The noncommutative fractional Fourier law in bounded and unbounded domains

Using the spectral theory on the $S$-spectrum it is possible to define the fractional powers of a large class of vector operators. This possibility leads to new fractional diffusion and evolution problems that are of particular interest for nonhomogeneous materials where the Fourier law is not simply the negative gradient operator but it is a nonconstant coefficients differential operator of the form $$ T=\sum_{\ell=1}^3e_\ell a_\ell(x)\partial_{x_\ell}, \ \ \ x=(x_1,x_2,x_3)\in \bar{\Omega}, $$ where, $\Omega$ can be either a bounded or an unbounded domain in $\mathbb{R}^3$ whose boundary $\partial\Omega$ is considered suitably regular, $\bar{\Omega}$ is the closure of $\Omega$ and $e_\ell$, for $\ell=1,2,3$ are the imaginary units of the quaternions $\mathbb{H}$. The operators $T_\ell:=a_\ell(x)\partial_{x_\ell}$, for $\ell=1,2,3$, are called the components of $T$ and $a_1$, $a_2$, $a_3: \bar{\Omega} \subset\mathbb{R}^3\to \mathbb{R}$ are the coefficients of $T$. In this paper we study the generation of the fractional powers of $T$, denoted by $P_{\alpha}(T)$ for $\alpha\in(0,1)$, when the operators $T_\ell$, for $\ell=1,2,3$ do not commute among themselves. To define the fractional powers $P_{\alpha}(T)$ of $T$ we have to consider the weak formulation of a suitable boundary value problem associated with the pseudo $S$-resolvent operator of $T$. In this paper we consider two different boundary conditions. If $\Omega$ is unbounded we consider Dirichlet boundary conditions. If $\Omega$ is bounded we consider the natural Robin-type boundary conditions associated with the generation of the fractional powers of $T$.

where, Ω can be either a bounded or an unbounded domain in R 3 whose boundary ∂Ω is considered suitably regular, Ω is the closure of Ω and e ℓ , for ℓ = 1, 2, 3 are the imaginary units of the quaternions H. The operators T ℓ := a ℓ (x)∂x ℓ , for ℓ = 1, 2, 3, are called the components of T and a1, a2, a3 : Ω ⊂ R 3 → R are the coefficients of T .
In this paper we study the generation of the fractional powers of T , denoted by Pα(T ) for α ∈ (0, 1), when the operators T ℓ , for ℓ = 1, 2, 3 do not commute among themselves. To define the fractional powers Pα(T ) of T we have to consider the weak formulation of a suitable boundary value problem associated with the pseudo S-resolvent operator of T . In this paper we consider two different boundary conditions. If Ω is unbounded we consider Dirichlet boundary conditions. If Ω is bounded we consider the natural Robin-type boundary conditions associated with the generation of the fractional powers of T represented by the operator 3 ℓ=1 a 2 ℓ (x)n ℓ (x)∂x ℓ + a(x)I, for x ∈ ∂Ω, where I is the identity operator, a : ∂Ω → R is a given function and n = (n1, n2, n3) is the outward unit normal vector to ∂Ω. The Robin-type boundary conditions associated with the generation of the fractional powers of T are, in general, different from the Robin boundary conditions associated to the heat diffusion problem which leads to operators of the type 3 ℓ=1 a ℓ (x)n ℓ (x)∂x ℓ + b(x)I, x ∈ ∂Ω. For this reason we also discuss the conditions on the coefficients a1, a2, a3 : Ω ⊂ R 3 → R of T and on the coefficient b : ∂Ω → R so that the fractional powers of T are compatible with the physical Robin boundary conditions for the heat equations.

Introduction
Fractional diffusion and fractional evolution equations take into account nonlocal phenomena giving a better description of the physical reality with respect to differential laws. The most successful variation of the heat equation that takes into account nonlocal effects is the fractional heat equation where the Laplace operator is replaced by the fractional Laplacian. There are several ways to define fractional powers of operators which are, in general, not equivalent. Using the spectral theory on the S-spectrum, see [3,16,17,22], a new class of fractional diffusion and evolution problems can be considered. In particular the S-spectrum approach to fractional diffusion problems 1 has been considered in [16] where the fractional powers of quaternionic operators are systematically treated.
