Multidimensional van der Corput-type estimates involving Mittag-Leffler functions

MULTIDIMENSIONAL VAN DER CORPUT-TYPE ESTIMATES INVOLVING

MITTAG-LEFFLER FUNCTIONS Michael Ruzhansky ^1,2, Berikbol T. Torebek ^1,3,4

Abstract

The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The general- isation is that we replace the exponential function with the Mittag-Leffler- type function, to study multidimensional oscillatory integrals appearing in the analysis of time-fractional evolution equations. More specifically, we study two types of integrals with functions Eα,β(iλφ(x)), x ∈ R^N and Eα,β(i^αλφ(x)), x ∈ R^N for the various range of α and β. Several gener- alisations of the van der Corput-type estimates are proved. As an appli- cation of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schr¨odinger equations are considered.

MSC 2010: Primary 42B20; Secondary 26D10, 33E12

Key Words and Phrases: van der Corput-type estimates; Mittag-Leffler function; time-fractional Schr¨odinger equation; time-fractional Klein-Gordon equation

Editorial Note: This paper has been presented at the online interna- tional conference “WFC 2020: Workshop on Fractional Calculus”, Ghent University, Belgium, 9–10 June 2020.

c 2020 Diogenes Co., Sofia

pp. 1663–1677 , DOI: 10.1515/fca-2020-0082

1. Introduction

One of the most important estimates in the theory of harmonic analysis is the van der Corput lemma, which is a decay estimate of the oscillatory in- tegrals. This estimate was obtained by the Dutch mathematician Johannes Gaultherus van der Corput [19] and named in his honour. Let us state the well-known classical van der Corput’s lemma:

• van der Corput lemma. Let φ be a real valued differentiable function such that the φ is monotonic and |φ(x)| ≥ 1, k ≥ 1,for all x∈(a, b),then

b

a

e^iλφ(x)ψ(x)dx

≤Cλ⁻¹, λ→ ∞, (1.1) whereC does not depend onλ.

A multidimensional version of van der Corput’s results would be of great value, but presents many difficulties. Let us consider an integral operator called an oscillatory integral defined by

I(λ) =

R^N

e^iλφ(x)ψ(x)dx, (1.2) where φ(x) and ψ(x) are two functions that map R^N to R and are called the phase and the amplitude, respectively, and λis a positive real number that can vary. It is well known that, if |∇φ| ≥1 on the support of ψ,then the following estimate is true

|I(λ)| ≤Cλ⁻¹. (1.3)

The decay rate here is sharp, but the constantCmay depend on phase and the van der Corput’s estimate does not scale well. Again, such an estimate is closely related to the problem of multilinear sublevel set estimates [10], one of the fundamental problems of harmonic analysis. Certain parame- ter dependent sublevel set estimates were obtained and used by Kamotski and Ruzhansky [7] in the analysis of elliptic and hyperbolic systems with multiplicities, to yield Sobolev space estimates for relevant classes of oscil- latory integrals and for the solutions of the hyperbolic systems of equations.

There are various versions of the van der Corput estimate but most with one-dimensional rate of decay. Furthermore, Carbery, Christ and Wright [2] and Ruzhansky [12] proposed multidimensional versions of the van der Corput lemma, in formulations where also the constant in the estimate is independent of the phase function.

Recently, the attention of many mathematicians has been attracted by various generalizations of Van der Corput-type estimates for integrals involving special functions [3, 13, 14, 20, 21, 22].

The main goal of the present paper is to obtain multidimensional van der Corput-type estimates for the integrals defined by

Iα,β¹ (λ) =

R^N

Eα,β(iλφ(x))ψ(x)dx, (1.4) and

Iα,β² (λ) =

R^N

Eα,β(i^αλφ(x))ψ(x)dx, (1.5) where 0< α, β∈R, φis a phase andψis an amplitude, andλis a positive real number that can vary. Here Eα,β(z) is the Mittag-Leffler function defined as (see e.g. [8, 5])

Eα,β(z) = ∞ k=0

z^k

Γ(αk+β), α >0, β ∈R, (1.6) for which we note that

E1,1(z) =e^z. (1.7)

This present paper is a continuation of [13, 14], where a variety of van der Corput type lemmas were obtained for the integrals (1.4) and (1.5) in the case N = 1.

