Bochner-Riesz means on symmetric spaces (Q1283238)

Suppose that \(G/K\) is a noncompact rank one Riemannian symmetric space of dimension \(d\). Denote by \(-\Delta_0\) the Laplace-Beltrami operator on \(G/K\), and by \(-\Delta\) its self-adjoint extension to \(L^2(G/K)\). Its spectral resolution is \(-\Delta= \int^\infty_{|\rho|^2} tdE(t)\), where the constant \(|\rho|^2\) depends on the geometry of \(G/K\). For every \(z\in\mathbb{C}\) with \({\mathfrak R}(z)\geq 0\) we define the Bochner-Riesz mean operators by \[ S^z_R f= \int^\infty_{|\rho|^2} \Biggl(1-{t\over R}\Biggr)^z_+ dE(t)f. \] Main result. Let \(0< \alpha<(d- 1)/2\) and \({2d\over d+ 2\alpha+ 1}< p\leq 2\). Then for every \(f\in L^p(K\setminus G/K)\), \(\lim_{R\to\infty} S^\alpha_R f(x)= f(x)\) a.e. This is proved by combining results of \textit{S. Giulini} and \textit{G. Mauceri} [Ann. Math. Pura Appl. (4) 159, 357-369 (1991; Zbl 0796.43007)] and \textit{C. Meaney} and \textit{E. Prestini} [J. Funct. Anal. 149, 277-304 (1997; Zbl 0883.43012)]. This result was proved by \textit{Y. Kanjin} [Ann. Sci. Kanazawa Univ. 25, 11-15 (1988)] in the case of Bochner-Riesz means of radial functions on the Euclidean space. Moreover, following a technique of Kanjin, it is shown that the range of indices is sharp.