Using these new techniques based on the S-spectrum we can generate the fractional Fourier laws starting from the differential Fourier law and the associated boundary conditions. This method has the advantage to modify only the Fourier law without changing the conservation of energy laws in the fractional heat equation for nonhomogeneous materials. To recall this method and its advantages we need some notation. An element in the quaternions H is of the form s = s 0 + s 1 e 1 + s 2 e 2 + s 3 e 3 , where s 0 , s ℓ are real numbers (ℓ = 1, 2, 3) and Re(s) := s 0 denotes the real part of s. The modulus of s is defined as |s| = (s 2 0 + s 2 1 + s 2 2 + s 2 3 ) 1/2 and the conjugate is given by s = s 0 − s 1 e 1 − s 2 e 2 − s 3 e 3 . In the sequel we will denote by S the unit sphere of purely imaginary quaternions, an element j in S is such that j 2 = −1.
With our approach Ω can be a either a bounded or an unbounded domain in R 3 whose boundary ∂Ω is sufficiently regular, Ω denotes the closure of Ω and e ℓ , for ℓ = 1, 2, 3, is an orthogonal basis for the quaternions H. We consider vector operators of the form where the components T ℓ of T , ℓ = 1, 2, 3, are defined by T ℓ := a ℓ (x)∂ x ℓ , x ∈ Ω, and we suppose that the coefficient a 1 , a 2 , a 3 : Ω ⊂ R 3 → R of T are not necessarily nonconstant. From the physical point of view the operator T , defined in (1.1), can represent the Fourier law for nonhomogeneous materials, but it can represent also different physical laws. Our goal is to generate the fractional powers of T when the operators T ℓ , for ℓ = 1, 2, 3 do not commute among themselves. The vector part of the fractional powers P α (T ), for α ∈ (0, 1), of T is called the fractional Fourier law associated with T .
It is important to observe that using the spectral theory on the S-spectrum to define the fractional powers of a vector operator T , one has to specify the boundary conditions associated with the operator T . When T is the Fourier law for the heat diffusion problems with the homogeneous Dirichlet boundary condition, there are not suppletive boundary conditions that are necessary to generate the fractional powers of T . In the case Ω is bounded we studied this problem in the papers [14,18,19]. In this paper we consider the case in which Ω is unbounded.
In the paper [20] we have studies the fractional powers of T with Robin-type boundary conditions, where Ω is bounded, and the components T ℓ , of T , for ℓ = 1, 2, 3 commute among themselves. It turns out that the Robin-type boundary conditions necessary to generate the fractional powers of T and the classical Robin boundary conditions of the heat equation are different. In this paper we study the generation of the fractional powers when the components T ℓ , of T , for ℓ = 1, 2, 3 do not commute among themselves and the relation between the two type Robin boundary conditions. In order to set the problem we need some results of the spectral theory on the S-spectrum. We will work in an Hilbert space but our techniques allow to define the fractional powers of operators in quaternionic Banach spaces.

Problems and main results on the fractional powers of vector operators
We consider a two-sided quaternionic Banach space V and we denote the set of closed quaternionic right linear operators on V by K(V ). The Banach space of all bounded right linear operators on V is indicated by the symbol B(V ) and is endowed with the natural operator norm. For T ∈ K(V ), we define the operator associated with the S-spectrum as: We define the S-resolvent set of T as ρ S (T ) := {s ∈ H : Q s (T ) is invertible and Q s (T ) −1 ∈ B(V )} and the S-spectrum of T as σ S (T ) := H \ ρ S (T ). The operator Q s (T ) −1 is called the pseudo S-resolvent operator. For s ∈ ρ S (T ), the left S-resolvent operator is defined as 2) and the right S-resolvent operator is given by The fractional powers of T , denoted by P α (T ), are defined as follows: for any j ∈ S, for α ∈ (0, 1) and v ∈ D(T ) we set where ds j = ds/j. These formulas are a consequence of the quaternionic version of the H ∞functional calculus based on the S-spectrum, see the book [16] for more details. For the generation of the fractional powers P α (T ) a crucial assumption on the S-resolvent operators is that, for s ∈ H\{0} with Re(s) = 0, the estimates hold with a constant Θ > 0 that does not depend on the quaternion s. It is important to observe that the conditions (2.6) assure that the integrals (2.4) and (2.5) 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 (2.4) or the one in (2.5). Moreover, they correspond to a modified version of Balakrishnan's formula that takes only spectral points with positive real part into account.
