

It is a remarkable consequence of Weierstrass’ construction of analytic func- tions with prescribed zeros that for each domain in the complex plane there exists an analytic function in that domain which is nowhere analytically ex- tendible. A domain in C ⁿ for which such a function exists is called a domain of holomorphy, or equivalently, a pseudoconvex domain. Thus, all planar do- mains are domains of holomorphy or, equivalently pseudoconvex domains.

The situation in higher dimensions changes dramatically. Studying power se- ries of functions defined in what we today call Hartogs domains Fritz Hartogs was the first to dicover domains in C ⁿ , n > 1, which are not pseudoconvex. In other words each holomorphic function in such a domain has compulsory an- alytic extension to a larger domain. Hartogs’ seminal paper lead to one of the fundamental problems in the theory of analytic functions of several complex variables, the so called Levi problem. This problem consists of describing do- mains of holomorphy in terms of geometric properties of its boundary. The Levi problem was first solved completely in the forties by Oka [23] for do- mains in C ² . In the fifties Bremermann, Norguet and Oka solved the Levi problem for domains in C ⁿ , n ≥ 3 [5][22][24].

For the solution of the Levi problem, the notion of plurisubharmonic func- tions was introduced by Oka in [23], and independently by Lelong in [18]. A [−∞,∞) -valued function u(z) defined in an open set Ω ⊂ C ⁿ is called plurisub- harmonic if it is upper semicontinuous, i.e. the sublevel sets {z ∈ Ω : u(z) < c}

are open for all real c, and if u satisfies the submean-value inequality u(z 0 ) ≤ 1

2π

_{2 π}

0 u(z 0 + ae ^{i θ} )dθ

for all z ₀ ,a ∈ C ⁿ such that the disk {z 0 +ta : |t| ≤ 1} is contained in Ω. If n = 1 the notion coincides with that of subharmonic functions of two real variables.

The above definition allows plurisubharmonic functions to be identically −∞.

In the case when a plurisubharmonic function u defined in Ω ⊂ C ⁿ is of class C ² , the definition given above is equivalent to the condition

∑ n j ,k=1

∂ ² u

∂z j ∂z k (z)a j a k ≥ 0, z ∈ Ω,a ∈ C ⁿ .

If the inequality above is strict for all a = 0 we say that u is a strictly plurisub-

harmonic function. If a plurisubharmonic function on a domain is not iden- tically −∞ we call it a non-trivial plurisubharmonic function. It turns out that the domains Ω of holomorphy are precisely those for which the function z → −logδ(z,∂Ω) is plurisubharmonic. (Here z ∈ Ω and δ(z,∂Ω) denotes the Euclidean distance from the point z to the boundary of Ω.) More general, by a theorem of Grauert a complex manifold is Stein if and only if it admits a strictly plurisubharmonic exhaustion function. (A Stein manifold is the coun- terpart of domains of holomorphy in the setting of manifolds, for a precise definition see [15])

It shall be mentioned in this context that there is also a partial differential equation approach to the problem of characterizing domains of holomorphy developed primarily by Morrey, Kohn and Hörmander. It turns out that a do- main Ω ⊂ C ⁿ is a domain of holomorphy if and only if the equation ∂g = f has a solution for each smooth ∂-closed differential form f of bidegree (0,q) (q = 1,...,n − 1). This problem, often called the “∂-problem”, turned out to be of great importance in itself and served as a bridge between the theory of several complex variables and partial differential equations. Also in this topic plurisubharmonic functions play a crucial role as weight functions in the L ² spaces which are used for finding solutions of the above mentioned differential equations. Note that weighted L ² -estimates still play a major role and continue to be extremely fruitful for applications within several complex variables and other fields like algebraic geometry.

The theory of plurisubharmonic functions (often called pluripotential the- ory) has a lot of further important applications besides the above mentioned ones. To list only some of them, plurisubharmonic functions are widely used in dynamics, often in form of the Green function of the set where the dynam- ics is well behaved. Plurisubharmonic Morse functions provide an important tool in holomorphic approximation, topology of Stein manifolds and others.

Although the fundament of pluripotential theory is developed there are still many interesting open problems in this area.

1.1 Properties of plurisubharmonic functions

Simple examples of plurisubharmonic functions are the convex functions and functions of the form log | f (z)|, where f (z) is analytic.

Vice versa, any plurisubharmonic function is related to functions of the lat- ter form. By a classical result of Bremermann every plurisubharmonic func- tion is given locally as the upper semicontinuous regularization u ^∗ (z) of

u (z) = lim

j →∞ sup 1

j log | f j (z)|

2

for some sequence { f j } of holomorphic functions. The upper semicontinuous regularization is defined by u ^∗ (z) = limsup _ξ→z u(ξ). Thus, plurisubharmonic functions are related to analytic functions. However while analytic functions are very "rigid", plurisubharmonic functions are more flexible and therefore provide a useful tool in several complex variables and other fields.

