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The purpose of this paper is to describe the smooth homogeneousCalderon-Zygmund operators for which the maximal singular integral T*f may becontrolled by the singular integral Tf. We consider two types of control. Thefirst is the L2 estimate of T*f by Tf, namely the estimate of the L2 norm ofT*f by a constant times the L2 norm of Tf. The second is the pointwise estimateof T*f(x) by a constant times M(Tf)(x), where M denotes the Hardy-Littlewoodmaximal operator. Notice that this is an improved variant of Cotlar'sinequality, because the term Mf(x) is missing on the right hand side. Our mainresult states that, for even operators, both are equivalent to a purelyalgebraic condition formulated in terms of the expansion of the kernel inspherical harmonics. The condition holds by higher order Riesz transforms,which then satisfy an improved version of Cotlar's inequality

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