A crucial problem is to determine the conditions on the coefficients a 1 , a 2 , a 3 : Ω ⊂ R 3 → R, of the operator T defined in (1.1), such that the purely imaginary quaternions are in the S-resolvent set ρ S (T ). This is a necessary condition, see formulas (2.4) and (2.5), since in the quaternionic case the map s → s α , for α ∈ (0, 1) is not defined for s ∈ (−∞, 0) and, unlike in the complex setting, it is not possible to choose different branches of s α in order to avoid this problem. For this reason it is of great importance to assume the condition Re(s) ≥ 0 that avoids the half real line (−∞, 0].
Regarding the boundary conditions of Robin-type, we will study the following problem associated with the fractional powers of the operator T .
where a : ∂Ω → R is a given function and n = (n 1 , n 2 , n 3 ) is the outward unit normal vector to ∂Ω. 3 (I) Determine the conditions on the coefficients a : ∂Ω → R, a 1 , a 2 , a 3 : Ω ⊂ R 3 → R such that the boundary value problem (2.7) has a unique solution in a suitable function space when Re(s) = 0. (II) Under the conditions in (I) determine the conditions on the coefficients such that the Sresolvent operators satisfy the estimates (2.6). (III) Consider the stationary heat equation for nonhomogeneous materials with Robin boundary conditions, for v : Ω → R, is given by where n = (n 1 , n 2 , n 3 ) is the outward unit normal vector to ∂Ω, and b : ∂Ω → R is a given continuous function. Determine the conditions on the coefficients a, b : ∂Ω → R, a 1 , a 2 , a 3 : Ω ⊂ R 3 → R such that the boundary condition in (2.8) implies the boundary condition in (2.7) (see Remark 2.2).
is associated with the boundary condition of problem (2.7) that naturally arise in the definition of the bilinear form associated with the existence of the pseudo S-resolvent operator as a bounded linear operator, while the operator in associated with the boundary condition of the problem (2.8) that naturally arises as a physical flux condition.
Remark 2.3. In the paper [20] we have investigated some possible solutions of the boundary value problem (2.7), in different Hilbert spaces, depending on the spectral parameter s ∈ H where the operator T = 3 ℓ=1 e ℓ T ℓ , defined in (1.1), has commuting components T ℓ , for ℓ = 1, 2, 3. Such analysis can be done also when the components T ℓ , for ℓ = 1, 2, 3 do not commute. In this paper we focus our attention on the spectral problem where s ∈ H and Re(s) = 0 because this is the case of interest for the definitions (2.4) with (2.5) so that we can generate the fractional powers of T .
Regarding the boundary Dirichlet conditions for the unbounded domains, we will study the following problem associated with the fractional powers of the operator T .
(2.9) (I) Determine the conditions on the coefficients a 1 , a 2 , a 3 : Ω ⊂ R 3 → R such that the boundary value problem (2.9) has a unique solution in a suitable function space when Re(s) = 0. (II) Under the conditions in (I) determine the conditions on the coefficients such that the Sresolvent operators satisfy the estimates (2.6).

2.1.
Summary of the main results of the paper. In Section 3 we give the weak formulation of Problems 2.1 and 2.4. In Section 4 we prove, under the condition a ∈ C 0 (∂Ω, R) and on the coefficients a 1 , a 2 , a 3 ∈ C 1 (Ω, R) of the operator T defined in (1.1), the existence and the uniqueness of the weak solutions of the problems and suitable estimates on the pseudo S-resolvent operators.
Precisely we summarize the results in the following points.
(A) The existence and uniqueness of the weak solution of Problem 2.1 is stated in Theorem 4.4 where we define the constants where · ∞ denotes the sup norm, and we assume where C P is the Poincaré-Wirtinger constant and C ∂Ω are a given constant that depends on ∂Ω.
