A subordination principle on Wright functions and regularized resolvent families

We obtain a vector-valued subordination principle for $(g_{\alpha}, g_{\alpha})$-regularized resolvent families which unified and improves various previous results in the literature. As a consequence we establish new relations between solutions of different fractional Cauchy problems. To do that, we consider scaled Wright functions which are related to Mittag-Leffler functions, the fractional calculus and stable L\'evy processes. We study some interesting properties of these functions such as subordination (in the sense of Bochner), convolution properties, and their Laplace transforms. Finally we present some examples where we apply these results.


Introduction
Conversely, given a convolution semigroup of sub-probability measures (µ t ) t>0 on [0, ∞), then there exists a unique Bernstein function f such that (1.1) holds true, see for example [26,Theorem 5.2]. The original subordination principle for stochastic processes in connection with diffusion equations and semigroups was introduced in [5]. In [6,Chapters 4.3,4.4] a detailed study of stochastic processes, their transition semigroups, generators and subordination results are developed. Now let A be a densely defined closed linear operator on a Banach space X which generates a C 0 -contraction semigroup (T (t)) t>0 ⊂ B(X). Then the solution of the first order abstract Cauchy problem u ′ (t) = Au(t), t > 0, is given by u(t) = T (t)x for t > 0. Now, suppose that (µ t ) t>0 is a vaguely continuous convolution semigroup of sub-probability measures on [0, ∞) with the corresponding Bernstein function f . Then the Bochner integral x ∈ X, t > 0, defines again a C 0 -contraction semigroup on X ([26, Proposition 12.1]). Then the semigroup (T f (t)) t>0 is called subordinate (in line with Bochner) to the semigroup (T (t)) t>0 with respect to the Bernstein function f . In particular, given 0 < α < 1 and dµ t (s) = f t,α (s)ds (where f t,α are the stable Lévy processes, see (3.2)) then is an analytic semigroup generated by −(−A) α , the fractional powers of the generator A according to Balakrishnan. For more details see [28,Chapter IX].
Other subordination formulae allow to define new families of operators from some previous ones by integration. Let A be the generator of a cosine function (C(t)) t>0 on a Banach space X (see definition in [1,Section 3.14]). Then A generates a holomorphic C 0 -semigroup (T (z)) z∈C+ of angle π 2 , given by 4z C(s)x ds, x ∈ X, z ∈ C + , ([1, Theorem 3.14.17]). Remember that the solution of the second order Cauchy problem    u ′′ (t) = Au(t), t > 0, is u(t) = C(t)x for t > 0 ([1, Section 3.14]). In [16], a two-kernel dependent family of strong continuous operators defined in a Banach space is introduced. This family allows us to consider in a unified treatment the notions of, among others, C 0 -semigroups of operators, cosine families, n-times integrated semigroups, resolvent families and k-generalized solutions. Let a ∈ L 1 loc (R + ) and k ∈ C(R + ). The family {S a,k (t)} t>0 ⊂ B(X) is a (a, k)-regularized resolvent family generated by A if the following conditions are fulfilled: S a,k (t) is strongly continuous for t > 0 and S a,k (0)x = k(0)x for all x ∈ X; S a,k (t)A ⊂ AS a,k (t), i.e., S a,k (t)A(x) ⊂ AS a,k (t)x for x ∈ D(A) and t > 0; and x ∈ X, t > 0, see [16,Definition 2.1]. In the case k(t) = 1, we obtain the resolvent families which are treated in detail in [25]; for k(t) = a(t) = 1, this family of operators is a C 0 -semigroup; and we also retrieve cosine functions for k(t) = 1 and a(t) = t, (t > 0). Subordination theorems for (a, k)-regularized resolvents have been considered in some different works. In [25,Section I.4] The main aim of this paper is to obtain subordination integral formulae to ( t α−1 Γ(α) , t β−1 Γ(β) )regularized resolvents (Theorem 4.5). To achieve this, we present a detailed presentation of Wright and Mittag-Leffler functions in Section 2, which includes some basic results and known connections of these functions and fractional differential equations.
In Section 3, we introduce a new family of bi-parameter special functions ψ α,β in two variables defined by scaling Wright functions (Definition 3.1). This family of functions ψ α,β plays a fundamental role in the subordination principle for ( t α−1 Γ(α) , t β−1 Γ(β) )-regularized resolvent families, see formula (4.3). Moreover, these functions satisfy a nice subordination formula, Theorem 3.6, which extends some known results for Wright M-function and stable Lévy processes, see Remark 3.7. In fact the algebraic nature (for convolution products) of these functions ψ α,β is shown in Proposition 3.4 and 3.5.
Finally, in Section 5 we present some comments, concrete examples and applications to fractional Cauchy problems which illustrate the main results of this paper . Notation. Let R + := [0, ∞), C + := {z ∈ C : Rz > 0}, and L 1 (R + ) be the Lebesgue Banach algebra of integrable function on R + with the usual convolution product The usual Laplace transform of a function f ,f , is defined bŷ for f ∈ L 1 (R + ). Let γ > 0, we denote by g γ (t) := t γ−1 Γ(γ) , t > 0, and g γ = 1 λ γ for λ ∈ C + . The set of continuous functions on R + such that lim t→∞ |f (t)| = 0 is denoted by C 0 (R + ), and the set of holomorphic functions on C + such that lim |z|→∞ |f (z)| = 0 by H 0 (C + ). We denote by X an abstract Banach space, B(X) the set of linear and bounded operators on the Banach space X, and C (∞) c (R + ; X) the set of functions of compact support and infinitely differentiable on R + into X.

