Wen Yuan

The main aim of this book is to provide an advanced textbook on functional analysis, focusing on the functional calculus of operators. Based on basic knowledge of functional analysis on metric spaces and linear bounded operators, this book systematically introduces the spectral theory of operators and the theory of operator semigroups, including Banach algebras, spectral theories of both bounded and unbounded linear operators, as well as semigroups of operators. They play important roles in various areas of modern mathematics such as harmonic analysis, partial differential equations, probability and statistics, engineering science, quantum physics, and statistical mechanics. This book provides rich examples and exercises, emphasizing the combination of the theory and examples to enhance readers' ability to apply functional analysis theory to solve related problems. Compared with many other functional analysis textbooks, this book focuses on the spectral theory and the operator semigroup theory, serving as an appropriate connection for functional analysis between undergraduate and graduate levels, making it an ideal textbook for both. It will also be a good reference book for researchers who are interested in modern analysis, such as functional analysis, harmonic analysis, and partial differential equations.

This monograph develops the Gaussian functional capacity theory with applications to restricting the Gaussian Campanato/Sobolev/BV space. Included in the text is a new geometric characterization of the Gaussian 1-capacity and the Gaussian Poincaré 1-inequality. Applications to function spaces and geometric measures are also presented. This book will be of use to researchers who specialize in potential theory, elliptic differential equations, functional analysis, probability, and geometric measure theory.

During the last 60 years the theory of function spaces has been a subject of growing interest and increasing diversity. Based on three formally different developments, namely, the theory of Besov and Triebel-Lizorkin spaces, the theory of Morrey and Campanato spaces and the theory of Q spaces, the authors develop a unified framework for all of these spaces. As a byproduct, the authors provide a completion of the theory of Triebel-Lizorkin spaces when p = 8.