# A Subordination Principle on Wright Functions and Regularized Resolvent Families

Authors: Luciano Abadias; Pedro J. Miana
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We obtain a vector-valued subordination principle forgα,gβ-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 Lévy 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.


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## Body

## 1. Introduction

A functionf:(0,∞)→R is a Bernstein function if f is of class C∞, f(λ)≥0 for all λ>0 and(1)-1n-1fn(λ)≥0,λ>0,n∈N.The celebrated Bochner subordination theorem characterizes Bernstein functions: given f, a Bernstein function, there exists a unique convolution semigroup of subprobability measures (μt)t>0 on [0,∞) such that (2)e-tf(λ)=∫0∞‍e-λsdμts,Rλ>0.Conversely, given a convolution semigroup of subprobability measures (μt)t>0 on [0,∞), then there exists a unique Bernstein function f such that (2) holds true; see, for example, [1, Theorem 5.2]. The original subordination principle for stochastic processes in connection with diffusion equations and semigroups was introduced in [2]. In [3, Chapters 4.3, and 4.4], a detailed study of stochastic processes, their transition semigroups, generators, and subordination results are developed.Now letA be a densely defined closed linear operator on a Banach space X which generates a C0-contraction semigroup (T(t))t>0⊂B(X). Then the solution of the first order abstract Cauchy problem (3)u′(t)=Au(t),t>0,u(0)=xis given by u(t)=T(t)x for t>0. Now, suppose that (μt)t>0 is a vaguely continuous convolution semigroup of subprobability measures on [0,∞) with the corresponding Bernstein function f. Then the Bochner integral (4)Tftx≔∫0∞‍Tsxdμts,x∈X,t>0,defines again a C0-contraction semigroup on X [1, Proposition 12.1]. Then the semigroup (Tf(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 dμt(s)=ft,α(s)ds (where ft,α are the stable Lévy processes, see (32)), then (5)Tαtx≔∫0∞‍ft,αsTsxds,x∈X,t>0,is an analytic semigroup generated by -(-A)α, the fractional powers of the generator A according to Balakrishnan. For more details, see [4, Chapter IX].Other subordination formulae allow defining new families of operators from some previous ones by integration. LetA be the generator of a cosine function (C(t))t>0 on a Banach space X (see definition in [5, Section 3.14]). Then A generates a holomorphic C0-semigroup (T(z))z∈C+ of angle π/2, given by (6)Tzx=1πz∫0∞‍e-s2/4zCsxds,x∈X,z∈C+,[5, Theorem 3.14.17]. Remember that the solution of the second order Cauchy problem (7)u′′(t)=Au(t),t>0,u(0)=x,u′(0)=0is u(t)=C(t)x for t>0 [5, Section 3.14].In [6], 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, C0-semigroups of operators, cosine families, n-times integrated semigroups, resolvent families, and k-generalized solutions. Let a∈Lloc⁡1(R+) and k∈C(R+). The family {Sa,k(t)}t>0⊂B(X) is a (a,k)-regularized resolvent family generated by A if the following conditions are fulfilled: Sa,k(t) is strongly continuous for t>0 and Sa,k(0)x=k(0)x for all x∈X; Sa,k(t)A⊂ASa,k(t), that is, Sa,k(t)A(x)⊂ASa,k(t)x, for x∈D(A) and t>0; and(8)Sa,ktx=ktx+A∫0t‍at-sSa,ksxds,x∈X,t>0;see [6, Definition 2.1]. In the case k(t)=1, we obtain the resolvent families which are treated in detail in [7]; for k(t)=a(t)=1, this family of operators is a C0-semigroup, and we also retrieve cosine functions for k(t)=1 and a(t)=t(t>0). Some spectral properties for (a,k)-regularized resolvent families are proved in [8].Subordination theorems for(a,k)-regularized resolvents have been considered in some different works. In [7, Section I.4], the use of the theory of Bernstein functions, completely positive functions, and the Laplace transform allows showing some subordination results for (a,1)-regularized resolvents [7, Theorem 4.1, Corollaries 4.4 and 4.5]. A nice subordination theorem for m-times integrated semigroup is proved in [6, Theorem 3.7]. In the case of (tα-1/Γ(α),1)-regularized functions, this subordination theorem is improved in [9, Theorem 3.1] and [10, Theorem 3.1] and an integral representation, similar to formula (6), is also proved involving Wright functions. In [11, Theorem 3.1], using holomorphic functional calculus, the authors prove a subordination result for (tα-1/Γ(α),1)-regularized resolvent families generated by fractional powers of closed operators, which extends both [4, Chapter IX, Section 11, Theorem 2] and [9, Theorem 3.1]. Finally, in [12, Theorem 2.8], a subordination principle for (a,k)-regularized resolvents, inspired in the original proof of [7, Theorem 4.1], is shown. In all these results, note that the subordination integral formula is only present in [9, Theorem 3.1] and [11, Theorem 3.1].The main aim of this paper is to obtain subordination integral formulae to(tα-1/Γ(α),tβ-1/Γ(β))-regularized resolvents (Theorem 12). 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 Section3, we introduce a new family of biparameter special functions ψα,β in two variables defined by scaling Wright functions (Definition 2). This family of functions ψα,β plays a fundamental role in the subordination principle for (tα-1/Γ(α),tβ-1/Γ(β))-regularized resolvent families; see formula (51). Moreover, these functions satisfy a nice subordination formula, Theorem 8, which extends some known results for Wright M-function and stable Lévy processes; see Remark 9. In fact the algebraic nature (for convolution products) of these functions ψα,β is shown in Propositions 5 and 7.In Section4, we consider (tα-1/Γ(α),tβ-1/Γ(β))-regularized resolvents in abstract Banach spaces. We prove the main subordination principle, Theorem 12, and some consequences in Remark 13 and Corollary 14.Finally, in Section5 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 L1(R+) be the Lebesgue Banach algebra of integrable function on R+ with the usual convolution product (9)f*gt=∫0t‍ft-sgsds,f,g∈L1R+,t≥0.The usual Laplace transform of a function f, f^, is defined by (10)f^λ=∫0∞‍fte-λtdt,λ∈C+,for f∈L1(R+). Let γ>0, and we denote gγ(t)≔tγ-1/Γ(γ), t>0, and gγ^(λ)=1/λγ for λ∈C+.The set of continuous functions onR+ such that limt→∞|f(t)|=0 is denoted as C0(R+), and the set of holomorphic functions on C+ such that lim|z|→∞|f(z)|=0 is denoted as H0(C+). We denote by X an abstract Banach space, B(X) the set of linear and bounded operators on the Banach space X, and Cc(∞)(R+;X) the set of functions of compact support and infinitely differentiable on R+ into X.


