In this thesis, we study two different kinds of approximation of plurisubharmonic functions. The first one is a Mergelyan type approximation for plurisubharmonic functions. That is, we study which domains in C^n have the property that every continuous plurisubharmonic function can be uniformly approximated with continuous and plurisubharmonic functions defined on neighborhoods of the domain. We will improve a result by Fornaess and Wiegerinck and show that domains with C^0-boundary have this property. We will also use the notion of plurisubharmonic functions on compact sets when trying to characterize those continuous and plurisubharmonic functions that can be approximated from outside. Here a new kind of convexity of a domain comes in handy, namely those domains in C^n that have a negative exhaustion function that is plurisubharmonic on the closure. For these domains, we prove that it is enough to look at the boundary values of a plurisubharmonic function to know whether it can be approximated from outside. The second type of approximation is the following: we want to approximate functions u that are defined on bounded hyperconvex domains Omega in C^n and have essentially boundary values zero and bounded Monge-Ampère mass, with increasing sequences of certain functions u_j that are defined on strictly larger domains. We show that for certain conditions on Omega, this is always possible. We also generalize this to functions with given boundary values. The main tool in the proofs concerning this second approximation is subextension of plurisubharmonic functions.

<p>In this thesis we study approximation of negative plurisubharmonic functions by functions defined on strictly larger domains. We show that, under certain conditions, every function <i>u</i> that is defined on a bounded hyperconvex domain Ω in C<i>n</i><i> </i>and has essentially boundary values zero and bounded Monge-Ampère mass, can be approximated by an increasing sequence of functions {<i>u</i><i>j</i>} that are defined on strictly larger domains, has boundary values zero and bounded Monge-Ampère mass. We also generalize this and show that, under the same conditions, the approximation property is true if the function u has essentially boundary values G, where G is a plurisubharmonic functions with certain properties. To show these approximation theorems we use subextension. We show that if Ω_1 and Ω_2 are hyperconvex domains in C<i>n</i> and if u is a plurisubharmonic function on Ω_1 with given boundary values and with bounded Monge-Ampère mass, then we can find a plurisubharmonic function û defined on Ω_2, with given boundary values, such that û <= u on Ω and with control over the Monge-Ampère mass of û.</p>

In this thesis we study approximation of negative plurisubharmonic functions by functions defined on strictly larger domains. We show that, under certain conditions, every function u that is defined on a bounded hyperconvex domain Ω in Cn and has essentially boundary values zero and bounded Monge-Ampère mass, can be approximated by an increasing sequence of functions {uj} that are defined on strictly larger domains, has boundary values zero and bounded Monge-Ampère mass. We also generalize this and show that, under the same conditions, the approximation property is true if the function u has essentially boundary values G, where G is a plurisubharmonic functions with certain properties. To show these approximation theorems we use subextension. We show that if Ω_1 and Ω_2 are hyperconvex domains in Cn and if u is a plurisubharmonic function on Ω_1 with given boundary values and with bounded Monge-Ampère mass, then we can find a plurisubharmonic function û defined on Ω_2, with given boundary values, such that û &lt;= u on Ω and with control over the Monge-Ampère mass of û.

In this thesis we focus on Dirichlet's problem for the complex Monge-Ampère equation. That is, for a given non-negative Radon measure µ we are interested in the conditions under which there exists a plurisubharmonic function u such that (ddcu)n=µ, where (ddc)n is the complex Monge-Ampère operator. If this function u exists, then can it be chosen with given boundary values? Is this solution uniquely determined within a given class of functions?

Some problems concerning the algebra of bounded holomorphic functions from bounded domains in Cn are solved. A bounded domain of holomorphy Q in C2 with nonschlicht i7°°- envelope of holomorphy is constructed and it is shown that there is a point in Q for which Gleason’s Problem for H°°(Q) cannot be solved. If A(f2) is the Banach algebra of functions holomorphic in the bounded domain Q in Cn and continuous on the boundary and if p is a point in Q, then the following problem is known as Gleason’s Problem for A(Q) : Is the maximal ideal in A(Q) consisting of functions vanishing at p generated by (Zl ~Pl) , ■■■ , (Zn - Pn) ? A sufficient condition for solving Gleason’s Problem for A(Q) for all points in Q is given. In particular, this condition is fulfilled by a convex domain Q with Lipi+£-boundary (0 &lt; e &lt; 1) and thus generalizes a theorem of S.L.Leibenzon. One of the ideas in the methods of proof is integration along specific polygonal lines. If Gleason’s Problem can be solved in a point it can be solved also in a neighbourhood of the point. It is shown, that the coefficients in this case depends holomorphically on the points. Defining a projection from the spectrum of a uniform algebra of holomorphic functions to Cn, one defines the fiber in the spectrum over a point as the elements in the spectrum that projects on that point. Defining a kind of maximum modulus property for domains in Cn, some problems concerning the fibers and the number of elements in the fibers in certain algebras of bounded holomorphic functions are solved. It is, for example, shown that the set of points, over which the fibers contain more than one element is closed. A consequence is also that a starshaped domain with the maximum modulus property has schlicht /y°°-envelope of holomorphy. These kind of problems are also connected with Gleason’s problem. A survey paper on general properties of algebras of bounded holomorphic functions of several variables is included. The paper, in particular, treats aspects connecting iy°°-envelopes of holomorphy and some areas in the theory of uniform algebras. / <p>Diss. (sammanfattning) Umeå : Umeå universitet, 1994, härtill 6 uppsatser</p> / digitalisering@umu

