On Sharp Estimates of Derivatives of Even Order

Abstract

The norms of embedding operators of Sobolev spaces $\Wo^n_2[0;1]\hookrightarrow\Wo^k_\infty[0;1]$ ($0⩽k⩽n-1$) are considered. The least possible quantities $A_{n,k}^2(x)$ in the inequalities $|f^{(k)}(x){|}^2⩽A_{n,k}^2(x)\Vert f^{(n)}{\Vert }_{L_2[0;1]}^2$ are studied. On the basis of the relations between the $A_{n,k}^2(x)$ and the antiderivatives of the Legendre polynomials, the properties of the maxima of the functions $A_{n,k}^2(x)$ are established. It is shown that, for all~$k$, the global maximum of the function $A_{n,k}^2$ 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.

9 pages, 2 figures