## 2. 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 has been partially considered in [13] and formulae (15) and (16) seem to be new.The Mittag-Leffler functions are defined by(11)Eα,β(z)≔∑n=0∞‍znΓαn+β,α,β>0,z∈C.We write Eα(z)≔Eα,1(z). The Mittag-Leffler functions satisfy the following fractional differential problems: (12)CDtαEα(ωtα)=ωEα(ωtα),RDtαtα-1Eα,αωtα=ωtα-1Eα,αωtα,for 0<α<1, under certain initial conditions, where DtαC and DtαR denote the Caputo and Riemann-Liouville fractional derivatives of order α, respectively; see Section 5 and [14, 15]. Their Laplace transform is (13)∫0∞‍e-λttβ-1Eα,β(ωtα)dt=λα-βλα-ω,Rλ>ω1/α,ω>0.For more details see [10, Section 1.3].Recently, the next algebraic property has been proved(14)∫tt+s‍-∫0s‍Eαωrαt+s-rαdr=α∫0t‍∫0s‍Eαωr1αEαωr2αt+s-r1-r21+αdr1dr2,t,s≥0,for 0<α<1 and ω∈C; see [13, Theorem 1]. In fact, a similar identity holds for generalized Mittag-Leffler function Eα,β with 0<α<1, β>α, and (15)∫tt+s‍-∫0s‍t+s-rβ-α-1Γβ-αrβ-1Eα,βωrαdr=αΓ1-α∫0t‍∫0s‍r1β-1Eα,βωr1αr2β-1Eα,βωr2αt+s-r1-r21+αdr1dr2,for t,s≥0 and ω>0. The proof of this result is a straightforward consequence of [16, Theorem 5]. In the case β=α for 0<α<1, the algebraic property is (16)t+sα-1Eα,αωt+sα=αΓ1-α∫0t‍∫0s‍r1α-1Eα,αωr1αr2α-1Eα,αωr2αt+s-r1-r21+αdr1dr2,which is a direct consequence of Theorems 2.1 and 2.2 of [17].The Wright function that we denote byWλ,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 [18]. This entire function is defined by the series representation, convergent in the whole complex plane: (17)Wλ,μz≔∑n=0∞‍znn!Γλn+μ,λ>-1,μ∈C.The equivalence between the above series and the following integral representations ofWλ,μ is easily proven by using the Hankel formula for the Gamma function: (18)Wλ,μz=12πi∫Ha‍σ-μeσ+zσ-λdσ,λ>-1,μ≥0,z∈C,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 [14, Appendix F]. It is clear that (19)ddzWλ,μ(z)=Wλ,λ+μ(z),z∈C.In addition, as discussed below, the following special cases are of considerable interest:(20)Mαz≔W-α,1-α-z,Fαz≔W-α,0-z,0<α<1,z∈C,interrelated through (21)Fαz=αzMαz,z∈C.The WrightM-function in two variables M is defined by (22)Mαs,t≔t-αMαst-α,t>0,s∈R.This function has been studied, for example, in [14, page 257] and [19, Section 6]; a subordination formula for time fractional diffusion process is given in [19, Formula (6.3)] and [14, Formula (F.55)]: for η,β∈(0,1), the following subordination formula holds true;(23)Mηβs,t=∫0∞‍Mηs,τMβτ,tdτ,t,s>0.This subordination formula had previously appeared in [10, Formula (3.28)].The deep connection between fractional differential equations (in space and in time) and Wright-type functions(Wλ,μ,Mα,Fα,Mα,…) has been studied in detail in [14, 19, 20].It is known that(24)Eα,α+βz=∫0∞‍eztW-α,β-tdt,z∈C,0<α<1,β≥0;that is, Eα,α+β(-(·)) is the Laplace transform of W-α,β(-(·)) in the whole complex plane; see [14, Formula (F.25)]. Then, observe that for 0<α<1(25)Eαz=∫0∞‍eztMαtdt,Eα,αz=∫0∞‍eztFαtdt,z∈C,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 (24) and (26)∫0∞‍e-ztWλ,μ±rdr=1zEλ,μ±1z,z>0,λ>0,[14, Formula (F.22)]. In the next proposition, we present some interesting properties of Wright functions. The next result extends the study which was done in [10, Chapter 1, page 14] for the case W-α,1-α with 0<α<1.Proposition 1.
Let0<α<1 and β≥0. Then the following properties hold:(i)
∫0∞‍tη-1/ΓηW-α,β-tdt=1/Γαη+β; η>0.(ii)
W-α,β(-t)≥0,  for t>0.Proof.
(i) Using the definition ofW-α,β, we have(27)∫0∞‍tη-1Γ(η)W-α,β(-t)dt=12πi∫Ha‍eσσ-β-αηdσ=gβ+αη(1),where we have applied the Fubini theorem and the Laplace transform of gη.
(ii) The positivity ofW-α,β(-t) follows from (24), the complete monotonicity of Eα,γ(-t) for t>0, 0<α<1, and γ≥α (see [14, Appendix E, Formula (E.32)]), and the Post-Widder inversion formula; see [10, Lemma 1.6].


