Classical Fine Potential Theory

Description

This comprehensive book explores the intricate realm of fine potential theory. Delving into the real theory, it navigates through harmonic and subharmonic functions, addressing the famed Dirichlet problem within finely open sets of R^n. These sets are defined relative to the coarsest topology on R^n, ensuring the continuity of all subharmonic functions. This theory underwent extensive scrutiny starting from the 1970s, particularly by Fuglede, within the classical or axiomatic framework of harmonic functions. The use of methods from fine potential theory has led to solutions of important classical problems and has allowed the discovery of elegant results for extension of classical holomorphic function to wider classes of “domains”. Moreover, this book extends its reach to the notion of plurisubharmonic and holomorphic functions within plurifinely open sets of C^n and its applications to pluripotential theory. These open sets are defined by coarsest topology that renders all plurisubharmonic functions continuous on C^n. The presentation is meticulously crafted to be largely self-contained, ensuring accessibility for readers at various levels of familiarity with the subject matter. Whether delving into the fundamentals or seeking advanced insights, this book is an indispensable reference for anyone intrigued by potential theory and its myriad applications. Organized into five chapters, the first four unravel the intricacies of fine potential theory, while the fifth chapter delves into plurifine pluripotential theory. Preface The Life and Work of Bent Fuglede Notations and Terminology Contents 1 Background in Potential Theory 1.1 Basics of Classical Potential Theory 1.2 Thin Sets and Fine Topology 1.3 Reduction and Sweeping of Functions 1.4 Sweeping of Measures. The Base Operation 1.5 Quasi-Topology and Fine Topology 2 Fundamentals of Fine Potential Theory 2.1 Finely Hyperharmonic and Finely Harmonic Functions 2.2 Reduction and Sweeping of Functions Relative to a Finely Open Set 2.3 The Fine Dirichlet Problem 2.4 Finely Superharmonic Functions and Fine Potentials 2.5 Localization in Fine Potential Theory. The Fine Green Kernel 2.6 Integral Representation of Fine Potentials 2.7 The Two-Dimensional Case N=2 2.8 Uniform Approximation by Harmonic or Subharmonic Functions 2.9 Invariant Functions 3 The Martin Boundary of a Fine Domain 3.1 Integral Representation in the Cone of Positive Finely Superharmonic … 3.2 Martin Compactification of a Fine Domain, the Martin … 3.3 The Fatou-Naïm-Doob Theorem for Finely Superharmonic Functions 3.4 Sweeping on Subsets of the Martin Space of a Fine Domain 3.5 Minimal Fine Topology 3.6 The Dirichlet Problem at the Martin Boundary of a Fine Domain 4 Further Developments 4.1 Polygonal Connectivity of Fine Domains 4.2 Finely Superharmonic Functions in Dirichlet Space 4.3 The Dirichlet Laplacian on Finely Open Sets 4.4 Finely Harmonic Morphisms 4.5 Finely Holomorphic Functions 5 Fine Complex Potential Theory 5.1 Background in Plurisubharmonic Functions Theory 5.1.1 Complex Differential Forms 5.1.2 Currents 5.1.3 Plurisubharmonic Functions 5.1.4 The Complex Monge-Ampère Operator 5.1.5 Monge-Ampère Capacity 5.2 The Pluri-Fine Topology on mathbbCn 5.3 Applications of the Pluri-Fine Topology to the Pluripotential Theory 5.4 Plurifinely Plurisubharmonic Functions 5.5 Plurifinely Holomorphic Functions 5.6 Biholomorphic Invariance 5.7 Local Approximation of calF-Plurisubharmonic Functions 5.8 The Monge-Ampère Operator for calF-Plurisubharmonic Functions 5.9 Maximal calF-Plurisubharmonic Functions 5.10 Maximal calF-Plurisubharmonic and the Monge-Ampère Operator Appendix An Overview of Further Results in Fine Potential Theory Appendix References -25pt Symbol Index Index