Harnack's inequality for fractional nonlocal equations

We prove interior Harnack's inequalities for solutions of fractional nonlocal equations. Our examples include fractional powers of divergence form elliptic operators with potentials, operators arising in classical orthogonal expansions and the radial Laplacian. To get the results we use an analytic method based on a generalization of the Caffarelli--Silvestre extension problem, the Harnack's inequality for degenerate Schr\"odinger operators proved by C. E. Guti\'errez, and a transference method. In this manner we apply local PDE techniques to nonlocal operators. On the way a maximum principle and a Liouville theorem for some fractional nonlocal equations are obtained.


Introduction
Very recently, a great deal of attention was given to nonlinear problems involving fractional integrodifferential operators. These problems arise in Physics (fluid dynamics, strange kinetics, anomalous transport) and Mathematical Finance (modeling with Lévy processes), among many other fields, see for instance [5,6,8,15,21,22] and the references therein. The main question is the regularity of solutions. One of the tools that plays a crucial role in the regularity theory of PDEs is Harnack's inequality, see for example [6,7,9,10,11,12,19,23,25,28].
In this paper we show interior Harnack's inequalities for solutions of nonlocal equations given by fractional powers of second order partial differential operators. The operators we consider are: • Divergence form elliptic operators L = − div(a(x)∇) + V (x) with bounded measurable coefficients a(x) and locally bounded nonnegative potentials V (x) defined on bounded domains; • Ornstein-Uhlenbeck operator O B and harmonic oscillator H B on R n ; • Laguerre operators L α , L ϕ α , L ℓ α , L ψ α and L L α on (0, ∞) n with α ∈ (−1, ∞) n ; • Ultraspherical operators L λ and l λ on (0, π) with λ > 0; • Laplacian on domains Ω ⊆ R n ; • Bessel operators ∆ λ and S λ on (0, ∞) with λ > 0. For the full description of the operators see Sections 3, 5 and 6. In general, all these operators L are nonnegative, self-adjoint and have a dense domain Dom(L) ⊂ L 2 (Ω, dη), where Ω ⊆ R n , n ≥ 1, is an open set and dη is some positive measure on Ω. In Section 2 we show how the fractional powers L σ , 0 < σ < 1, can be defined by using the spectral theorem.
Theorem A (Harnack's inequality for fractional equations). Let L be any of the operators listed above and 0 < σ < 1. Let O be an open and connected subset of Ω and fix a compact subset K ⊂ O. There exists a positive constant C, depending only on σ, n, K and the coefficients of L such that for all functions f ∈ Dom(L), f ≥ 0 in Ω, such that L σ f = 0 in L 2 (O, dη). Moreover, f is a continuous function in O.
Theorem A is new, except for three cases: the Laplacian on R n ([7, Theorem 5.1] and [16, p. 266]), the Laplacian on the one-dimensional torus [19,Theorem 6.1] and the harmonic oscillator [23, Theorem 1.2]. Harnack's inequality is well-known for divergence form Schrödinger operators with locally bounded potentials [12], see also [9,11,28]. For the non-divergence form operators listed above the result can be obtained by using our transference method of Section 4. Very recently a Harnack's inequality for the fractional Laplacian with lower order terms was proved in [25].
A novel proof of Harnack's inequality for the fractional Laplacian was given by L. Caffarelli and L. Silvestre by using the extension problem in [7]. Let us briefly explain it here. Consider f : R n → R as in the hypotheses of Theorem A. Let u(x, y) be the extension of f to the upper half space R n+1 + obtained by solving div(y 1−2σ ∇u) = 0, in R n × (0, ∞); u(x, 0) = f (x), on R n .
Letũ(x, y) = u(x, |y|), y ∈ R, be the reflection of u to R n+1 . The hypothesis (−∆) σ f = 0 in O implies that y 1−2σ u y (x, y) → 0 as y → 0 + , for all x ∈ O. This is used to show thatũ is a weak solution of the degenerate elliptic equation with A 2 weight div(|y| 1−2σ ∇ũ) = 0, in O × (−R, R) ⊂ R n+1 , for some R > 0. Recall that a nonnegative function ω on R n is an Then the theory of degenerate elliptic equations by E. Fabes, C. Kenig and R. Serapioni in [10] says thatũ satisfies an interior Harnack's inequality and it is locally Hölder continuous, thus f (x) =ũ(x, 0) has the same properties. The idea of [7] was also exploited in [23] for the case of the fractional harmonic oscillator (−∆+|x| 2 ) σ on R n , under the additional assumption f ∈ C 2 . In [23] a generalization of the extension problem was proved that applies to a general class of differential operators, and it was used to get the result for the harmonic oscillator. We observe that, instead of the theory of [10], Harnack's inequality for degenerate Schrödinger operators of C. E. Gutiérrez [12] had to be applied.
To get Harnack's inequalities for fractional powers of the operators listed above we push further the Caffarelli-Silvestre ideas. We proceed in two steps. First we use two tools: the extension problem of [23] and Harnack's inequality for degenerate Schrödinger operators of C. E. Gutiérrez [12]. These are enough to get Theorem 3.2, from which the result for divergence form elliptic operators with potentials and some Schrödinger operators from orthogonal expansions is deduced. Secondly, we apply systematically a transference method that permits us to derive the results for other operators involving terms of order one and in non-divergence form. The transference method is inspired in ideas from Harmonic Analysis of orthogonal expansions, where it is used to transfer L p boundedness of operators, see for example [1,2,14]. In that case, the dimension, the underlying measure and the parameters that define the operators play a significant role. Here we can obtain our estimates without any restrictions on dimensions or parameters.
Let us remark that in Theorem A we require the condition f ≥ 0 all over Ω, which is needed to ensure that the solution to the extension problem u is nonnegative in Ω × (0, ∞). In fact, u can be given in terms of the solution e −tL f of the L-heat diffusion equation, see Theorem 2.2 below, so we only would need the condition e −tL f ≥ 0 in O. Certainly it is sufficient to assume that e −tL is positivity-preserving (see (2.2) below), but this hypothesis is not strictly necessary.
As a by-product of our method, we obtain a Liouville theorem for fractional powers of divergence form elliptic operators on R n , see Remark 3.3. We also get a maximum and comparison principle for general fractional operators, see Remark 2.1.
In Section 2 we present the definition of fractional powers of differential operators, we get maximum and comparison principles and we state the extension problem of [23]. The method of reflections for proving Harnack's inequality for divergence form elliptic Schrödinger operators is given in Section 3. The transference method is explained in Section 4.
The rest of the paper is concerned with the proof of Theorem A in each case. As the reader may notice, we have two sets of applications of our method: operators with discrete spectrum and operators with continuous spectrum. In the first set we have divergence form elliptic operators in bounded domains and classical operators related to orthogonal expansions in possibly unbounded domains (Sections 3 and 5). In the second set (Section 6) we have the Laplacian (Fourier transform) and the Bessel operator (Hankel transform), that generalizes the radial Laplacian.
We will present most of the results about Harnack's inequalities in the case when the sets K and O in Theorem A are balls inside Ω. In that situation the constant C does not depend on the radius of the balls. By the standard covering argument [11,Theorem 2.5] the general result can be easily deduced.