## 3. Scaled Wright Functions

In this section, we introduce two-parameter Wright functions in Definition2, which we call scaled Wright functions. This class of functions includes the Wright M-function introduced in [19, Formula (6.2)] and also considered in [14, Formula (F.51)] and stable Lévy processes. They satisfy important properties (Theorem 3 and Proposition 5) and a subordination principle (Theorem 8) and play a crucial role in this paper.Definition 2.
For0<α<1 and β≥0, we define the function ψα,β in two variables by (28)ψα,βt,s≔tβ-1W-α,β-st-α,t>0,s∈C.Note that, using the change of variablez=σ/t, we get the integral representation (29)ψα,βt,s=12πi∫Ha‍z-βetz-szαdz,t,s>0.The functionψα,β is considered in the literature in some particular cases.(i)
Forβ=1-α, (30)ψα,1-αt,s=t-αMαst-α=Mαs,t=φt,αs,t,s>0,where Mα(s,t) is the Wright M-function in two variables studied in [14, page 257] and φt,α(s) is considered in [10, Theorem 3.1]; for α=1/2, (31)ψ1/2,1/2t,s=1πte-s2/4t,t,s>0;see [14, Appendix F, Formula (F.16)].(ii)
Forβ=0,(32)ψα,0t,s=12πi∫Ha‍etz-szαdz≕fs,αt,t,s>0,is the stable Lévy process of order α; see Section 1, [2], and [4, Chapter IX], in particular (33)ψ1/2,0t,s=12πt-3/2se-s2/4t,t,s>0.In the next theorem, we join some properties which are satisfied by functionsψα,β.Theorem 3.
Let0<α<1 and β≥0; one has the following. (i)
ψα,β(t,s)≥0,  for t,s>0.(ii)
∫0∞‍e-λtψα,β(t,s)dt=λ-βe-λαs,  for s,λ>0.(iii)
∫0∞‍eλsψα,β(t,s)ds=tα+β-1Eα,α+β(λtα),  for t>0, λ∈C.(iv)
∬0∞e-λse-μtψα,βt,sdsdt=1/μβ(μα+λ)  for t>0, λ,μ>0.(v)
ψα,β+γ(t,s)=(gγ*ψα,β(·,s))(t),  for t,s,γ>0.(vi)
∫0∞‍gη(s)ψα,β(t,s)ds=gαη+β(t),  for t,η>0.Proof.
(i) It is clear by Definition2 and Proposition 1. (ii) It is easy to see that ψα,β(t,s) is of exponential growth in t. Let λ>max⁡(a,b)>0, (where a and b are involved in the definition of Hankel path Ha). Then, by Cauchy theorem of residue, (34)∫0∞‍e-λtψα,β(t,s)dt=12πi∫Ha‍z-βe-szαz-λdz=λ-βe-sλα.(iii) Using (24), the result is direct by a change of variable. (iv) We combine parts (ii) and (iii) to obtain the equality. (v) It is clear using Laplace transform and (i). (vi) It is clear by a change of variable and applying Proposition 1(i).We combine Theorem3(ii) and (iii) and formula (13) to get the following corollary.Corollary 4.