Some elementary properties of plurisubharmonic functions follow immediately from the definition given above. The set of plurisubharmonic functions form a convex cone i.e au (z) + bw(z) is plurisubharmonic if a and b are positive constants and u (z) and w(z) are plurisubharmonic functions. The maximum of a finite family of plurisubharmonic functions is a plurisubharmonic function. The decreasing limit of plurisubharmonic functions is again a plurisubharmonic function. On the other hand, if {u _α (z)}

is an arbitrary family of plurisubharmonic functions which are uniformly bounded from above the function u (z) := sup _α u _α (z) is in general not plurisubharmonic, but it is not hard to prove that the upper semicontinuous regularization u ^∗ (z) of u(z) is plurisubharmonic. Bremermann’s theorem cited above is one of the situations where functions of the just described form arise naturally. Another such situation is Bremermann’s definition of the Dirichlet problem in the multi-dimensional complex setting using Perron’s method for plurisubharmonic functions, or more generally, the definition of any other extremal plurisubharmonic function. We are able to approximate each plurisubharmonic function u in Ω by forming the convolution of u with standard smoothing kernels. In this way we obtain a decreasing sequence u j of plurisubharmonic functions, defined in any fixed relatively compact subdomain of Ω, which is of class C ^∞ and tends to u as j → ∞.

Plurisubharmonic functions are biholomorpically invariant. This means that if f : U → Ω is analytic and has an analytic inverse and u is plurisubharmonic in Ω, then u ◦ f is plurisubharmonic in U. Hence the plurisubharmonic func- tions are well defined on any complex analytic manifold.

1.2 Pluripolar sets

One of the most important class of sets which appear naturally in the theory of analytic functions are the analytic varieties, which locally are described as the common zero set of a finite family of analytic functions. More precisely, a subset E of a domain Ω is said to be an analytic variety (or an analytic set) in Ω if for each point z 0 ∈ Ω there exists a connected neighborhood U ⊂ Ω of z 0

and a finite family { f 1 (z),..., f k (z)} of analytic functions in U such that

E ∩U = {z ∈ U : f 1 (z) = ··· = f k (z) = 0}.

If E =U then max(log| f 1 (z)|,...,log| f k (z)|) is a non-trivial plurisubharmonic function on U which equals −∞ on E ∩U. A subset E ⊂ Ω of a domain in C ⁿ is called pluripolar in Ω if for each point z 0 ∈ E there exists a connected neighborhood U ⊂ Ω of z 0 and a non-trivial plurisubharmonic function u (z) defined in U such that E ∩U ⊂ u ⁻¹ (−∞).

We saw that any analytic variety E in a domain Ω, E = Ω, is a pluripolar set.

The definition of pluripolar sets is given by local conditions. By a remarkable theorem of Josefson [16] which answers a question posed by Lelong there is an equivalent global condition. Any pluripolar set E ⊂ Ω is a subset of the −∞

-locus of a plurisubharmonic function defined in the whole C ⁿ . Hence we may think of pluripolar sets as subsets of the −∞-locus of some globally defined plurisubharmonic function.

Hence, in particular, for any analytic variety in a domain Ω there is a plurisubharmonic function in Ω which equals −∞ on this set. Note that with- out further conditions there is no global description of an analytic variety by analytic functions as in the formula above. Pluripolar sets play an im- portant role in several complex variables and in related fields. For instance, the pluripolar sets are removable singularities for plurisubharmonic functions which are bounded from above. Moreover let {u _α (z)} be a family of plurisub- harmonic functions which are uniformly bounded from above. Consider the previously mentioned functions u (z) := sup α u _α (z) and the upper semicontin- uous regularization u ^∗ of u. The set where {u(z) < u ^∗ (z)} is called negligible.

It was proved in [2] by Bedford and Taylor that such sets are pluripolar.

1.3 Complete pluripolar sets

If a subset E ⊂ Ω equals the exact −∞-locus of a plurisubharmonic function defined in Ω then E is called complete pluripolar in Ω. These sets are of particular interest in pluripotential theory. For instance, in [30] Siu has observed that Bishop’s removable singularity theorem for analytic varieties with finite volume applies also to complete pluripolar sets i.e. the following Theorem is proved.

Theorem. Let U be an open subset of C ⁿ and u a plurisubhar- monic function on U. Suppose the set E of all points z of U with u(z) = −∞ is closed. Let A be an analytic variety of U
E of pure dimension.

If the volume of A is finite, then the closure of A in U is an analytic variety of U.

This result has been extended to the case of positive closed cur- rents in works of El Mir [11] and Sibony [28]. In each of these cases it is essential that E is complete pluripolar in Ω.

4

Simple examples of complete pluripolar sets are graphs of entire functions;

if f (z) is holomorphic in C ⁿ then the plurisubharmonic function log | f (z)−w|

shows that the graph Γ f (C ⁿ ) of f (z) over C ⁿ ,

Γ f (C ⁿ ) = {(z,w) ∈ C ^{n +1} : z ∈ C ⁿ ,w = f (z)},

is a complete pluripolar set. But not only complex analytic manifolds of co- dimension one appear as complete pluripolar sets.