Under the above conditions the boundary value Problem (2.7) has a unique weak solution u ∈ H(Ω, (B) In the case we work on unbounded domains the weak solution to Problems 2.4 is stated in Theorem 4.8, i.e., the boundary value Problem (2.9) has a unique weak solution u ∈ H 1 0 (Ω, H), for s ∈ H \ {0} with Re(s) = 0 when we assume Observe that the condition M < +∞, in the case of unbounded domain, is a consequence of the the Sobolev-Gagliardo-Nirenberg inequality.
(C) In both cases (A) and (B) we proved the following estimates where c > 0 is a given constant, b s (u, u) is the bilinear form associated with the weak formulation of the problems and the estimates hold for all s ∈ H \ {0} with Re(s) = 0 .
(D) In Section 5, based on the estimates in point (C), we prove the estimates (2.6) for the Sresolvent operators and we define the fractional powers of T using formula (2.4) or equivalently using (2.5).
(E) Finally, consider the point (III) of the Problem 2.1. Suppose that there exists a constant µ such that the functions a 1 , a 2 , a 3 satisfy the conditions and the coefficients a and b are such that Observe that, using (2.10) and (2.11), for x ∈ ∂Ω, we have

The weak formulation of the Problems 2.1 and 2.4
In the following Ω can be either a bounded or an unbounded domain of R 3 according to the problem that we will consider. The boundary ∂Ω of Ω is assumed to be of class C 1 even though for some lemmas in the sequel the conditions on the open set Ω can be weakened. We define The space L 2 with the scalar product: ∈ Ω is a Hilbert space. We furthermore introduce the quaternionic Sobolev space is the set of real-valued infinitely differentiable functions with compact support on Ω. If u ∈ H 1 then ∂ x ℓ (u j ) = g ℓj for ℓ = 1, 2, 3, and j= 0, 1, 2, 3. With the quaternionic scalar product we have that H 1 (Ω, H) becomes a quaternionic Hilbert space and the norm is defined by As usual the space H 1 0 (Ω, H) is the closure of the space C ∞ 0 (Ω, H) in H 1 (Ω, H) with respect to the norm · H 1 . Now we give to the problems (2.7) and (2.9) the weak formulations in order to apply the Lax-Milgram lemma in the space H 1 (Ω, H) and H 1 0 (Ω, H), respectively. From the Definition 1.1 of the operator T , we have is the vector part. We consider the bilinear form Integrating by parts we obtain where dS(x) is the infinitesimal surface area of ∂Ω. If we use the boundary condition in (2.7), i.e., Instead, if we use the boundary condition in (2.9), we obtain Relying on the above considerations we can give the following two definitions.
Let Ω be a bounded domain in R 3 with the boundary ∂Ω of class C 1 , let a ∈ C 0 (∂Ω, R) and a 1 , a 2 , a 3 ∈ C 1 (Ω, R). We define the bilinear form: for all functions u, v ∈ H 1 (Ω, H).

Definition 3.2.
Let Ω be either a bounded or an unbounded domain in R 3 with the boundary ∂Ω of class C 1 , let a 1 , a 2 , a 3 ∈ C 1 (Ω, R). We define the bilinear form: In this section we prove existence and uniqueness of the weak solutions of Problems 2.1 and 2.4 (Definition 3.3), using Lax-Milgram lemma. Moreover, we need crucial estimates on the S-resolvent operators in order to define the fractional powers of the operator T .
To prove existence and uniqueness of the weak solutions it will be sufficient to show that the bilinear forms b s (·, ·), in Definition 3.1 or Definition 3.2, are continuous in H 1 (Ω, H) and they are coercive in an appropriate closed subspace of H 1 (Ω, H) where the choice of these subspaces depend on the boundary conditions of the problems.
First we prove the continuity while the coercivity will be proved in Section 4.1 for the first problem and in Section 4.2 for the second one. As a direct consequence of the coercivity, we will prove an L 2 estimate for the weak solution u that belongs to a subspace of H 1 (Ω, H) and also we will prove an L 2 estimate for the term T (u). These L 2 estimates will be crucial in order to prove the boundedness of the pseudo S-resolvent operator Q s (T ) and the estimates (2.6).
