Analysis in Banach Spaces: Volume III: Harmonic Analysis and Spectral Theory

This third volume of Analysis in Banach Spaces provides a systematic treatment of Banach space-valued singular integrals, Fourier transforms, and function spaces, further developing and ramifying the theory of functional calculus and describing applications to maximal regularity of evolution equations. It is an invaluable reference for researchers interested in functional analysis, harmonic analysis, and the operator-theoretic approach.

Format: Hardback
Length: 826 pages
Publication date: 08 December 2023
Publisher: Springer International Publishing AG

This third volume of Analysis in Banach Spaces presents a comprehensive and systematic exploration of Banach space-valued singular integrals, Fourier transforms, and function spaces. Building upon the foundations established in Volume II, it delves deeper into the theory of functional calculus, extending its applications to address the problem of maximal regularity in evolution equations. The exposition offers a unified treatment of a vast array of results, many of which were previously accessible only in the form of research papers. By employing modern techniques suitable for vector-valued analysis, the book presents some classical topics in a fresh and innovative manner. Thanks to its accessible writing style, accompanied by comprehensive and detailed proofs, this book serves as an invaluable resource for researchers engaged in functional analysis, harmonic analysis, and the operator-theoretic approach to deterministic and stochastic evolution equations.

Introduction:
This third volume of Analysis in Banach Spaces continues the series' mission to provide a comprehensive and systematic treatment of various topics in functional analysis and its applications. In this installment, we focus on the study of Banach space-valued singular integrals, Fourier transforms, and function spaces. These subjects play pivotal roles in modern mathematics and have numerous applications in fields such as physics, engineering, and mathematics.

Chapter 1:
In Chapter 1, we introduce the fundamental concepts and definitions related to Banach space-valued singular integrals. We begin by discussing the definition of a Banach space and its basic properties, such as the norm and the dual space. We then explore the concept of a singular integral, which is a generalization of the classical integral to Banach spaces. We introduce the notion of a measure and discuss its role in the theory of singular integrals.

Chapter 2:
In Chapter 2, we delve into the theory of Fourier transforms. We begin by discussing the definition of Fourier transforms and their properties, such as the Parseval theorem, the inversion formula, and the Fourier transform of a function. We then explore the Fourier transform of functions on a compact metric space and its applications to the study of partial differential equations.

Chapter 3:
In Chapter 3, we introduce the concept of function spaces and their applications to the study of functional analysis. We begin by discussing the definition of a function space and its basic properties, such as the norm and the dual space. We then explore the theory of linear operators and their properties, such as the norm, the adjoint, and the spectrum. We also discuss the concept of Banach algebras and their applications to the study of functional equations.

Chapter 4:
In Chapter 4, we explore the theory of functional calculus. We begin by discussing the definition of functional calculus and its basic properties, such as the evaluation and differentiation of functions. We then introduce the concept of a functional equation and discuss its properties and solutions. We also explore the theory of partial differential equations and their solutions using functional calculus techniques.

Chapter 5:
In Chapter 5, we discuss the applications of functional calculus to the study of maximal regularity of evolution equations. We begin by discussing the definition of maximal regularity and its properties. We then introduce the concept of a maximal regularity problem and discuss its mathematical formulation. We explore the theory of evolution equations and their solutions using functional calculus techniques.

Chapter 6:
In Chapter 6, we summarize the main results and conclusions of the book. We summarize the main ideas discussed in each chapter and highlight the key contributions made to the field of functional analysis. We also discuss the open problems and future research directions in the area.

Conclusion:
In conclusion, this third volume of Analysis in Banach Spaces offers a comprehensive and systematic treatment of Banach space-valued singular integrals, Fourier transforms, and function spaces. It further develops and ramifies the theory of functional calculus from Volume II and describes applications of these new notions and tools to the problem of maximal regularity of evolution equations. The exposition provides a unified treatment of a large body of results, much of which has previously only been available in the form of research papers. Some of the more classical topics are presented in a novel way.

Weight: 1430g
Dimension: 235 x 155 (mm)
ISBN-13: 9783031465970
Edition number: 1st ed. 2023