Almost Diagonal Operators and Atomic and Molecular Decompositions

Abstract

In the first part of this chapter we shall show that under certain restrictions on the parameters our spaces $A_{p, q}^{s, \tau }({\mathbb{R}}^ n)$ allow characterizations by smooth molecules and smooth atoms. This opens the door to the discretization of our distribution spaces $A_{p, q}^{s, \tau }({\mathbb{R}}^n)$. Afterwards, based on these discretizations, we are able to compare the classes $A_{p, q}^{s, \tau }({\mathbb{R}}^n)$ with the Besov-Morrey spaces $\mathcal{N}^s_{pqu} (\mathbb{R}^n)$ (see item (xxv) in Sect. 1.3) and the Triebel-Lizorkin-Morrey spaces $\mathcal{E}^s_{pqu} (\mathbb{R}^n)$ (see item (xxv) in Sect. 1.3).