On Sharp Estimates of Derivatives of Even Order

Abstract

The norms of embedding operators of Sobolev spaces $\Wo^n_2[0;1]\hookrightarrow\Wo^k_\infty[0;1]$ ($0\leqslant k\leqslant n-1$) are considered. The least possible quantities $A^2_{n,k}(x)$ in the inequalities $|f^{(k)}(x)|^2\leqslant A^2_{n,k}(x)\|f^{(n)}\|^2_{L_2[0;1]}$ are studied. On the basis of the relations between the $A^2_{n,k}(x)$ and the antiderivatives of the Legendre polynomials, the properties of the maxima of the functions $A^2_{n,k}(x)$ are established. It is shown that, for all~$k$, the global maximum of the function $A^2_{n,k}$ on the closed interval $[0;1]$ is the maximum point nearest to the midpoint of the interval; in particular, for even~$k$, $x=1/2$ is such a point. For the parameter~$k$ of even order, an explicit formula for the norms of the embedding operators is obtained.