For real curves it is in general difficult to determine whether they are pluripolar or even complete pluripolar. In [6] Diederich and Fornaess construct a real C ^∞ curve in the unit ball B ² ⊂ C ² which is complete pluripolar in B ² . On the other hand, in [7] they find a smooth curve which is not pluripolar.

In general the structure of complete pluripolar sets can be rather compli- cated. In [32] Wermer gave an example of a compact set K ⊂ ∂D × C whose polynomial hull

K  = {z ∈ C ⁿ : |p(z)| ≤ max

K |p| for all polynomials p}

contains no one-dimensional complex analytic manifold and the projection of K onto the first coordinate plane equals  D. In [19] Levenberg showed that the compact set K in Wermer’s example can be constructed such that  K
K is complete pluripolar in D × C.

The above considerations motivate the following problems.

Problem. Characterize complete pluripolar sets and understand their structure.

Problem. If a pluripolar set is not complete pluripolar, on which sets all plurisubharmonic functions equal −∞ if they are so on the original set? What is the mechanism of this “propagation”?

In this thesis we study the described problems.

2. Description of the results of the thesis

2.1 Pluripolar completeness of graphs and of closed curves

Here we discuss natural questions related to the first of the above mentioned problems. The first of these questions concerns pluripolar completeness of closed curves in C ⁿ , the second concerns pluripolar completeness of graphs of complex-valued functions over subsets of one coordinate plane. Our first result of this thesis answers the following question which was left open in the paper [20] by Levenberg, Martin and Poletsky.

Question 1. Does there exist a C ^∞ function f : ∂D → C whose graph is complete pluripolar in C ² ?

This question naturally arose in [20] where the authors construct analytic functions in D whose graphs are complete pluripolar in C ² . They prove the existence of functions f that are analytic in D and of class C ^∞ on D such that the graph of f over D is complete pluripolar in C ² . The difficulty for obtaining smooth complete pluripolar graphs over ∂D is that the corresponding function on ∂D cannot be the boundary values of functions who are analytic in D.

In Paper I we show that the answer to Question 1 is affirmative by giving an explicit example of a smooth function f : ∂D → C and a plurisubharmonic function u which equals −∞ exactly on the graph of f . It turns out that as in [20] a further development of an idea of Sadullaev [26] can be applied. The function f is defined by a suitable lacunary Fourier series. The logarithm of the modulus of suitably chosen partial sums of this series are building blocks for the construction of the plurisubharmonic function u. In fact, by using the ideas of this construction we are able to prove a much stronger result:

Theorem 1 For any closed subset F of the complex plane there exists a con- tinuous function on it whose graph is complete pluripolar in C ² .

Theorem 1 is proved in Paper I. The technique used for constructing the

function f and a plurisubharmonic function u which equals −∞ exactly on

the graph of f is similar to the one described above. We consider a locally finite covering of C
F consisting of open disks D j and functions f j defined by lacunary Fourier series which converges exactly outside D _j . As described above, the logarithm of the modulus of suitably chosen partial sums of the Fourier series of the functions f j are used for the construction of u. The proof of Theorem 1 is technically not easy.

It is a natural question to ask how “often” it may happen that graphs over ∂D (or general subsets of C ⁿ ) are complete pluripolar. We are able to prove that functions with the property described in Question 1 are not rare. In fact, they are dense in the C ^k topology in the space of C ^k functions on

∂D. That is, we prove the following Theorem

Theorem 2 Let k be an integer, k ≥ 0, g(z) ∈C ^k (∂D) and ε > 0. Then there ex- ists a function f (z) ∈ C ^∞ (∂D) such that the graph of f (z) is complete pluripo- lar in C ² and || f (z) − g(z)|| C

< ε.

Using a classical result of J. Wermer concerning the polynomial hull of closed real analytic curves in C ⁿ and a result on approximation of biholomor- phic mappings by automorphisms of C ⁿ due to Forstneric and Rosay we are able to extend Theorem 2 to arbitrary closed curves instead of graphs. In Paper I we prove the following Theorem.

Theorem 3 Let k be an integer, k ≥ 0, γ a closed C ^k curve in C ⁿ and ε > 0.

Then there exists an embedded closed C ^∞ curve γ 1 in C ⁿ such that ||γ(z) − γ 1 (z)|| C

< ε and γ 1 is complete pluripolar in C ⁿ .

One might expect that non-pluripolar sets should be “big” in some sense.

However, in [7] Diederich and Fornaess construct a smooth real curve in C ² which is non-pluripolar. Thus it seems difficult in general to character- ize smooth curves which are pluripolar or complete pluripolar respectively.

It follows from the construction in [7] that in each open ball in C ² there ex- ists a non-pluripolar curve. Moreover, the construction of the non-pluripolar curve in [7] can be done in such a way that this curve is the graph of a smooth function γ(z) defined on ∂D.

Thus the property of being a pluripolar, complete pluripolar, or a non- pluripolar closed curve in C ⁿ is in general not stable under small perturba- tions.

