Developments in Mathematics

This book is a self-contained textbook of the theory of Besov spaces and Triebel-Lizorkin spaces. All that is required of readers is that they be familiar with integration theory.Illustrations are included to show how these spaces are defined, which is very complicated. Owing to that complexity, many definitions are required in order to describe it. At the outset, the necessary terminology is provided, and the theory of distributions, L^p spaces, the Hardy-Littlewood maximal operator, and the singular integral operators are called upon. One of the highlights is that the proof of the Sobolev embedding theorem is extremely simple. There are two types for each function space: a homogeneous one and an inhomogeneous one. The theory of function spaces, which readers usually learn in a standard course, can be readily applied to the inhomogeneous one. However, that theory is not sufficient for a homogeneous space; it needs to be reinforced with some knowledge of the theory of distributions. This topic, however subtle, is also covered in the book. Additionally, related function spaces--namely, Hardy spaces, bounded mean oscillation spaces, and Holder continuous spaces--are defined and discussed, and it is shown that they are special cases of Besov spaces and Triebel-Lizorkin spaces.Besov spaces and Triebel-Lizorkin spaces enjoy helpful structural features in that atomic and related decompositions enable the investigation of the function spaces in a systematic manner. One of the benefits of atomic decomposition is that it can be considered the pointwise product of two functions. Because Besov spaces and Triebel-Lizorkin spaces are made up of distributions, this problem is a very subtle one. However, atomic decomposition paves a smooth path to overcoming this difficultly.Finally, the book is oriented toward applications to partial differential equations and problems of harmonic analysis. These include a priori estimates of elliptic differential equations, the T1 theorem, pseudo-differential operators, the generator of semi-group and spaces on domains, and the Kato problem. As well, various function spaces are introduced to overcome the shortcomings of Besov spaces and Triebel-Lizorkin spaces.