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

Abstract

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}$$\\begin{aligned} (\\mathcal {E}_{\\varvec{\\phi },\\alpha }f)(x)=-\\mathcal {F}^{-1}_{\\zeta \\rightarrow x}\\left( \\left[ \\max \\{1,|\\varvec{\\phi }(||\\zeta ||_{p})|\\} \\right] ^{-\\alpha }\\widehat{f}(\\zeta )\\right) , \ ext { } x\\in {\\mathbb {Q}}_{p}^{n}, \\ \\ \\alpha \\in \\mathbb {R}, \\end{aligned}$$\\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.