Tempered Homogeneous Function Spaces

If one tries to transfer assertions for the inhomogeneous spaces $A_{p,q}^s({\mathbb{R}}^n)$, $A\in \{B,F\}$, appropriately to their homogeneous counterparts $\stackrel{\ast }{A}{}_{p,q}^s({\mathbb{R}}^n)$ within the framework of the dual pairing $(S({\mathbb{R}}^n),S^{\prime }({\mathbb{R}}^n))$ then it is hard to make a mistake as long as the parameters $p,q,s$ are restricted by $0<p,q\le \infty$ and, in particular, $n(\frac{1}{p}–1)<s<\frac{n}{p}$. It is the main aim of these notes to say what this means.

This book is addressed to graduate students and mathematicians having a working knowledge of basic elements of the theory of function spaces, especially of type $B_{p,q}^s$ and $F_{p,q}^s$.