Fischer Decomposition and Cauchy-Kovalevskaya extension in fractional Clifford analysis: the Riemann-Liouville case

Abstract

In this paper we present the basic tools of a fractional function theory in higher dimensions by means of a fractional correspondence to the Weyl relations via fractional Riemann-Liouville derivatives. A Fischer decomposition, Almansi decomposition, fractional Euler and Gamma operators, monogenic projection, and basic fractional homogeneous powers will be constructed. Moreover, we establish the fractional Cauchy-Kovalevskaya extension ($FCK$-extension) theorem for fractional monogenic functions defined on $\BR^d$. Based on this extension principle, fractional Fueter polynomials, forming a basis of the space of fractional spherical monogenics, i.e. fractional homogeneous polynomials, are introduced. We studied the connection between the $FCK$-extension of functions of the form $\x ~P_l$ and the classical Gegenbauer polynomials. Finally we present an example of an $FCK$-extension.