Abstract
We prove affirmatively the one-dimensional case of a conjecture of Stein regarding the $L^p$-boundedness of the Polynomial Carleson operator for $1\lt p\lt \infty $. \par Our proof relies on two new ideas: (i) we develop a framework for \emph higher-order wave-packet analysis that is consistent with the time-frequency analysis of the (generalized) Carleson operator, and (ii) we introduce a \emph local analysis adapted to the concepts of mass and counting function, which yields a new tile discretization of the time-frequency plane that has the major consequence of eliminating the exceptional sets from the analysis of the Carleson operator. As a further consequence, we are able to deliver the full $L^p$-boundedness range and prove directly—without interpolation techniques—the strong $L^2$ bound for the (generalized) Carleson operator, answering a question raised by C. Fefferman.
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C. Muscalu, T. Tao, and C. Thiele, "$L^p$ estimates for the biest. I. The Walsh case," Math. Ann., vol. 329, iss. 3, pp. 401-426, 2004.
@ARTICLE{MTT1,
author = {Muscalu, Camil and Tao, Terence and Thiele, Christoph},
title = {{$L^p$} estimates for the biest. {I}. {T}he {W}alsh case},
journal = {Math. Ann.},
fjournal = {Mathematische Annalen},
volume = {329},
year = {2004},
number = {3},
pages = {401--426},
issn = {0025-5831},
mrclass = {42B25 (42B30)},
mrnumber = {2127984},
mrreviewer = {Donald Krug},
doi = {10.1007/s00208-004-0518-1},
url = {https://doi.org/10.1007/s00208-004-0518-1},
zblnumber = {1073.42009},
} -
[MTT2]
C. Muscalu, T. Tao, and C. Thiele, "$L^p$ estimates for the biest. II. The Fourier case," Math. Ann., vol. 329, iss. 3, pp. 427-461, 2004.
@ARTICLE{MTT2,
author = {Muscalu, Camil and Tao, Terence and Thiele, Christoph},
title = {{$L^p$} estimates for the biest. {II}. {T}he {F}ourier case},
journal = {Math. Ann.},
fjournal = {Mathematische Annalen},
volume = {329},
year = {2004},
number = {3},
pages = {427--461},
issn = {0025-5831},
mrclass = {42B25 (42B30)},
mrnumber = {2127985},
mrreviewer = {Donald Krug},
doi = {10.1007/s00208-003-0508-8},
url = {https://doi.org/10.1007/s00208-003-0508-8},
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} -
[NRW1]
A. Nagel, N. Rivière, and S. Wainger, "On Hilbert transforms along curves," Bull. Amer. Math. Soc., vol. 80, pp. 106-108, 1974.
@ARTICLE{NRW1,
author = {Nagel, Alexander and Rivi{è}re, N{é}stor and Wainger, Stephen},
title = {On {H}ilbert transforms along curves},
journal = {Bull. Amer. Math. Soc.},
fjournal = {Bulletin of the Amer. Math. Soc.},
volume = {80},
year = {1974},
pages = {106--108},
issn = {0002-9904},
mrclass = {44A25 (42A40)},
mrnumber = {0450899},
mrreviewer = {S. Izumi},
doi = {10.1090/S0002-9904-1974-13374-4},
url = {https://doi.org/10.1090/S0002-9904-1974-13374-4},
zblnumber = {0293.44002},
} -
[NRW2]
A. Nagel, N. M. Rivière, and S. Wainger, "On Hilbert transforms along curves. II," Amer. J. Math., vol. 98, iss. 2, pp. 395-403, 1976.
