Abstract
In this paper we provide a proof of the Carleson $\varepsilon ^2$-conjecture. This result yields a characterization (up to exceptional sets of zero length) of the tangent points of a Jordan curve in terms of the finiteness of the associated Carleson $\varepsilon ^2$-square function.
-
[BL]
M. Badger and S. Lewis, "Local set approximation: Mattila-Vuorinen type sets, Reifenberg type sets, and tangent sets," Forum Math. Sigma, vol. 3, p. 24, 2015.
@ARTICLE{BL,
author = {Badger, Matthew and Lewis, Stephen},
title = {Local set approximation: {M}attila-{V}uorinen type sets, {R}eifenberg type sets, and tangent sets},
journal = {Forum Math. Sigma},
fjournal = {Forum of Mathematics. Sigma},
volume = {3},
year = {2015},
pages = {Paper no. e24, 63},
mrclass = {49Q15 (28A75 35R35 49J52)},
mrnumber = {3482273},
mrreviewer = {Stefan Steinerberger},
doi = {10.1017/fms.2015.26},
url = {https://doi.org/10.1017/fms.2015.26},
zblnumber = {1348.49042},
} -
[Bishop-conjectures]
C. J. Bishop, "Some questions concerning harmonic measure," in Partial Differential Equations with Minimal Smoothness and Applications, Springer, New York, 1992, vol. 42, pp. 89-97.
@INCOLLECTION{Bishop-conjectures,
author = {Bishop, Christopher J.},
title = {Some questions concerning harmonic measure},
booktitle = {Partial Differential Equations with Minimal Smoothness and Applications},
venue = {{C}hicago, {IL},
1990},
series = {IMA Vol. Math. Appl.},
volume = {42},
pages = {89--97},
publisher = {Springer, New York},
year = {1992},
mrclass = {30C85 (30C35 30C62 31A15)},
mrnumber = {1155854},
mrreviewer = {T. J. Lyons},
doi = {10.1007/978-1-4612-2898-1\_7},
url = {https://doi.org/10.1007/978-1-4612-2898-1_7},
zblnumber = {0792.30005},
} -
[Bishop-Jones-schwartzian]
C. J. Bishop and P. W. Jones, "Harmonic measure, $L^2$ estimates and the Schwarzian derivative," J. Anal. Math., vol. 62, pp. 77-113, 1994.
@ARTICLE{Bishop-Jones-schwartzian,
author = {Bishop, Christopher J. and Jones, Peter W.},
title = {Harmonic measure, {$L^2$} estimates and the {S}chwarzian derivative},
journal = {J. Anal. Math.},
fjournal = {Journal d'Analyse Mathématique},
volume = {62},
year = {1994},
pages = {77--113},
issn = {0021-7670},
mrclass = {30C85 (30C62 31A15)},
mrnumber = {1269200},
mrreviewer = {Juha Heinonen},
doi = {10.1007/BF02835949},
url = {https://doi.org/10.1007/BF02835949},
zblnumber = {0801.30024},
} -
[David-vitus]
G. David, "Unrectifiable $1$-sets have vanishing analytic capacity," Rev. Mat. Iberoamericana, vol. 14, iss. 2, pp. 369-479, 1998.
@ARTICLE{David-vitus,
author = {David, Guy},
title = {Unrectifiable {$1$}-sets have vanishing analytic capacity},
journal = {Rev. Mat. Iberoamericana},
fjournal = {Revista Matem\'{a}tica Iberoamericana},
volume = {14},
year = {1998},
number = {2},
pages = {369--479},
issn = {0213-2230},
mrclass = {42B20 (30E20)},
mrnumber = {1654535},
mrreviewer = {Joan Verdera},
doi = {10.4171/RMI/242},
url = {https://doi.org/10.4171/RMI/242},
zblnumber = {0913.30012},
} -
[DS1] G. David and S. Semmes, Singular Integrals and Rectifiable Sets in ${\bf R}^n$: Beyond Lipschitz Graphs, Math. Soc. France, Paris, 1991, vol. 193.
