1.1. The $S$-functional calculus

An element in the quaternions $\mathbb {H}$ is of the form $s=s_0+s_1e_1+s_2e_2+s_3e_3$, where $s_0$, $s_\ell$ are real numbers ($\ell =1,2,3$), ${\rm Re}(s):=s_0$ denotes the real part of $s$ and $e_\ell$, for $\ell =1,2,3$, are the imaginary units which satisfy the relations: $e_1^{2}=e_2^{2}=e_3^{2}=e_1e_2e_3=-1$. The modulus of $s$ is defined as $|s|=(s_0^{2}+s_1^{2}+s_2^{2}+s_3^{2})^{1/2}$ and the conjugate is given by $\overline {s}=s_0-s_1e_1-s_2e_2-s_3e_3$. In the sequel, we will denote by $\mathbb {S}$ the unit sphere of purely imaginary quaternions, an element $j$ in $\mathbb {S}$ is such that $j^{2}=-1$. We consider a two-sided quaternionic Banach space $V$ and we denote the set of closed densely defined quaternionic right linear operators on $V$ by $\mathcal {K}(V)$. The Banach space of all bounded right linear operators on $V$ is indicated by the symbol $\mathcal {B}(V)$ and is endowed with the natural operator norm. For $T\in \mathcal {K}(V)$, we define the operator associated with the $S$-spectrum as:

(1.1)\begin{equation} \mathcal{Q}_{s}(T) := T^{2} - 2\mathrm{Re}(s)T + |s|^{2}\mathcal{I}, \quad \text{for }s\in\mathbb{H} \end{equation}

where $\mathcal {Q}_{s}(T):\mathcal {D}(T^{2})\to V$, where $\mathcal {D}(T^{2})$ is the domain of $T^{2}$. We define the $S$-resolvent set of $T$ as

\begin{align*} \rho_S(T):= \{ s\in\mathbb{H}: \mathcal{Q}_{s}(T) \text{ is invertible and } \mathcal{Q}_{s}(T)^{{-}1}\in\mathcal{B}(V)\} \end{align*}

and the $S$-spectrum of $T$ as

\begin{align*} \sigma_S(T):=\mathbb{H}\setminus\rho_S(T). \end{align*}

The operator $\mathcal {Q}_{s}(T)^{-1}$ is called the pseudo $S$-resolvent operator. For $s\in \rho _S(T)$, the left $S$-resolvent operator is defined as

(1.2)\begin{equation} S_L^{{-}1}(s,T):= \mathcal{Q}_s(T)^{{-}1}\overline{s} -T\mathcal{Q}_s(T)^{{-}1} \end{equation}

and the right $S$-resolvent operator is given by

(1.3)\begin{equation} S_R^{{-}1}(s,T):={-}(T-\mathcal{I} \overline{s})\mathcal{Q}_s(T)^{{-}1}. \end{equation}

The fractional powers of an operator $T$ such that $j\mathbb {R}\subset \rho _S(T)$ for any $j\in \mathbb {S}$, are denoted by $P_{\alpha }(T)$ and are defined as follows. For any $j\in \mathbb {S}$, for $\alpha \in (0,1)$ and $v\in \mathcal {D}(T)$, we set

(1.4)\begin{equation} P_{\alpha}(T)v := \displaystyle\frac{1}{2\pi} \displaystyle\int_{{-}j\mathbb{R}} S_L^{{-}1}(s,T)\,ds_j\, s^{\alpha-1} T v, \end{equation}

or

(1.5)\begin{equation} P_{\alpha}(T)v := \displaystyle\frac{1}{2\pi} \displaystyle\int_{{-}j\mathbb{R}} s^{\alpha-1} \,ds_j\, S_R^{{-}1}(s,T) T v, \end{equation}

where $ds_j=ds/j$. These formulas are a consequence of the quaternionic version of the $H^{\infty }$-functional calculus based on the $S$-spectrum, again see [Reference Colombo and Gantner16] for details. For the generation of the fractional powers $P_{\alpha }(T)$, a crucial assumption on the $S$-resolvent operators is that, for $s\in \mathbb {H}\setminus \{0\}$ with ${\rm Re}(s)=0$, the estimates

(1.6)\begin{equation} \left\|S_L^{{-}1}(s,T)\right\|_{\mathcal{B}(V)} \leq \displaystyle\frac{\Theta}{|s|}\quad\text{and}\quad \left\|S_R^{{-}1}(s,T)\right\|_{\mathcal{B}(V)} \leq \frac{\Theta}{|s|}, \end{equation}

hold with a constant $\Theta >0$ that does not depend on $s$. It is important to observe that the conditions (1.6) assure that the integrals (1.4) and (1.5) are convergent and so the fractional powers are well defined.

