On radial Fourier multipliers and almost everywhere convergence

We study almost everywhere (a.e.) convergence on L-p and Lorentz spaces L-p,L-q, for variants of Riesz means at the critical index lambda(p) = d(1/2 - 1/p) - 1/2 for p > 2d/(d - 1). For the classical Riesz means S-t(lambda(p)), we show a.e. convergence for f is an element of L-p,L-1. We derive more general results for radial and quasi-radial Fourier multipliers and associated maximal functions, acting on L-2 spaces with power weights and their interpolation spaces. We also include a characterization of boundedness of such multiplier transformations on weighted L-2 spaces, and a sharp endpoint bound for Stein's square function associated with the Riesz means.