@ARTICLE{NRW2,
author = {Nagel, Alexander and Rivi{è}re, N{é}stor M. and Wainger, Stephen},
title = {On {H}ilbert transforms along curves. {II}},
journal = {Amer. J. Math.},
fjournal = {American Journal of Mathematics},
volume = {98},
year = {1976},
number = {2},
pages = {395--403},
issn = {0002-9327},
mrclass = {44A25 (42A40)},
mrnumber = {0450900},
mrreviewer = {S. Izumi},
doi = {10.2307/2373893},
url = {https://doi.org/10.2307/2373893},
zblnumber = {0334.44012},
} -
[NSWvarcurv]
A. Nagel, E. M. Stein, and S. Wainger, "Hilbert transforms and maximal functions related to variable curves," in Harmonic Analysis in Euclidean Spaces, 1979, pp. 95-98.
@INPROCEEDINGS{NSWvarcurv,
author = {Nagel, Alexander and Stein, Elias M. and Wainger, Stephen},
title = {Hilbert transforms and maximal functions related to variable curves},
booktitle = {Harmonic Analysis in {E}uclidean Spaces},
venue = {{P}roc. {S}ympos. {P}ure {M}ath., {W}illiams {C}oll., {W}illiamstown, {M}ass., 1978, {P}art 1},
series = {Proc. Sympos. Pure Math., XXXV, Part},
pages = {95--98},
publisher = {Amer. Math. Soc., Providence, R.I.},
year = {1979},
mrclass = {42B25},
mrnumber = {0545242},
mrreviewer = {Hans P. Heinig},
doi = {10.1090/pspum/035.1},
url = {https://doi.org/10.1090/pspum/035.1},
zblnumber = {0463.42008},
} -
[Neu]
J. von Neumann, "Proof of the quasi-ergodic hypothesis," Proc. Nat. Acad. Sci. U.S.A., vol. 18, iss. 1, pp. 70-82, 1932.
@ARTICLE{Neu,
author = {von Neumann, J.},
title = {Proof of the quasi-ergodic hypothesis},
journal = {Proc. Nat. Acad. Sci. U.S.A.},
volume = {18},
number = {1},
pages = {70--82},
year = {1932},
doi = {10.1073/pnas.18.1.70},
url = {https://doi.org/10.1073/pnas.18.1.70},
zblnumber = {58.1271.03},
} -
[PhSt1]
D. H. Phong and E. M. Stein, "Singular integrals related to the Radon transform and boundary value problems," Proc. Nat. Acad. Sci. U.S.A., vol. 80, iss. 24, , Phys. Sci., pp. 7697-7701, 1983.
@ARTICLE{PhSt1,
author = {Phong, D. H. and Stein, E. M.},
title = {Singular integrals related to the {R}adon transform and boundary value problems},
journal = {Proc. Nat. Acad. Sci. U.S.A.},
fjournal = {Proceedings of the National Academy of Sciences of the United States of America},
volume = {80},
year = {1983},
number = {24, , Phys. Sci.},
pages = {7697--7701},
issn = {0027-8424},
mrclass = {42B20 (42B25 43A80 44A15 45E99)},
mrnumber = {0728667},
mrreviewer = {Steven George Krantz},
doi = {10.1073/pnas.80.24.7697},
url = {https://doi.org/10.1073/pnas.80.24.7697},
zblnumber = {0567.42010},
} -
[PhSt2]
D. H. Phong and E. M. Stein, "Hilbert integrals, singular integrals, and Radon transforms. I," Acta Math., vol. 157, iss. 1-2, pp. 99-157, 1986.
@ARTICLE{PhSt2,
author = {Phong, D. H. and Stein, E. M.},
title = {Hilbert integrals, singular integrals, and {R}adon transforms. {I}},
journal = {Acta Math.},
fjournal = {Acta Mathematica},
volume = {157},
year = {1986},
number = {1-2},
pages = {99--157},
issn = {0001-5962},
mrclass = {42B20 (35N15 47G05)},
mrnumber = {0857680},
mrreviewer = {Gerald B. Folland},
doi = {10.1007/BF02392592},
url = {https://doi.org/10.1007/BF02392592},
zblnumber = {0622.42011},
} -
[PT]
M. Pramanik and E. Terwilleger, "A weak $L^2$ estimate for a maximal dyadic sum operator on ${\Bbb R}^n$," Illinois J. Math., vol. 47, iss. 3, pp. 775-813, 2003.
