The book deals with the two scales Bsp,q and Fsp,q of spaces of distributions, where -?<s<? and 0<p,q??, which include many classical and modern spaces, such as H”lder 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 Rn 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 Rn. 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.
From the reviews:
"There is good reason to think of this book as Volume 1 in a series of (by now) celebrated books on the theory of function spaces treated in a systematic way. It turned out that for many beginners in this part of harmonic analysis this book is still the natural choice to start with. ... this book can still be recommended for people who start their work in function spaces of rather general type and who have some working knowledge in functional analysis ... ." (Dorothee Haroske, Zentralblatt MATH, Vol. 1235, 2012)
I. Function Spaces and Elliptic Differential Equations.- 1. Spaces of Entire Analytic Functions.- 2. Function Spaces on Rn .- 3. Function Spaces on Domains.- 4. Regular Elliptic Differential Equations.- II. Further Types of Functions Spaces.- 5. Homogeneous Function Spaces.- 6. Ultra-Distributions and Weighted Spaces of Entire Analytic Functions.- 7. Weighted Function Spaces on Rn.- 8. Weighted Function Spaces on Domains and Degenerate Elliptic Differential Equations.- 9. Periodic Function Spaces.- 10. Further Types of Function Spaces.- References.- Index.