The following interesting observation was made by Levenberg, Martin and Poletsky in [20]. There exists an analytic function f in D which extends continuously to D so that the graph of f over D is not a pluripolar set. In their construction of the function f it is not clear whether the regularity of f could be strengthen without loosing the above described property. In other words the following question was left open.

8

Question 2. Does there exist an analytic function in D which is of class C ^∞ on D and so that the graph of f over D is not a pluripolar set.

In Paper III we show that the answer to this question is affirmative by showing that a nonpluripolar piece of the above mentioned nonpluripolar graph of γ(z) over ∂D (compare [7]) coincides with the graph of a C ^∞ function on D which is analytic in D.

2.2 Propagation of pluripolar sets

If a pluripolar set E ⊂ Ω is not complete pluripolar then any plurisubhar- monic function u in Ω which equals −∞ on E is necessarily equal to −∞ on a strictly larger set. For instance, if we consider a proper relatively open subset E of a connected one-dimensional analytic manifold M embedded in a domain Ω ⊂ C ⁿ , then any plurisubharmonic function defined in Ω which equals −∞

on E equals −∞ on the whole of M. In other words, the property of plurisub- harmonic functions to be −∞ on a non-complete pluripolar set “propagates”

to a larger set. By a slight abuse of language we will say that non-complete pluripolar sets “propagate”. This propagation property is reflected in the no- tion of the pluripolar hull E _Ω ^∗ of a pluripolar subset E (with respect to Ω). It is defined by the following formula

E _Ω ^{∗ de f} = ^ {z ∈ Ω : u(z) = −∞},

where the intersection is taken over all plurisubharmonic functions in Ω which equal −∞ on E. It shall be mentioned here that the pluripolar hull of a pluripo- lar set is in general not a complete pluripolar subset of C ⁿ [21]. However, in [33] Zeriahi showed that if the pluripolar hull of a pluripolar set E is simul- taneously a G _δ and F _σ subset, then the mentioned pluripolar hull is complete pluripolar.

It is in general hard to describe the pluripolar hull of a pluripolar set.

There is a certain analogy to the problem of describing the polynomial

hull of a compact set. Note first that a one-dimensional analytic manifold

with boundary (bordered Riemann surface) is contained in the polynomial

hull of a compact set if its boundary is contained in the compact, while a

one-dimensional analytic manifold V is contained in the pluripolar hull of a

compact set if a subset E of V of positive logarithmic capacity is contained

in the compact. Roughly, the mechanism of propagation in the first case is

the maximum modulus principle, in the second case it is the two-constant

theorem. This observation has been formalized in the general case by

Poletsky [25]. The following criteria describe the polynomial (the pluripolar

hull, respectively) of a compact.

Theorem. Let K be a compact set and let D be a Runge neighbor- hood of K. Fix z 0 ∈ D. Then z 0 belongs to the polynomial hull of K if and only if, for any open set V ⊂ D containing K and for any ε > 0, there exists an analytic disk g : D → D in D with g(0) = z 0 and

m 1 ({t ∈ [0,2π] : g(e ^it ) ∈ V}) > 2π − ε.

Here g : D → D means that g is analytic in D and continuous in D and m 1 (·) denotes linear measure. The theorem is proved in [25].

Theorem. Let K be a compact pluripolar set. Fix z 0 ∈ C ⁿ . Then z 0

belongs to the pluripolar hull of K (relative to C ⁿ ) if and only if there exists C > 0 and a sequence of uniformly bounded analytic discs g j : D → C ⁿ with g(0) = z 0 such that, for any open neighborhood V of K, there exists j 0 such that, for all j ≥ j 0 ,

m 1 ({t ∈ [0,2π] : g j (e ^it ) ∈ V}) > C.

The theorem is proved in [21].

It is well known that there are compact sets with non-trivial polynomial hull without analytic structure (i.e. there is no relatively open subset of an analytic manifold which is contained in the non-trivial part of the polynomial hull, see for instance [31] and the example of Wermer described above). Still, the non-trivial part of the polynomial hull has some structural properties which can be described in the language of currents [8] [9]. Moreover, restrictions of analytic functions to such sets satisfy the local maximum modulus principle [13]. The expectation is that also the non-trivial parts of pluripolar hulls have some interesting structural properties.

A first step towards the understanding of the phenomenon of propagation of

pluripolar sets is to study pluripolar hulls of suitable classes of pluripolar sets

for which the mechanism of propagation seems more transparent. The expec-

tation is that this is the case for graphs of analytic functions over domains in

the complex plane. If a function f has analytic continuation to a larger domain

then the graph of the analytic continuation is contained in the pluripolar hull

of the original graph. Therefore it seems natural to relate non-triviality of the