@BOOK{DS1,
author = {David, Guy and Semmes, S.},
title = {Singular Integrals and Rectifiable Sets in {${\bf R}^n$}: {B}eyond {L}ipschitz Graphs},
series = {Astérisque},
volume = {193},
publisher = {Math. Soc. France, Paris},
year = {1991},
pages = {152},
issn = {0303-1179},
mrclass = {42B20 (42B25)},
mrnumber = {1113517},
mrreviewer = {Stephen Buckley},
zblnumber = {0743.49018},
} -
[DS2]
G. David and S. Semmes, Analysis of and on uniformly rectifiable sets, Amer. Math. Soc., Providence, RI, 1993, vol. 38.
@BOOK{DS2,
author = {David, Guy and Semmes, Stephen},
title = {Analysis of and on uniformly rectifiable sets},
series = {Math. Surv. Monogr.},
volume = {38},
publisher = {Amer. Math. Soc., Providence, RI},
year = {1993},
pages = {xii+356},
isbn = {0-8218-1537-7},
mrclass = {28A75 (30C65 30E20 42B20 42B25)},
mrnumber = {1251061},
mrreviewer = {Christopher Bishop},
doi = {10.1090/surv/038},
url = {https://doi.org/10.1090/surv/038},
zblnumber = {0832.42008},
} -
[GM]
J. B. Garnett and D. E. Marshall, Harmonic Measure, Cambridge Univ. Press, Cambridge, 2005, vol. 2.
@BOOK{GM,
author = {Garnett, John B. and Marshall, Donald E.},
title = {Harmonic Measure},
series = {New Math. Monogr.},
volume = {2},
publisher = {Cambridge Univ. Press, Cambridge},
year = {2005},
pages = {xvi+571},
isbn = {978-0-521-47018-6; 0-521-47018-8},
mrclass = {31-02 (31A15)},
mrnumber = {2150803},
mrreviewer = {Christopher Bishop},
doi = {10.1017/CBO9780511546617},
url = {https://doi.org/10.1017/CBO9780511546617},
zblnumber = {1077.31001},
} -
[Jerison-Kenig]
D. S. Jerison and C. E. Kenig, "Boundary behavior of harmonic functions in nontangentially accessible domains," Adv. in Math., vol. 46, iss. 1, pp. 80-147, 1982.
@ARTICLE{Jerison-Kenig,
author = {Jerison, David S. and Kenig, Carlos E.},
title = {Boundary behavior of harmonic functions in nontangentially accessible domains},
journal = {Adv. in Math.},
fjournal = {Advances in Mathematics},
volume = {46},
year = {1982},
number = {1},
pages = {80--147},
issn = {0001-8708},
mrclass = {31B25 (42B25)},
mrnumber = {0676988},
mrreviewer = {Yves Meyer},
doi = {10.1016/0001-8708(82)90055-X},
url = {https://doi.org/10.1016/0001-8708(82)90055-X},
zblnumber = {0514.31003},
} -
[Jo]
P. W. Jones, "Rectifiable sets and the traveling salesman problem," Invent. Math., vol. 102, iss. 1, pp. 1-15, 1990.
@ARTICLE{Jo,
author = {Jones, Peter W.},
title = {Rectifiable sets and the traveling salesman problem},
journal = {Invent. Math.},
fjournal = {Inventiones Mathematicae},
volume = {102},
year = {1990},
number = {1},
pages = {1--15},
issn = {0020-9910},
mrclass = {26B15 (05C38 28A75 30E10 42C99 90C10)},
mrnumber = {1069238},
mrreviewer = {Pertti Mattila},
doi = {10.1007/BF01233418},
url = {https://doi.org/10.1007/BF01233418},
zblnumber = {0731.30018},
} -
[Kenig-Toro-AENS]
C. E. Kenig and T. Toro, "Poisson kernel characterization of Reifenberg flat chord arc domains," Ann. Sci. École Norm. Sup. (4), vol. 36, iss. 3, pp. 323-401, 2003.
