The derivatives and integrals of fractional order on \(a\)-adic groups (Q1310525)

\textit{W.-X. Zheng} [Rocky Mt. J. Math. 15, 801-815 (1985; Zbl 0608.41023)] introduced the concept of a derivative for functions defined on a local field and developed parts of the theory for such derivatives. In the paper under review Zheng's theory is extended by considering functions on an \(a\)-adic group \(G_ a\), where \(a = \{a_ n: n\in \mathbb{Z}\}\) is a sequence of natural numbers \(\geq 2\). The author defines derivatives and integrals of fractional order both in the pointwise sense and in a norm sense for functions in \(X\), where \(X = C(G_ a)\) or \(L^ p(G_ a)\), \(1 \leq p\leq \infty\), and proves a number of standard results for such derivatives and integrals. As an application it is shown that, in case \(\text{sup}\{a_ j: j \in \mathbb{Z}\} < \infty\), a function \(f\in X\) is strongly differentiable of order \(\alpha > 0\) if and only if \(f\) can be expressed as the Bessel potential of order \(\alpha\) of a function \(g \in X\).