For0<α,  γ<1, one denotes by Ψγ,α the function given by(35)Ψγ,αt,s≔∫0∞‍ψγ,0t,uψα,0s,udu,t,s>0.Then Ψγ,α(t,s)=Ψα,γ(s,t), and(36)∫0∞‍e-λtΨγ,α(t,s)dt=sα-1Eα,α(-λγsα),s>0,∬0∞e-μse-λtΨγ,αt,sdtds=1λγ+μα,for μ,λ>0.Note that(37)Ψ1/2,1/2t,s=12πt+s3/2,t,s>0.The following key lemma includes the particular caseα=β=1/2 and 0<η proved in [21, Lemma 1].Proposition 5.
For0<α<1, β≥0, and 0<η, the following identity holds: (38)ψα,β+αηt,u=∫u∞‍gηs-uψα,β(t,s)dsfor t,u>0.Proof.
Note thatψα,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 (39)∫0∞‍e-λt∫u∞‍gηs-uψα,βt,sdsdt=λ-β∫u∞‍gηs-ue-λαsds=λ-βλ-αηe-λαu,with λ>0, where we have applied Theorem 3(ii) and [15, Chapter II, Formula (5.11)].
In the left side, we also apply the Laplace transform in the variablet to get that (40)∫0∞‍e-λtψα,β+αηt,udt=λ-β+αηe-λαu,u>0,with λ>0, where we have used Theorem 3(ii).Remark 6.
Foru=0 in Proposition 5, we obtain the equality (41)ψα,β+αηt,0=∫0∞‍gηsψα,βt,sds=gαη+βt,t>0,proved in Theorem 3(vi).Finally, we show an algebraic identity which satisfies functionsψα,1-α.Proposition 7.
Take0<α<1, β>α, and t,s>0. Then (i)
∫tt+s‍-∫0s‍((t+s-r)β-α-1/Γβ-α)ψα,β-αr,udr = α/Γ1-α∫0t‍∫0s‍(ψα,β-αr1,·*ψα,β-αr2,·u/(t+s-r1-r2)1+α)dr1dr2;(ii)
ψα,0(t+s,u)=(α/Γ1-α)∫0t‍∫0s‍((ψα,0(r1,(·))*ψα,0(r2,(·)))u/(t+s-r1-r2)1+α)dr1dr2.Proof.
Note thatψα,β-α^(r,(·))(λ)=rβ-1Eα,β(-λrα), for β≥α,  r>0, and λ∈C; see Theorem 3(iii). Then we apply the Laplace transform in the variable u in both equalities to get identities (15) and (16). The injectivity of Laplace transform allows us to finish the proof.To finish this section, we prove a subordination formula for functionsψα,β which expands some well-known results.Theorem 8.
For0<α,δ<1, β≥α, and δ≥γ, the following identity holds:(42)ψαγ,β-α+αδ-γt,s=∫0∞‍ψα,β-αt,rψγ,δ-γr,sdr,t,s>0.Proof.
To show this theorem, we apply the Laplace transform in both variables(t,s), the so-called double Laplace transform, Fubini theorem, Theorem 3(ii) and (iii), and finally formula (13) to get that (43)∬0∞e-λt-μs∫0∞‍ψα,β-αt,rψγ,δ-γr,sdrdsdt=∫0∞‍λα-βe-λαrrγ-1Eγ,δ-μrγdr=λαγ-β-α+αγλαγ+μ,for Rλ,Rμ>0. Due to Theorem 3(iv) and the uniqueness of the double Laplace transform (see, e.g., [22, page 346]), we conclude the equality.Remark 9.
In the case thatβ=δ=1, we obtain formula (23). For α=β and γ=δ, we get the following subordination formula for stable Lévy processes: (44)fs,αγt=∫0∞‍fr,αtfs,γrdr,s,t>0,for 0<α, γ<1. Finally, for α=γ=1/2 and β=δ=1, we obtain that (45)ψ1/4,3/4t,s=1πt∫0∞‍e-r2/4t-s2/4rdrr,t,s>0,where we use equality (31).


