Abstract
The purpose of this paper is to prove that, given a dynamical system (X,M,μ, τ) and 0 < q < 1, the Lorentz spaces L1,q(μ) satisfy the so-called Return Times Property for the Tail, contrary to what happens in the case q = 1. In fact, we consider a more general case than in previous papers since we work with a σ-finite measure μ and a transformation τ which is only Cesàro bounded. The proof uses the extrapolation theory of Rubio de Francia for one-sided weights. These results are of independent interest and can be applied to many other situations.
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The first author is supported by MTM2013-40985-P, MTM2016-75196-P (MINECO/FEDER, UE) and 2014SGR289.
The second and third authors are supported by grants MTM2011-28149-C02-02 and MTM2015-66157-C2-2-P (MINECO/FEDER, UE) and grant FQM-354 of the Junta de Andaluca.
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Carro, M.J., Lorente, M. & Martín-Reyes, F.J. A counting problem in ergodic theory and extrapolation for one-sided weights. JAMA 134, 237–254 (2018). https://doi.org/10.1007/s11854-018-0008-0
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DOI: https://doi.org/10.1007/s11854-018-0008-0