# American Institute of Mathematical Sciences

April  2018, 38(4): 2065-2078. doi: 10.3934/dcds.2018084

## The return times property for the tail on logarithm-type spaces

 Departament de Matemàtiques i Informàtica, Universitat de Barcelona, 08007 Barcelona, Spain

Received  April 2017 Revised  October 2017 Published  January 2018

Fund Project: The authors were supported by grants MTM2016-75196-P (MINECO/FEDER, UE) and 2014SGR289

Given a dynamical system
 $(Ω,Σ,μ, τ)$
with
 $μ$
a non-atomic probability measure and
 $τ$
an invertible measure preserving ergodic transformation, we prove that the maximal operator, considered by I. Assani, Z. Buczolich and R. D. Mauldin in 2005,
 ${N^*}f\left( x \right) = \mathop {\sup }\limits_{\alpha > 0} \alpha \# \left\{ {k \ge 1:\frac{{\left| {f\left( {{\tau ^k}x} \right)} \right|}}{k} > \alpha } \right\}$
satisfies that
 ${N^*}:\left[ {L \log_3 L (μ)} \right] \longrightarrow L^{1, ∞}(μ)$
is bounded where the space
 $\left[ {L \log_3 L (μ)} \right]$
is defined by the condition
 $\Vert f\Vert_{\left[ {L \log_3 L (μ)} \right]} = ∈t_0^1 \frac{\sup\limits_{t≤q y}tf_μ^*(t)}{y} \log_3 \frac 1y dy < ∞,$
with
 $\log_3 x = 1+\log_+\log_+\log_+ x$
and
 $f^*_μ$
the decreasing rearrangement of
 $f$
with respect to
 $μ$
. This space is near
 $L \log_3 L (μ)$
, which is the optimal Orlicz space on which such boundedness can hold. As a consequence, the space
 $\left[ {L \log_3 L (μ)} \right]$
satisfies the Return Times Property for the Tail; that is, for every
 $f∈\left[ {L \log_3 L (μ)} \right]$
, there exists a set
 $X_0$
so that
 $μ(X_0) = 1$
and, for all
 $x_0∈ X_0$
, all dynamical systems
 $(Y,\mathcal{C},ν, S)$
and all
 $g∈ L^1(ν)$
, the sequence
 $R_ng(y) = \frac1nf(τ^nx_0)g(S^ny) \overset{n\to∞}\longrightarrow 0,\;\;\;\;\;\; ν\text{-a.e. } y∈ Y.$
Citation: María Jesús Carro, Carlos Domingo-Salazar. The return times property for the tail on logarithm-type spaces. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 2065-2078. doi: 10.3934/dcds.2018084
##### References:
 [1] N. Yu Antonov, Convergence of Fourier series, East J. Approx., 2 (1996), 187-196. Google Scholar [2] J. Arias-de-Reyna, Pointwise convergence of Fourier series, J. London Math. Soc., 65 (2002), 139-153. doi: 10.1112/S0024610701002824. Google Scholar [3] I. Assani, Strong laws for weighted sums of independent identically distributed random variables, Duke Math. J., 88 (1997), 217-246. Google Scholar [4] I. Assani, Convergence of the p-Series for stationary sequences, New York J. Math., 3A (1997), 15-30. Google Scholar [5] I. Assani, Z. Buczolich and R. D. Mauldin, An L1 counting problem in ergodic theory, J. Anal. Math., 95 (2005), 221-241. doi: 10.1007/BF02791503. Google Scholar [6] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988. Google Scholar [7] J. Bourgain, Return time sequences of dynamical systems, IHES, Preprint, 1988.Google Scholar [8] J. Bourgain, Temps de retour pour des systèmes dynamiques, D. R. Acad. Sci. Paris Sér. I Math, 306 (1988), 