Some problems concerning holomorphic continuation of the class of bounded holo­morphic functions from bounded domains in Cn that are domains of holomorphy are solved. A bounded domain of holomorphy Ω in C2 with nonschlicht H°°-envelope of holomorphy is constructed and it is shown that there is a point in D for which Glea­son’s Problem for H°°(Ω) cannot be solved. Furthermore a proof of the existence of a bounded domain of holomorphy in C2 for which the volume of the H°°-envelope of holomorphy is infinite is given. The idea of the proof is to put a family of so-called ”Sibony domains” into the unit bidisk by a packing procedure and patch them together by thin neighbourhoods of suitably chosen curves. If H°°(Ω) is the Banach algebra of bounded holomorphic functions on a bounded domain Ω in Cn and if p is a point in Ω, then the following problem is known as Gleason’s Problem for Hoo(Ω) : Is the maximal ideal in H°°(Ω) consisting of functions vanishing at p generated by (z1 -p1) , ... ,   (zn - pn) ? A sufficient condition for solving Gleason’s Problem for 77°° (Ω) for all points in Ω is given. In particular, this condition is fulfilled by a convex domain Ω with Lip1+e boundary (0 &lt; e &lt; 1) and thus generalizes a theorem of S.L.Leibenson. It is also proved that Gleason’s Problem can be solved for all points in certain unions of two polydisks in C2. One of the ideas in the methods of proof is integration along specific polygonal lines. Certain properties of some open sets defined by global plurisubharmonic func­tions in Cn are studied. More precisely, the sets Du = {z e Cn : u(z) &lt; 0} and Eh = {{z,w) e Cn X C : h(z,w) &lt; 1} are considered where ti is a plurisubharmonic function of minimal growth and h≠0 is a non-negative homogeneous plurisubharmonic function. (That is, the functions u and h belong to the classes L(Cn) and H+(Cn x C) respectively.) It is examined how the fact that Eh and the connected components of Du are H°°-domains of holomorphy is related to the structure of the set of disconti­nuity points of the global defining functions and to polynomial convexity. Moreover it is studied whether these notions are preserved under a certain bijective mapping from L(Cn) to H+(Cn x C). Two counterexamples are given which show that polynomial convexity is not preserved under this bijection. It is also proved, for example, that if Du is bounded and if the set of discontinuity points of u is pluripolar then Du is of type H°°. A survey paper on general properties of envelopes of holomorphy is included. In particular, the paper treats aspects of the theory for the bounded holomorphic functions. The results for the bounded holomorphic functions are compared with the corresponding ones for the holomorphic functions. / digitalisering@umu.se

Cette thèse porte sur la théorie du pluripotentiel et des problèmes d'équidistribution. Elle consiste en 4 chapitres. Le premier chapitre se consarce à l'étude de la régularité de la solution de l'équation de Monge-Ampère complexe sur une variété kahlérienne compacte X. Plus précisement, à l'aide des outils de la géométrie Cauchy-Riemann, on montre que la dernière équation possède une (unique) solution holdérienne pour une large classe géométrique de mesures de probabilités supportées par des sous-variétés réelles de X. Dans le chapitre 2, on étudie l'intersection des courants positifs fermés de grand bidegré. On y prouve que le produit extérieur de deux courants positifs fermés dont l'un possède un superpotentiel continu est positif fermé. Ceci généralise un résultat classique pour les courants de bidegré (1,1). Les deux chapitres suivants sont des applications de la théorie du pluripotentiel à des problèmes d'équidistribution. Dans le chapitre 3, on donne une vitesse explicite de convergence pour l'équidistribution des points de Fekete dans un compact K de l'espace euclidien à bord lisse par morceaux vers la mesure d'équilibre de K. Ici, les points de Fekete sont des bons points dans le problème d'interpolation d'une fonction continue sur K par des polynômes. Un tel contrôle de vitesse est crucial en pratique qu'on utilise les points de Fekete. La thèse se termine par le chapitre 4 où on prouve un analogue de la loi de Weyl pour les résonances d'un opérateur de Schodinger générique sur l'espace euclidien de dimension impair. Les résonances sont des objets centraux dans l'étude des opérateurs de Schrodinger. Elles jouent un rôle similaire à celui des valeurs propres dans le cadre compact. / This thesis concerns the pluripotential theory and equidistribution problems. It consists of 4 chapters. The first chapter is dedicated to the study of the regularity of the solution of the complexe Monge-Ampère equation on a compact Kahler manifold X. More precisely, using tools from the Cauchy-Riemann geometry, we prove that the last equation possesses a unique Holder continuous solution for a large geometric class of probability measures supported on real submanifolds of X. In the chapter 2, we study the intersecton of positive closed currents of higher bidegree. We prove there that the wedge product of two such currents one of which has a continuous superpotential est closed and positive. This property generalises a classical result for currents of bidegree (1,1). The next two chapters are applications of the pluripotential theory to equidistribution problems. In the chapter 3, we give an explicit speed of convergence for the equidistribution of Fekete's points in a compact subset K of the Euclidean space with piecewise smooth boundary toward the equilibrium measure of K. Here, the Fekete's points are good points for the interpolation problem of continuous functions by polynomials on K. A such control of speed is crucial in practice when ones use Fekete's points. The thesis is ended by the chapter 4 where we prove an analogue of Weyl's law for the resonances of a generic Schrodinger operator on an Euclidean space of odd dimension. The resonances are central objects in the research of Schrodinger operators. They play a similar role to that of eigenvalues in the compact setting.

