Geodesic
On the Trace-Class Property of Hankel Operators Arising in the Theory of the Korteweg--de Vries Equation
Matematičeskie zametki, Tome 104 (2018) no. 3, pp. 374-395

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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. 
@article{MZM_2018_104_3_a4,
     author = {S. M. Grudsky and A. V. Rybkin},
     title = {On the {Trace-Class} {Property} of {Hankel} {Operators} {Arising} in the {Theory} of the {Korteweg--de} {Vries} {Equation}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {374--395},
     publisher = {mathdoc},
     volume = {104},
     number = {3},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_104_3_a4/}
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S. M. Grudsky; A. V. Rybkin. On the Trace-Class Property of Hankel Operators Arising in the Theory of the Korteweg--de Vries Equation. Matematičeskie zametki, Tome 104 (2018) no. 3, pp. 374-395. http://geodesic.mathdoc.fr/item/MZM_2018_104_3_a4/