Spaces of Sobolev type with positive smoothness on \(\mathbb R^n\), embeddings and growth envelopes
The author investigates three partly different approaches to Triebel-Lizorkin spaces on \(\mathbb R\). The first approach is the Fourier-analytic one, the second is by differences, and the third by subatomic decompositions. As it has been known before, these approaches coincide for a wide range of the parameters \(s,p\) and \(q\). However, if either \(0< p<1\) or \(0 < q < 1\) and \(s>0\) is small, then they do not. But also in those situations they show the same embedding behaviour into Lorentz spaces, so they differ not much. To a certain extent, this is a surprise. The author also discusses Jawerth-Franke type embeddings and growth envelopes for all three scales.
scientific article; zbMATH DE number 5704729