Such integrals as in (1.4) and (1.5) arise in the study of decay estimates of solutions of the time-fractional Schr¨odinger and the time-fractional wave equations (for example see [4, 6, 9, 16, 17, 18]). In Section 4 we will give several immediate applications of the obtained estimates to time-fractional Schr¨odinger and time-fractional Klein-Gordon equations.

As the Mittag-Leffler functions have a very rich analytic structure, and, philosophically, give a generalisation of the usual exponent, the obtained estimate will also involve the dependence on the parametersα and β. 1.1. Preliminaries. We will often make use of the following estimate.

Proposition 1.1 ([11]). If0< α <2, β is an arbitrary real number, μis such thatπα/2< μ <min{π, πα}, then there isC1, C2 >0,such that we have

|Eα,β(z)| ≤ C

1 +|z|, z ∈C, μ≤ |arg(z)| ≤π, (1.8) and

|Eα,β(z)| ≤C1(1 +|z|)^(1−β)/αexp(Re(z^1/α)) + C2

1 +|z|, z∈C, |arg(z)| ≤μ. (1.9)

Proposition 1.2 ([1]). The following optimal estimates are valid for the real-valued Mittag-Leffler function

1

1 + Γ(1−α)x ≤Eα,1(−x)≤ 1

1 + _Γ(1+α)¹ x, x≥0, 0< α <1; (1.10) 1

1 +

Γ(1−α) Γ(1+α)x

2 ≤Γ(α)Eα,α(−x)

≤ 1

1 +

Γ(1+α) Γ(1+2α)x

2, x≥0, 0< α <1; (1.11)

1

1 + ^Γ(β_Γ(β)^−α)x ≤Γ(β)Eα,β(−x)

≤ 1

1 + _Γ(β+α)^Γ(β) x, x≥0, 0< α≤1, β > α. (1.12) Proposition 1.3. ([13]) Let α, β >0 and φ:R^N → C. Then for all λ∈Cwe have

Eα,β(iλφ(x)) =E2α,β

−λ²φ²(x) +iλφ(x)E2α,β+α

−λ²φ²(x) . (1.13) 2. Van der Corput-type estimates for the integral I_α,β¹ (λ) In this section we consider the integral operator defined by

Iα,β¹ (λ) =

R^N

Eα,β(iλφ(x))ψ(x)dx,

where 0 < α ≤ 1, β > 0, φ is a phase and ψ is an amplitude, and λ is a positive real number that can vary. We are interested in particular in the behavior of I_α,β¹ (λ) when λ is large, as for small λ the integral is just bounded.

Theorem 2.1. Let φ : R^N → R be a measurable function and let ψ∈L¹(R^N).If 0< α < 1, β >0,and m= ess inf_x∈RN|φ(x)|>0,then we have the estimate

|Iα,β¹ (λ)| ≤ M ψ L¹(R^N)

1 +mλ , λ≥0, (2.1)

where M does not depend onφ, ψ and λ.

P r o o f. As for smallλthe integral (1.4) is just bounded, we give the proof forλ≥1.Letφ:R^N →Rbe a measurable function andψ∈L¹(R^N). Then

|I_α,β¹ (λ)|=

R^N

Eα,β(iλφ(x))ψ(x)dx ≤

R^N

|Eα,β(iλφ(x))| |ψ(x)|dx.

Using formula (1.13) and estimate (1.9) we have that

|Eα,β(iλφ(x))| ≤E2α,β

−λ²φ²(x) +λ|φ(x)|E2α,α+β

−λ²φ²(x)

≤ C

1 +λ²φ²(x) + Cλ|φ(x)|

1 +λ²φ²(x) ≤C 1 +λ|φ(x)|

1 +λ²φ²(x).

