On the non-uniform estimate of the rate of convergence in the local limit theorem in the case of a stable limit distribution

Let $p_n(x)$ be a probability density function of normalized and centered sum of $n$ i. i. d. random variables belonging to the domain of attraction of the stable distribution $G$ of index $\alpha$, $0<\alpha\le 2$, $\alpha\ne 1$. Let $p(x)$ be a probability density function of $G$. It is proved that under certain conditions the relation
$$ \lim_{n\to\infty}|x|^\delta|p_n(x)-p(x)|=0,\qquad 0\le\delta<\alpha\ne 1, $$
holds uniformly in $x$, $-\infty<x<\infty$.