We recall that the bilinear form for all u, v ∈ H 1 (Ω, H).
We note that the constant C(s) depends on s ∈ H but does not depend on u and v ∈ H 1 (Ω, H). The continuity of the bilinear forms b s (u, v) can be obtained in a similar way as described in [20] and in [18]. For the bilinear form (3.1), we need suitable estimates of the boundary term Lemma 4.1. Let u ∈ H 1 (Ω, H) and let Ω be a bounded domain with ∂Ω is of class C 1 . Furthermore let a ∈ C 0 (∂Ω, R), then we have where C ∂Ω is the constant in formula (4.1).
Proof. It follows from the scalar valued case see [10, p.315], precisely, suppose that u ∈ H 1 (Ω, R) and Ω is a bounded domain in R 3 with boundary ∂Ω of class C 1 . Then u| ∂Ω ∈ H 1/2 (∂Ω), and there exists a positive constant C ∂Ω such that  Let Ω be a bounded domain in R 3 with boundary ∂Ω of class C 1 . Assume that a ∈ C 0 (∂Ω, R) and a 1 , a 2 , a 3 ∈ C 1 (Ω, R). Then the terms in the bilinear form 8 b s (·, ·) defined in (3.1) satisfy the estimates: 4) while for the boundary term in (3.1) the following inequality holds: We adapt to the quaternionic setting the Poincaré-Wirtinger's inequality (see for example [24, p.275]).

Corollary 4.3.
Let Ω be a bounded domain in R 3 with boundary ∂Ω of class C 1 and let u ∈ H(Ω, H). Then we have u 2 L 2 (Ω;H) ≤ C 2 P u 2 H for any u ∈ H, where C P is the Poincaré-Wirtinger constant in (4.7).
Proof. Under the above hypotheses on the bounded domain Ω in R 3 the Poincaré-Wirtinger inequality claims that for all u ∈ H 1 (Ω, R) the following inequality holds: where C P does not depend on u. The quaternionic case follows from estimate (4.7). 9 Theorem 4.4.
Let Ω be a bounded domain in R 3 with boundary ∂Ω of class C 1 . Assume that a ∈ C 0 (∂Ω, R) and let T be the operator defined in (1.1) with coefficients a 1 , a 2 , a 3 ∈ C 1 (Ω, R). Define the following constants: where · ∞ denotes the sup norm and C ∂Ω is the constant in Theorem 4.1. Moreover, assume that where C P is the constant in (4.7). Then: (I) The boundary value Problem (2.7) has a unique weak solution u ∈ H(Ω, H), for s ∈ H \ {0} with Re(s) = 0, and u, u)). (4.10) (II) Moreover, we have the following estimate Proof.
Step (I). To prove the existence and uniqueness of the solution for the weak solution, it is sufficient to prove the coercivity of the bilinear form b s (·, ·), in Definition 3.1, in since the continuity is proved in Proposition 4.2. First we write explicitly Re b js 1 (u, u), where we have set s = js 1 , for s 1 ∈ R and j ∈ S: By the Cauchy-Schwartz inequality and Lemma 4.1, we have Since By the hypothesis, we know that K Ω := C T − C ′ T C P − K a,Ω 1 + C 2 P > 0, thus the following estimates hold: (4.14) In particular the inequality (4.14) implies the inequality (4.10), while the inequality (4.13) implies the coercivity of b js 1 (·, ·) and, by the Lax-Milgram Lemma, we have that for any w ∈ L 2 (Ω, H) there exists a unique u w ∈ H such that b js 1 (u w , v) = w, v L 2 , for all v ∈ H and for all s 1 ∈ R.
Step (II). What remains to prove is the inequality (4.11). Starting from (3.1) and applying the Cauchy-Schwartz inequality, Lemma 4.1 for the boundary term, Corollary 4.3 for the term u L 2 and the inequality we have: Collecting the term 3 ℓ=1 a ℓ ∂ x ℓ u 2 L 2 , observing that, with some computations, we have: and since the condition (4.9) holds, we get the desired inequality (4.11): where we have set Although the technique for proving Theorem 4.4 is different from the technique used in Theorem 4.1 of [14], we note that the condition (4.9) differs from the condition in Theorem 4.1 of [14] for some terms that arise since in this article we supposed the components of T are non commutative and a Robin-type condition on the boundary of Ω instead of a Dirichlet condition.

