Spaces on ${\mathbb{R}^n}$: the general case

Abstract

By Theorem 2.9, the spaces $B_{pq}^s({\mathbb{R}^n})$ and $F_{pq}^s({\mathbb{R}^n})$ introduced in Definition 2.6 coincide with the well-established spaces usually denoted in this way. In particular, we have the following lifting property: Let $\sigma \in \mathbb{R}$ and

$$ {I_\sigma }:f \mapsto {\left( {id - \Delta } \right)^{\frac{\sigma }{2}}}f,\Delta = \sum\limits_{j = 1}^n {\frac{{{\partial ^2}}}{{\partial x_j^2}}} ; $$

(3.1)

then I_σ is not only an isomorphic map from S(ℝⁿ) onto itself, and from S’(ℝⁿ) onto itself, but also, in obvious notation,

$$ {I_\sigma }B_{pq}^s\left( {{\mathbb{R}^n}} \right) = B_{pq}^{s - \sigma }\left( {{\mathbb{R}^n}} \right)and {I_\sigma }F_{pq}^s\left( {{\mathbb{R}^n}} \right) = F_{pq}^{s - \sigma }\left( {{\mathbb{R}^n}} \right), $$

(3.2)

at least, so far as all the spaces involved fit in Definition 2.6. We refer to [Triß], 2.3.8. Of course, one could use (3.2) to introduce the spaces $B_{pq}^s({\mathbb{R}^n})$ and $F_{pq}^s({\mathbb{R}^n})$ also for those values of s which are not covered so far. But in order to be consistent we prefer a definition extending 2.6. For this purpose one has first to adapt Definition 2.4 to this more general situation.