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Estimates of Derivatives in Sobolev Spaces in Terms of Hypergeometric Functions

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Abstract

The paper deals with sharp estimates of derivatives of intermediate order \(k\le n-1\) in the Sobolev space \(\mathring W^n_2[0;1]\), \(n\in\mathbb N\). The functions \(A_{n,k}(x)\) under study are the smallest possible quantities in inequalities of the form

$$|y^{(k)}(x)|\le A_{n,k}(x)\|y^{(n)}\|_{L_2[0;1]}.$$

The properties of the primitives of shifted Legendre polynomials on the interval \([0;1]\) are used to obtain an explicit description of these functions in terms of hypergeometric functions. In the paper, a new relation connecting the derivatives and primitives of Legendre polynomials is also proved.

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Authors and Affiliations

  1. Lomonosov Moscow State University, Moscow, 119991, Russia

    T. A. Garmanova

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  1. T. A. Garmanova
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Correspondence to T. A. Garmanova.

Additional information

Translated from Matematicheskie Zametki, 2021, Vol. 109, pp. 500-507 https://doi.org/10.4213/mzm13040.

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Garmanova, T.A. Estimates of Derivatives in Sobolev Spaces in Terms of Hypergeometric Functions. Math Notes 109, 527–533 (2021). https://doi.org/10.1134/S0001434621030214

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  • DOI: https://doi.org/10.1134/S0001434621030214

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