Abstract
We study the behavior of the ergodic singular integral τ associated to a nonsingular measurable flow {τ:t ∈ ℝ} 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 {τt:t>-0} is Cesàrobounded,f and τf are integrable functions, then the truncations of the singular integral converge to τf 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.
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Communicated by Aline Bonami
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.
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Lorente, M. The convergence inL 1 of singular integrals in harmonic analysis and ergodic theory. The Journal of Fourier Analysis and Applications 5, 617–638 (1999). https://doi.org/10.1007/BF01257195
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DOI: https://doi.org/10.1007/BF01257195