Summability almost everywhere of Fourier series in \(L_ p\) with respect to eigenfunctions (Q791800)

Let \(A(D)=\sum_{| \alpha | =m}a_{\alpha}D^{\alpha}\) be a differential operator with real coefficients; here \(D^{\alpha}=\frac{(- i\partial)^{| \alpha |}}{\partial x^{\alpha_ 1}_ 1...\partial x^{\alpha_ n}_ n}\). Define, as usual, the Riesz means of \(f\in L_ p({\mathbb{R}}^ n)\) with compact support by the formula \[ \sigma^ s_{\lambda}(x,f)=(2\pi)^{-N}\int_{A(\xi)<\lambda}(1- \frac{A(\xi)}{\lambda})^ s\hat f(\xi)l^{i<x-y,\xi>}d\xi \] where \(A(\xi)=\sum_{| \alpha | =m}a_{\alpha}x^{\alpha}\) and \(\hat f\) is the Fourier transform of f. Theorem: If \(1<p\leq 2\) and \(s>(n-1)(1/p-1/2)\) then \(\lim_{\lambda \to \infty}\sigma^ s_{\lambda}(x,f)=f(x)\) almost everywhere.