4.2.
Weak solution of the Problem 2.4. In Theorem 4.4 we prove the invertibility of the operator Q s (T ). So we adapt the strategy explained in Theorem 4.4 in the case of an unbounded domain under the more restrictive hypothesis according to that the coefficients of the first derivatives in the operator Q s (T ) are supposed to be in L 3 (Ω, H). We need a couple of lemmas, that are well known to adapt the Sobolev-Gagliardo-Nirenberg inequality to the quaternions.
We recall formula (5) in Theorem 8.8 p.212 in [10] and we give a sketch of the proof for the sake of completeness.
Lemma 4.5. For any u ∈ W 1,1 (R), we have Proof. We prove the statement for v ∈ C 1 0 (R), the general case will follow from the fact that C 1 0 (R) is dense in W 1,1 (R). We have: thus we can conclude that If v ∈ W 1,1 (R) then there exists a sequence v j ∈ C 1 0 (R) such that v j W 1,1 −→ v. Inequality (4.17) implies the convergence of the sequence to v in L ∞ (R). Thus the estimate (4.17) holds true for any v ∈ W 1,1 (R).
The following lemma can be proved for R n even though we will consider the case R 3 . It is Lemma 9.4 p.278 in [10] and we give a sketch of the proof. Lemma 4.6. Let F i ∈ L n−1 (R n , R) for i = 1, . . . , n such that F i does not depend on x i . Then Proof. The proof follows by an induction argument. The case n = 2 is a consequence of the following fact: Now we suppose that we have proved the statement in the case n = k − 1 when k > 2 is an integer. By the Hölder inequality we have (4.18) By induction we obtain (4.19) 12 Integrating over x 1 the inequality (4.18) and using the inequality (4.19), we have which concludes the proof.
So we finally have the Sobolev-Gagliardo-Nirenberg inequality for the quaternions obtained by adapting Theorem 9.9, p.278 in [10] and using the above lemmas.
Lemma 4.7. For any u ∈ H 1 (R n , H), we have u ∈ L 2n n−2 (R n , H) and the following estimate holds true Proof. We can suppose u ∈ C 1 0 (R n , H). First we observe that: Hölder inequality The above chain of inequalities can be summarized by the following inequality

13
Thus we conclude that If u ∈ H 1 (R n , H) then there exists a sequence u j ∈ C 1 0 (R n , H) such that u j H 1 −→ u. Inequality (4.21) implies the convergence of the sequence to u in L 2n n−2 (R n , H). Thus the estimate (4.21) holds true for any u ∈ H 1 (R n , H).
