Fractional powers of higher order vector operators on bounded and unbounded domains

TL;DR

This paper develops a method to generate fractional powers of certain quaternionic differential operators of order m on bounded and unbounded domains using the $H^$-functional calculus, even when components do not commute.

Contribution

It introduces conditions for generating fractional powers of higher order quaternionic differential operators on various domains, extending previous methods to non-commuting components.

Findings

Established sufficient conditions for fractional powers of quaternionic operators.

Extended the $H^$-functional calculus to higher order quaternionic operators.

Applied the theory to operators on bounded and unbounded domains.

Abstract

Using the $H^\infty$-functional calculus for quaternionic operators, we show how to generate the fractional powers of some densely defined differential quaternionic operators of order $m\geq 1$, acting on the right linear quaternionic Hilbert space $L^2(\Omega,\mathbb C\otimes\mathbb H)$. The operators that we consider are of the type $$ T=i^{m-1}\left(a_1(x) e_1\partial_{x_1}^{m}+a_2(x) e_2\partial_{x_2}^{m}+a_3(x) e_3\partial_{x_3}^{m}\right), \ \ \ x=(x_1,\, x_2,\, x_3)\in \overline{\Omega}, $$ where $\overline{\Omega}$ is the closure of either a bounded domain $\Omega$ with $C^1$ boundary, or an unbounded domain $\Omega$ in $\mathbb R^3$ with a sufficiently regular boundary which satisfy the so called property $(R)$, $\{e_1,\, e_2,\, e_3\}$ is an orthonormal basis for the imaginary units of $\mathbb H$, $a_1,\,a_2,\, a_3: \overline{\Omega} \subset\mathbb{R}^3\to \mathbb{R}$ are the coefficients of $T$. In particular it will be given sufficient conditions on the coefficients of $T$ in order to generate the fractional powers of $T$, denoted by $P_{\alpha}(T)$ for $\alpha\in(0,1)$, when the components of $T$, i.e. the operators $T_l:=a_l\partial_{x_l}^m$, do not commute among themselves.