BOCHNER-RIESZ MEANS OF FUNCTIONS IN WEAK-L(P)

The Bochner-Riesz means of order delta greater-than-or-equal-to 0 for suitable test functions on R(N) are defined via the Fourier transform by (S(R)(delta)f)(xi) = (1 - \xi\2/R2)+(delta)f(xi). We show that the means of the critical index delta = N/P - N + 1/2, 1 < p < 2N/N + 1, do not map L(p,infinity)(R(N)) into L(p,infinity) (R(N)), but they map radial functions of L(p,infinity) (R(N)) into L(p,infinity) (R(N)). Moreover, if f is radial and in the L(p,infinity) (R(N)) closure of test functions, S(R)(delta)f (x) converges, as R --> + infinity, to f(x) in norm and for almost every x in R(N). We also observe that the means of the function absolute value of x-N/p, which belongs to L(p,infinity) (R(N)) but not to the closure of test functions, converge for no x.