Fractional powers of higher order vector 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 $m\ge 1$, 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.

arXiv admin note: text overlap with arXiv:2010.04688