Exact estimates of high-order derivatives in Sobolev spaces

Abstract

The paper describes the splines $Q_{n,k}(x,a)$, which for an arbitrary point $a\in (0;1)$ and an arbitrary function $y\in {\mathring{W}}_p^n[0;1]$ set the relations $y^{(k)}(a)={\int }_0^1{y}^{(n)}(x)Q_{n,k}^{(n)}(x,a)dx$. The relation of the $L_{p^{\prime }}[0;1]$ norm minimization for $Q_{n,k}^{(n)}$ ($1/p+1/p^{\prime }=1$) with the problem of the best estimates of derivatives of $y^{(k)}(a)⩽A_{n,k,p}(a)\Vert y^{(n)}{\Vert }_{L_p[0;1]}$, and also with the problem of finding the exact embedding constants of the Sobolev space ${\mathring{W}}_p^n[0;1]$ into the space ${\mathring{W}}_{\infty }^k[0;1]$, $n\in \mathbb{N}$, $k=0,1,\dots ,n-1$. Exact embedding constants are found for $k=n-1$ and $p=\infty$, as well as for all $n\in \mathbb{N}$, $k=0,1,\dots ,n-1$ and $p=1$.

arXiv comment2 figures, in Russian language