## 4. Subordination Principle for Regularized Resolvent Families

In the following we consider that the operatorA is a densely defined closed linear operator on a Banach space X. Let α,β>0. A family {Sα,β(t)}t>0⊂B(X) is a (gα,gβ)-regularized resolvent family generated by A if the following conditions are satisfied.(a)
Sα,β(t) is strongly continuous for t>0 and limt→0+Sα,β(t)x/gβ(t)=x for all x∈X.(b)
Sα,β(t)A⊂ASα,β(t); that is, Sα,β(t)A(x)⊂ASα,β(t)x for x∈D(A) and t>0.(c)
The integral equation(46)Sα,β(t)x=tβ-1Γ(β)x+A∫0t‍t-sα-1ΓαSα,βsxdsholds for x∈X and t>0.This family of operators was formerly introduced for general kernels (a,k) in [6, Definition 2.1]. The above definition is also considered for α>0 and β≥1 in [23], for 0<α=β<1 in [17], and for α>0 and β=1 in [9, Definition 2.1].The reason why we do not consider the value ofSα,β(·) at 0 in condition (a) (compare with [6, Definition 2.1, Condition (R1)]) is that the function t↦gβ(t) has a singularity at 0 if 0<β<1.LetS:(0,∞)→B(X) be a strongly continuous operator family such that S(·)x∈Lloc⁡1(R+,X), for any x∈X. The operator family {S(t)}t>0 is said to be Laplace-transformable if there exists ω∈R such that the Laplace transform of S, (47)S^λx=∫0∞‍e-λtStxdt,Rλ>ω,converges for x∈X; see, for example, [5, Definition 3.1.4]. If A generates a (gα,gβ)-regularized resolvent family {Sα,β(t)}t>0 such that S(·)x∈Lloc⁡1(R+,X) for x∈X and is Laplace transformable of parameter ω, we write A∈Cα,β(ω). We denote Cα,β≔⋃‍{Cα,β(ω);ω≥0}. The case of (gα,1)-regularized resolvent families exponentially bounded, Sα,1(t)≤Mewt for t>0, is considered in [11, Definition 2.5] and [10, Definition 2.4]; in this case, Cα(ω)≔Cα,1(ω).The next theorem characterizes the Laplace transform of(gα,gβ)-regularized family and extends [23, Theorem 3.11] and the proof is similar to the proof of [6, Proposition 3.1].Theorem 10.
Letα,β>0. Then A∈Cα,β(ω) if and only if (ωα,∞)⊂ρ(A) and there exists a strongly continuous function Sα,β(·):(0,∞)→B(X), locally integrable, lim⁡t→0+(Sα,β(t)x/gβ(t))=x for all x∈X, and Laplace transformable such that (48)∫0∞‍e-λtSα,βtxdt=λα-βλα-A-1x,λ>ω,for all x∈X. Furthermore, the family {Sα,β(t)}t>0 is the (gα,gβ)-regularized resolvent family generated by A.Example 11.
For0<α<1 and β≥α, the family {Sα,β(t)}t>0 defined by (49)Sα,βtfs≔ψα,β-αt,·*fs=∫0s‍f(s-u)ψα,β-α(t,u)du,hhhhihs∈R+,f∈L1(R+),is a (gα,gβ)-regularized resolvent family generated by A=-d/dt on the Banach space L1(R+). First note that the family is well defined since ψα,β-αt,·1=gβ(t) for t>0; see Theorem 3(vi). In addition, by Theorem 3(ii), we have that (50)∫0∞‍e-λtSα,βtfsdt=∫0s‍fs-u∫0∞‍e-λtψα,β-αt,udtdu=λα-β∫0s‍e-λαufs-udu=λα-βλα+ddt-1fs,s∈R+.The strong continuity at the origin is a simple check, and the reader can follow the steps on the proof of Theorem 12; in particular, the stable Lévy processes (f(·),α(t))t>0 are convolution (gα,gα)-regularized resolvent families. Similarly, for λ∈C, Mittag-Leffler functions (tβ-1Eα,β(-λtα))t≥0 are (gα,gβ)-regularized resolvent families generated by Af(z)=-λf(z) on the Banach space C0(R+) (or H0(C+)); see formula (13). These two families of functions are canonical examples of (gα,gβ)-regularized resolvent families.The next theorem is the main one of this paper.Theorem 12.