Mittag-Leffler and Wright functions
In this section we present definitions and basic properties of Mittag-Leffler and Wright functions. The algebraic structure of those functions have been partially considered in [24] and formulae (2.2) and (2.3) seems to be new.
The Mittag-Leffler functions are defined by We write E α (z) := E α,1 (z). The Mittag-Leffler functions satisfy the following fractional differential problems for 0 < α < 1, under certain initial conditions, where C D α t and R D α t denote the Caputo and Riemann-Liouville fractional derivatives of order α respectively, see section 5 and [19,22]. Their Laplace transform is For more details see [4,Section 1.3].
Recently the next algebraic property has been proved for 0 < α < 1 and ω ∈ C, see [24, Theorem 1]. In fact, a similar identity holds for generalized Mittag-Leffer function E α,β with 0 < α < 1, β > α and for t, s ≥ 0 and ω > 0. The proof of this result is a straightforward consequence of the [14,Theorem 5]. In the case β = α for 0 < α < 1, the algebraic property is which is a direct consequence of Theorem 2.1 and Theorem 2.2 of [20]. The Wright function, that we denote by W λ,µ , was introduced and investigated by E. Maitland Wright in a series of notes starting from 1933 in the framework of the theory of partitions, see [27]. This entire function is defined by the series representation, convergent in the whole complex plane, The equivalence between the above series and the following integral representations of W λ,µ is easily proven by using the Hankel formula for the Gamma function, where Ha denotes the Hankel path defined as a contour that begins at t = −∞ − ia (a > 0), encircles the branch cut that lies along the negative real axis, and ends up at t = −∞ + ib (b > 0), for more details see [19,Appendix F]. It is clear that In addition, as discussed below, the following special cases are of considerable interest: This function has been studied, for example, in [19, p. 257] and [18, Section 6]; a subordination formula for time fractional diffusion process is given in [18,Formula (6.3)] and [19, (F.55)]: for η, β ∈ (0, 1), the following subordination formula holds true for 0 < s, t, This subordination formula had previously appeared in [4,Formula (3.28)].
It is known that where both functions are related to the solutions of the fractional differential problems mentioned above. Nice connections between Mittag-Leffler functions and Wright functions are obtained by the Laplace transform, see formula (2.5) and In the next proposition, we present some interesting properties of Wright functions. The next result extends the study which was done in [4, Chapter 1, p.14] for the case W −α,1−α with 0 < α < 1.
where we have applied the Fubini theorem and the Laplace transform of g η .