483-485. Google Scholar [9] J. Bourgain, Pointwise ergodic theorems for arithmetic sets, With an appendix by the author, H. Furstenberg, Y. Katznelson, and D. S. Ornstein, Inst. Hautes Études Sci. Publ. Math., 69 (1989), 5-45. Google Scholar [10] M. J. Carro, New extrapolation estimates, J. Funct. Anal., 174 (2000), 155-166. doi: 10.1006/jfan.2000.3568. Google Scholar [11] M. J. Carro and C. Domingo-Salazar Endpoint estimates for Rubio de Francia operators, Trans. Amer. Math. Soc. , (2017), to appear.Google Scholar [12] M. J. Carro, L. Grafakos and J. Soria, Weighted weak-type (1, 1) estimates via Rubio de Francia extrapolation, J. Funct. Anal., 269 (2015), 1203-1233. doi: 10.1016/j.jfa.2015.06.005. Google Scholar [13] M. J. Carro, M. Lorente and F. J. Martín-Reyes, A counting problem in ergodic theory and extrapolation for one-sided weights, Journal d'Analyse Mathématique, (2016), to appear.Google Scholar [14] M. J. Carro and P. Tradacete, Extrapolation on $L^{p,∞}(μ)$, J. Funct. Anal., 265 (2013), 1840-1869. Google Scholar [15] H. H. Chung, R. Hunt and D. S. Kurtz, The Hardy-Littlewood maximal function on $L(p,q)$ spaces with weights, Indiana Univ. Math. J., 31 (1982), 109-120. doi: 10.1512/iumj.1982.31.31012. Google Scholar [16] C. Demeter, M. T. Lacey, T. Tao and C. Thiele, Breaking the duality in the return times theorem, Duke Math. J., 143 (2008), 281-355. doi: 10.1215/00127094-2008-020. Google Scholar [17] C. Demeter and A. Quas, Weak-L1 estimates and ergodic theorems, New York J. Math., 10 (2004), 169-174. Google Scholar [18] L. Ephremidze, The rearrangement inequality for the ergodic maximal function, Georgian Math. J., 8 (2001), 727-732. Google Scholar [19] R. Kerman and A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math., 71 (1982), 277-284. Google Scholar [20] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226. doi: 10.1090/S0002-9947-1972-0293384-6. Google Scholar [21] K. Noonan, Birkhoff's theorem and the Return Times for the Tail, Master Thesis, University of North Carolina, 2002. Google Scholar [22] P. Ortega, Weights for the ergodic maximal operator and a.e. convergence of the ergodic averages for function in Lorentz spaces, Tohoku Math. J., 45 (1993), 437-446. doi: 10.2748/tmj/1178225894. Google Scholar [23] J. L. Rubio de Francia, Factorization theory and $A_p$ weights, Amer. J. Math., 106 (1984), 533-547. doi: 10.2307/2374284. Google Scholar [24] D. J. Rudolph, A joinings proof of Bourgain's return time theorem, Ergodic Theory Dynam. Systems, 14 (1994), 197-203. Google Scholar [25] D. J. Rudolph, Fully generic sequences and a multiple-term return-times theorem, Invent. Math., 131 (1998), 199-228. Google Scholar [26] E. M. Stein and N. J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc., 140 (1969), 35-54. doi: 10.1090/S0002-9947-1969-0241685-X. Google Scholar [27] S. Yano, Notes on Fourier analysis XXIX: An extrapolation theorem, J. Math. Soc. Japan, 3 (1951), 296-305. doi: 10.2969/jmsj/00320296. Google Scholar