Let0<η1≤2,  0<η2, and ω≥0. If A∈Cη1,η2(ω) generates a (gη1,gη2)-regularized resolvent family {Sη1,η2(t)}t>0, then A∈Cαη1,αη2+β(ω1/α) generates the following (gαη1,gαη2+β)-regularized resolvent family: (51)Sαη1,αη2+βtx≔∫0∞‍ψα,βt,sSη1,η2sxds,t>0,x∈X,for 0<α<1 and β≥0. Moreover the following equality holds: (52)Sαη1,αη2+βtx=gβ*Sαη1,αη2tx,t>0,x∈X,for β>0.Proof.
First we show that{Sαη1,αη2+β(t)}t>0 is Laplace transformable of parameter ω1/α. Note that using Theorem 3(ii) we get that (53)∫0∞‍e-λtSαη1,αη2+β(t)xdt=∫0∞‍∫0∞‍e-λtψα,βt,sdtSη1,η2(s)xds=λ-β∫0∞‍e-λαsSη1,η2(s)xds=λαη1-(αη2+β)λαη1-A-1x,for λ>ω1/α, and (ωη1,∞)=((ω1/α)αη1,∞)⊂ρ(A).
The family{Sαη1,αη2+β(t)}t>0 is strongly continuous on (0,∞) and now we prove the strong continuity at the origin. Let x∈X, and then (54)Sαη1,αη2+β(t)x-gαη2+β(t)xgαη2+β(t)≤∫0∞‍ψα,β(t,s)gαη2+β(t)Sη1,η2(s)x-gη2(s)xds=Γαη2+β∫0∞‍tα-αη2W-α,β-uhhhhhhhhhhhhh·Sη1,η2(utα)x-gη2(utα)xdu=Γαη2+β∫0∞‍gη2uW-α,β-ugη2utαhhhhhhhhhhhhh·Sη1,η2utαx-gη2utαxdu,where we have used Theorem 3(vi). We apply the dominated convergence theorem to the above term and Proposition 1(i) to conclude that(55)Sαη1,αη2+β(t)x-gαη2+β(t)xgαη2+β(t)⟶0,t⟶0+,since Sη1,η2(t) is a (gη1,gη2)-regularized resolvent family. Finally, by Theorem 10, we obtain that the family {Sαη1,αη2+β(t)}t>0 is a (gαη1,gαη2+β)-regularized resolvent family generated by A. The proof of equality (52) is a straightforward consequence of Theorem 3(v).Remark 13.
Note that the above subordination theorem extends some subordination results which have been considered in this paper and previous papers.(i)
Now, we consider the family of functions(ψγ,δ-γ(t,(·)))t>0 which is a (gγ,gδ)-regularized resolvent family for 0<γ<1 and δ≥γ (see Example 11). Then we apply Theorem 12 for 0<α<1 and β>0, and we obtain the formula (56)ψαγ,β+αδ-γt,·=∫0∞‍ψα,βt,sψγ,δ-γs,·ds,t>0,which is shown in Theorem 8.(ii)
If0<η1≤2 and η2=1 in Theorem 12, we retrieve the subordination principle for (gαη1,1)-regularized resolvent families given in [9, Theorem 3.1] and [10, Theorem 3.1] for 0<α<1 and β=1-α.(iii)
By [12, Theorem 2.8(i)], given a (gη1,gη2)-regularized resolvent family generated by A, we obtain a (gγ1,gγ2)-regularized resolvent family generated by A, where 0<γ1<η1 and γ2=1-γ1/η1+(γ1/η1)η2. This is a particular case of Theorem 12 for α=γ1/η1 and β=1-γ1/η1.Corollary 14.
Let0<η≤2 and 0<α<1. If {Sη,η(t)}t>0 is a (gη,gη)-regularized resolvent family generated by A∈Cη,η(ω), then A∈Cαη,αη(ω1/α) generates the following (gαη,gαη)-regularized resolvent family: (57)Sαη,αηtx≔∫0∞‍ψα,0t,sSη,ηsxds,t>0,x∈X.In particular, we have the following remarkable particular cases.(i)
IfA generates a C0-semigroup {T(t)}t>0, that is, a (g1,g1)-regularized resolvent family, satisfying T(t)≤Meωt, t≥0, then A∈Cα,α(ω1/α) generates the following (gα,gα)-regularized resolvent family: (58)Sα,αtx≔∫0∞‍ψα,0t,sTsxds,t>0,x∈X.(ii)
Letβ∈(1,2] and {Sβ,β(t)}t>0 be a (gβ,gβ)-regularized resolvent family generated by A∈Cβ,β(ω). Then A∈C1,1(ωβ) generates the following C0-semigroup: (59)T(t)x≔∫0∞‍ψ1/β,0t,sSβ,βsxds,t>0,x∈X.In the case that A is the generator of an exponentially bounded sine function S(t)≤Meωt,  (t≥0) (see details in [5, Section 3.15]), that is, (S(t))t>0 is a (g2,g2)-regularized resolvent family, then A generates the following C0-semigroup: (60)T(t)x≔12πt-3/2∫0∞‍se-s2/4tSsxds,t>0,x∈X,with exponential bound ω2, which is proved in [24, Theorem 5.2], and see also [25, Proposition 2.5 and Theorem 2.6].


