Two variants of fractional powers of Hankel–Clifford transformations and pseudo-differential operators

Abstract

Two variants of fractional powers of Hankel–Clifford transformations of order \(\nu \ge 0\) with parameter \(\alpha \) are introduced and studied their properties. The operational formulas are developed. We study pseudo-differential operators associated with the symbol \(a(x,y)\) and their integral representations. Some properties of Sobolev type spaces are studied. Fractional powers of Hankel–Clifford transformations are used in the solution of partial differential equation involving Kepinski type operators \(\Delta _{1,\nu ,\mu ,\alpha }^{*}\) and \(\Delta _{2,\nu ,\mu ,\alpha }^{*}\).