Exact estimates of high-order derivatives in Sobolev spaces

TL;DR

This paper derives exact estimates for high-order derivatives in Sobolev spaces using spline functions, connecting norm minimization with embedding constants, and provides precise constants for specific cases.

Contribution

It introduces spline-based relations for derivative estimates and computes exact Sobolev embedding constants for various orders and norms.

Findings

Exact embedding constants for $k=n-1$, $p= abla$

Spline relations for derivative estimation

Connections between norm minimization and Sobolev embeddings

Abstract

The paper describes the splines $Q_{n,k}(x,a)$, which for an arbitrary point $a\in(0;1)$ and an arbitrary function $y\in\mathring{W}^n_p[0;1]$ set the relations $y^{(k)}(a)=\int_0^1 y^{(n)}(x)Q^{(n)}_{n,k}(x,a)dx$. The relation of the $L_{p'}[0;1]$ norm minimization for $Q^{(n)}_{n,k}$ ($1/ p+1/p'=1$) with the problem of the best estimates of derivatives of $y^{(k)}(a)\leqslant A_{n,k,p}(a)\|y^{(n)}\|_{L_p[0;1]}$, and also with the problem of finding the exact embedding constants of the Sobolev space $\mathring{W}^n_p[0;1]$ into the space $\mathring{W}^k_\infty[0;1]$, $n\in\mathbb{N}$, $k=0,1,\ldots, n-1$. Exact embedding constants are found for $k=n-1$ and $p=\infty$, as well as for all $n\in\mathbb{N}$, $k=0,1,\ldots, n-1$ and $p=1$.