@ARTICLE{PT,
author = {Pramanik, Malabika and Terwilleger, Erin},
title = {A weak {$L^2$} estimate for a maximal dyadic sum operator on {${\Bbb R}^n$}},
journal = {Illinois J. Math.},
fjournal = {Illinois Journal of Mathematics},
volume = {47},
year = {2003},
number = {3},
pages = {775--813},
issn = {0019-2082},
mrclass = {42B30 (47B38)},
mrnumber = {2007237},
mrreviewer = {Christoph M. Thiele},
doi = {10.1215/ijm/1258138194},
url = {https://doi.org/10.1215/ijm/1258138194},
zblnumber = {1040.42014},
} -
[P]
V. I. Prohorenko, "Divergent Fourier series," Mat. Sb. (N.S.), vol. 75 (117), pp. 185-198, 1968.
@ARTICLE{P,
author = {Prohorenko, V. I.},
title = {Divergent {F}ourier series},
journal = {Mat. Sb. (N.S.)},
volume = {75 (117)},
year = {1968},
pages = {185--198},
mrclass = {42.11},
mrnumber = {0223815},
mrreviewer = {I. M. Sheffer},
doi = {10.1070/SM1968v004n02ABEH002786},
url = {https://doi.org/10.1070/SM1968v004n02ABEH002786},
zblnumber = {},
} -
[RS1]
F. Ricci and E. M. Stein, "Oscillatory singular integrals and harmonic analysis on nilpotent groups," Proc. Nat. Acad. Sci. U.S.A., vol. 83, iss. 1, pp. 1-3, 1986.
@ARTICLE{RS1,
author = {Ricci, F. and Stein, E. M.},
title = {Oscillatory singular integrals and harmonic analysis on nilpotent groups},
journal = {Proc. Nat. Acad. Sci. U.S.A.},
fjournal = {Proceedings of the National Academy of Sciences of the United States of America},
volume = {83},
year = {1986},
number = {1},
pages = {1--3},
issn = {0027-8424},
mrclass = {22E30 (42B20 43A80)},
mrnumber = {0822187},
doi = {10.1073/pnas.83.1.1},
url = {https://doi.org/10.1073/pnas.83.1.1},
zblnumber = {0583.43010},
} -
[RS2]
F. Ricci and E. M. Stein, "Harmonic analysis on nilpotent groups and singular integrals. I. Oscillatory integrals," J. Funct. Anal., vol. 73, iss. 1, pp. 179-194, 1987.
@ARTICLE{RS2,
author = {Ricci, Fulvio and Stein, E. M.},
title = {Harmonic analysis on nilpotent groups and singular integrals. {I}. {O}scillatory integrals},
journal = {J. Funct. Anal.},
fjournal = {Journal of Functional Analysis},
volume = {73},
year = {1987},
number = {1},
pages = {179--194},
issn = {0022-1236},
mrclass = {42B20 (22E30 43A80 58G15)},
mrnumber = {0890662},
mrreviewer = {Detlef H. Müller},
doi = {10.1016/0022-1236(87)90064-4},
url = {https://doi.org/10.1016/0022-1236(87)90064-4},
zblnumber = {0622.42010},
} -
[RS3]
F. Ricci and E. M. Stein, "Harmonic analysis on nilpotent groups and singular integrals. III. Fractional integration along manifolds," J. Funct. Anal., vol. 86, iss. 2, pp. 360-389, 1989.