(2.2) Asφandψdo not depend onλ,andm= ess inf_x∈RN|φ(x)|>0,then from (2.2) we have

|Iα,β¹ (λ)| ≤

R^N

|Eα,β(iλφ(x))| |ψ(x)|dx≤C

R^N

1 +λ|φ(x)|

1 +λ²φ²(x)|ψ(x)|dx

≤2C

R^N

1 +λ|φ(x)|

(1 +λ|φ(x)|)² |ψ(x)|dx≤C1

R^N

|ψ(x)|

1 +λ|φ(x)|dx≤ M ψ L¹(R^N)

1 +mλ .

The proof is complete. 2

Now we find upper and lower bounds in estimates for I_α,β^1,Ω(λ) =

Ω

Eα,β(iλφ(x))ψ(x)dx, (2.3) where Ω⊂R^N is a bounded domain, 0< α≤ ¹₂ and β ≥2α.

Theorem 2.2. Let 0 < α ≤ 1/2, β > 2α. Let φ ∈ L^∞(Ω) be a real-valued function and let ψ∈L^∞(Ω).

Suppose that m1 = inf

x∈Ω|φ(x)|> 0 and m2 = inf

x∈Ω|ψ(x)|>0. Then we have

|I_α,β^1,Ω(λ)| ≤K1|Ω| ψ L^∞(Ω)

1 +λ φ L^∞(Ω)

1 +k1λ²m²₁ , λ≥0, (2.4) where |Ω|is a Lebesgue measure ofΩ,

K1 = max 1

Γ(β), 1 Γ(α+β)

and

k1 = min

Γ(β)

Γ(2α+β), Γ(α+β) Γ(3α+β)

.

We also have

|I_α,β^1,Ω(λ)| ≥ m2|Ω| Γ(α+β)

λm1

1 +^Γ(β_Γ(α+β)^−α)λ² φ ²_L∞(Ω)

, λ≥0. (2.5)

If m1 = inf

x∈Ω|φ(x)|= 0 and m2 = inf

x∈Ω|ψ(x)|>0.Then we have

|I_α,β^1,Ω(λ)| ≥ m2|Ω| Γ(β)

1

1 +^Γ(β_Γ(β)^−2α)λ² φ ²_L∞(Ω)

, λ≥0. (2.6)

P r o o f. First, we prove estimate (2.4). Let φ ∈ L^∞(Ω) and ψ ∈ L^∞(Ω). Then

|I_α,β^1,Ω(λ)| ≤

Ω

|Eα,β(iλφ(x))| |ψ(x)|dx.

Using formula (1.13) and estimate (1.9) we have that

|Eα,β(iλφ(x))| ≤E2α,β

−λ²φ²(x) +λ|φ(x)|E2α,α+β

−λ²φ²(x) . (2.7) The properties of functions φandψ,and the use of estimate (1.12) lead to the result

|I_α,β^1,Ω(λ)| ≤

Ω

|Eα,β(iλφ(x))| |ψ(x)|dx

≤

Ω

E2α,β

−λ²φ²(x) +λ|φ(x)|E2α,α+β

−λ²φ²(x)

|ψ(x)|dx

≤ ψ L^∞(Ω)

Γ(β)

Ω

1

1 + _Γ(2α+β)^Γ(β) λ²φ²(x)dx + ψ L^∞(Ω)

Γ(α+β)

Ω

λ|φ(x)|

1 +_Γ(3α+β)^Γ(α+β)λ²φ²(x)dx

≤ ψ L^∞max 1

Γ(β), 1 Γ(α+β)

×

×

⎡

⎣

Ω

1

1 + _Γ(2α+β)^Γ(β) λ²φ²(x)dx+

Ω

λ|φ(x)|

1 + _Γ(3α+β)^Γ(α+β)λ²φ²(x)dx

⎤

⎦

≤ |Ω| ψ L^∞max 1

Γ(β), 1 Γ(α+β)

