Estimates for the maximal singular integral in terms of the singular integral: the case of even kernels

Abstract

Let T be a smooth homogeneous Calder on-Zygmund singular integral operator in R n . In this paper we study the problem of controlling the maximal singular integral T ? f by the singular integral Tf. The most basic form of control one may consider is the estimate of the L 2 (R n ) norm of T ? f by a constant times the L 2 (R n ) norm of Tf. We show that if T is an even higher order Riesz transform, then one has the stronger pointwise inequality T ? f(x) C M(Tf)(x), where C is a constant and M is the Hardy-Littlewood maximal operator. We prove that the L 2 estimate of T ? by T is equivalent, for even smooth homogeneous Calder on-Zygmund operators, to the pointwise inequality between T ? and M(T ). Our main result characterizes the L 2 and pointwise inequalities in terms of an algebraic condition expressed in terms of the kernel ( x) jxjn of T , where is an