p-adic integral transforms on compact subgroups of \({\mathbb{C}}_ p\)
Let \(X^ m:=X_ d\times {\mathbb{Z}}_ p^{m-1}\) where \((d,p)=1\), \(X_ d:=\lim_{\overset \leftarrow N}{\mathbb{Z}}/dp^ N{\mathbb{Z}}\). Three types of p-adic integral transforms are considered. (1) For a \({\mathbb{C}}_ p\)-valued bounded measure \(\mu\) on \(X^ m\) and a representation \(\rho\) of \(X^ m\) in \({\mathbb{C}}_ p^{\times}\) the inverse Mellin transform \[ f_{\mu}(\rho)=\int_{X^ m}\rho (x) d\mu (x), \] (2) (Amice-Vélu) For any continuous function \(f: X^ m\to {\mathbb{C}}_ p\) the transform \[ g(z)=\int_{X^ m}f(x) d\mu_ z(x) \] (in the open unit polydisc, \(g(z)=\sum f(n) z^ n).\) (3) For \(\mu\), \(\rho\) as in (1) the Stieltjes transform \[ \psi_{\rho,\mu}(z)=\int_{X^ m}\frac{d\mu (x)}{\prod^{m}_{j=1}(1-z_ j \rho_ j(x))}. \] It is shown (Theorem 4) that these transforms are related by \[ \int_{X^ m}f_{\mu}(\rho^ y) d\mu_ z(y)=\psi_{\rho,\mu}(z)\quad (where\quad \rho^ y(x):=\rho (xy)). \] Examples and applications are given. In particular, a connection between (3) and the transform \(\phi (z)=\int_{\sigma}(z-x)^{-1} d\mu (x)\) (where \(\sigma\) is a compact subset of \({\mathbb{C}}_ p\) and \(z\in {\mathbb{C}}_ p\setminus \sigma )\), considered by Barsky and Vishik, is established. New results on \(\phi\) are obtained for the case where \(\sigma\) is a compact subgroup of \({\mathbb{C}}_ p^{\times}\) by applying Theorem 4.
scientific article; zbMATH DE number 3929124