⎡⎣ 1

1+_Γ(2α+β)^Γ(β) λ²m²₁+ λ φ L^∞

1+_Γ(3α+β)^Γ(α+β)λ²m²₁

⎤

⎦

≤ |Ω| ψ L^∞max 1

Γ(β), 1 Γ(α+β)

1 +λ φ L^∞

1 + min

_Γ(β)

Γ(2α+β),_Γ(3α+β)^Γ(α+β)

λ²m²₁. Now, we prove the lower bound estimate (2.6). As |x+iy| ≥ |x|,we obtain

|Eα,β(iλφ(x))| ≥E2α,β

−λ²φ²(x) . (2.8) Hence, by estimate (1.12), it follows that

|I_α,β^1,Ω(λ)| ≥m2

Ω

E2α,β

−λ²φ²(x) dx

≥ m2

Γ(β)

Ω

1

1 + ^Γ(β_Γ(β)^−2α)λ²φ²(x)dx

≥ m2|Ω|

Γ(β)

1

1 +^Γ(β_Γ(β)^−2α)λ² φ ²_L∞(Ω)

.

Now, we prove the lower bound estimate (2.5). As|x+iy| ≥ |y|,we obtain

|Eα,β(iλφ(x))| ≥λ|φ(x)|E2α,α+β

−λ²φ²(x) . (2.9) Hence, by estimate (1.12), it follows

|I_α,β^1,Ω(λ)| ≥m2λ

Ω

|φ(x)|E2α,α+β

−λ²φ²(x) dx

≥ m2λ Γ(α+β)

Ω

|φ(x)|

1 +^Γ(β_Γ(α+β)^−α)λ²φ²(x)dx

≥ m2|Ω| Γ(α+β)

λm1

1 + ^Γ(β_Γ(α+β)^−α)λ² φ ²_L∞(Ω)

.

The proof is complete. 2

Theorem 2.3. Let 0 < α < 1/2, β = 2α. Let φ ∈ L^∞(Ω) be a real-valued function and let ψ∈L^∞(Ω).

Let m1 = inf

x∈Ω|φ(x)| > 0 and m2 = inf

x∈Ω|ψ(x)| > 0, then we have the following estimates

|Iα,2α^1,Ω (λ)| ≤K|Ω| ψ L^∞(Ω)

×1 +λ φ L^∞(Ω)

1 +

Γ(1+2α)

Γ(1+4α)λ² φ ²_L∞(Ω)

1 + min Γ(3α)

Γ(5α),

Γ(1+2α) Γ(1+4α)

λ²m²1

2 , λ≥0, (2.10)

and

|I_α,2α^1,Ω (λ)| ≥ m2|Ω| Γ(3α)

λm1

1 + _Γ(3α)^Γ(α)λ² φ ²_L∞(Ω)

, λ≥0, (2.11) where K= max

1

Γ(2α),_Γ(3α)¹

. If m1 = inf

x∈Ω|φ(x)|= 0 and m2 = inf

x∈Ω|ψ(x)|>0,then we have

|Iα,2α^1,Ω (λ)| ≥ m2|Ω|

Γ(2α) 1

1 +

Γ(1−2α)

Γ(1+2α)λ² φ ²_L∞(Ω)

2, λ≥0. (2.12)

P r o o f. First, we prove the estimate (2.10). Let φ ∈ L^∞(Ω) and ψ∈L^∞(Ω).Then

|I_α,2α^1,Ω (λ)| ≤

Ω

|Eα,2α(iλφ(x))| |ψ(x)|dx.

