The Spaces $B^s _{pq}$ and $F^s _{pq}$ :Definitions and Characterizations

Abstract

It was the aim of the first chapter to look at function spaces from a historical point of view and to convince the reader that the two scales $B^s _{pq}$ and $F^s _{pq}$ occupy the very heart of the theory of function spaces. In the present chapter and the following one we develop systematically the technical part of the theory of the spaces $B^s _{pq}$ and $F^s _{pq}$ on $\mathbb{R}^n$ as it stands at the end of the eighties. The remaining chapters rest on these fundamentals. Our recent approach, compared with earlier ones, for example in [Triß], not to speak about [Triα], can be described as follows. We give rather general and, unfortunately, highly technical characterizations of the spaces $F^s _{pq}$ and $B^s _{pq}$ which more or less cover desirable concrete characterizations, for example, via differences or derivatives of functions, or via harmonic or thermic extensions, and which also provide the basis for later applications. This will be done in 2.4 for $F^s _{pq}$ and in 2.5 for $B^s _{pq}$, always with a preference of the more complicated spaces $F^s _{pq}$. In this sense, 2.4 and 2.5 may be considered as the heart of the present chapter. Characterizations of some spaces $F^s _{pq}$ and $B^s _{pq}$ in terms of differences of functions, or as harmonic or thermic extensions are known, for example, we derived them in [Triß] by rather specific and sometimes quite tricky arguments. Now we return in 2.6 to this subject, but this time as consequences of the characterizations in 2.4 and 2.5.