Fractional operators and extension problem
Along this paper all the operators will verify the following General assumption. By L = L x we denote a nonnegative self-adjoint second order partial differential operator with dense domain Dom(L) ⊂ L 2 (Ω, dη) ≡ L 2 (Ω).
Here Ω is an open subset of R n , n ≥ 1, and dη is a positive measure on Ω. The operator L acts in the variables x ∈ R n .
The Spectral Theorem can be applied to an operator L as in the general assumption, see [20,Chapter 13]. Given a real measurable function φ on [0, ∞), the operator φ(L) is defined as In this paper we are going to use: • The heat-diffusion semigroup generated by L, defined as φ(L) = e −tL , t ≥ 0. For f ∈ L 2 (Ω), we have that v = e −tL f solves the evolution equation v t = −Lv, for t > 0. Moreover, e −tL f L 2 (Ω) ≤ f L 2 (Ω) , for all t ≥ 0, and e −tL f → f in L 2 (Ω) as t → 0 + . • The fractional powers of L, given by φ(L) = L σ , with domain Dom(L σ ) ⊃ Dom(L). When f ∈ Dom(L σ ) we have L σ e −tL f = e −tL L σ f . If f ∈ Dom(L) then Lf, f = L 1/2 f 2 L 2 (Ω) , where ·, · denotes the inner product in L 2 (Ω). Also, for f ∈ Dom(L), where Γ is the Gamma function, see for example [29, p. 260].