@ARTICLE{RS3,
author = {Ricci, Fulvio and Stein, Elias M.},
title = {Harmonic analysis on nilpotent groups and singular integrals. {III}. {F}ractional integration along manifolds},
journal = {J. Funct. Anal.},
fjournal = {Journal of Functional Analysis},
volume = {86},
year = {1989},
number = {2},
pages = {360--389},
issn = {0022-1236},
mrclass = {22E30 (42B20 47G05)},
mrnumber = {1021141},
mrreviewer = {Detlef H. Müller},
doi = {10.1016/0022-1236(89)90057-8},
url = {https://doi.org/10.1016/0022-1236(89)90057-8},
zblnumber = {0684.22006},
} -
@ARTICLE{Rie,
author = {Riesz, Marcel},
title = {Sur les fonctions conjuguées},
journal = {Math. Z.},
fjournal = {Mathematische Zeitschrift},
volume = {27},
year = {1928},
number = {1},
pages = {218--244},
issn = {0025-5874},
mrclass = {DML},
mrnumber = {1544909},
doi = {10.1007/BF01171098},
url = {https://doi.org/10.1007/BF01171098},
zblnumber = {53.0259.02},
} -
@ARTICLE{Rot,
author = {Roth, K. F.},
title = {On certain sets of integers},
journal = {J. London Math. Soc.},
fjournal = {The Journal of the London Mathematical Society},
volume = {28},
year = {1953},
pages = {104--109},
issn = {0024-6107},
mrclass = {10.0X},
mrnumber = {0051853},
mrreviewer = {P. Erdős},
doi = {10.1112/jlms/s1-28.1.104},
url = {https://doi.org/10.1112/jlms/s1-28.1.104},
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} -
[sj3]
P. Sjölin, "An inequality of Paley and convergence a.e. of Walsh-Fourier series," Ark. Mat., vol. 7, pp. 551-570, 1969.
@ARTICLE{sj3,
author = {Sj{ö}lin, Per},
title = {An inequality of {P}aley and convergence a.e. of {W}alsh-{F}ourier series},
journal = {Ark. Mat.},
fjournal = {Arkiv för Matematik},
volume = {7},
year = {1969},
pages = {551--570},
issn = {0004-2080},
mrclass = {42.16},
mrnumber = {0241885},
mrreviewer = {R. A. Hunt},
doi = {10.1007/BF02590894},
url = {https://doi.org/10.1007/BF02590894},
zblnumber = {0169.08203},
} -
[sj2]
P. Sjölin, "Convergence almost everywhere of certain singular integrals and multiple Fourier series," Ark. Mat., vol. 9, pp. 65-90, 1971.
@ARTICLE{sj2,
author = {Sj{ö}lin, Per},
title = {Convergence almost everywhere of certain singular integrals and multiple {F}ourier series},
journal = {Ark. Mat.},
fjournal = {Arkiv för Matematik},
volume = {9},
year = {1971},
pages = {65--90},
issn = {0004-2080},
mrclass = {42A92},
mrnumber = {0336222},
mrreviewer = {W. C. Connett},
doi = {10.1007/BF02383638},
url = {https://doi.org/10.1007/BF02383638},
zblnumber = {0212.41703},
} -
[Sjoconv]
P. Sjölin, "Convolution with oscillating kernels," Indiana Univ. Math. J., vol. 30, iss. 1, pp. 47-55, 1981.
@ARTICLE{Sjoconv,
author = {Sj{ö}lin, Per},
title = {Convolution with oscillating kernels},
journal = {Indiana Univ. Math. J.},
fjournal = {Indiana University Mathematics Journal},
volume = {30},
year = {1981},
number = {1},
pages = {47--55},
issn = {0022-2518},
mrclass = {42B30 (44A35)},
mrnumber = {0600031},
mrreviewer = {C. Carton-Lebrun},
doi = {10.1512/iumj.1981.30.30004},
url = {https://doi.org/10.1512/iumj.1981.30.30004},
zblnumber = {0419.47020},
} -
[SS]
