Almost everywhere convergence of Riesz means on certain noncompact symmetric spaces (Q1191345)

Let \(G/K\) be a non-compact Riemannian symmetric space and \(\Delta\) the selfadjoint extension on \(L^ 2(G/K)\) of the Laplace-Beltrami operator on \(G/K\). If \(-\int^ \infty_{\|\rho\|^ 2} tdE_ t\) is the spectral resolution of \(\Delta\), one defines the Riesz means \(S^ z_ R\) for \(\text{Re} z\geq 0\) and \(R\geq\|\rho\|^ 2\) (\(\rho\) is the halfsum of positive roots) by \[ S^ z_ R=-\int^ \infty_{\|\rho\|^ 2}\bigl(1-{t\over R}\bigr)^ z_ + dE_ t. \] They are given by convolution with \(C^ \infty\)-kernels \(s^ z_ R\) whose spherical Fourier transform is \(\widetilde{s}^ z_ R(\lambda)=\bigl(1-{\|\rho\|^ 2 +\|\lambda\|^ 2\over R}\bigr)^ z_ +\). Using the explicit inversion formulas for the spherical transform in the rank one and the complex case the authors give estimates for \(s^ z_ R\) which imply almost everywhere convergence \(S^ z_ R f\to f\) for \(f\in L^ p\), \(1\leq p\leq 2\) and large enough \(\text{Re} z\).