On Hardy-Bessel potential spaces over the ring of integers in a local field (Q1329212)

The Hardy-Bessel potential space \(F^ \alpha_ p\) is defined as the class of distributions \(f\) such that \(g_ \alpha \in L^ p\) with the Littlewood-Paley type function \(g_ \alpha\). Then the author gives two equivalent quasi-norms on \(F^ \alpha_ p\) by the Bessel potential \(J^ \alpha\) and the area function \(S^ \alpha_ r\). The topic of this note is very interesting, not only because the technique used for a local field is quite different from that for \(\mathbb{R}^ n\), but also because the topic on a local field has importance for some applications.