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##### References:
 [1] N. Yu Antonov, Convergence of Fourier series, East J. Approx., 2 (1996), 187-196. Google Scholar [2] J. Arias-de-Reyna, Pointwise convergence of Fourier series, J. London Math. Soc., 65 (2002), 139-153. doi: 10.1112/S0024610701002824. Google Scholar [3] I. Assani, Strong laws for weighted sums of independent identically distributed random variables, Duke Math. J., 88 (1997), 217-246. Google Scholar [4] I. Assani, Convergence of the p-Series for stationary sequences, New York J. Math., 3A (1997), 15-30. Google Scholar [5] I. Assani, Z. Buczolich and R. D. Mauldin, An L1 counting problem in ergodic theory, J. Anal. Math., 95 (2005), 221-241. doi: 10.1007/BF02791503. Google Scholar [6] C. Bennett and R. Sharpley, Interpolation of Operators, Academic Press, 1988. Google Scholar [7] J. Bourgain, Return time sequences of dynamical systems, IHES, Preprint, 1988.Google Scholar [8] J. Bourgain, Temps de retour pour des systèmes dynamiques, D. R. Acad. Sci. Paris Sér. I Math, 306 (1988), 483-485. Google Scholar [9] J. Bourgain, Pointwise ergodic theorems for arithmetic sets, With an appendix by the author, H. Furstenberg, Y. Katznelson, and D. S. Ornstein, Inst. Hautes Études Sci. Publ. Math., 69 (1989), 5-45. Google Scholar [10] M. J. Carro, New extrapolation estimates, J. Funct. Anal., 174 (2000), 155-166. doi: 10.1006/jfan.2000.3568. Google Scholar [11] M. J. Carro and C. Domingo-Salazar Endpoint estimates for Rubio de Francia operators, Trans. Amer. Math. Soc. , (2017), to appear.Google Scholar [12] M. J. Carro, L. Grafakos and J. Soria, Weighted weak-type (1, 1) estimates via Rubio de Francia extrapolation, J. Funct. Anal., 269 (2015), 1203-1233. doi: 10.1016/j.jfa.2015.06.005. Google Scholar [13] M. J. Carro, M. Lorente and F. J. Martín-Reyes, A counting problem in ergodic theory and extrapolation for one-sided weights, Journal d'Analyse Mathématique, (2016), to appear.Google Scholar [14] M. J. Carro and P. Tradacete, Extrapolation on $L^{p,∞}(μ)$, J. Funct. Anal., 265 (2013), 1840-1869. Google Scholar [15] H. H. Chung, R. Hunt and D. S. Kurtz, The Hardy-Littlewood maximal function on $L(p,q)$ spaces with weights, Indiana Univ. Math. J., 31 (1982), 109-120. doi: 10.1512/iumj.1982.31.31012. Google Scholar [16] C. Demeter, M. T. Lacey, T. Tao and C. Thiele, Breaking the duality in the return times theorem, Duke Math. J., 143 (2008), 281-355. doi: 10.1215/00127094-2008-020. Google Scholar [17] C. Demeter and A. Quas, Weak-L1 estimates and ergodic theorems, New York J. Math., 10 (2004), 169-174. Google Scholar [18] L. Ephremidze, The rearrangement inequality for the ergodic maximal function, Georgian Math. J., 8 (2001), 727-732. Google Scholar [19] R. Kerman and A. Torchinsky, Integral inequalities with weights for the Hardy maximal function, Studia Math., 71 (1982), 277-284. Google Scholar [20] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc., 165 (1972), 207-226. doi: 10.1090/S0002-9947-1972-0293384-6. Google Scholar [21] K. Noonan, Birkhoff's theorem and the Return Times for the Tail, Master Thesis, University of North Carolina, 2002. Google Scholar [22] P. Ortega, Weights for the ergodic maximal operator and a.e. convergence of the ergodic averages for function in Lorentz spaces, Tohoku Math. J., 45 (1993), 437-446. doi: 10.2748/tmj/1178225894. Google Scholar [23] J. L. Rubio de Francia, Factorization theory and $A_p$ weights, Amer. J. Math., 106 (1984), 533-547. doi: 10.2307/2374284. Google Scholar [24] D. J. Rudolph, A joinings proof of Bourgain's return time theorem, Ergodic Theory Dynam. Systems, 14 (1994), 197-203. Google Scholar [25] D. J. Rudolph, Fully generic sequences and a multiple-term return-times theorem, Invent. Math., 131 (1998), 199-228. Google Scholar [26] E. M. Stein and N. J. Weiss, On the convergence of Poisson integrals, Trans. Amer. Math. Soc., 140 (1969), 35-54. doi: 10.1090/S0002-9947-1969-0241685-X. Google Scholar [27] S. Yano, Notes on Fourier analysis XXIX: An extrapolation theorem, J. Math. Soc. Japan, 3 (1951), 296-305. doi: 10.2969/jmsj/00320296. Google Scholar
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