The convergence in L1 of singular integrals in harmonic analysis and ergodic theory

We study the behavior of the ergodic singular integral T associated to a nonsingular measurable pow {tau(t) : t is an element of R} on a finite measure space and a Calderon-Zygmund kernel with support in (0, infinity). We show that if the flow preserves the measure or with more generality, if the flow is such that the semipow {tau(t) : t greater than or equal to 0} is Cesaro-bounded, f and Tf are integrable functions, then the truncations of the singular integral converge to Tf nor only in the a.e. sense but also in the L-1-norm. To obtain this result we study the problem for the singular integrals in the real line and in the setting of the weighted L-1-spaces.