El Kadiri M., Fuglede B. Classical Fine Potential Theory

Springer Nature Singapore, 2025. — 433 p.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
Background in Potential Theory
Basics of Classical Potential Theory
Thin Sets and Fine Topology
Reduction and Sweeping of Functions
Sweeping of Measures The Base Operation
Quasi-Topology and Fine Topology
Fundamentals of Fine Potential Theory
Finely Hyperharmonic and Finely Harmonic Functions
Reduction and Sweeping of Functions Relative to a Finely Open Set
The Fine Dirichlet Problem
Finely Superharmonic Functions and Fine Potentials
Localization in Fine Potential Theory The Fine Green Kernel
Integral Representation of Fine Potentials
The Two-Dimensional Case N=2
Uniform Approximation by Harmonic or Subharmonic Functions
Invariant Functions
The Martin Boundary of a Fine Domain
Integral Representation in the Cone of Positive Finely Superharmonic …
Martin Compactification of a Fine Domain, the Martin …
The Fatou-Naïm-Doob Theorem for Finely Superharmonic Functions
Sweeping on Subsets of the Martin Space of a Fine Domain
Minimal Fine Topology
The Dirichlet Problem at the Martin Boundary of a Fine Domain
Further Developments
Polygonal Connectivity of Fine Domains
Finely Superharmonic Functions in Dirichlet Space
The Dirichlet Laplacian on Finely Open Sets
Finely Harmonic Morphisms
Finely Holomorphic Functions
Fine Complex Potential Theory
Background in Plurisubharmonic Functions Theory
Complex Differential Forms
Currents
Plurisubharmonic Functions
The Complex Monge-Ampère Operator
Monge-Ampère Capacity
The Pluri-Fine Topology on mathbbCn
Applications of the Pluri-Fine Topology to the Pluripotential Theory
Plurifinely Plurisubharmonic Functions
Plurifinely Holomorphic Functions
Biholomorphic Invariance
Local Approximation of calF-Plurisubharmonic Functions
The Monge-Ampère Operator for calF-Plurisubharmonic Functions
Maximal calF-Plurisubharmonic Functions
Maximal calF-Plurisubharmonic and the Monge-Ampère Operator
Appendix An Overview of Further Results in Fine Potential Theory
Appendix References
Symbol Index
Index