For the definition of the fractional powers of the operator $T$, we can use equivalently the integral representation in (1.4) or the one in (1.5). Moreover, they correspond to a modified version of Balakrishnan's formula that takes only spectral points with positive real part into account.

1.2. Operators of order one vs operators of order $m>1$

In some of our previous papers, we have defined fractional powers of operators of first order such as

\begin{align*} T:=\begin{pmatrix} a_1(x)\partial_{x_1}\\a_2(x)\partial_{x_2}\\a_3(x)\partial_{x_3}. \end{pmatrix} \end{align*}

acting on functions $u:\Omega \subset \mathbb {R}^{3}\to \mathbb {R}$ belonging to $H^{1}_0(\Omega,\mathbb {R})\subset L^{2}(\Omega,\mathbb {R})$ where $\Omega$ is a (possibly) unbounded domain with $C^{1}$ boundary. To use the $S$-functional calculus, first we identify the gradient operator with the quaternionic gradient operator

\begin{align*} \begin{pmatrix} a_1(x)\partial_{x_1}\\a_2(x)\partial_{x_2}\\a_3(x)\partial_{x_3} \end{pmatrix}\equiv e_1 a_1(x)\partial_{x_1}+e_2a_2(x)\partial_{x_2}+e_3a_3(x)\partial_{x_3} \end{align*}

and we consider the operator $\mathcal {Q}_{s}(T)$ defined in a weak sense over $H^{1}_0(\Omega,\mathbb {H})\subset L^{2}(\Omega, \mathbb {H})$. It is important to observe that the above identification has some consequence on the bilinear form $b_s(u,v)=\langle \mathcal {Q}_s(T)(u), v\rangle _\mathbb {H}$, where $\langle a,b \rangle _\mathbb {H}:=\overline a b$ for all $a,\, b\in \mathbb {H}$. Indeed, performing an integration by parts, we have (see in the following):

\begin{align*} b_s(u,u)=\sum_{l=1}^{3}\|a_l(x)\partial_{x_l}u\|^{2}+|s|^{2}\|u\|^{2}+\text{ other terms} \end{align*}

where the “other terms” are the scalar products of the first derivatives of $u$ with $u$ multiplied by the derivative of the coefficients. In the formula, we indicate that $b_s(u,u)$ contains two positive terms: the $L^{2}$-norm of $u$ and the $L^{2}$-norm of the first derivative of $u$ multiplied by the coefficients $a_j$'s. This fact allows us to determine some conditions on the coefficients $a_j$'s in order to guarantee the continuity and the coercivity of $b_s(\cdot,\cdot )$ and, moreover, the uniform estimates for the $S$-resolvent operator.

In this paper, we consider vector operators of order $m>1$ and $m\in \mathbb {N}$ of the type

\begin{align*} T:=\begin{pmatrix} a_1(x)\partial_{x_1}^{m}\\a_2(x)\partial_{x_2}^{m}\\a_3(x)\partial_{x_3}^{m} \end{pmatrix}. \end{align*}

If we consider the previous identification

\begin{align*} T := e_1 a_1(x)\partial_{x_1}^{m}+e_2a_2(x)\partial_{x_2}^{m}+e_3a_3(x)\partial_{x_3}^{m} , \end{align*}

and we consider the operator $\mathcal {Q}_s(T)$ defined in a weak sense over $H^{m}_0(\Omega,\mathbb {H})\subset L^{2}(\Omega, \mathbb {H})$, we have to distinguish the cases of $m$ odd or even. If $m$ is odd using $m$-times an argument of integration by parts we obtain

\begin{align*} b_s(u,u)=\sum_{l=1}^{3}\|a_l(x)\partial_{x_l}^{m}u\|^{2}+|s|^{2}\|u\|^{2}+\text{ other terms} \end{align*}

