Spectral synthesis in the Sobolev space associated with integral metric

Yu. V. Netrusov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 22, Tome 217 (1994), pp. 92-111

Résumé

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.