Almost everywhere convergence for modified Bochner-Riesz means at the critical index for \(p\geq2\) (Q505541)

The classical Bochner-Riesz operators were introduced by Solomom Bochner in 1936. Various properties of these have been investigated by several authors since then. For example, studies related to \(L^p\) almost everywhere convergence and maximal operators have been carried out by Tao, Ashurov and others for \(p<2\) and by Carbery, Christ and others for \(p\geq 2\). In this paper, modified Bochner-Riesz multipliers introduced by Seeger in 1987 are studied and boundedness for a maximal modified Bochner-Riesz operator between weighted \(L^2\) spaces is obtained. Here methods developed by \textit{A. Carbery} et al. [J. Lond. Math. Soc., II. Ser. 38, No. 3, 513--524 (1988; Zbl 0631.42004)] are followed, taking into account the necessity to deal with nonhomogeneous weights. As a consequence of this boundedness result, the author obtains sufficient conditions for a.e. convergence of the modified Bochner-Riesz means on \(\mathbb{R}^n\) at the critical index \(p_{\lambda}=2n/(n-2\lambda -1)\).