and the bilinear form still contains two positive terms: the $L^{2}$-norm of $u$ and the $L^{2}$-norm of the derivatives of order $m$. If we try to compute the bilinear form in the same way when $m$ is even, we obtain

\begin{align*} b_s(u,u)={-}\sum_{l=1}^{3}\|a_l(x)\partial_{x_l}^{m}u\|^{2}+|s|^{2}\|u\|^{2}+\text{ other terms} \end{align*}

losing the positivity of the first term. One way to overcome this problem is to identify $T$ with

\begin{align*} e_1 a_1(x)\partial_{x_1}^{m}+e_2a_2(x)\partial_{x_2}^{m}+e_3a_3(x)\partial_{x_3}^{m} \end{align*}

if $m$ is odd and with

\begin{align*} i(e_1 a_1(x)\partial_{x_1}^{m}+e_2a_2(x)\partial_{x_3}^{m}+e_3a_3(x)\partial_{x_3}^{m}) \end{align*}

if $m$ is even where $i$ is the imaginary unit of $\mathbb {C}$. In other words, we have complexified the coefficients of the quaternionic gradient operator and the operator $T$ is identified with the quaternionic gradient operator with real coefficients if $m$ is odd or with the quaternionic gradient operator with purely imaginary coefficients if $m$ is even. For the precise definitions of $b_s(\cdot,\cdot )$ and of the quaternionic Hilbert space $\mathbb {C}\otimes \mathbb {H}$, see § 2. In light of these considerations, we give the following definition.

Definition 1.2 Let $\Omega$ be a $C^{1}$-domain in $\mathbb {R}^{3}$, bounded or unbounded, and let $a_l: \overline \Omega \to \mathbb {R}$ for $l=1,\, 2,\, 3$ be $C^{m}(\overline \Omega )$ functions. We define in a classical way over $C^{m}(\overline \Omega, \mathbb {C}\otimes \mathbb {H})$ the operator

\begin{align*} 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}. \end{align*}

1.3. $\Omega$ bounded vs $\Omega$ unbounded

We will treat separately the cases of $\Omega$ bounded and of $\Omega$ unbounded. The unbounded case is more complicated as explained at the end of this section and needs some more constraints on the shape of $\Omega$ that we now introduce. For the following definition, we shall utilize $n$-dimensional spherical coordinates $(r,\omega )$ where $r\geq 0$ is the distance from the origin, $\omega \in S_{n-1}$, and $S_{n-1}$ denotes the sphere in $\mathbb {R}^{n}$ (see [Reference Elcrat and Lean26]; in our case, however, $n=3$).

Definition 1.3 An open set $\Omega \subset \mathbb {R}^{3}$ is said to have the property $(R)$ if there exists $P\in \mathbb {R}^{3}\setminus \overline \Omega$ such that every ray through $P$ has the intersection with $\Omega$ which is either empty or an infinite interval. More precisely, for each $\omega \in S_2$ set

\begin{align*} \begin{cases} f(\omega):=\inf \{r\geq 0:\, P+r\omega\in\Omega\} & \textrm{if }\{P+r\omega:r\geq 0\}\cap \Omega\neq \emptyset\\ f(\omega):=\infty & \textrm{if } \{P+r\omega:r\geq 0\}\cap \Omega= \emptyset. \end{cases} \end{align*}

We are assuming that if $f(\omega )\neq \infty$, then $P+r\omega \in \Omega$ for all $r\in (f(\omega ),\infty )$.

Examples of unbounded domains, which satisfy the property $(R)$ are: $\Omega :=\{x\in \mathbb {R}^{3}:\, |x-P|>M\}$ and $\Omega :=\{x\in \mathbb {R}^{3} :\, \langle P-x,v\rangle >0\}$ where $v\in \mathbb {R}^{3}$ is a vector, $P\in \mathbb {R}^{3}$ is a point and $M>0$ is a positive constant (here $\langle \cdot,\cdot \rangle$ is the standard scalar product of $\mathbb {R}^{3}$). We are ready to formulate in the precise way the problems that we have to solve.

