In this thesis, we give a new definition of Hardy Spaces on hyperconvex domains in terms of Monge-Amp`ere measures which unifies the Hardy spaces on polydiscs and balls. Also we survey Monge-Amp`ere operators and Monge- Amp`ere measures.

Thesis (MSc (Mathematics))--University of Stellenbosch, 2010. / ENGLISH ABSTRACT: One of the early results that we encounter in Analysis is that every metric space admits a completion, that is a complete metric space in which it can be densely embedded. We present in this work a new construction which appears to be more general and yet has nice properties. These spaces subsequently called hyperconvex spaces allow one to extend nonexpansive mappings, that is mappings that do not increase distances, disregarding the properties of the spaces in which they are defined. In particular, theorems of Hahn-Banach type can be deduced for normed spaces and some subsidiary results such as fixed point theorems can be observed. Our main purpose is to look at the structures of this new type of “completion”. We will see in particular that the class of hyperconvex spaces is as large as that of complete metric spaces. / AFRIKAANSE OPSOMMING: Een van die eerste resultate wat in die Analise teegekom word is dat enige metriese ruimte ’n vervollediging het, oftewel dat daar ’n volledige metriese ruimte bestaan waarin die betrokke metriese ruimte dig bevat word. In hierdie werkstuk beskryf ons sogenaamde hiperkonvekse ruimtes. Dit gee ’n konstruksie wat blyk om meer algemeen te wees, maar steeds gunstige eienskappe het. Hiermee kan nie-uitbreidende, oftewel afbeeldings wat nie afstande rek nie, uitgebrei word sodanig dat die eienskappe van die ruimte waarop dit gedefinieer is nie ’n rol speel nie. In die besonder kan stellings van die Hahn- Banach-tipe afgelei word vir genormeerde ruimtes en sekere addisionele ressultate ondere vastepuntstellings kan bewys word. Ons hoofdoel is om hiperkonvekse ruimtes te ondersoek. In die besonder toon ons aan dat die klas van alle hiperkonvekse ruimtes net so groot soos die klas van alle metriese ruimtes is.

<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 est consacrée à l'étude de la régularité des solutions des équations de Monge-Ampère complexes ainsi que des équations hessiennes complexes dans un domaine borné de Cn. Dans le premier chapitre, on donne des rappels sur la théorie du pluripotentiel. Dans le deuxième chapitre, on étudie le module de continuité des solutions du problème de Dirichlet pour les équations de Monge-Ampère lorsque le second membre est une mesure à densité continue par rapport à la mesure de Lebesgue dans un domaine strictement hyperconvexe lipschitzien. Dans le troisième chapitre, on prouve la continuité hölderienne des solutions de ce problème pour certaines mesures générales. Dans le quatrième chapitre, on considère le problème de Dirichlet pour les équations hessiennes complexes plus générales où le second membre dépend de la fonction inconnue. On donne une estimation précise du module de continuité de la solution lorsque la densité est continue. De plus, si la densité est dans Lp , on démontre que la solution est Hölder-continue jusqu'au bord. / In this thesis we study the regularity of solutions to the Dirichlet problem for complex Monge-Ampère equations and also for complex Hessian equations in a bounded domain of Cn. In the first chapter, we give basic facts in pluripotential theory. In the second chapter, we study the modulus of continuity of solutions to the Dirichlet problem for complex Monge-Ampère equations when the right hand side is a measure with continuous density with respect to the Lebesgue measure in a bounded strongly hyperconvex Lipschitz domain. In the third chapter, we prove the Hölder continuity of solutions to this problem for some general measures. In the fourth chapter, we consider the Dirichlet problem for complex Hessian equations when the right hand side depends on the unknown function. We give a sharp estimate of the modulus of continuity of the solution as the density is continuous. Moreover, for the case of Lp-density we demonstrate that the solution is Hölder continuous up to the boundary.