论文标题
Sobolev空间中高级导数的确切估计值
Exact estimates of high-order derivatives in Sobolev spaces
论文作者
论文摘要
本文描述了键$ q_ {n，k}（x，a）$，对于任意点$ a \ in（0; 1）$和任意函数$ y \ in \ mathring {w}^n_p [0; 1] y^{（n）}（x）q^{（n）} _ {n，k}（x，a）dx $。 $ l_ {p'} [0; 1] $最小化的关系，$ q^{（n）} _ {n，k} $（$ 1/ p+1/ p'= 1 $）以及$ y^{（k）}（a）\ leqslantant的最佳估计问题的问题a_ {n，k，p}（a）\ | y^{（n）} \ | _ {l_p [0; 1]} $，也存在解决Sobolev space $ \ Mathring的精确嵌入常数{W}^n_p [0; 1] $ n \ in \ mathbb {n} $，$ k = 0,1，\ ldots，n-1 $。找到$ k = n-1 $和$ p = \ infty $的精确嵌入常数，以及所有$ n \ in \ mathbb {n} $，$ k = 0,1，\ ldots，n-1 $和$ p = 1 $。
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 $p=1$.