pluripolar hull of such graphs to various kinds of generalized analytic con-

tinuation. In [20] it is proved that analytic functions in D defined by suitable

lacunary Taylor series have complete pluripolar graphs. (The lacunary condi-

tion is much stronger than Hadamard lacunarity.) In particular, the pluripolar

hull of such graphs is trivial, i.e consists of the graph itself, and therefore

by Zeriahi’s theorem the graph is complete pluripolar. Since such functions

10

provide canonical examples of analytic functions in D which do not possess analytic continuation to any larger domain, it was conjectured in [20] that an- alytic continuation of a function is roughly the only reason for the pluripolar hull of its graph to be non-trivial. This conjecture was disproved by Edigarian and Wiegerinck in [10] where they construct a counterexample. In [29] Siciak noticed that the function in [10] admits pseudocontinuation across a subset E of the circle of positive measure. This means the following. Let Ω 1 ⊂ D and Ω 2 ⊂ {z ∈ C : |z| > 1} be domains such that there exists a nonempty open arc I ⊂ ∂D such that Ω 1 ∪ I ∪ Ω 2 is a domain. A function f in Ω 1 has pseudo- continuation F (z) across E if F (z) is meromorphic in Ω 2 and the angular limits of f (z) and F (z) exist and are equal for all points of E. Siciak showed that the pluripolar hull of the graph of f over D contains the graph of the pseudocontinuation. He also noticed that if an analytic function f (z) in D ad- mits pseudocontinuation through a set E of positive measure on the circle and the graph of the non-tangential limits is in the pluripolar hull of the graph of f over D then also the graph of the pseudocontinuation is in the mentioned pluripolar hull. It is important here that E has positive measure. In this paper Siciak also showed that the existence of pseudocontinuation of the function f (z) is not necessary for non-triviality of the pluripolar hull of the graph of f . In this thesis we relate non-triviality of the pluripolar hull of graphs of ana- lytic functions in D to the notion of fine analytic continuation described below which seems better related to the mentioned question. Indeed, fine analytic continuation of functions is sufficient for non-triviality of the pluripolar hull of its graph. Under additional assumptions it is also necessary (Theorem 7).

In all known examples non-triviality of the pluripolar hull is related to either fine analytic continuation or, maybe, to some generalization of it, which, intu- itively, might be understood as “branched fine analytic continuation”.

The detailed description is the following. Recall that the fine topology on the complex plane was introduced by Cartan (see e.g. [4]) as the weakest topology on C for which all subharmonic functions are continuous. A neigh- borhood basis of a point in this topology consists of sets which differ from a Euclidean neighborhood of this point by a set which is thin at this point. Thin sets were introduced by Brelot. A set F ⊂ C is thin at a point z 0 , if either z 0 is not in its closure F or z ₀ ∈ F and there exists a subharmonic function u(z) in a neighborhood of z ₀ such that lim _{z ∈F,z→z}

u(z) < u(z 0 ). One can always choose u(z) in such a way that the limit on the left equals −∞.

By a closed fine neighborhood V of p we mean a connected closed set which

has the form B
U for some connected closed neighborhood B of p and an

open set U ⊂ C which is thin at p. Note that U can be taken simply connected

(but in general not connected). Since subharmonic functions are upper semi-

continuous, each fine neighborhood of p contains a closed fine neighborhood

of p. A set is finely open if it contains a fine neighborhood of each of its points.

We will say that a continuous function on a subset S of C has the Mergelyan property if it can be approximated uniformly on S by analytic functions in a neighborhood of S. This notion is in view of Mergelyan’s Theorem which states that for compact sets K with finitely many components of the complement all continuous functions on K which are holomorphic in the interior of K have this property. It is convenient to introduce the following “semilocal” notion. A continuous function F (z) on a closed fine neighborhood V of a point p ∈ C is called finely analytic at p (on V) if F (z) can be approximated uniformly on V by analytic functions F n (z) in a neighborhood U (F n ) of V. The notion of finely analytic functions is well known and used for functions which have the Mergelyan property on finely open sets [12]. In this thesis we give the following definition of fine analytic continuation.

Definition. Suppose f (z) is analytic in D. Let p be a point on ∂D.

We say that f (z) has fine analytic continuation F (z) at p if there exists a closed fine neighborhood V of p and a finely analytic function F (z) at p on V such that F | _D∩V = f _D∩V .

Note that this describes a local property of a function. In paper II we require for simplicity that V is of the form B
U for some connected closed neighborhood B of p and an open set U ⊂ C
D which is thin at p.

The theorems stated in paper II holds as well with the slightly more general defintion given above. We prove the following result.

Theorem 4 Let f be analytic in D and let p ∈ ∂D. Suppose f has fine analytic continuation F at p to a closed fine neighborhood V of p. Then there exists another closed fine neighborhood V 1 ⊂ V of p, such that the graph of F over V 1 is contained in the pluripolar hull of the graph of f over D.

Theorem 4 says that fine analytic continuation of f at a point p ∈ ∂D is suffi- cient for non-triviality of the pluripolar hull of the graph of f . In view of this Theorem we are able to give new examples of non-extendible analytic func- tions in the unit disk so that the pluripolar hull of their graphs are non-trivial.

For instance, in paper II we give an example of a univalent function in D which extends smoothly up to the boundary of D with this property. We are also able to give an example of a non-extendible analytic function f with the property that the pluripolar hull of the graph of f has infinitely many sheets. Indepen- dently, in [34] Zwonek has constructed an analytic function in D which does not have analytic extension across ∂D for which the pluripolar hull of its graph has at least two sheets over D.