Harnack's inequality for fractional Schrödinger operators
In this section we consider a uniformly elliptic Schrödinger operator of the form Here a = (a ij ) is a symmetric matrix of real-valued measurable coefficients such that µ −1 |ξ| 2 ≤ a(x)ξ · ξ ≤ µ|ξ| 2 , for some constant µ > 0, for almost every x ∈ Ω and for all ξ ∈ R n . The potential V is a locally bounded function on Ω.
Here Ω can be an unbounded set. We assume that L satisfies the general assumption at the beginning of Section 2, with dη(x) = dx, the Lebesgue measure. The Thenũ verifies the degenerate Schrödinger equation in the weak sense inB : Proof. Let ϕ ∈ C ∞ c (B). Take any 0 < δ < R. Since u is a solution of the extension equation in (2.4) for L, for any fixed y ∈ (δ, R), we have Recall that we are assuming that u ∈ C ∞ ((0, R) : Dom(L)). By integrating the last identity in y, applying Fubini's theorem and integration by parts, From here we get We are ready to prove thatũ is a weak solution of (3.1) inB. We have to check that By using (3.2), As δ → 0 + , the first and second terms above tend to zero because of (I). Also the fourth term goes to zero because V (x)ũ|y| 1−2σ ∈ L 1 loc . Since ∇ x u(x, y) L 2 (BR(x0),dx) remains bounded as y → 0 + , for any small δ > 0 there exists a constant c > 0 such that if |y| < δ then ∇ x u(x, y) L 2 (BR(x0),dx) ≤ c. This property and (I) imply that the third term above tends to zero as δ → 0 + . Theorem 3.2 (Harnack's inequality for L σ ). Let L be as above. Assume that the heat-diffusion semigroup e −tL is positivity-preserving, see remains bounded as y → 0 + , where u is a solution to the extension problem (2.4) for L and f . There exist constants R 0 < R and C depending only on n, σ, µ, and V , but not on f , such that, In order to prove Theorem 3.2 we use Theorem 3.1 and the following version of Gutiérrez's Harnack inequality for degenerate Schrödinger equations. Consider a degenerate Schrödinger equation of the form whereã = (ã ij ) is an N × N symmetric matrix of real-valued measurable coefficients such that λ −1 ω(X)|ξ| 2 ≤ã(X)ξ · ξ ≤ λω(X)|ξ| 2 , for some λ > 0, for almost every X ∈ R N and for all ξ ∈ R N . The function ω is an A 2 weight. The potentialṼ satisfiesṼ /ω ∈ L p ω locally, for some large p = p N,ω . Let O be any open bounded subset of R N . Then there exist positive constants r 0 , γ depending only on λ, N , ω, O andṼ such that if v is any nonnegative weak solution of (3.3) in O then for every ball B r with B 8r ⊂ O and 0 < r ≤ r 0 we have As a consequence, v is continuous in O. See [12].
, by (2.6) and the hypothesis on ∇ x u, we see that u satisfies the conditions of Theorem 3.1. Now, equation (3.1) is a degenerate Schrödinger equation with A 2 weight ω(x, y) = |y| 1−2σ and potentialṼ = |y| 1−2σ V (x) such thatṼ /ω ∈ L p ω locally for all p sufficiently large. By Gutiérrez's result just explained above, Harnack's inequality forũ holds. By restrictingũ to y = 0 we get Harnack's inequality for f . Moreover,ũ is continuous in B R (x 0 ) and thus f .
3.1. The case of nonnegative potentials. Under the additional assumptions that Ω is a bounded set and that the potential V is a nonnegative function in Ω, we can prove Theorem A for L σ . In this case the domain of L is Dom(L) = W 1,2 0 (Ω) and it is known that e −tL is positivity-preserving, see [9, Denote by u the solution of the extension problem for f as in Theorem 2.2. By virtue of Theorem 3.2, to prove Harnack's inequality for L σ we just have to verify that u satisfies condition (II) of Theorem 3.1. As f ∈ W 1,2 0 (Ω), by the ellipticity condition, (for the last equality see Section 2). Now, since u ∈ C 2 ((0, ∞) : W 1,2 0 (Ω)), ∇ x u(x, y) is well defined and belongs to L 2 (Ω, dx) for each y > 0. We can apply (2.3), (3.4) and the properties of the heatdiffusion semigroup e −tL stated at the beginning of Section 2 to get Thus ∇ x u(x, y) L 2 (BR(x0),dx) remains bounded as y → 0 + and (II) in Theorem 3.1 is valid. Hence Theorem A is proved for this case. Observe that, in particular, Theorem A is valid for the Laplacian in bounded domains with Dirichlet boundary conditions.

