Spectral synthesis in the Sobolev space associated with integral metric
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 22, Tome 217 (1994), pp. 92-111
Citer cet article
Voir la notice du chapitre de livre provenant de la source Math-Net.Ru
The aim of this paper is to prove Theorem A. Theorem A. Let $l\in\mathbb N$, $A\subset\mathbb R^n$. Then the following two conditions are equivalent: 1) for any $\varepsilon>0$ there exist a function $f_\varepsilon$ and an open set $G\supset A$ such that $$ \operatorname{supp}f_\varepsilon\subset\mathbb R^n\setminus G,\qquad\|f-f_\varepsilon\|_{W^l_1}\le\varepsilon; $$ 2) for any $\alpha=(\alpha_1,\dots,\alpha_n)\in\{0,1,2,\dots,\}^n$, $|\alpha|=\alpha_1+\dots+\alpha_n, there exists a set $E_\alpha$ with the following properties: a) if $n\le l-|\alpha|$ then $E_\alpha=A$; b) if $n>l-|\alpha|$ then the Hausdorff measure of order $n-l+|\alpha|$ of set $A\setminus E_\alpha$ is equal to zero; c) for any point $x\in E_\alpha$ the following relation holds: $$ \lim_{a\to0}a^{-n}\int_{D(x,a)}|D^\alpha f(y)|\,dy=0, $$ where $D(x,a)$ is the ball of radius $a>0$ centered at $x\in\mathbb R^n$. Some generalizations of this result are also proved. Bibliography: 9 titles.