Let Ω be an unbounded domain in R 3 with boundary ∂Ω of class C 1 . Let T be the operator defined in (1.1) with coefficients a 1 , a 2 , a 3 ∈ C 1 (Ω, R). Suppose that and where K 3 = 4 is the constant in Lemma 4.7 for n = 3. Then: (I) The boundary value Problem (2.9) has a unique weak solution u ∈ H 1 0 (Ω, H), for s ∈ H \ {0} with Re(s) = 0, and (II) Moreover, we have the following estimate Proof. In order to use the Lax-Milgram Lemma to prove the existence and the uniqueness of the solution for the weak formulation of the problem, it is sufficient to prove the coercivity of bilinear form b s (·, ·) in Definition 3.2 since the continuity can be proved with similar computations as in Proposition 4.2. First we write explicitly Re b js 1 (u, u), where we have set s = js 1 , for s 1 ∈ R and j ∈ S: (4.26) We observe that since u ∈ H 1 0 (Ω, H), we can extend u by 0 outside Ω and we still have u ∈ H 1 (R n , H). For a general function u ∈ L 2 (Ω, H), we definẽ 14 Thus we have By the Hölder inequality, Lemma 4.7 (for the case n = 3) and hypothesis (4.22), we have that: Hölder inequality The above chain of inequalities can be summarized by the following inequality (4.27) Finally using the inequality (4.27) in (4.26), we obtain By the hypothesis (4.23) we know that thus the quadratic form b js 1 (·, ·) is coercive for every s 1 ∈ R and the following estimates hold: (4.28) In particular we have Re b js 1 (u, u) ≥ s 2 1 u 2 L 2 . (4.29) As a consequence the inequality (4.29) implies the inequality (4.10). The inequality (4.28) implies the coercivity of b js 1 (·, ·) and, by the Lax-Milgram Lemma, we have that for any w ∈ L 2 (Ω, H) there exists u w ∈ H 1 0 (Ω, H), for s 1 ∈ R \ {0} and j ∈ S, such that b js 1 (u w , v) = w, v L 2 , for all v ∈ H 1 0 (Ω, H). What remains to prove is the inequality (4.11). Starting from (3.2), applying the inequality (4.27) and observing that we have: C T and this concludes the proof. Remark 4.9. As we have mentioned in the introduction the case Ω bounded with homogeneous Dirichlet boundary conditions has already been investigated in our previous work. Above we have treated the case when Ω is unbounded. In the case Ω is bounded the condition (4.22) is not required.

5.
The estimates for the S-resolvent operators and the fractional powers of T After we prove existence and uniqueness results for the weak solutions of the problems we discussed in the previous sessions we can give meaning to the boundary condition using classical results on regularity of elliptic equations up to the boundary. In the case of Robin boundary conditions this requires the assumptions that the boundary has to be more regular, in the case of second order operators the boundary has to be of class C 2 if we want to have solutions in H 2 . In fact we can speak of the normal derivative ∂ ν u of a function u ∈ H 2 (Ω, R) (more in general we can set this problem in W 2,p for 1 ≤ p < ∞), we set ∂ n u := (∇u)| ∂Ω · n, where n is the unit normal vector to ∂Ω. This has meaning since (∇u)| ∂Ω ∈ L 2 (∂Ω) for Ω ⊂ R N bounded. For the regularity of the Neumann problem see p.299 in [10]. Using the estimates in Theorem 4.4 for the case of the Robin-type boundary conditions or estimate in Theorem 4.8, for the case of the Dirichlet boundary conditions in unbounded domains, we can now show in both cases that the S-resolvent operator of T decays fast enough along the set of purely imaginary quaternions.
holds. Moreover, the S-resolvent operators satisfy the estimates for any s = js 1 , for s 1 ∈ R \ {0} and j ∈ S, with a constant Θ that does not depend on s.
The estimates (5.2) follow from the estimate (4.11) (respectively (4.25)). Indeed we have C T u w 2 (4.11) (respectively (4.25)) ≤ Re(b js 1 (u w , u w )) ≤ |b js 1 (u w , u w )| ≤ | w, u w L 2 | ≤ w L 2 u w L 2 Thanks to the above results, we are now ready to establish our main statement. Proof. The right S-resolvent equation implies and so As α ∈ (0, 1), the estimate (5.2) now yields We conclude this paper with some comments. (I) In the literature there are several non linear models that involve the fractional Laplacian and even the fractional powers of more general elliptic operators, see for example, the books [11,31].
(II) The S-spectrum approach to fractional diffusion problems used in this paper is a generalization of the method developed by Balakrishnan, see [5], to define the fractional powers of a real operator A. In the paper [15] following the book of M. Haase, see [27], has been developed the theory on fractional powers of quaternionic linear operators, see also [2,13].
(III) The spectral theorem on the S-spectrum is also an other tool to define the fractional powers of vector operators, see [1] and for perturbation results see [12].
(IV) An historical note on the discovery of the S-resolvent operators and of the S-spectrum can be found in the introduction of the book [17].
The most important results in quaternionic operators theory based on the S-spectrum and the associated theory of slice hyperholomorphic functions are contained in the books [4,3,16,17,21,22,25,26], for the case on n-tuples of operators see [23].
(V) Our future research directions will consider the development of ideas from one and several complex variables, such as in [6,7,8,9,28,30,29] to the quaternionic setting.