Let $a$ be a positive elliptic operator with constant coefficients, and let $\Omega$ be a region in $R^l$ . We consider the operator $a$ on $C^\infty_0(\Omega)$ , and we let $\hat a$ be an extension of this operator with a positive lower bound. Let $\{E_\lambda\}$ denote the spectral family of the operator $\hat a$ . The operator $E_\lambda$ or its Riesz mean $E^a_\lambda$ will be considered on functions $f\in L^p(\Omega)$ , $1\leqslant p<\infty$ , such that $\operatorname{supp}f\subseteq\Omega_0$ , where $\Omega_0$ is a region with compact closure in $\Omega$ . We will study the norm of the operator $ E_\lambda\colon L_p(\Omega_0)\to L_p(\Omega_0)$ . We obtain definitive results when the point $(p,\alpha)$ lies in one of the three regions: \begin{gather}
\left{(p,\alpha):1\leqslant p\leqslant\frac{2l}{l+1},0\leqslant\alpha\leqslant\alpha_p=\frac lp-\frac{l+1}2\right},\
\left{(p,\alpha):\frac{2l}{l-1}\leqslant p\leqslant\frac{2l}{l-1},\alpha=0\right},\
\left{(p,\alpha):1\leqslant p\leqslant2,\alpha>(l-1)\biggl(\frac1p-\frac12\biggr)\right}.
\end{gather}For$1\leqslant p\leqslant\frac{2l}{l+1}$,$\alpha=\alpha_p=\frac lp-\frac{l+1}2$we construct an example of a function for which the Riesz mean of order$\alpha_p$of its spectral expansion diverges almost everywhere. For$\frac{2l}{l+1}<p<2$,$\alpha=0$we construct an analogous example for multiple Fourier series expansions.Bibliography: 26 titles.