P. Sjölin and F. Soria, "Remarks on a theorem by N. Yu. Antonov," Studia Math., vol. 158, iss. 1, pp. 79-97, 2003.
@ARTICLE{SS,
author = {Sj{ö}lin, Per and Soria, Fernando},
title = {Remarks on a theorem by {N}. {Y}u. {A}ntonov},
journal = {Studia Math.},
fjournal = {Studia Mathematica},
volume = {158},
year = {2003},
number = {1},
pages = {79--97},
issn = {0039-3223},
mrclass = {42A20 (42B25 42B35)},
mrnumber = {2014553},
mrreviewer = {Elijah Liflyand},
doi = {10.4064/sm158-1-7},
url = {https://doi.org/10.4064/sm158-1-7},
zblnumber = {1053.42005},
} -
[So1]
F. Soria, "Note on differentiation of integrals and the halo conjecture," Studia Math., vol. 81, iss. 1, pp. 29-36, 1985.
@ARTICLE{So1,
author = {Soria, Fernando},
title = {Note on differentiation of integrals and the halo conjecture},
journal = {Studia Math.},
fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
volume = {81},
year = {1985},
number = {1},
pages = {29--36},
issn = {0039-3223},
mrclass = {42B25 (42B05)},
mrnumber = {0818168},
doi = {10.4064/sm-81-1-29-36},
url = {https://doi.org/10.4064/sm-81-1-29-36},
zblnumber = {0526.28003},
} -
[So2]
F. Soria, "On an extrapolation theorem of Carleson-Sjölin with applications to a.e. convergence of Fourier series," Studia Math., vol. 94, iss. 3, pp. 235-244, 1989.
@ARTICLE{So2,
author = {Soria, Fernando},
title = {On an extrapolation theorem of {C}arleson-{S}j{ö}lin with applications to a.e. convergence of {F}ourier series},
journal = {Studia Math.},
fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
volume = {94},
year = {1989},
number = {3},
pages = {235--244},
issn = {0039-3223},
mrclass = {42A50},
mrnumber = {1019791},
mrreviewer = {Colin C. Graham},
doi = {10.4064/sm-94-3-235-244},
url = {https://doi.org/10.4064/sm-94-3-235-244},
zblnumber = {0684.42001},
} -
[s1]
E. M. Stein, "On limits of sequences of operators," Ann. of Math. (2), vol. 74, pp. 140-170, 1961.
@ARTICLE{s1,
author = {Stein, E. M.},
title = {On limits of sequences of operators},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {74},
year = {1961},
pages = {140--170},
issn = {0003-486X},
mrclass = {42.11},
mrnumber = {0125392},
mrreviewer = {R. P. Boas},
doi = {10.2307/1970308},
url = {https://doi.org/10.2307/1970308},
zblnumber = {0103.08903},
} -
[Stepois]
E. M. Stein, "Maximal functions. I. Spherical means," Proc. Nat. Acad. Sci. U.S.A., vol. 73, iss. 7, pp. 2174-2175, 1976.
@ARTICLE{Stepois,
author = {Stein, Elias M.},
title = {Maximal functions. {I}. {S}pherical means},
journal = {Proc. Nat. Acad. Sci. U.S.A.},
fjournal = {Proceedings of the National Academy of Sciences of the United States of America},
volume = {73},
year = {1976},
number = {7},
pages = {2174--2175},
issn = {0027-8424},
mrclass = {42A40 (43A85)},
mrnumber = {0420116},
mrreviewer = {Alberto Torchinsky},
doi = {10.1073/pnas.73.7.2174},
url = {https://doi.org/10.1073/pnas.73.7.2174},
zblnumber = {0332.42018},
} -
[s2] E. M. Stein, "Oscillatory integrals related to Radon-like transforms," in Proceedings of the Conference in Honor of Jean-Pierre Kahane, 1995, pp. 535-551.