In the examples described above, the non-trivial part of the mentioned

pluripolar hulls contain a one-dimensional analytic manifold. However, this

12

is not always the case. In paper II we give an example of a function f for which the non-trivial part of the pluripolar hull of the graph of f is contained in the graph of a continuous function g defined on a “swiss cheese” S. This means that S is a connected closed set of positive measure without interior points. Consequently no one-dimensional analytic manifold is contained in the non-trivial part of the pluripolar hull of the graph of f . Even more can be said in this context: in paper IV we give an example of an analytic function f in D with the following property. The projection onto the first coordinate plane of the closure of the pluripolar hull of the graph of f equals C and the part of the pluripolar hull contained outside D × C contains no one-dimensional analytic manifold. The construction of f is made in such a way that the non-trivial part of the pluripolar hull of the graph of f is contained in a Wermer-like set like described earlier.

It is a natural question to ask whether fine analytic continuation of a func- tion is also necessary for non-triviality of the pluripolar hull of its graph. In paper II we prove the following two theorems in this direction.

Theorem 5 Let f be an analytic function in D whose graph is not complete pluripolar in C ² . Then the projection of the pluripolar hull of the graph of f onto the first coordinate plane contains a fine neighborhood S of a point p ∈ ∂D.

Theorem 6 Let f be analytic in a domain D ⊂ C. Then the projection of the pluripolar hull of the graph of f onto the first coordinate plane is open in the fine topology.

In particular, the pluripolar hull of the graph of an analytic function f in D cannot differ from the graph itself by a small but non-empty set. Moreover if the graph of such a function is not complete pluripolar then its pluripolar hull projects onto a fine neighborhood of a point on the unit circle. Difficulties for understanding fine analytic continuation properties of the function occur in view of possible multi-sheetedness. It is not known whether the pluripolar hull of the graph always contains the graph of a continuous extension of f over a fine neighborhood of a point on the circle. However, if the pluripolar hull contains the graph of a C ² extension of the function f over such a set (instead of a continuous extension) then in fact f has fine analytic extension.

This is the statement of Theorem 7 below (see paper III ). We do not know whether C ² can be replaced by "continuous". It shall be mentioned here that in [27] Shcherbina proved that if a continuous function f on a domain D has a pluripolar graph then f is in fact analytic in D.

Theorem 7 Let g(z) be analytic in D. Suppose there exists a fine neighborhood

S of a point p ∈ ∂D and a function f (z) of class C ² on S with g(z) = f (z) for

each z ∈ D ∩ S. If the graph of f over S is contained in the pluripolar hull of the graph of g over D then g(z) has fine analytic continuation f (z) at p.

Here “ f (z) is of class C ² on S” means that f (z) is C ² in the sense of Whitney on each compact subset K ⊂ S. This means that for all non-negative integers k,l with k + l ≤ 2 there exist functions { f _(k,l) (z)} defined on S satisfying the following compatibility condition. f _(0,0) (z) = f (z) and if z and z 0 are any two points in S then

f _(k,l) (z) = ∑

k

+l

≤2−(k+l)

f _(k,l)+(k

,l

) (z 0 )

k  !l  ! (z − z 0 ) ^k

(z − z 0 ) ^l

+ R _(k,l) (z,z 0 ) where R _(k,l) (z,z 0 ) = o(|z − z 0 | ^{2 −(k+l)} ) for z → z 0 .

14

Acknowledgments

First of all I would like to express my gratitude to my adviser Prof. Burglind

Juhl-Jöricke for providing me with ideas and thoughts. She has given me in-

valuable advice and suggestions throughout the work with this thesis. Sec-

ondly I would like to thank Prof. Maceij Klimek for suggestions and reading

of parts of my manuscripts. I would also like to thank Prof. Dietrich von Rosen

for support while I still was a lecturer at SLU. Finally I would like to take

this opportunity to thank friends and colleagues with whom I have exchanged

mathematical thoughts concerning matters in this text.

Swedish summary

Pluripolära mängder och pluripolära höljen

Det är en anmärkningsvärd konsekvens av Weierstrass konstruktion av ana- lytiska funktioner med en given nollställemängd, att det på varje öppet sam- manhängande område i det komplexa talplanet finns en analytisk funktion som saknar analytisk fortsättning till något större område. Ett öppet sammanhän- gande område i C ⁿ där det finns en analytisk funktion som ovan, kallas för ett holomorfiområde eller ett pseudokonvext område. Alltså är varje öppen sam- manhängande mängd i det komplexa talplanet ett holomorfiområde. I högre dimensioner är situationen annorlunda. Där kan det mycket väl hända att alla analytiska funktioner i ett öppet sammanhängande område nödvändigtvis har analytisk fortsättning till ett strikt större område. Denna observation ledde till ett av de fundamentala problemen i teorin för analytiska funktioner av flera variabler, det så kallade Levi-problemet. Detta problem innebär att beskriva holomorfiområden i termer av geometriska egenskaper hos ränderna till så- dana områden. Det första arbetet inom detta icke-triviala problem gjordes av Hartogs i [14] där han studerade serier av analytiska funktioner i vad vi idag kallar Hartogsområden. Men Levi-problemet blev inte löst fullständigt förrän på fyrtiotalet av Oka [23] för områden i C ² . På femtiotalet löste Bremermann, Norguet och Oka Levi-problemet för områden i C ⁿ , n ≥ 3.([5][22][24]).

