Exact estimates of high-order derivatives in Sobolev spaces

2022/08/26 by Garmanova, T. A., Sheipak, I. A.
#41A44 #41A50 #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2208.12791

Abstract

The paper describes the splines Q_n,k(x,a), which for an arbitrary point a∈(0;1) and an arbitrary function y∈\mathringWⁿ_p[0;1] set the relations y^(k)(a)=∫₀¹ y⁽ⁿ⁾(x)Q⁽ⁿ⁾_n,k(x,a)dx. The relation of the L_p'[0;1] norm minimization for Q⁽ⁿ⁾_n,k (1/ p+1/p'=1) with the problem of the best estimates of derivatives of y^(k)(a)\leqslant A_n,k,p(a)‖y⁽ⁿ⁾‖_Lp[0;1], and also with the problem of finding the exact embedding constants of the Sobolev space \mathringWⁿ_p[0;1] into the space \mathringW^k_∞[0;1], n∈ℕ, k=0,1,…, n-1. Exact embedding constants are found for k=n-1 and p=∞, as well as for all n∈ℕ, k=0,1,…, n-1 and p=1.

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