Elliptic pseudodifferential equations and Sobolev spaces over \(p\)-adic fields (Q975812)

The authors consider pseudodifferential operators on spaces of complex-valued functions on \(\mathbb Q_p^n\), where \(\mathbb Q_p\) is the field of \(p\)-adic numbers. The symbols have the form \(|f(\xi_1,\ldots ,\xi_n)|_p^\alpha\), where \(\alpha >0\) and \(f\) is a homogeneous polynomial. Such an operator is called \textit{elliptic} if \(f\) vanishes only at the origin. For elliptic operators, the authors find fundamental solutions and prove that the operators act as isomorphisms on a scale of Sobolev type spaces. Note that a more general class of \(p\)-adic elliptic pseudodifferential operators was introduced and studied by \textit{S.\,Haran} [Ann.\ Inst.\ Fourier 43, No.\,4, 997--1053 (1993; Zbl 0974.22009)]. Nevertheless, the theory developed in the paper under review is more explicit and may be useful in future research.