Abstract

The norms of embedding operators \(\mathring{W}^n_2[0,1]\hookrightarrow\mathring{W}^k_\infty[0,1]\) (\(0\leqslant k\leqslant n-1\)) of Sobolev spaces are considered. The least possible values of \(A^2_{n,k}(x)\) in the inequalities \(|f^{(k)}(x)|^2\leqslant A^2_{n,k}(x)\|f^{(n)}\|^2_{L_2[0,1]}\) (\(f\in \mathring{W}^n_2[0,1]\)) are studied. On the basis of relations between the functions \(A^2_{n,k}(x)\) and primitives of the Legendre polynomials, properties of the maxima of the functions \(A^2_{n,k}(x)\) are determined. It is shown that, for any \(k\), the points of global maximum of the function \(A^2_{n,k}\) on the interval \([0,1]\) is the point of local maximum nearest to the midpoint of this interval; in particular, for even \(k\), such a point is \(x=1/2\). For all even \(k\), explicit expressions for the norms of embedding operators are found.

中文翻译：

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关于 Sobolev 空间中偶数阶导数的尖锐估计

摘要

嵌入算子的范数\(\mathring{W}^n_2[0,1]\hookrightarrow\mathring{W}^k_\infty[0,1]\) ( \(0\leqslant k\leqslant n-1\ ) ) 的 Sobolev 空间被考虑。不等式中\(A^2_{n,k}(x)\)的最小可能值\(|f^{​​(k)}(x)|^2\leqslant A^2_{n,k}( x)\|f^{(n)}\|^2_{L_2[0,1]}\) ( \(f\in \mathring{W}^n_2[0,1]\) ) 进行了研究。根据函数\(A^2_{n,k}(x)\)与勒让德多项式的原语之间的关系，函数\(A^2_{n,k}(x) \)确定。表明，对于任何\(k\)，函数\(A^2_{n,k}\)的全局最大值点在区间\([0,1]\) 上是离该区间中点最近的局部最大值点；特别是，即使是\(k\)，这样的点是\(x=1/2\)。对于所有偶数\(k\)，可以找到嵌入运算符范数的显式表达式。