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

In this thesis we studythe problem of controlling the maximal singular integralT^*f by the singular integral Tf. The most basic formof control one may consider is the estimate of the L^2(R^n)norm of Tf. We show that if T is an even higher order Riesz transform, then one has the strongerpointwise inequality T^*f(x) < = CM(Tf)(x), where C is a constant and Mis the Hardy-Littlewood maximal operator. We prove that L^2 estimateof T^*f by T is equivalent , for even smooth homogeneous Calderon-Zygmund operators, to the pointwise inequality between T^* and M(T).Our main result characterizes the L^2 and pointwise inequalities in terms of analgebraic condition expressed in terms of the kernel of T. Let \Omega= \Sigma P_j the expansion of \Omega in spherical harmonics P_j.Then our characterizing condition states that T is of the form R \circ U,where U is an invertible operator in Aand R is a higher order Riesz transform associated with a homogeneous harmonic polynomialP which divides each P_j in the ring of polynomials in n variables with real coefficients.of degree j.