För att lösa Levi-problemet introducerades de plurisubharmoniska funktion- erna av Oka i [23], och oberoende av Lelong i [18]. En [−∞,∞)-värd funktion u (z) definierad i ett öppet område Ω ⊂ C ⁿ kallas plurisubharmonisk om den är uppåt halvkontinuerlig, det vill säga {z ∈ Ω : u(z) < c} är en öppen mängd för alla reella tal c, och om u uppfyller medelvärdesolikheten

u(z 0 ) ≤ 1 2 π

_{2 π}

0 u(z 0 + ae ^{i θ} )dθ

för alla z ₀ ,a ∈ C ⁿ så att skivan {z 0 +ta : |t| ≤ 1} är innehållen i Ω. Om n = 1 så är definitionen ovan ekvivalent med definitionen av subharmoniska funk- tioner av två reella variabler. Med denna definition är även funktioner som är identiskt lika med −∞ plurisubharmoniska. Om en plurisubharmonisk funk- tion definierad i ett öppet sammanhängande område inte är identiskt lika med

−∞ säger vi att funktionen är en icke-trivial plurisubharmonisk funktion. Det

visar sig att holomorfiområdena i C ⁿ är precis de som uppfyller att funktio-

nen z → −logδ(z,∂Ω) är plurisubharmonisk. (Här gäller att z ∈ Ω och med δ(z,∂Ω) menas det Euklidiska avståndet från punkten z till randen av Ω.)

Det ska nämnas i detta sammanhang att det också finns ett sätt att beskriva holomorfiområden med hjälp av partiella differentialekvationer, utvecklat mestadels av Morrey, Kohn och Hörmander. Det visar sig att ett öppet sammanhängande område Ω ⊂ C ⁿ är ett holomorfiområde om och endast om ekvationen ∂g = f har en lösning för varje slät ∂-sluten differentialform f av bi-grad (0,q) (q = 1,...,n − 1). Detta problem -”∂-problemet”

visade sig vara extremt användbart och knöt samman teorin för analytiska funktioner av flera variabler och teorin för partiella differentialekvationer.

Även här spelar de plurisubharmoniska funktionerna en viktig roll som viktfunktioner i de L ² -rum som används för att hitta lösningar till de ovan nämnda differentialekvationerna. Det är värt att notera att de viktade L ² -uppskattningarna fortfarande är viktiga och användbara i tillämpningar inom teorin för analytiska funktioner och andra områden inom matematiken som till exempel algebraisk geometri.

Teorin som behandlar plurisubharmoniska funktioner (ofta kallad pluripo- tentialteori) har många viktiga tillämpningar förutom de som nämnts ovan.

Till exempel, plurisubharmoniska funktioner används flitigt inom komplex dynamik, ofta i form av Green-funktioner av mängder där dynamiken beter sig “snällt”. Plurisubharmoniska Morse-funktioner är ett användbart verktyg för att beskriva topologin hos Stein-mångfalder. Fastän pluripotentialteorin är väl utvecklad återstår många intressanta problem att lösa inom detta område.

En av de viktigaste och mest naturliga mängderna i teorin för analytiska funktioner av flera variabler är de analytiska mängderna, som lokalt beskrivs som den gemensamma nollställemängden till en ändlig familj av analytiska funktioner. För att vara exakt, en delmängd E av ett öppet sammanhängande område Ω kallas en analytisk mängd om det för varje z 0 ∈ Ω finns en sam- manhängande omgivning U ⊂ Ω av z 0 och en ändlig familj { f 1 (z),..., f k (z)}

av analytiska funktioner definierade i U så att

E ∩U = {z ∈ U : f 1 (z) = ··· = f k (z) = 0}.

Om E = U så gäller att w(z) = max(log| f 1 (z)|,...,log| f k (z)|) är en icke-trivial plurisubharmonisk funktion i U som är lika med −∞ på E ∩U.

