On~summability and convergence of eigenfunction expansions of a~diffe\-rential operator

K. I. Babenko

Sbornik. Mathematics, Tome 20 (1973) no. 2, pp. 157-211

Résumé

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 $\frac{2l}{l+1}$, $\alpha=0$ we construct an analogous example for multiple Fourier series expansions. Bibliography: 26 titles.