@INPROCEEDINGS{s2,
author = {Stein, Elias M.},
title = {Oscillatory integrals related to {R}adon-like transforms},
booktitle = {Proceedings of the {C}onference in {H}onor of {J}ean-{P}ierre {K}ahane},
venue = {{O}rsay, 1993},
journal = {J. Fourier Anal. Appl.},
fjournal = {The Journal of Fourier Analysis and Applications},
year = {1995},
number = {Special Issue},
pages = {535--551},
issn = {1069-5869},
mrclass = {42B20 (35S30 42B25)},
mrnumber = {1364908},
mrreviewer = {Andreas Seeger},
zblnumber = {0971.42009},
} -
[STeWA]
E. M. Stein and S. Wainger, "The estimation of an integral arising in multiplier transformations," Studia Math., vol. 35, pp. 101-104, 1970.
@ARTICLE{STeWA,
author = {Stein, Elias M. and Wainger, Stephen},
title = {The estimation of an integral arising in multiplier transformations},
journal = {Studia Math.},
fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
volume = {35},
year = {1970},
pages = {101--104},
issn = {0039-3223},
mrclass = {47.70 (42.00)},
mrnumber = {0265995},
mrreviewer = {S. Izumi},
doi = {10.4064/sm-35-1-101-104},
url = {https://doi.org/10.4064/sm-35-1-101-104},
zblnumber = {0202.12401},
} -
[sw1]
E. M. Stein and S. Wainger, "Problems in harmonic analysis related to curvature," Bull. Amer. Math. Soc., vol. 84, iss. 6, pp. 1239-1295, 1978.
@ARTICLE{sw1,
author = {Stein, Elias M. and Wainger, Stephen},
title = {Problems in harmonic analysis related to curvature},
journal = {Bull. Amer. Math. Soc.},
fjournal = {Bulletin of the Amer. Math. Soc.},
volume = {84},
year = {1978},
number = {6},
pages = {1239--1295},
issn = {0002-9904},
mrclass = {42B20 (28A15)},
mrnumber = {0508453},
mrreviewer = {Alberto Torchinsky},
doi = {10.1090/S0002-9904-1978-14554-6},
url = {https://doi.org/10.1090/S0002-9904-1978-14554-6},
zblnumber = {0393.42010},
} -
[sw]
E. M. Stein and S. Wainger, "Oscillatory integrals related to Carleson’s theorem," Math. Res. Lett., vol. 8, iss. 5-6, pp. 789-800, 2001.
@ARTICLE{sw,
author = {Stein, Elias M. and Wainger, Stephen},
title = {Oscillatory integrals related to {C}arleson's theorem},
journal = {Math. Res. Lett.},
fjournal = {Mathematical Research Letters},
volume = {8},
year = {2001},
number = {5-6},
pages = {789--800},
issn = {1073-2780},
mrclass = {42B20 (47G10)},
mrnumber = {1879821},
mrreviewer = {B. S. Rubin},
doi = {10.4310/MRL.2001.v8.n6.a9},
url = {https://doi.org/10.4310/MRL.2001.v8.n6.a9},
zblnumber = {0998.42007},
} -
[Str]
R. S. Strichartz, "Singular integrals supported on submanifolds," Studia Math., vol. 74, iss. 2, pp. 137-151, 1982.
@ARTICLE{Str,
author = {Strichartz, Robert S.},
title = {Singular integrals supported on submanifolds},
journal = {Studia Math.},
fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica},
volume = {74},
year = {1982},
number = {2},
pages = {137--151},
issn = {0039-3223},
mrclass = {42B20 (44A15 58G99)},
mrnumber = {0679830},
doi = {10.4064/sm-74-2-137-151},
url = {https://doi.org/10.4064/sm-74-2-137-151},
zblnumber = {0501.43007},
} -
[Sz1]
E. Szemerédi, "On sets of integers containing no four elements in arithmetic progression," Acta Math. Acad. Sci. Hungar., vol. 20, pp. 89-104, 1969.
