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

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

María Jesús Carro and  Carlos Domingo-Salazar

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
$R_ng(y) = \frac1nf(τ^nx_0)g(S^ny) \overset{n\to∞}\longrightarrow 0,\;\;\;\;\;\; ν\text{-a.e. } y∈ Y.$
Keywords: Return times theorem, Muckenhoupt weights, Yano's extrapolation, Rubio de Francia's extrapolation.
Mathematics Subject Classification: Primary: 37A05, 42B99; Secondary: 46E30.
Citation: María Jesús Carro, Carlos Domingo-Salazar. The return times property for the tail on logarithm-type spaces. Discrete & Continuous Dynamical Systems, 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

show all references

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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