How To Measure Smoothness

Abstract

This book deals with spaces of functions and distributions of Holder, Sobolev, Hardy, Campanato, Besov type and their descendants. So it is quite clear that we restrict our attention also in this introductory chapter to spaces of these types. There is hardly any doubt that these types of spaces were and are the backbone (in the widest sense of the word) of the theory of function spaces from the very beginning up to our time. On the other hand, in the last few decades many other types of function spaces have been treated extensively, some of them closely connected with the spaces considered here (anisotropic and weighted spaces, spaces with dominating mixed derivatives, spaces on non-smooth domains and on general structures), while others have been based on different principles (Lorentz and Orlicz spaces, etc.). The following books cover a large variety of different aspects of the theory of function spaces in the widest sense: [Sob4, Nik2, BIN, KJF, Pee6, Ste1, StW2, Ada, Maz, MaS, JoW, Triα, Tris, ScT]. Finally, we refer to the recent surveys [BKLN, KuN1] (1988) concentrating mainly, but not exclusively, on developments in the Soviet Union.