Van der Corput lemmas for Mittag-Leffler functions. II. $\alpha$-directions

The paper is devoted to study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory integrals appearing in the analysis of time-fractional partial differential equations. More specifically, we study integral of the form $I_{\alpha,\beta}(\lambda)=\int_\mathbb{R}E_{\alpha,\beta}\left(i^\alpha\lambda \phi(x)\right)\psi(x)dx,$ for the range $0<\alpha\leq 2,\,\beta>0$. This extends the variety of estimates obtained in the first part, where integrals with functions $E_{\alpha,\beta}\left(i \lambda \phi(x)\right)$ have been studied. Several generalisations of the van der Corput lemmas are proved. As an application of the above results, the generalised Riemann-Lebesgue lemma, the Cauchy problem for the time-fractional Klein-Gordon and time-fractional Schr\"{o}dinger equations are considered.


Introduction
In this paper we continue the study of oscillatory-type integrals involving Mittag-Leffler functions E α,β initiated in [RT20].In the case of α = β = 1, we have E 1,1 (z) = e z , thus reducing the integral to the classical question of decay of oscillatory integrals.
Indeed, the estimate obtained by the Dutch mathematician Johannes Gaultherus van der Corput [vdC21] and named in his honour, following Stein [St93], can be stated as follows: • van der Corput lemma.Suppose φ is a real-valued and smooth function in for k = 1 and φ ′ is monotonic, or k ≥ 2.Here C does not depend on λ.Various generalisations of the van der Corput lemmas have been investigated over the years [Gr05, SW70, St93, PS92, PS94, Rog05, Par08, Xi17].Multidimensional analogues of the van der Corput lemmas were studied in [BG11, CCW99, CLTT05, GPT07, PSS01, KR07], while in [Ruz12] the multi-dimensional van der Corput lemma was obtained with constants independent of the phase and amplitude.
The main goal of the present paper is to study van der Corput lemmas for the oscillatory integral defined by where 0 < α < 2, β > 0, φ is a phase and ψ is an amplitude, and λ is a positive real number that can vary.Here E α,β (z) is the Mittag-Leffler function defined as (see e.g.[KST06,GKMR14]) with the property that E 1,1 (z) = e z .
Here we can point out already one extension of (1.1) in view of (1.3), namely, an extension (in Theorem 3.5) to the range 0 < α < 2 in the form b a E α,α (i α λφ(x)) ψ(x)dx ≤ Cλ −1/k , λ → ∞, (1.4) for k = 1 and φ ′ is monotonic, or k ≥ 2. This present paper is a continuation of [RT20], where a variety of van der Corput type lemmas were obtained for the integral defined by where 0 < α < 1, β > 0. As we see above, the integral (1.5) is different from the integral (1.2), since in (1.5) there is a purely imaginary number i before the phase function, and in (1.2) the fractional power of the imaginary number, i.e. i α .In addition, the asymptotic behavior of the Mittag-Leffler function in these cases is also different, yielding different decay rates.Such integrals as in (1.2) arise in the study of decay estimates of solutions of the time-fractional Schrödinger and the time-fractional wave equations (for example see [DX08,Gr19,Nab04,SZ20]).In Section 4 we will give several immediate applications of the obtained estimates to time-fractional Klein-Gordon and Schrödinger equations.
As in the case of (1.5) studied in [RT20], we find that the decay rates of (1.2) as λ → ∞ depend not only on the assumptions on the phase but also on the ranges of parameters α and β.We also obtain more results in the case of bounded intervals.For the convenience of the reader, let us briefly summarise the results of this paper, distinguishing between different sets of assumptions: van der Corput lemmas on R: consider I α,β defined by (1.2).
for k = 1 and φ ′ is monotonic, or k ≥ 2.Here M k does not depend on λ.
where M does not depend on λ.We will often make use of the following estimate.

Van der Corput lemma in R
In this section we consider I α,β defined by (1.2), that is, As for small λ the integral (1.2) is just bounded, we consider the case λ ≥ 1.
Proof.Let φ : R → R be a measurable function and and Re(iλ 1/α (φ(x)) 1/α ) = 0, then using estimate (1.6) we have that As φ and ψ do not depend on λ, and m = ess inf x∈R |φ(x)| > 0, then for β ≥ α + 1 we have In the case 1 < β < α + 1 we have that The cases (i) and (ii) are proved.Now we will prove the case (iii).Applying the asymptotic estimate (see [KST06, page 43]) Here M 3 is a constant that does not depend on λ.The proof is complete.
It is easy to see from estimate (1.6) that, for β = 1, the function E α,1 (•) does not decrease, and it will be only bounded function.In this case, the method of proving the Theorem 2.1 does not suitable.Below we give an estimate for the integral (1.2) in the case β = 1.
Theorem 2.2.Let φ : R → R be an invertible and differentiable function, and let where M does not depend on φ, ψ and λ.

