An exact inequality of different metrics is obtained for discrete Luxemburg norms in a finite-dimensional space. As a consequence, using this inequality, an inequality of different metrics is proved for Luxemburg norms on functions for which there is an upper bound for the norm of a derivative in terms of the norm of the function, and an alternative proof is presented for S.M. Nikol’skii’s inequality of different metrics for norms of a trigonometric polynomial in Orlicz spaces.
Keywords: inequality of different metrics, discrete Luxemburg norm, trigonometric polynomial, Orlicz space
Received April 11, 2024
Revised August 27, 2024
Accepted September 2, 2024
Funding Agency: This work was performed as a part of the research conducted in the Ural Mathematical Center and supported by the Ministry of Education and Science of the Russian Federation (agreement no. № 075-02-2024-1377).
Aleksandr Dmitrievich P’yankov, Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: sascha.pyankow@mail.ru
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Cite this article as: A.D. P’yankov. Inequality of different metrics for discrete Luxemburg norms in finite-dimensional spaces. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2024, vol. 30, no. 4, pp. 212–223.