## 5. Applications, Examples, and Final Comments

In this last section we present some applications of our results to fractional Cauchy problems and classical semigroups in Lebesgue spaces. Now we recall some basic definitions on fractional Cauchy problems. Letf∈Cc∞R+;X; we call Riemann-Liouville fractional integral of f of order γ>0, I-γf to the function given by (61)I-γft≔gγ*ft=∫0t‍t-sγ-1Γγfsds,t≥0,and the Riemann-Liouville fractional derivative of f of order γ>0 is given by (62)RDtγft≔dndtnI-n-γft,t≥0,and with n=[γ]+1. Also, we consider the Caputo fractional derivative of f of order γ>0,(63)CDtγf(t)≔gn-γ*f(n)(t)=I-(n-γ)f(n)(t),t≥0,with n=γ+1; see, for example [14, 15] and [11, Section 4]. Note that, in the above definitions, the function f can be taken in a larger space than Cc(∞)(R+) where the definitions make sense.


### 5.1. Fractional Powers in Fractional Cauchy Problems

The next results extend [11, Theorem 4.9(a), (c)]: for 0<α<1, the solutions of Caputo fractional problems are (gα,1)-regularized resolvent families (see [10, Definition 2.3]) which may be obtained by integration from (gα,gα)-regularized resolvent families by (52). Note that these are solutions of Riemann-Liouville fractional problems; see [17, Theorem 1.1].Theorem 15.
Letα,γ∈0,1 and A be the generator of a uniformly bounded C0-semigroup {T(t)}t>0⊂B(X).(i)
The fractional Cauchy problem(64)RDtαv(t)=Av(t),t>0,(g1-α*v)(0)=x∈D(A)is well-posed and its unique solution is given by (65)vt=∫0∞‍ψα,0t,sTsxds,t>0.(ii)
The fractional Cauchy problem(66)RDtγv(t)=--Aαv(t),t>0,g1-γ*v0=x∈D(A)is well-posed and its unique solution is given by (67)v(t)=∫0∞‍Ψγ,αt,sTsxds,t>0.Proof.
(i) By Corollary14(i), the operator A generates a (gα,gα)-regularized resolvent family which provides the solution of fractional Cauchy problem (64); see [17, Theorem 1.1]. (ii) By [4, Chapter IX], the operator -(-A)α generates a C0-semigroup {T(α)(t)}t>0 given by (5). By part (i), Fubini theorem, and the definition of Ψγ,α in Corollary 4, we conclude the proof.


### 5.2. Convolution Semigroups onLp(Rn)

Let{T(t)}t≥0 be a uniformly bounded convolution semigroup in Rn generated by A; that is, (68)Ttfx=kt*fx=∫Rn‍ktx-yfydy,t>0,x∈Rn.Two well-known examples of convolution semigroups are Gaussian and Poisson semigroups, gt(x)=(1/(4πt)n/2)e-|x|2/4t and pt(x)=(Γ(n+1)/2/π(n+1)/2)(t/(t2+|x|2)(n+1)/2), whose generators are the Laplacian Δ and -(-Δ)1/2, respectively; see, for example, [4, Chapter IX]. Then, by Theorem 15(i), the solution of the Riemann-Liouville fractional diffusion problem of order 0<α<1(69)RDtαut,x=Aut,x,t>0,g1-α*u·,x(0)=fxis given by (70)u(t,x)=∫Rn‍∫0∞‍ψα,0t,sksx-ydsfydy.In addition, to the particular case of the Laplacian, we obtain the solution of the Caputo fractional diffusion problem of order 0<α<1, (71)CDtαvt,x=Δvt,x,t>0,v(0,x)=f(x),(considered in [11, Example 4.13]) which is given by v(t,x)=∫0t‍g1-α(t-s)u(s,x)ds, for t>0.