En delmängd E ⊂ Ω av ett öppet sammanhängande område i C ⁿ kallas

pluripolär i Ω om det för varje z 0 ∈ E finns en sammanhängande omgivning

U ⊂ Ω av z 0 och en icke-trivial plurisubharmonisk funktion u definierad i U

sådan att E ∩ U ⊂ u ⁻¹ (−∞). Funktionen w(z) definierad ovan visar att de

analytiska mängderna E ⊂ Ω, E = Ω är pluripolära mängder. Definitionen

av pluripolära mängder är given i termer av lokala villkor. Tack vare en

anmärkningsvärd sats bevisad av Josefson [16] finns det även ett ekvivalent

globalt villkor; varje pluripolär mängd E ⊂ Ω är delmängd till −∞-lokuset

18

till en icke-trivial plurisubharmonisk funktion definierad i hela C ⁿ . Därför kan vi tänka på pluripolära mängder som delmängder av −∞-lokuset till en globalt definierad plurisubharmonisk funktion. Speciellt gäller att om E är en analytisk mängd i ett öppet sammanhängande område Ω så finns en plurisubharmonisk funktion i Ω som är lika med −∞ på E. Pluripolära mängder spelar en avgörande roll inom pluripotentialteorin och närliggande områden. Till exempel är de pluripolära mängderna hävbara singulariteter för plurisubharmoniska funktioner som är begränsade uppifrån. Om en delmängd E ⊂ Ω är exakt lika med −∞-lokuset till en plurisubharmonisk funktion definierad i Ω så kallas E för fullständigt pluripolär i Ω. Dessa mängder är av speciellt intresse i pluripotentialteorin. Till exempel kan nämnas att Siu har visat i [30] att Bishops sats om hävbara singulariteter för analytiska mängder med ändlig volym också gäller för fullständigt pluripolära mängder, det vill säga han bevisar följande sats.

Sats. Låt U vara en öppen delmängd av C ⁿ och u en plurisubhar- monisk funktion på U. Antag att mängden E bestående av alla punkter z i U där u(z) = −∞ är sluten. Låt A vara en analytisk mängd i U
E av enhetlig dimension. Om A:s volym är ändlig så gäller att tillslutningen av A i U är en analytisk mängd i U.

Detta resultat har utökats till att gälla positiva slutna strömmar i arbeten av El Mir [11] och Sibony [28]. I var och en av de ovan nämnda situationerna är det avgörande att E är fullständigt pluripolär i Ω. Exempel på fullständigt pluripolära mängder är grafer av hela funktioner; om f (z) är analytisk i C ⁿ så är den plurisubharmoniska funktionen log | f (z) − w| lika med −∞ exakt på f (z):s graf Γ f (C ⁿ ) över C ⁿ ,

Γ f (C ⁿ ) = {(z,w) ∈ C ^{n +1} : z ∈ C ⁿ ,w = f (z)}.

Därför är Γ f (C ⁿ ) en fullständigt pluripolär mängd. Men inte enbart (n − 1)- dimensionella analytiska mångfalder är fullständigt pluripolära mängder. För reella kurvor är det i allmänhet svårt att avgöra huruvida de är pluripolära eller fullständigt pluripolära. I [6] konstruerar Diederich och Fornaess en reell C ^∞ -kurva i enhetsbollen B ² ⊂ C ² som är fullständigt pluripolär i B ² . I allmän- het kan strukturen hos pluripolära mängder vara komplicerad. I [32] visade Wermer ett exempel på en kompakt mängd K ⊂ ∂D × C vars polynomiella hölje

K  = {z ∈ C ⁿ : |p(z)| ≤ max

K |p| för alla polynom p}

inte innehåller någon en-dimensionell analytisk mångfald samt att projektio-

nen av  K ner på det första koordinatplanet är lika med D. I [19] visade Leven-

berg att den kompakta mängden K i Wermers exempel kan konstrueras så att K 
K är fullständigt pluripolär i D × C.

De ovan beskrivna fenomenen motiverar följande problem, som vi studerar i denna avhandling.

Problem. Karaktärisera fullständigt pluripolära mängder och förstå deras struktur.

Problem. Om en pluripolär mängd inte är fullständigt pluripolär, på vilken mängd är alla plurisubharmoniska funktioner lika med −∞ om de är så på den ursprungliga mängden? Vad är den underliggande mekanismen bakom denna “pluripolära fortsättning”?

Vi visar att i någon mening så finns det "många" fullständigt pluripolära mängder. Till exempel visar vi att släta slutna kurvor som är fullständigt pluripolära är täta i rummet av slutna C ^k -kurvor i C ⁿ (här är k ett godtyckligt naturligt tal). Vi visar också att på varje sluten delmängd av det komplexa talplanet finns det en kontinuerlig funktion vars graf är fullständigt pluripolär i C ² .

Vi studerar också pluripolära höljen av pluripolära mängder. Speciellt stud- erar vi pluripolära höljen av grafer av funktioner som är analytiska i enhetsski- van. Vi relaterar fin analytisk fortsättning av en funktion till pluripolära höljet av dess graf. Vi visar att om en funktion har fin analytisk fortsättning så är det pluripolära höljet till dess graf icke-trivialt. Det är naturligt att fråga sig huruvida implikationen åt andra hållet också är giltig. Vi kan inte ge ett full- ständigt svar i denna avhandling, men vi visar följande steg i denna riktning.

Om en funktion f som är analytisk i enhetskivan har en graf vars pluripolära hölje innehåller grafen av en slät fortsättning av f till en fin omgivning av en punkt på enhetscirkeln, så gäller att f har fin analytisk fortsättning.

20

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