Theory of Function Spaces

The book deals with the two scales B^s_p,q and F^s_p,q of spaces of distributions, where ‑∞<s<∞ and 0<p,q≤∞, which include many classical and modern spaces, such as Hölder spaces, Zygmund classes, Sobolev spaces, Besov spaces, Bessel-potential spaces, Hardy spaces and spaces of BMO-type. It is the main aim of this book to give a unified treatment of the corresponding spaces on the Euclidean n-space R_n in the framework of Fourier analysis, which is based on the technique of maximal functions, Fourier multipliers and interpolation assertions. These topics are treated in Chapter 2, which is the heart of the book. Chapter 3 deals with corresponding spaces on smooth bounded domains in R_n. These results are applied in Chapter 4 in order to study general boundary value problems for regular elliptic differential operators in the above spaces. Shorter Chapters (1 and 5-10) are concerned with: Entire analytic functions, ultra-distributions, weighted spaces, periodic spaces, degenerate elliptic differential equations.

The book deals with the two scales B^s_p,q and F^s_p,q of spaces of distributions, where ‑∞<s<∞ and 0<p,q≤∞, which include many classical and modern spaces, such as Hölder spaces, Zygmund classes, Sobolev spaces, Besov spaces, Bessel-potential spaces, Hardy spaces and spaces of BMO-type. It is the main aim of this book to give a unified treatment of the corresponding spaces on the Euclidean n-space R_n in the framework of Fourier analysis, which is based on the technique of maximal functions, Fourier multipliers and interpolation assertions. These topics are treated in Chapter 2, which is the heart of the book. Chapter 3 deals with corresponding spaces on smooth bounded domains in R_n. These results are applied in Chapter 4 in order to study general boundary value problems for regular elliptic differential operators in the above spaces. Shorter Chapters (1 and 5-10) are concerned with: Entire analytic functions, ultra-distributions, weighted spaces, periodic spaces, degenerate elliptic differential equations.

------ Reviews

It is written in a concise but well readable style. (…) This book can be best recommended to researchers and advanced students working on functional analysis or functional analytic methods for partial differential operators or equations.

- ZentralblattMATH

The noteworthy new items in the book are: the use of maximal functions, treatment of BMO spaces, treatment of Beurling ultradistributions as well as addition of new results, too numerous to mention, obtained within the last 7 years or so.

- Mathematical Reviews