On spectral decompositions of functions in $H_p^\alpha$

The paper is devoted to a study of the spectral resolutions $E_\lambda f$ and their Riesz means $E_\lambda^sf$, corresponding to selfadjoint extensions of elliptic differential operators $A(x,D)$ of order $m$ in an $N$-dimensional domain $G$. It is proved that if $f$ belongs to the Nikol'skii class $\overset\circ H{}_p^\alpha(G)$ and has compact support in $G$, then for
$$ \alpha>0,\quad s\geqslant0,\quad\alpha+s\geqslant\frac{N-1}2,\quad p\alpha>N $$
the Riesz means $E_\lambda^sf$ converge for $\lambda\to\infty$ to $f$ uniformly on each compact set $K\subset G$.
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