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