@ARTICLE{Sz1,
author = {Szemerédi, E.},
title = {On sets of integers containing no four elements in arithmetic progression},
journal = {Acta Math. Acad. Sci. Hungar.},
fjournal = {Acta Mathematica. Academiae Scientiarum Hungaricae},
volume = {20},
year = {1969},
pages = {89--104},
issn = {0001-5954},
mrclass = {10.68},
mrnumber = {0245555},
mrreviewer = {H. Halberstam},
doi = {10.1007/BF01894569},
url = {https://doi.org/10.1007/BF01894569},
zblnumber = {0175.04301},
} -
[Sz2]
E. Szemerédi, "On sets of integers containing no $k$ elements in arithmetic progression," Acta Arith., vol. 27, pp. 199-245, 1975.
@ARTICLE{Sz2,
author = {Szemerédi, E.},
title = {On sets of integers containing no {$k$} elements in arithmetic progression},
journal = {Acta Arith.},
fjournal = {Polska Akademia Nauk. Instytut Matematyczny. Acta Arithmetica},
volume = {27},
year = {1975},
pages = {199--245},
issn = {0065-1036},
mrclass = {10L10},
mrnumber = {0369312},
mrreviewer = {S. L. G. Choi},
doi = {10.4064/aa-27-1-199-245},
url = {https://doi.org/10.4064/aa-27-1-199-245},
zblnumber = {0303.10056},
} -
[Tao]
T. Tao, "Norm convergence of multiple ergodic averages for commuting transformations," Ergodic Theory Dynam. Systems, vol. 28, iss. 2, pp. 657-688, 2008.
@ARTICLE{Tao,
author = {Tao, Terence},
title = {Norm convergence of multiple ergodic averages for commuting transformations},
journal = {Ergodic Theory Dynam. Systems},
fjournal = {Ergodic Theory and Dynamical Systems},
volume = {28},
year = {2008},
number = {2},
pages = {657--688},
issn = {0143-3857},
mrclass = {37A30 (28D05 47A35)},
mrnumber = {2408398},
mrreviewer = {Nikos Frantzikinakis},
doi = {10.1017/S0143385708000011},
url = {https://doi.org/10.1017/S0143385708000011},
zblnumber = {1181.37004},
} -
[Wa]
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@ARTICLE{Wa,
author = {Walsh, Miguel N.},
title = {Norm convergence of nilpotent ergodic averages},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {175},
year = {2012},
number = {3},
pages = {1667--1688},
issn = {0003-486X},
mrclass = {37A30 (37A05)},
mrnumber = {2912715},
mrreviewer = {Nikos Frantzikinakis},
doi = {10.4007/annals.2012.175.3.15},
url = {https://doi.org/10.4007/annals.2012.175.3.15},
zblnumber = {1248.37008},
} -
[Zieg]
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@ARTICLE{Zieg,
author = {Ziegler, Tamar},
title = {Universal characteristic factors and {F}urstenberg averages},
journal = {J. Amer. Math. Soc.},
fjournal = {Journal of the Amer. Math. Soc.},
volume = {20},
year = {2007},
number = {1},
pages = {53--97},
issn = {0894-0347},
mrclass = {37A30 (28D05 37A25)},
mrnumber = {2257397},
mrreviewer = {Randall McCutcheon},
doi = {10.1090/S0894-0347-06-00532-7},
url = {https://doi.org/10.1090/S0894-0347-06-00532-7},
zblnumber = {1198.37014},
} -
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@MISC{Zo,
author = {Zorin-Kranich, P.},
title = {Maximal polynomial modulations of singular integrals},
arxiv = {1711.03524},
year = {2017},
zblnumber = {},
} -
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@BOOK{Zyg,
author = {Zygmund, A.},
title = {Trigonometric Series. {V}ol. {I},
{II}},
series = {Cambridge Math. Library},
edition = {{T}hird},
note = {with a foreword by Robert A. Fefferman},
publisher = {Cambridge Univ. Press, Cambridge},
year = {2002},
pages = {xii; Vol. I: xiv+383 pp.; Vol. II: viii+364},
isbn = {0-521-89053-5},
mrclass = {01A75 (42-02)},
mrnumber = {1963498},
zblnumber = {1084.42003},
}