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Multidimensional van der Corput-Type Estimates Involving Mittag-Leffler Functions

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Abstract

The paper is devoted to study multidimensional van der Corput-type estimates for the intergrals involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Lefflertype 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 ∈ RN and Eα,β (iαλφ(x)), x ∈ RN for the various range of α and β. Several generalisations of the van der Corput-type estimates are proved. As an application of the above results, the Cauchy problem for the multidimensional time-fractional Klein-Gordon and time-fractional Schrödinger equations are considered.

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Authors and Affiliations

  1. Dept. of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281 Building S8 B, 9000, Ghent, Belgium

    Michael Ruzhansky & Berikbol T. Torebek

  2. School of Mathematical Sciences, Queen Mary University of London, London, UK

    Michael Ruzhansky

  3. Al—Farabi Kazakh National University, 71 Al—Farabi ave., Almaty, 050040, Kazakhstan

    Berikbol T. Torebek

  4. Institute of Mathematics and Mathematical Modeling, 125 Pushkin str., Almaty, 050010, Kazakhstan

    Berikbol T. Torebek

Authors
  1. Michael Ruzhansky
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  2. Berikbol T. Torebek
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Correspondence to Michael Ruzhansky.

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Ruzhansky, M., Torebek, B.T. Multidimensional van der Corput-Type Estimates Involving Mittag-Leffler Functions. Fract Calc Appl Anal 23, 1663–1677 (2020). https://doi.org/10.1515/fca-2020-0082

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  • DOI: https://doi.org/10.1515/fca-2020-0082

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