Remark 3.3 (Liouville theorem for fractional divergence form elliptic operators).
Let Ω = R n and The following Liouville theorem is true: If f ≥ 0 on R n and L σ f = 0 in L 2 (R n ) then f must be a constant function. Indeed, for this f , the reflectionũ of u is a nonnegative weak solution of (3.1) with V ≡ 0 in R n+1 , soũ is constant and therefore f is a constant function. Here we have applied the Liouville theorem for degenerate elliptic equations in divergence form with A 2 weights, which is a simple consequence of Harnack's inequality of [10].
Remark 3.4. Since our method is based on Gutiérrez's result [12], we are not able to get the exact dependence on σ of the constant C in Harnack's inequality of Theorem 3.2.

Transference method for Harnack's inequality
In this section we assume that L satisfies the general assumptions of Section 2. We explain in detail a general method to transfer Harnack's inequality from L σ to another operatorL σ related to L. This method will be useful when considering differential operators arising in classical orthogonal expansions and also for the Bessel operator.
Firstly, by a change of measure, we have the following trivial result.
is also an orthonormal system in L 2 (Ω, dη(x)). Next we set up the notation for the change of variables.
Now we are in position to describe the transference method. By using the definition above and Lemma 4.1 we construct a new differential operator. This new operator will be nonnegative and self-adjoint in L 2 (Ω, dη(x)), whereΩ = h(Ω) and dη( If E is the resolution of the identity of L then the resolution of the identityĒ of (U • W ) •L verifies Therefore if f ∈ Dom(L σ ) then we see that the fractional powers ofL satisfȳ Proof. Let f ∈ Dom(L σ ), f ≥ 0, such thatL σ f = 0 in L 2 (Ō, dη), for some open setŌ ⊂Ω. Take a compact setK ⊂Ō. We want to see that there is a constant C depending onK andL σ such that Observe that, by the definition of dη and since dη(x) = η(x) dx, and (U • W )f ∈ Dom(L) is nonnegative. By the assumption on L σ , there exists C depending on h −1 (K) and L σ such that sup This in turn implies (4.1) as desired.

Classical orthogonal expansions
In this section we consider operators L (as in the general assumptions of Section 2) for which there exists a family {ϕ k } k∈N n 0 of eigenfunctions of L, with associated nonnegative eigenvalues {λ k } k∈N n 0 , namely, Lϕ k (x) = λ k ϕ k (x), such that {ϕ k } is an orthonormal basis of L 2 (Ω, dη). In all our examples, the eigenvalues will satisfy the following: there exists a constant c ≥ 1 such that λ k ∼ |k| c , for any k = (k 1 , . . . , k n ) ∈ N n 0 , |k| = k 1 + · · · + k n . We also suppose that the eigenfunctions ϕ k are in C 2 (Ω) and that their derivatives satisfy the following local estimate. For any compact subset K ⊂ Ω and any multi-index β ∈ N n 0 , |β| ≤ 2, there exist ε = ε K,β ≥ 0 and a constant C = C K,β such that for any k ∈ N n 0 . For f ∈ L 2 (Ω, dη) the heat-diffusion semigroup can be written as e −tL f (x) = ∞ |k|=0 e −tλ k c k ϕ k (x). For 0 < σ < 1, the domain of L σ is given as Dom(L σ ) = {f ∈ L 2 (Ω, dη) : . Under these assumptions we can show that the solution u of the extension problem is classical. To this end, let K be any compact subset of Ω. First we show that the series that defines e −tL f (x) is uniformly convergent in K ×(0, T ), for every T > 0. Indeed, by applying that λ k ∼ |k| c , estimate (5.1), the inequality s ρ e −s ≤ C ρ e −s/2 (valid for s, ρ > 0 and some constant C ρ > 0) and Cauchy-Schwartz's inequality, and the uniform convergence follows. As a consequence, u in (2.3) is well defined, for by the estimate above, for any x ∈ K and y > 0, for some function F = F (y). This estimate also implies that in the first identity of (2.3) we can interchange the integration in t with the summation that defines e −tL f (x) to get By using (5.1) and the same arguments as above, it is easy to see that this series defines a function in C 2 (Ω) ∩ C 1 (0, ∞). Moreover, since each term of the series in (5.2) satisfies equation (2.4) in the classical sense, we readily see that u is a classical solution to (2.4).
Next we will present the concrete applications. We will take advantage of well-known formulas, see for instance [1,2], to apply our transference method to get Harnack's inequality for operators of classical orthogonal expansions which are not of the form considered in Section 3. A remarkable advantage of the transference method is that we do not need to check that the semigroup e −tL is positivity-preserving.