<p>In the theory of holomorphic functions of one complex variable it is often useful to study subharmonic functions. The subharmonic can be described using the Laplace operator. When one studies holomorphic functions of several complex variables one should study the plurisubharmonic functions instead. Here the complex Monge--Ampère operator has a role similar to that of the Laplace operator in the theory of subharmonic functions. The complex Monge--Ampère operator is nonlinear and therefore it is not as well understood as the Laplace operator. We consider two types of boundary value problems for the complex Monge--Ampere equation in certain pseudoconvex domains. In this thesis the right-hand side in the Monge--Ampère equation will always be smooth, strictly positive and meet a monotonicity condition. The first type of boundary value problem we consider is a Dirichlet problem where we look for plurisubharmonic solutions which are zero on the boundary of the domain. We show that this problem has a unique smooth solution if the domain has a smooth bounded plurisubharmonic exhaustion function which is globally Lipschitz and has Monge--Ampère mass larger than one everywhere. We obtain some results on which domains have such a bounded exhaustion function. The second type of boundary value problem we consider is a boundary blow-up problem where we look for plurisubharmonic solutions which tend to infinity at the boundary of the domain. Here we also assume that the right-hand side in the Monge--Ampère equation satisfies a growth condition. We study this problem in strongly pseudoconvex domains with smooth boundary and show that it has solutions which are Hölder continuous with arbitrary Hölder exponent α, 0 ≤ α < 1. We also show a uniqueness result. A result on the growth of the solutions is also proved. This result is used to describe the boundary behavior of the Bergman kernel.</p>

In the theory of holomorphic functions of one complex variable it is often useful to study subharmonic functions. The subharmonic can be described using the Laplace operator. When one studies holomorphic functions of several complex variables one should study the plurisubharmonic functions instead. Here the complex Monge--Ampère operator has a role similar to that of the Laplace operator in the theory of subharmonic functions. The complex Monge--Ampère operator is nonlinear and therefore it is not as well understood as the Laplace operator. We consider two types of boundary value problems for the complex Monge--Ampere equation in certain pseudoconvex domains. In this thesis the right-hand side in the Monge--Ampère equation will always be smooth, strictly positive and meet a monotonicity condition. The first type of boundary value problem we consider is a Dirichlet problem where we look for plurisubharmonic solutions which are zero on the boundary of the domain. We show that this problem has a unique smooth solution if the domain has a smooth bounded plurisubharmonic exhaustion function which is globally Lipschitz and has Monge--Ampère mass larger than one everywhere. We obtain some results on which domains have such a bounded exhaustion function. The second type of boundary value problem we consider is a boundary blow-up problem where we look for plurisubharmonic solutions which tend to infinity at the boundary of the domain. Here we also assume that the right-hand side in the Monge--Ampère equation satisfies a growth condition. We study this problem in strongly pseudoconvex domains with smooth boundary and show that it has solutions which are Hölder continuous with arbitrary Hölder exponent α, 0 ≤ α &lt; 1. We also show a uniqueness result. A result on the growth of the solutions is also proved. This result is used to describe the boundary behavior of the Bergman kernel.

Cette thèse étudie les équidistributions de zéros de sections holomorphesaléatoires de fibrés en droites pour les mesures modérées. Elle consiste en deuxparties.Dans la première partie, nous construisons une famille étendue de mesuressingulières modérées sur des espaces projectifs. Ces mesures sont générées pardes fonctions quasi-plurisousharmoniques avec les potentiels höldériens.Le deuxième partie traite une propriété d' équidistribution dans un contextegénéral. Nous établissons un théorème d'équidistribution dans le cas dequelques fibrés en droites gros munis de métriques singulières. Une vitesse deconvergence précise pour l'équidistribution est obtenue. / This thesis investigates the equidistributions of zeros of random holomorphic sections of line bundles for moderate measures. It consists of two parts. In the first part, we construct a large family of singular moderate measures on projective spaces. These measures are generated by quasi-plurisubharmonic functions with Holder potentials.The second part deals with an equidistribution property in general settings. We establish an equidistribution theorem in the case of several big line bundles endowed with singular metrics. A precise convergence speed for the equidistribution is obtained.