Van der Corput lemma in finite interval
In this section we consider integral (1.2) in the finite interval (3.1) Since I α,β (λ) is bounded for small λ, further we can assume that λ ≥ 1.

Proof. Proof of (i)
where C is an arbitrary constant independent of φ and λ.
where M 1 is the arbitrary constant independent of λ.
Let us prove the estimate (3.2) by the induction method for k ≥ 2. We assume that (3.2) is true for k ≥ 2. And assuming |φ (k+1) (x)| ≥ 1, for all x ∈ I, we prove the estimate (3.2 Further, we will write

By inductive hypothesis
As If 0 < α < 2 and 1 < β < α + 1, then by (1.6) we have where C is an arbitrary constant independent of φ and λ.Let c = a, then by φ ∈ C 1 (I) and |φ ′ (x)| ≥ 1, we have , where M 2 is the arbitrary constant independent of λ.
Let us prove the estimate (3.3) by induction method on k ≥ 2. We assume that (3.3) holds for k ≥ 2. Assuming |φ (k+1) (x)| ≥ 1, for all x ∈ I, we prove the estimate k+1) we obtain the estimate (3.3) for k + 1, which proves the result.The cases when c = a or c = b can be proved similarly.

By inductive hypothesis
where M k does not depend on λ; (ii): for 0 < α < 2 and 1 < β < α + 1 we have where M k does not depend on λ.
Proof.We write (3.1) as where Let 0 < α < 2 and β ≥ α + 1. Integrating by parts and applying the estimate of part (i) of Theorem 3.1 we obtain The case (ii) can be proved similarly by applying results of part (ii) of Theorem 3.1.
for k = 1 and φ ′ is monotonic, or k ≥ 2.Here M k does not depend on λ.
We note that the classical van der Corput lemma (1.1) is covered by (3.7) with α = 1.
Proof.First we will prove the case k = 1.Let 0 < α < 2, λ ≥ 1 and let φ has one zero c ∈ [a, b].Let us consider the integral

Then integrating by parts gives
As φ ′ is monotonic and φ ′ (x) ≥ 1 for all x ∈ [a, b], then 1 φ ′ is also monotonic, and has a fixed sign.Hence estimate (1.6) and φ(c Here M 1 does not depended on λ.We prove (3.7 Taking ǫ = λ − 1 2 we obtain the estimate (3.7) for k = 2.We prove the case k ≥ 2 by induction method.Let (3.7) is true for k, and suppose
Below we show that if φ ′ is not monotonic, then to obtain estimate (3.7) when k = 1, it is necessary to increase the smoothness of function φ.
where M does not depend on λ.
Proof.Suppose that φ ∈ C 2 (I) and |φ ′ (x)| ≥ 1 for all x ∈ I, then from (3.8) we have Since φ ∈ C 2 (I) and |φ ′ (x)| ≥ 1 for all x ∈ I, then the function will be continuous and bounded, and therefore by (1.6) we have where , C ≥ |E α,1 (z)| and M is a constant independent of λ.
Theorem 3.5.Let 0 < α < 2 and let φ be a real-valued function such that φ ∈ for k = 1 and φ ′ is monotonic, or k ≥ 2.Here M k does not depend on λ.
Theorem 3.5 can be proved similarly as Theorem 3.1.The case of α = 1 corresponds to the classical van der Corput lemma (1.1).Also, for k = 1, Theorem 3.5 holds if we replace the condition that φ ′ is monotonic by φ ∈ C 2 (I).

Applications
In this section we give some applications of van der Corput lemmas involving Mittag-Leffler function.
Applying the Fourier transform F to problem (4.1)-(4.3)with respect to space variable x yields The general solution of equation (4.4) can be represented as where C 1 (ξ) and C 2 (ξ) are unknown coefficients.Then by initial conditions (4.5)-(4.6)we have û By applying the inverse Fourier transform F −1 we have where ψ(ξ) = 1 π R e −iyξ ψ(y)dy.

4. 1 .
Applications to the fractional evolution equations.4.1.1.Decay estimates for the time-fractional Klein-Gordon equation.Consider the time-fractional Klein-Gordon equation