This article is cited in 4 papers

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

S. M. Grudsky^a, A. V. Rybkin<sup>b

a</sup> Centro de Investigación y de Estudios Avanzados del Instituto Politécnico Nacional
^b University of Alaska Fairbanks

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.

Keywords: Hankel operator, trace-class operator, Korteweg–de Vries equation, inverse problem method.

UDC: 517.957+517.984.2

Received: 05.02.2018

DOI: 10.4213/mzm12112

 English version: Mathematical Notes, 2018, 104:3, 377–394

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