Using formula (1.13), we have that

|Eα,2α(iλφ(x))| ≤E2α,2α

−λ²φ²(x) +λ|φ(x)|E2α,3α

−λ²φ²(x) . (2.13) The properties of functions φ and ψ, using estimates (1.11) and (1.12), imply

|I_α,2α^1,Ω (λ)| ≤

Ω

|Eα,2α(iλφ(x))| |ψ(x)|dx

≤

Ω

E2α,2α

−λ²φ²(x) +λ|φ(x)|E2α,3α

−λ²φ²(x)

|ψ(x)|dx

≤ ψ L^∞(Ω)

Γ(2α)

Ω

1 1 +

Γ(1+2α)

Γ(1+4α)λ²φ²(x) 2dx

+ ψ L^∞(Ω)

Γ(3α)

Ω

λ|φ(x)|

1 + ^Γ(3α)_Γ(5α)λ²φ²(x)dx

≤ ψ L^∞(Ω)K

Ω

1 1 +

Γ(1+2α)

Γ(1+4α)λ²φ²(x) 2dx

+ ψ L^∞(Ω)K

Ω

λ|φ(x)|

1 + ^Γ(3α)_Γ(5α)λ²φ²(x)dx

≤ K|Ω| ψ L^∞(Ω)

1 +

Γ(1+2α) Γ(1+4α)λ²m²1

2

+K|Ω| ψ L^∞(Ω)

λ φ L^∞(Ω)

1 +

Γ(1+2α)

Γ(1+4α)λ² φ ²_L∞(Ω)

1 +^Γ(3α)_Γ(5α)λ²m²₁ 1 +

Γ(1+2α) Γ(1+4α)λ²m²₁

≤K|Ω| ψ L^∞(Ω)

1 +λ φ L^∞(Ω)

1 +

Γ(1+2α)

Γ(1+4α)λ² φ ²_L∞(Ω)

1 + min Γ(3α)

Γ(5α),

Γ(1+2α) Γ(1+4α)

λ²m²₁

2 ,

where K= max 1

Γ(2α),_Γ(3α)¹

.

Now, we prove estimate (2.12). We have

|Eα,2α(iλφ(x))| ≥E2α,2α

−λ²φ²(x) . (2.14) Hence, by estimate (1.11), it follows that

|I_α,2α^1,Ω (λ)| ≥m2

Ω

E2α,2α

−λ²φ²(x) dx

≥ m2

Γ(2α)

Ω

1 1 +

Γ(1−2α)

Γ(1+2α)λ²φ²(x) 2dx

≥ m2|Ω| Γ(2α)

1 1 +

Γ(1−2α)

Γ(1+2α)λ² φ ²_L∞(Ω)

2.

The estimate (2.11) can be proved similarly as estimate (2.5) by replacing β = 2α. In fact,

|I_α,2α^1,Ω (λ)| ≥m2λ

Ω

|φ(x)|E2α,3α

−λ²φ²(x) dx

≥ m2λ Γ(3α)

Ω

|φ(x)|

1 + _Γ(3α)^Γ(α)λ²φ²(x)dx

≥ m2|Ω| Γ(3α)

λm1

1 +_Γ(3α)^Γ(α)λ² φ ²_L∞(Ω)

.

The proof is complete. 2 3. Van der Corput-type estimates for the integral I_α,β² (λ).

In this section we consider I_α,β² defined by (1.4), that is, Iα,β² (λ) =

R^N

Eα,β(i^αλφ(x))ψ(x)dx.

As for smallλthe integral (1.4) is just bounded, we consider the caseλ≥1. Theorem 3.1. Let φ:R^N →Rbe a measurable function and let ψ∈ L¹(R^N). Suppose that 0< α ≤2, β > 1,and m= ess inf_x∈RN|φ(x)|>0, then

(i): for0< α <2 andβ ≥α+ 1we have

|Iα,β² (λ)| ≤ M1

1 +λm ψ L¹(R^N), λ≥1, (3.1) whereM1 does not depend onφ, ψ andλ;

(ii): for0< α <2 and 1< β < α+ 1we have

|I_α,β² (λ)| ≤ M2

(1 +λm)^β−1α ψ L¹(R^N), λ≥1, (3.2) whereM2 does not depend onφ, ψ andλ;

(iii): forα= 2 and 1< β <3we have

|I2,β² (λ)| ≤ M3

(1 +λm)^β−12 ψ L¹(R^N), λ≥1, (3.3) whereM3 does not depend onφ, ψ andλ.

