On Sharp Estimates of Even-Order Derivatives in Sobolev Spaces

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.