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Well-posedness of a model for the growth of tree stems and vines
The return times property for the tail on logarithm-type spaces
Departament de Matemàtiques i Informàtica, Universitat de Barcelona, 08007 Barcelona, Spain |
$(Ω,Σ,μ, τ)$ |
$μ$ |
$τ$ |
${N^*}f\left( x \right) = \mathop {\sup }\limits_{\alpha > 0} \alpha \# \left\{ {k \ge 1:\frac{{\left| {f\left( {{\tau ^k}x} \right)} \right|}}{k} > \alpha } \right\}$ |
${N^*}:\left[ {L \log_3 L (μ)} \right] \longrightarrow L^{1, ∞}(μ)$ |
$\left[ {L \log_3 L (μ)} \right]$ |
$\Vert f\Vert_{\left[ {L \log_3 L (μ)} \right]} = ∈t_0^1 \frac{\sup\limits_{t≤q y}tf_μ^*(t)}{y} \log_3 \frac 1y dy < ∞,$ |
$\log_3 x = 1+\log_+\log_+\log_+ x$ |
$f^*_μ$ |
$f$ |
$μ$ |
$L \log_3 L (μ)$ |
$\left[ {L \log_3 L (μ)} \right]$ |
$f∈\left[ {L \log_3 L (μ)} \right]$ |
$X_0$ |
$μ(X_0) = 1$ |
$x_0∈ X_0$ |
$(Y,\mathcal{C},ν, S)$ |
$g∈ L^1(ν)$ |
$R_ng(y) = \frac1nf(τ^nx_0)g(S^ny) \overset{n\to∞}\longrightarrow 0,\;\;\;\;\;\; ν\text{-a.e. } y∈ Y.$ |
References:
[1] |
N. Yu Antonov,
Convergence of Fourier series, East J. Approx., 2 (1996), 187-196.
|
[2] |
J. Arias-de-Reyna,
Pointwise convergence of Fourier series, J. London Math. Soc., 65 (2002), 139-153.
doi: 10.1112/S0024610701002824. |
[3] |
I. Assani,
Strong laws for weighted sums of independent identically distributed random variables, Duke Math. J., 88 (1997), 217-246.
|
[4] |
I. Assani,
Convergence of the p-Series for stationary sequences, New York J. Math., 3A (1997), 15-30.
|
[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. |
[6] |
C. Bennett and R. Sharpley,
Interpolation of Operators, Academic Press, 1988. |
[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.
|
[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.
|
[10] |
M. J. Carro,
New extrapolation estimates, J. Funct. Anal., 174 (2000), 155-166.
doi: 10.1006/jfan.2000.3568. |
[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. |
[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.
|
[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. |
[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. |
[17] |
C. Demeter and A. Quas,
Weak-L1 estimates and ergodic theorems, New York J. Math., 10 (2004), 169-174.
|
[18] |
L. Ephremidze,
The rearrangement inequality for the ergodic maximal function, Georgian Math. J., 8 (2001), 727-732.
|
[19] |
R. Kerman and A. Torchinsky,
Integral inequalities with weights for the Hardy maximal function, Studia Math., 71 (1982), 277-284.
|
[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. |
[21] |
K. Noonan,
Birkhoff's theorem and the Return Times for the Tail, Master Thesis, University of North Carolina, 2002. |
[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. |
[23] |
J. L. Rubio de Francia,
Factorization theory and $ A_p$ weights, Amer. J. Math., 106 (1984), 533-547.
doi: 10.2307/2374284. |
[24] |
D. J. Rudolph,
A joinings proof of Bourgain's return time theorem, Ergodic Theory Dynam. Systems, 14 (1994), 197-203.
|
[25] |
D. J. Rudolph,
Fully generic sequences and a multiple-term return-times theorem, Invent. Math., 131 (1998), 199-228.
|
[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. |
[27] |
S. Yano,
Notes on Fourier analysis XXIX: An extrapolation theorem, J. Math. Soc. Japan, 3 (1951), 296-305.
doi: 10.2969/jmsj/00320296. |
show all references
References:
[1] |
N. Yu Antonov,
Convergence of Fourier series, East J. Approx., 2 (1996), 187-196.
|
[2] |
J. Arias-de-Reyna,
Pointwise convergence of Fourier series, J. London Math. Soc., 65 (2002), 139-153.
doi: 10.1112/S0024610701002824. |
[3] |
I. Assani,
Strong laws for weighted sums of independent identically distributed random variables, Duke Math. J., 88 (1997), 217-246.
|
[4] |
I. Assani,
Convergence of the p-Series for stationary sequences, New York J. Math., 3A (1997), 15-30.
|
[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. |
[6] |
C. Bennett and R. Sharpley,
Interpolation of Operators, Academic Press, 1988. |
[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.
|
[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.
|
[10] |
M. J. Carro,
New extrapolation estimates, J. Funct. Anal., 174 (2000), 155-166.
doi: 10.1006/jfan.2000.3568. |
[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. |
[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.
|
[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. |
[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. |
[17] |
C. Demeter and A. Quas,
Weak-L1 estimates and ergodic theorems, New York J. Math., 10 (2004), 169-174.
|
[18] |
L. Ephremidze,
The rearrangement inequality for the ergodic maximal function, Georgian Math. J., 8 (2001), 727-732.
|
[19] |
R. Kerman and A. Torchinsky,
Integral inequalities with weights for the Hardy maximal function, Studia Math., 71 (1982), 277-284.
|
[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. |
[21] |
K. Noonan,
Birkhoff's theorem and the Return Times for the Tail, Master Thesis, University of North Carolina, 2002. |
[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. |
[23] |
J. L. Rubio de Francia,
Factorization theory and $ A_p$ weights, Amer. J. Math., 106 (1984), 533-547.
doi: 10.2307/2374284. |
[24] |
D. J. Rudolph,
A joinings proof of Bourgain's return time theorem, Ergodic Theory Dynam. Systems, 14 (1994), 197-203.
|
[25] |
D. J. Rudolph,
Fully generic sequences and a multiple-term return-times theorem, Invent. Math., 131 (1998), 199-228.
|
[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. |
[27] |
S. Yano,
Notes on Fourier analysis XXIX: An extrapolation theorem, J. Math. Soc. Japan, 3 (1951), 296-305.
doi: 10.2969/jmsj/00320296. |
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