P r o o f. Let φ:R^N →R be a measurable function andψ ∈L¹(R^N). As

|arg(i^αλφ(x))|= πα 2 , and

Re(iλ^1/α(φ(x))^1/α) = 0, then using estimate (1.9) we have that

|Iα,β² (λ)| ≤

R^N

|Eα,β(i^αλφ(x))| |ψ(x)|dx

≤C1

R^N

(1 +λ|φ(x)|)^(1−β)/α|ψ(x)|dx

+C2

R^N

|ψ(x)| 1 +λ|φ(x)|dx.

As φand ψ do not depend on λ,and m = ess inf_x∈RN|φ(x)|>0,then for β≥α+ 1 we have

|I_α,β² (λ)| ≤

R^N

|Eα,β(i^αλφ(x))| |ψ(x)|dx

≤max{C1, C2}

R^N

|ψ(x)| 1 +λ|φ(x)|dx

≤ M1

1 +λm ψ L¹(R^N). In the case 1< β < α+ 1 we have that

|Iα,β² (λ)| ≤

R^N

|Eα,β(i^αλφ(x))| |ψ(x)|dx

≤max{C1, C2}

R^N

|ψ(x)|

(1 +λ|φ(x)|)^β−1α dx

≤ M2

(1 +λm)^β−1α ψ L¹(R^N). The cases (i) and (ii) are proved.

Now we will prove the case (iii). Applying the asymptotic estimate (see [8, page 43])

E2,β(z) = 1

2z^(1−β)/2

e^√z+e^{−√z−πi(1−β)sign(argz)}

− N k=1

z^−k

Γ(β−2k) +O 1

z^N+1

, |z| → ∞, |arg(z)| ≤π, we have

|I2,β² (λ)| ≤

R^N

|E2,β(−λφ(x))| |ψ(x)|dx

≤M3λ^(1−β)/2

R^N

|φ(x)|^(1−β)/2|ψ(x)|dx

≤M3m^(1−β)/2λ^(1−β)/2

R^N

|ψ(x)|dx

≤ M3 ψ L¹(R^N)

(1 +mλ)^(β−1)/2.

HereM3 is a constant that does not depend onλ.The proof is complete.

2 4. Applications

In this section we give some applications of multidimensional van der Corput-type estimates involving Mittag-Leffler function.

4.1. Decay estimates for the time-fractional Schr¨odinger equation.

Consider the time-fractional Schr¨odinger equation

iD^α0+,tu(t, x)−Δ_xu(t, x) +μu(t, x) = 0, t >0, x∈R^N, (4.1) with Cauchy data

u(0, x) =ψ(x), x∈R^N, (4.2) where λ, μ >0 and

D^α0+,tu(t, x) = 1 Γ(1−α)

t

0

(t−s)^−αus(s, x)ds is the Caputo fractional derivative of order 0< α <1.

By using the direct and inverse Fourier and Laplace transforms, we can obtain a solution to problem (4.1)-(4.2) in the form

u(t, x) =

R^N

e^ixξEα,1

i(|ξ|²+μ)t^α ψˆ(ξ)dξ, (4.3) where ˆψ(ξ) = _(2π)¹_N

R^N e^−iyξψ(y)dy. Suppose that ψ ∈ L¹(R^N) and ˆψ ∈ L¹(R^N).As

ξ∈Rinf^N(|ξ|²+μ)>0,

then using Theorem 2.1 we obtain the dispersive estimate u(t,·) _L∞(R^N)≤C(1 +t)^−α ψ ˆ L¹(R^N), t≥0.