5.1.
Ornstein-Uhlenbeck operator and harmonic oscillator. In [13], Gutiérrez dealt with the Ornstein-Uhlenbeck operator where B is an n × n positive definite symmetric matrix. The operator O B is positive and symmetric in L 2 (R n , dγ B (x)), where dγ B (x) = (det B) n/2 π −n/2 e −Bx·x dx is the B-Gaussian measure. Let us consider the eigenvalue problem O B w = λw, with boundary conditions w(x) = O(|x| k ), for some k ≥ 0 as |x| → ∞. Firstly, let us assume that the matrix B is diagonal, which means that It is not difficult to see that in this case the eigenfunctions w are the multidimensional Hermite polynomials defined by . . . , d n ), where H ki is the one-dimensional Hermite polynomial of degree k i , see [13]. For the general case, since B is a positive definite symmetric matrix, there exists an orthogonal matrix A such that ABA t = D, where A t is the transpose of A. Then the eigenfunctions become H B k (x) = H D k (Ax). Let us also consider the harmonic oscillator where D is a matrix as above, with zero boundary condition at infinity. Under these assumptions H D is positive and symmetric in L 2 (R n , dx). It is well known that the multidimensional Her- The Hermite functions form an orthonormal basis of L 2 (R n , dx). Observe that we may also consider since it has the same eigenfunctions as H D with eigenvalues 2(k · d) ≥ 0. We can also put a more general matrix B in the place of D; we will prove Harnack's inequality for it by using the transference method. The potential here is V (x) = |Dx| 2 , which is a locally bounded function on R n . By Mehler's formula [13,24,26], e −tHD is positivity-preserving. In [26], it is shown that there exists C such that h D k L ∞ (R n ,dx) ≤ C for all k. Using the relation where e i is the i-th coordinate vector in N n 0 , we see that (5.1) is valid for h D k (x). Therefore the solution u to the extension problem given in (2.3) for H D is a classical solution. Let . We have to verify that ∇ x u(x, y) L 2 (BR(x0),dx) remains bounded as y → 0 + . In fact, we will have ∇ x u(x, 0) L 2 (BR(x0),dx) = ∇ x f (x) L 2 (BR(x0),dx) . Indeed, as we can write f = ∞ |k|=0 c k h D k , by (5.2) and the identity for the derivatives of the Hermite functions h D k given above, Observe that the term in parenthesis above is uniformly bounded in y and, since we readily see that it converges to 0 when y → 0 + . Moreover, as f ∈ Dom(H D ), Hence, by dominated convergence in (5.3), we get that See also [2]. It can be easily checked, as done for (H D ) σ above, that the operator (H D −

Laguerre operators.
We suggest the reader to check [1,14,17,24,26] for the proof of the basics about Laguerre expansions we use here. Let us consider the system of multidimensional Laguerre polynomials L α k (x), where k ∈ N n 0 , α = (α 1 , · · · , α n ) ∈ (−1, ∞) n and x ∈ (0, ∞) n . It is well known that the Laguerre polynomials form a complete orthogonal system in L 2 ((0, ∞) n , dγ α (x)), where dγ α (x) = x α1 1 e −x1 dx 1 · · · x αn n e −xn dx n . We denote byL α k the orthonormalized Laguerre polynomials. The polynomialsL α k are eigenfunctions of the Laguerre differential operator There are several systems of Laguerre functions. We first prove Harnack's inequality for the operator L ϕ α (related to the system ϕ α k below) and then we apply the transference method of Section 4 to get the result for the remaining systems.

