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On the Trace-Class Property of Hankel Operators Arising in the Theory of the Korteweg–de Vries Equation

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Abstract

The trace-class property of Hankel operators (and their derivatives with respect to the parameter) with strongly oscillating symbol is studied. The approach used is based on Peller’s criterion for the trace-class property of Hankel operators and on the precise analysis of the arising triple integral using the saddle-point method. Apparently, the obtained results are optimal. They are used to study the Cauchy problem for the Korteweg–de Vries equation. Namely, a connection between the smoothness of the solution and the rate of decrease of the initial data at positive infinity is established.

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Authors and Affiliations

  1. Centro de Investigación y de Estudios Avanzados del Instituto Politécnico, Nacional, 07360, Mexico

    S. M. Grudsky

  2. University of Alaska, Fairbanks, 757500, USA

    A. V. Rybkin

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  1. S. M. Grudsky
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  2. A. V. Rybkin
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Correspondence to S. M. Grudsky.

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Original Russian Text © S. M. Grudsky, A. V. Rybkin, 2018, published in Matematicheskie Zametki, 2018, Vol. 104, No. 3, pp. 374–395.

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Grudsky, S.M., Rybkin, A.V. On the Trace-Class Property of Hankel Operators Arising in the Theory of the Korteweg–de Vries Equation. Math Notes 104, 377–394 (2018). https://doi.org/10.1134/S0001434618090067

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