Abstract
In this paper we consider four operators arising in harmonic analysis in R', n > 1. They share one feature: each is known to be bounded on LP for all 1 < p < o, but for each it has been an open question whether a weak type (1, 1) bound also holds. In two cases we show that the weak (1, 1) bound does hold, and for the others we establish weaker estimates which are still sharper than the LP boundedness. Although the four operators are quite distinct, they all involve convolution kernels which are related to the singular integrals of Calderon and Zygmund, but which lack the smoothness required for the usual analysis. In each case this defect is mitigated by the presence of some form of curvature. Consider the Bochner-Riesz means
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