4.2. Decay estimates for the time-fractional Klein-Gordon equa- tion. Consider the time-fractional Klein-Gordon equation

D0+,t^α u(t, x) +i^αΔ_xu(t, x)−i^αμu(t, x) = 0, t >0, x∈R^N, (4.4) with initial data

I0+,t^2−αu(0, x) = 0, x∈R^N, (4.5)

∂tI_0+,t^2−αu(0, x) =ψ(x), x∈R^N, (4.6) where μ >0,and

I0+,t^α u(t, x) = 1 Γ(α)

t

0

(t−s)^α−1u(s, x)ds

and

D^α0+,tu(t, x) = 1 Γ(2−α)∂t²

t

0

(t−s)^1−αu(s, x)ds

is the Riemann-Liouville fractional integral and derivative of order 1< α≤ 2.

If α= 2,then from (4.4) we obtain a classical Klein-Gordon equation.

Applying the Fourier transform F to problem (4.4)-(4.6) with respect to space variable xyields

D^α0+,tuˆ(t, ξ)−i^α(|ξ|²+μ)ˆu(t, ξ) = 0, t >0, ξ∈R^N, (4.7) I0+,t^2−αuˆ(0, ξ) = 0, ξ∈R^N, (4.8)

∂tI_0+,t^2−αuˆt(0, ξ) = ˆψ(ξ), ξ∈R^N, (4.9) due to F {Δ_xu(t, x)}=−|ξ|²uˆ(t, ξ). The general solution of equation (4.7) can be represented as

uˆ(t, ξ) =C1(ξ)t^α−1Eα,α

i^α(|ξ|²+μ)t^α +C2(ξ)t^α−2Eα,α−1

i^α(|ξ|²+μ)t^α , whereC1(ξ) andC2(ξ) are unknown coefficients. Then by initial conditions (4.8)-(4.9) we have

uˆ(t, ξ) = ˆψ(ξ)t^α−1Eα,α

i^α(|ξ|²+μ)t^α . By applying the inverse Fourier transform F⁻¹ we have

u(t, x) =

R^N

e^ixξt^α−1Eα,α

i^α(|ξ|²+μ)t^α ψˆ(ξ)dξ, (4.10) where ˆψ(ξ) = _(2π)¹_N

R^N e^−iyξψ(y)dy.

Suppose thatψ∈L¹(R^N) and ˆψ∈L¹(R^N).As inf

ξ∈R^N(|ξ|²+μ)>0,then using Theorem 3.1 (ii) we obtain the dispersive estimate

u(t,·) _L∞(R^N)≤Ct^α−1(1 +t^α)^1−αα ψ ˆ L¹(R^N)

≤Ct^α−1(1 +t)^1−α ψ ˆ L¹(R^N), t >0. Acknowledgements

The authors were supported in parts by the FWO Odysseus Project 1 grant G.0H94.18N: Analysis and Partial Differential Equations. The first author was supported in parts by the EPSRC grant EP/R003025/1. The second author was supported by the Ministry of Education and Science of the Republic of Kazakhstan Grant AP08052046. No new data was collected or generated during the course of research.

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1 Dept. of Mathematics: Analysis, Logic and Discrete Mathematics Ghent University, Krijgslaan 281

Building S8 B 9000 Ghent, BELGIUM

2 School of Mathematical Sciences, Queen Mary University of London London, UNITED KINGDOM

e-mail: michael.ruzhansky@ugent.be

3 Al–Farabi Kazakh National University

71 Al–Farabi ave., Almaty, 050040, KAZAKHSTAN

4 Institute of Mathematics and Mathematical Modeling 125 Pushkin str., Almaty, 050010, KAZAKHSTAN e-mail: berikbol.torebek@ugent.be (Corresponding author)

Received: June 9, 2020

Please cite to this paper as published in:

Fract. Calc. Appl. Anal., Vol. 23, No 6 (2020), pp. 1663–1677, DOI: 10.1515/fca-2020-0082