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Exact constants in generalized inequalities for intermediate derivatives

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Abstract

Consider the Sobolev space W n2 (ℝ+) on the semiaxis with norm of general form defined by a quadratic polynomial in derivatives with nonnegative coefficients. We study the problem of exact constants A n,k in inequalities of Kolmogorov type for the values of intermediate derivatives |f (k)(0)| ≤ A n,k ‖f‖. In the general case, the expression for the constants A n,k is obtained as the ratio of two determinants. Using a general formula, we obtain an explicit expression for the constants A n,k in the case of the following norms:

$$ \left\| f \right\|_1^2 = \left\| f \right\|_{L_2 }^2 + \left\| {f^{(n)} } \right\|_{L_2 }^2 and\left\| f \right\|_2^2 = \sum\limits_{l = 0}^n {\left\| {f^{(l)} } \right\|_{L_2 }^2 } . $$

In the case of the norm ‖ · ‖1, formulas for the constants A n,k were obtained earlier by another method due to Kalyabin. The asymptotic behavior of the constants A n,k is also studied in the case of the norm ‖ · ‖2. In addition, we prove a symmetry property of the constants A n,k in the general case.

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Authors and Affiliations

  1. Donetsk National University, Donetsk, Ukraine

    A. A. Lunev & L. L. Oridoroga

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  1. A. A. Lunev
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  2. L. L. Oridoroga
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Correspondence to A. A. Lunev.

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Original Russian Text © A. A. Lunev, L. L. Oridoroga, 2009, published in Matematicheskie Zametki, 2009, Vol. 85, No. 5, pp. 737–744.

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Lunev, A.A., Oridoroga, L.L. Exact constants in generalized inequalities for intermediate derivatives. Math Notes 85, 703–711 (2009). https://doi.org/10.1134/S0001434609050101

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  • DOI: https://doi.org/10.1134/S0001434609050101

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