### 5.3. Multiplication Families onC0(Rn)

Let{T(t)}t≥0 be a multiplication semigroup in Rn generated by qx; that is,(72)Ttfx=etqxfx,x∈Rn,t>0.Some examples are q(x)=-4π2|x|2,-2π|x|,-log⁡(1+4π2|x|2), treated in [26]. Then, by Theorem 15(i), the solution of the Riemann-Liouville fractional diffusion problem of order 0<α<1(73)RDtαut,x=q(x)ut,x,t>0,g1-α*u·,x0=fx,x∈Rn,is given by (74)ut,x=∫0∞‍ψα,0(t,s)esq(x)f(x)ds=tα-1Eα,αqxtαfx,t>0,x∈Rn,where we have applied Theorem 3(iii). Even more, by Theorem 15(ii), the solution of the Riemann-Liouville fractional diffusion problem of order 0<α, γ<1, (75)RDtγut,x=--qxαu(t,x),t>0,(g1-γ*u(·,x))(0)=f(x)is given by (76)ut,x=∫0∞‍Ψγ,αt,sesqxfxds=tγ-1Eγ,γ--qxαtγfx,t>0,where we have applied Theorem 3(ii) and (iii).


## 5.1. Fractional Powers in Fractional Cauchy Problems

The next results extend [11, Theorem 4.9(a), (c)]: for 0<α<1, the solutions of Caputo fractional problems are (gα,1)-regularized resolvent families (see [10, Definition 2.3]) which may be obtained by integration from (gα,gα)-regularized resolvent families by (52). Note that these are solutions of Riemann-Liouville fractional problems; see [17, Theorem 1.1].Theorem 15.
Letα,γ∈0,1 and A be the generator of a uniformly bounded C0-semigroup {T(t)}t>0⊂B(X).(i)
The fractional Cauchy problem(64)RDtαv(t)=Av(t),t>0,(g1-α*v)(0)=x∈D(A)is well-posed and its unique solution is given by (65)vt=∫0∞‍ψα,0t,sTsxds,t>0.(ii)
The fractional Cauchy problem(66)RDtγv(t)=--Aαv(t),t>0,g1-γ*v0=x∈D(A)is well-posed and its unique solution is given by (67)v(t)=∫0∞‍Ψγ,αt,sTsxds,t>0.Proof.
(i) By Corollary14(i), the operator A generates a (gα,gα)-regularized resolvent family which provides the solution of fractional Cauchy problem (64); see [17, Theorem 1.1]. (ii) By [4, Chapter IX], the operator -(-A)α generates a C0-semigroup {T(α)(t)}t>0 given by (5). By part (i), Fubini theorem, and the definition of Ψγ,α in Corollary 4, we conclude the proof.


## 5.2. Convolution Semigroups onLp(Rn)

Let{T(t)}t≥0 be a uniformly bounded convolution semigroup in Rn generated by A; that is, (68)Ttfx=kt*fx=∫Rn‍ktx-yfydy,t>0,x∈Rn.Two well-known examples of convolution semigroups are Gaussian and Poisson semigroups, gt(x)=(1/(4πt)n/2)e-|x|2/4t and pt(x)=(Γ(n+1)/2/π(n+1)/2)(t/(t2+|x|2)(n+1)/2), whose generators are the Laplacian Δ and -(-Δ)1/2, respectively; see, for example, [4, Chapter IX]. Then, by Theorem 15(i), the solution of the Riemann-Liouville fractional diffusion problem of order 0<α<1(69)RDtαut,x=Aut,x,t>0,g1-α*u·,x(0)=fxis given by (70)u(t,x)=∫Rn‍∫0∞‍ψα,0t,sksx-ydsfydy.In addition, to the particular case of the Laplacian, we obtain the solution of the Caputo fractional diffusion problem of order 0<α<1, (71)CDtαvt,x=Δvt,x,t>0,v(0,x)=f(x),(considered in [11, Example 4.13]) which is given by v(t,x)=∫0t‍g1-α(t-s)u(s,x)ds, for t>0.


## 5.3. Multiplication Families onC0(Rn)

Let{T(t)}t≥0 be a multiplication semigroup in Rn generated by qx; that is,(72)Ttfx=etqxfx,x∈Rn,t>0.Some examples are q(x)=-4π2|x|2,-2π|x|,-log⁡(1+4π2|x|2), treated in [26]. Then, by Theorem 15(i), the solution of the Riemann-Liouville fractional diffusion problem of order 0<α<1(73)RDtαut,x=q(x)ut,x,t>0,g1-α*u·,x0=fx,x∈Rn,is given by (74)ut,x=∫0∞‍ψα,0(t,s)esq(x)f(x)ds=tα-1Eα,αqxtαfx,t>0,x∈Rn,where we have applied Theorem 3(iii). Even more, by Theorem 15(ii), the solution of the Riemann-Liouville fractional diffusion problem of order 0<α, γ<1, (75)RDtγut,x=--qxαu(t,x),t>0,(g1-γ*u(·,x))(0)=f(x)is given by (76)ut,x=∫0∞‍Ψγ,αt,sesqxfxds=tγ-1Eγ,γ--qxαtγfx,t>0,where we have applied Theorem 3(ii) and (iii).



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