A. Kassymov, “Some weak geometric inequalities for the Riesz potential”, Eurasian Math. J., 2020, Volume 11, Number 3,Pages <nobr>42

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Some weak geometric inequalities for the Riesz potential

A. Kassymov^abc

^a Al-Farabi Kazakh National University, 71 Al-Farabi Ave, 050040 Almaty, Kazakhstan
^b Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, S8 Building, Ghent, Belgium
^c Institute of Mathematics and Mathematical Modeling, 125 Pushkin St, 050010 Almaty, Kazakhstan

Abstract: In the present paper, we prove that the first eigenvalue of the Riesz potential is weakly maximised in a quasi-ball among all Haar measurable sets on homogeneous Lie groups. It is an analogue of the classical Rayleigh–Faber–Krahn inequality for the Riesz potential. We also prove a weak version of the Hong–Krahn–Szegö inequality for the Riesz potential on homogeneous Lie groups.

Keywords and phrases: convolution operators, Riesz potential, Rayleigh–Faber–Krahn inequality, Hong–Krahn–Szegö inequality, homogeneous Lie group.

MSC: 35P99, 47G40

Received: 18.06.2019

Language: English

DOI: 10.32523/2077-9879-2020-11-3-42-50