The convergence inL 1 of singular integrals in harmonic analysis and ergodic theory

We study the behavior of the ergodic singular integral associated to a nonsingular measurable flow {:t } on a finite measure space and a Calderón—Zygmund kernel with support in (0, ). We show that if the flow preserves the measure or, with more generality, if the flow is such that the semiflow {_t:t>-0} is Cesàrobounded,f and f are integrable functions, then the truncations of the singular integral converge to f not only in the a.e. sense but also in the L¹-norm. To obtain this result we study the problem for the singular integrals in the real line and in the setting of the weighted L¹-spaces.This research has been partially supported by a D.G.I.C.Y.T. grant (PB94-1496), a D.G.E.S. grant (PB97-1097) and Junta de Andalucía.