Introduction, heuristics, and preliminaries

Abstract

The function spaces B s pq and F s pq on ℝn and on domains with respect to the full range of the parameters $$ s \in \mathbb{R}{\text{, }}0 < p \leqslant \infty , 0 < q \leqslant \infty , $$ (1.1) were introduced between 1959 and 1975. They cover many well-known classical concrete function spaces having their own history. In 1.2 we give a corresponding short list. These two scales of spaces and their special cases attracted a lot of attention and have been treated systematically, with numerous applications given. We mention in particular the following books, reflecting also the development of this theory: [Sob50], [Nik77] (first edition 1969), [Ste70], [BIN75], [Pee76], [Triα] (1978), [Triβ] (1983) and [Triγ] (1992). Special aspects but related to our intentions have been studied in [Maz85] (Sobolev spaces), [Zie89] (Sobolev spaces), and [ST87] (periodic spaces, anisotropic spaces and spaces with dominating mixed derivatives). The two surveys [BKLN88] and [KuN88] cover in particular the Russian literature. More recent developments of the spaces B s pq and F s pq may be found in [ET96], [RuS96], [AdH96], and [Triδ] (1997). We refer the interested reader to Chapter 1 in [Triγ]: in this survey we tried to describe the historical roots and how the diverse ideas developed over the years (up to 1990). There one finds also many references to the original papers.KeywordsSobolev SpaceFunction SpaceHolomorphic FunctionAtomic DecompositionHistorical RootThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.