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.