5.2.1.
Laguerre functions ϕ α k . This multidimensional system in L 2 ((0, ∞) n , dµ 0 (x)), where dµ 0 (x) = dx 1 · · · dx n , is given as a tensor product ϕ α k (x) = ϕ α1 k1 (x 1 ) · · · ϕ αn kn (x n ), where each factor ϕ αi ki ( ki (x 2 i ). The functions ϕ α k are eigenfunctions of the differential operator namely, Clearly, the functions ϕ α k are locally bounded in (0, ∞) n . Observe that Therefore, (5.1) holds for this system and we get that the solution u in (2.3) of the extension problem for L ϕ α is classical. Moreover, it can be easily seen from [24, p. 102] that e −tL ϕ α is positivity-preserving. Let us prove Theorem A for (L ϕ α ) σ . We can do this as we did for (H D ) σ above by following the reasoning line by line, but with some modifications as follows.
, and let u be the corresponding solution to the extension problem. By (5.7) and a similar argument for that of H σ we can check that ∇ x u(x, y) L 2 (BR(x0),dµ0(x)) converges to ∇ x f L 2 (BR(x0),dµ0(x)) , as y → 0 + . Moreover, the potential in (5.5) is locally bounded. Hence, by Theorem 3.2, f satisfies Harnack's inequality and it is continuous.
Note that the same arguments above can be used for (L ϕ α − α+1 2 ) σ instead of (L ϕ α ) σ , so it also satisfies Theorem A.

Laplacian and Bessel operators
In this section we will prove Theorem A for the fractional powers of the Bessel operator. This operator is a generalization of the radial Laplacian. For the sake of completeness and to show how the proof works, we present first the case of the fractional Laplacian on R n , for which the more familiar Fourier transform applies.
The main difference with respect to the examples given before is that these operators have a continuous spectrum and the Fourier and Hankel transforms come into play. where f denotes the Fourier transform: The eigenfunctions of −∆, indexed by the continuous parameter ξ, are ϕ ξ (x) = e −ix·ξ , x ∈ R n , and (−∆)ϕ ξ (x) = |ξ| 2 ϕ ξ (x). Note that for any compact subset K ⊂ R n and any multi-index β ∈ N n 0 , |β| ≤ 2, we have For any f ∈ L 2 (K, dx), the heat semigroup is defined by e t∆ f (x) = 1 (2π) n/2 the integral that defines e t∆ f (x) is absolutely convergent in K × (0, T ) with T > 0. Moreover, e t∆ is positivity-preserving in the sense of (2.2) because it is given by convolution with the Gauss-Weierstrass kernel. Note that, in this spectral language, Dom( Let us show Theorem A for (−∆) σ . Assume that f ∈ W 2,2 (R n ), f ≥ 0 and (−∆) σ f = 0 in L 2 (B R , dx), for some ball B R ⊂ R n . By Theorem 3.2, we just must check that ∇ x u(x, y) L 2 (BR,dx) remains bounded as y → 0 + . To that end, observe that for any x ∈ B R and y > 0, by (6.2), for some function F (y). This means that we can interchange integrals in u to get (6.3) u(x, y) = y 2σ 4 σ Γ(σ)(2π) n/2 R n c ξ (f )ϕ −ξ (x) ∞ 0 e −t|ξ| 2 e − y 2 4t dt t 1+σ dξ.
We also consider the Bessel operator which is positive and symmetric in L 2 ((0, ∞), dx). Observe that the potential V (x) = λ 2 −λ x 2 is a locally bounded function. If we let ψ λ ξ (x) = x λ ϕ λ ξ (x) then S λ ψ λ ξ (x) = ξ 2 ψ λ ξ (x), see [3]. The Hankel transform is a unitary transformation in L 2 ((0, ∞), dx), see [27,Chapter 8]. On the other hand, it is known that for any compact subset K ⊂ (0, ∞) and k ∈ N 0 , there exist a nonnegative number ε = ε K,k and a constant C = C K,k such that ψ λ ξ (x) L ∞ (K,dx) ≤ C, and d k dx k ψ λ ξ (x) L ∞ (K,dx) ≤ C|ξ| ε , see [17]. Therefore parallel to the case of the Laplacian we can define the heat semigroup as so the integral that defines e −tS λ f (x) is absolutely convergent in K × (0, T ) with T > 0. Since e −tS λ is positivity-preserving (see [3]), we can follow step by step the arguments we gave for the case of the classical Laplacian to derive Theorem A for the operator (S λ ) σ . In order to get Theorem A for (∆ λ ) σ we apply the transference method. Indeed, an obvious modification of Lemma 4.1 is applied with M (x) = x λ to get (∆ λ ) σ = U −1 • (S λ ) σ • U .