An inequality of different metrics in the generalized Lorentz space

G. A. Akishev

Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 24 (2018) no. 4, pp. 5-18

Résumé

The main goal of the paper is to prove the Jackson-Nikol'skii inequality for multiple trigonometric polynomials in the generalized Lorentz space $L_{\psi,\theta}(\mathbb{T}^{m})$. In the first section we give definitions of a symmetric space of functions, a fundamental function, and the Boyd index of a space. In particular, we define the generalized Lorentz and Lorentz-Zygmund spaces. In addition, definitions of a weakly varying function and of the Lorentz-Karamata space are given. In the second section we prove an analog of the inequality of different metrics for multiple trigonometric polynomials in generalized Lorentz spaces $L_{\psi,\theta}(\mathbb{T}^{m})$ with identical Boyd indices but different fundamental functions. In the Lorentz-Karamata space, the order-exact Jackson-Nikol'skii inequality for multiple trigonometric polynomials is obtained.

Détail

Mots-clés : Lorentz-Karamata space
Keywords: Jackson-Nikol'skii inequality, trigonometric polynomial.

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