p-adic Bessel α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-potentials and some of their applications

In this article, we will study a class of pseudo-differential operators on p-adic numbers, which we will call p-adic Bessel α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-potentials. These operators are denoted and defined in the form (Eϕ,αf)(x)=-Fζ→x-1max{1,|ϕ(||ζ||p)|}-αf^(ζ),x∈Qpn,α∈R,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} (\mathcal {E}_{\varvec{\phi },\alpha }f)(x)=-\mathcal {F}^{-1}_{\zeta \rightarrow x}\left( \left[ \max \{1,|\varvec{\phi }(||\zeta ||_{p})|\} \right] ^{-\alpha }\widehat{f}(\zeta )\right) , \text { } x\in {\mathbb {Q}}_{p}^{n}, \ \ \alpha \in \mathbb {R}, \end{aligned}$$\end{document}where f is a p-adic distribution and max{1,|ϕ(||ζ||p)|}-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ \max \{1,|\varvec{\phi }(||\zeta ||_{p})|\}\right] ^{-\alpha }$$\end{document} is the symbol of the operator. We will study some properties of the convolution kernel (denoted as Kα\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K_{\alpha }$$\end{document}) of the pseudo-differential operator Eϕ,α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {E}_{\varvec{\phi },\alpha }$$\end{document}, α∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in \mathbb {R}$$\end{document}; and demonstrate that the family (Kα)α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(K_{\alpha })_{\alpha >0}$$\end{document} determines a convolution semigroup on Qpn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {Q}_{p}^{n}$$\end{document}. Furthermore, we will introduce new types of Feller semigroups, and explore new Markov processes and non-homogeneous initial value problems on p-adic numbers.