Spectral synthesis in the Sobolev space associated with integral metric

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 $\alpha=(\alpha_1,\dots,\alpha_n)\in\{0,1,2,\dots,\}^n$, $|\alpha|=\alpha_1+\dots+\alpha_n<l$, 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.