Besov Spaces and Lizorkin-Triebel Spaces

Abstract

In this chapter we shall apply the general theory developed in Chapter 2 to the function spaces known as Besov, and Lizorkin-Triebel spaces. In Section 4.1 we define the Besov spaces B^p,q _α , and present their theory in a way that suits our purposes. In Section 4.2 the Lizorkin—Triebel spaces F^p,q _α are defined, and then most of the section is devoted to a proof of the fact that this scale of spaces contains the spaces L^{α p}, in fact F^p,2 _α =L^α,p for 1 < p < ∞. This result will be proved by means of a multiplier theorem of S. G. Mikhlin, whose proof is also included. In Section 4.3 we continue the presentation of these spaces in a way parallel to Section 4.1. This involves proving a rather deep theorem of J. Peetre. Now the stage is set for our application of the general nonlinear potential theory, which takes its beginning in Section 4.4. The short Section 4.5 is devoted to an important inequality of Th. H. Wolff. In Section 4.6, which is independent of most of the preceding theory in this chapter, we give a representation of the Besov, and Lizorkin—Triebel spaces by means of “smooth atoms”. In Section 4.7 we apply this representation to formulate an “atomic” nonlinear potential theory, which among other things gives a new way of viewing the Wolff inequality. Finally, in Section 4.8 we use the atomic representation to give a characterization of L^{α, p} by means of a local approximation property. This result implies Strichartz’ theorem, Theorem 3.5.6, whose proof was previously postponed.