Scaled Wright functions
In this section, we introduce two-parameter Wright functions in Definition 3.1, which we call scaled Wright functions. This class of functions includes the Wright M-function introduced in [18,Formula (6.2)] and also considered in [19, (F.51)] and stable Lévy processes. They satisfy important properties (Theorem 3.2 and Proposition 3.4), a subordination principle (Theorem 3.6) and play a crucial role in this paper.
Definition 3.1. For 0 < α < 1 and β ≥ 0, we define the function ψ α,β in two variables by Note that using the change of variable z = σ t , we get the integral representation The function ψ α,β is considered in the literature in some particular cases: is the stable Lévy process of order α, see introduction, [5] and [28, Chapter IX], in particular In the next proposition, we join some properties which are verified by functions ψ α,β .
Proposition 3.4. For 0 < α < 1, β ≥ 0, and 0 < η, the following identity holds Proof. Note that ψ α,β is a Laplace transformable function and locally integrable in two variables. We apply the Laplace transform in variable t to get in the right side In the left side, we also apply the Laplace transform in the variable t to get that ∞ 0 e −λt ψ α,β+αη (t, u) dt = λ −(β+αη) e −λ α u , u > 0, with λ > 0, where we have used Theorem 3.2 (ii).
To finish this section, we prove a subordination formula for functions ψ α,β which expands some well-known results.

Subordination principle and spectral inclusions for regularized resolvent families
In the following we consider that the operator A is a densely defined closed linear operator on a Banach space X. Let α, β > 0. A family {S α,β (t)} t>0 ⊂ B(X) is a (g α , g β )-regularized resolvent family generated by A if the following conditions are satisfied.
(a) S α,β (t) is strongly continuous for t > 0 and lim holds for x ∈ X and t > 0.
Proof. Take x ∈ D(A). By (2.1) and Theorem 4.1 we have that and we obtain (4.1) by the uniqueness of Laplace transform. The proof of (4.2) follows since A is a closed densely defined operator.
For a closed operator A, we denote by σ(A), σ p (A), σ r (A) and σ a (A) the spectrum, point spectrum, residual spectrum and approximate point spectrum of A respectively. For more details see [11,Chapter IV]. The proof of the next theorem is inspired in the proof of [15,Theorem 3.2] and as some parts run parallel we skip them.
The next theorem is the main one of this paper.
Proof. First we show that {S αη1,αη2+β (t)} t>0 is Laplace transformable of parameter ω 1 α . Note that using Proposition 3.2 (ii) we get that The family {S αη1,αη2+β (t)} t>0 is strongly continuous on (0, ∞) and now we prove the strong continuity at the origin. Let x ∈ X, then where we have used Theorem 3.2 (vi). We apply the dominated convergence theorem to the above term and we conclude that is a (g η1 , g η2 )-regularized resolvent family and Proposition 2.1 (i). Finally, by Theorem 4.1, we obtain that the family {S αη1,αη2+β (t)} t>0 is a (g αη1 , g αη2+β )-regularized resolvent family generated by A. The proof of the equality (4.4) is a straightforward consequence of Theorem 3.2 (v).
(i) The fractional Cauchy problem is well-posed and its unique solution is given by is well-posed and its unique solution is given by Proof. In addition, to the particular case of the Laplacian, we obtain the solution of the Caputo fractional diffusion problem of order 0 < α < 1,    C D α t v(t, x) = ∆v(t, x), t > 0, v(0, x) = f (x), (considered in [9,Example 4.13]) is given by v(t, x) = t 0 g 1−α (t − s)u(s, x) ds, for t > 0. 5.3. Multiplication families on C 0 (R n ). Let {T (t)} t≥0 be a multiplication semigroup in R n generated by q(x), that is, Some examples are q(x) = −4π 2 |x| 2 , −2π|x|, − log(1 + 4π 2 |x| 2 ), treated in [7]. Then by Theorem 5.1 (i), the solution of the Riemann-Liouville fractional diffusion problem of order 0 < α < 1    R D α t u(t, x) = q(x)u(t, x), t > 0, (g 1−α * u(·, x))(0) = f (x), x ∈ R n , is given by where we have applied Theorem 3.2 (iii). Even more, by Theorem 5.1 (ii), the solution of the Riemann-Liouville fractional diffusion problem of order 0 < α, γ < 1    R D γ t u(t, x) = −(−q(x)) α u(t, x), t > 0, Conflict of Interests. The authors declare that there is no conflict of interests regarding the publication of this paper.