Problem 1.4 Let $\Omega \subset \mathbb {R}^{3}$ be with $C^{1}$ boundary, which is either bounded or unbounded and satisfying the property $(R)$. Let $F:\Omega \to \mathbb {C}\otimes \mathbb {H}$ be a given $L^{2}$-function and denote by $u:\Omega \to \mathbb {C}\otimes \mathbb {H}$ the unknown function satisfying the boundary value problem:

(1.7)\begin{equation} \begin{cases} \mathcal{Q}_{s}(T) (u)=F\\ \partial^{\mathbf{b}} u(x)=0 & \forall \, \mathbf{b}\in\mathbb{N}^{3}_0\text{ such that } |\mathbf{b}|\leq m-1\,{\rm and}\, x\in \partial\Omega, \end{cases} \end{equation}

where ${\bf b} =(b_1,b_2,b_3)$ and $\partial ^{\mathbf {b}}=\partial ^{b_1}_{x_1}\partial ^{b_2}_{x_2}\partial ^{b_3}_{x_3}$. Determine the conditions on the coefficients $a_1, a_2, a_3 : \Omega \to \mathbb {R}$ such that the boundary value problem has a unique solution in a suitable function space and, moreover, the $L^{2}$-estimates (1.6) for the $S$-resolvent operators hold.

Remark 1.5 If $\Omega$ is bounded and its boundary is of class $C^{m}$, the previous boundary value problem is equivalent to

(1.8)\begin{equation} \begin{cases} \mathcal{Q}_{s}(T) (u)=F\\ \partial_{\nu}^{j} u(x)=0 & \text{for }0\leq j\leq m-1\text{ and } x\in \partial\Omega, \end{cases} \end{equation}

where, $\nu$ is the normal vector field pointing outside $\partial \Omega$ (see Theorem 7.41 in [Reference Renardy and Rogers33]).

We are going to solve the previous problem by the use of the Lax–Milgram Lemma. In particular, to solve the boundary value problem (1.7), we want to find some conditions on the coefficients $a_j$'s in a such way that the continuity and the coercivity of the quadratic form $b_s(u,v)$ associated with $\mathcal {Q}_s(T)$ hold (see Definition 2.3). The need of proving also the estimates (1.6) makes the assumptions on the coefficients $a_l$'s stronger than the usual one that we have to require for the coercivity of $b_s(\cdot,\cdot )$, since we can not rely on the term $|s|^{2}\|u\|^{2}$ of $b_s(\cdot,\cdot )$. For this reason, the other positive term in $b_s(u,u)$, that is $\sum _{l=1}^{3}\|a_l\partial _{x_l}^{m} u\|$, has to control the $L^{2}$-norm of $u$ and the $L^{2}$-norms of all the partial derivatives of $u$ up to order $m-1$. In § 3, when $\Omega$ is a bounded domain of $\mathbb {R}^{3}$, through an iterated use of Poincaré's inequality, we will show that the conditions:

• • $|a_j| \gg \max (C_\Omega ^{m}, C_\Omega )$;

• • $|a_j| \gg |\partial ^{\beta } a_j|$ for any $|\beta |< m$,

are sufficient to solve the Problem 1.4. Here and in what follows, $C_\Omega$ denotes the Poincaré constant of $\Omega$. When $\Omega$ is an unbounded domain of $\mathbb {R}^{3}$ and $m=1$, the role of the Poincaré inequalities is replaced by the Gagliardo–Nirenberg estimates and a condition of integrability on the first derivatives of the coefficients is sufficient to get the coercivity of $b_s(\cdot,\cdot )$ (see [Reference Colombo, Deniz-Gonzales and Pinton24]).

When $\Omega$ is an unbounded domain of $\mathbb {R}^{3}$ and the order of the operator $T$ is greater than $1$, the Gagliardo–Nirenberg estimates can not be used in an iterated way as for Poincaré's inequality. In § 2, we propose one way to overcome this problem by the use of a weighted Poincaré's inequality on some unbounded domains under an exponential decay condition at infinity of the coefficients $a_j$'s (see Theorem 4.3).

In this paper, the definition of the fractional quaternionic operators is based on the $S$-spectrum approach to fractional diffusion problems see [Reference Alpay, Colombo, Qian and Sabadini2, Reference Colombo and Gantner13, Reference Colombo and Gantner14]. This method is a generalization of the Balakrishnan's method to construct the fractional power of a real operator, see [Reference Balakrishnan5]. There are also several nonlinear models that involve the fractional power of scalar elliptic operators, see for example [Reference Bucur and Valdinoci11, Reference Vazquez34].