Bochner-Riesz means of functions in weak-L p

Abstract

The Bochner-Riesz means of order δ≥0 for suitable test functions on ℝN are defined via the Fourier transform by\((S_R^\delta f)(\xi ) = (1 - |\xi |^2 /R^2 )^\delta + \hat f(\xi )\). We show that the means of the critical index\(\delta = \frac{N}{p} - \frac{{N + 1}}{2},1< p< \frac{{2N}}{{N + 1}}\), do not mapLp,∞(ℝN) intoLp,∞(ℝN), but they map radial functions ofLp,∞(ℝN) intoLp,∞(ℝN). Moreover, iff is radial and in theLp,∞(ℝN) closure of test functions,SRδf(x) converges, asR→+∞, tof(x) in norm and for almost everyx in ℝN. We also observe that the means of the function|x|−N/p, which belongs toLp,∞(ℝN) but not to the closure of test functions, converge for nox.