Bochner-Riesz means of functions in weak-Lp

The Bochner-Riesz means of order δ≥0 for suitable test functions on ℝN are defined via the Fourier transform by {Mathematical expression}. We show that the means of the critical index {Mathematical expression}, do not map Lp,∞(ℝN) into Lp,∞(ℝN), but they map radial functions of Lp,∞(ℝN) into Lp,∞(ℝN). Moreover, if f is radial and in the Lp,∞(ℝN) closure of test functions, SRδf(x) converges, as R→+∞, to f(x) in norm and for almost every x in ℝN. We also observe that the means of the function |x|-N/p, which belongs to Lp,∞(ℝN) but not to the closure of test functions, converge for no x.