Van der Corput lemmas for Mittag-Leffler functions. II. α–directions

Michael Ruzhansky,Berikbol T Torebek

doi:10.1016/j.bulsci.2021.103016

Abstract

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 Iα,β(λ)=∫REα,β(iαλϕ(x))ψ(x)dx, for the range 0<α≤2,β>0. This extends the variety of estimates obtained in the first part, where integrals with functions Eα,β(iλϕ(x)) 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 Schrödinger equations are considered.

Highlights

The paper is devoted to study analogues of the van der Corput lemmas involving Mittag-Leffler functions
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

Summary

Van der Corput lemma on finite intervals

We assume that (3.2) is true for k = 1 and let |φ (x)| ≥ 1, for all x ∈ I, we prove the estimate (3.2) for k = 2. Let us prove the estimate (3.3) by induction method on k ≥ 2. Taking = λ k+1 (1 + λ) α(k+1) we obtain the estimate (3.3) for k + 1, which proves the result. Integrating by parts and applying the estimate of part (i) of Theorem 3.1 we obtain log(1 + λ) |Iα,β(λ)| ≤ Mk (1 + λ)1/k |ψ(b)| + |ψ (x)|dx. The case (ii) can be proved by applying results of part (ii) of Theorem 3.1.

Applications

Conclusion