@ARTICLE{Kenig-Toro-AENS,
author = {Kenig, Carlos E. and Toro, Tatiana},
title = {Poisson kernel characterization of {R}eifenberg flat chord arc domains},
journal = {Ann. Sci. \'{E}cole Norm. Sup. (4)},
fjournal = {Annales Scientifiques de l'\'{E}cole Normale Supérieure. Quatrième Série},
volume = {36},
year = {2003},
number = {3},
pages = {323--401},
issn = {0012-9593},
mrclass = {31B25 (31A15 31B15 35R35)},
mrnumber = {1977823},
mrreviewer = {Luca Lorenzi},
doi = {10.1016/S0012-9593(03)00012-0},
url = {https://doi.org/10.1016/S0012-9593(03)00012-0},
zblnumber = {1027.31005},
} -
[Leger]
J. C. Léger, "Menger curvature and rectifiability," Ann. of Math. (2), vol. 149, iss. 3, pp. 831-869, 1999.
@ARTICLE{Leger,
author = {Léger, J. C.},
title = {Menger curvature and rectifiability},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {149},
year = {1999},
number = {3},
pages = {831--869},
issn = {0003-486X},
mrclass = {49Q15},
mrnumber = {1709304},
mrreviewer = {Karsten Grosse-Brauckmann},
doi = {10.2307/121074},
url = {https://doi.org/10.2307/121074},
zblnumber = {0966.28003},
} -
[Mattila-book]
P. Mattila, Geometry of Sets and Measures in Euclidean Spaces. Fractals and Rectifiability, Cambridge Univ. Press, Cambridge, 1995, vol. 44.
@BOOK{Mattila-book,
author = {Mattila, Pertti},
title = {Geometry of Sets and Measures in {E}uclidean Spaces. Fractals and Rectifiability},
series = {Cambridge Stud. Adv. Math.},
volume = {44},
publisher = {Cambridge Univ. Press, Cambridge},
year = {1995},
pages = {xii+343},
isbn = {0-521-46576-1; 0-521-65595-1},
mrclass = {28A75 (49Q20)},
mrnumber = {1333890},
mrreviewer = {Harold Parks},
doi = {10.1017/CBO9780511623813},
url = {https://doi.org/10.1017/CBO9780511623813},
zblnumber = {0819.28004},
} -
[NToV]
F. Nazarov, X. Tolsa, and A. Volberg, "On the uniform rectifiability of AD-regular measures with bounded Riesz transform operator: the case of codimension 1," Acta Math., vol. 213, iss. 2, pp. 237-321, 2014.
@ARTICLE{NToV,
author = {Nazarov, Fedor and Tolsa, Xavier and Volberg, Alexander},
title = {On the uniform rectifiability of {AD}-regular measures with bounded {R}iesz transform operator: the case of codimension 1},
journal = {Acta Math.},
fjournal = {Acta Mathematica},
volume = {213},
year = {2014},
number = {2},
pages = {237--321},
issn = {0001-5962},
mrclass = {42B20 (31B10)},
mrnumber = {3286036},
mrreviewer = {Gerald B. Folland},
doi = {10.1007/s11511-014-0120-7},
url = {https://doi.org/10.1007/s11511-014-0120-7},
zblnumber = {1311.28004},
} -
[Naber-Valtorta]
A. Naber and D. Valtorta, "Rectifiable-Reifenberg and the regularity of stationary and minimizing harmonic maps," Ann. of Math. (2), vol. 185, iss. 1, pp. 131-227, 2017.
@ARTICLE{Naber-Valtorta,
author = {Naber, Aaron and Valtorta, Daniele},
title = {Rectifiable-{R}eifenberg and the regularity of stationary and minimizing harmonic maps},
journal = {Ann. of Math. (2)},
fjournal = {Annals of Mathematics. Second Series},
volume = {185},
year = {2017},
number = {1},
pages = {131--227},
issn = {0003-486X},
mrclass = {58E20 (53C43)},
mrnumber = {3583353},
mrreviewer = {Andreas Gastel},
doi = {10.4007/annals.2017.185.1.3},
url = {https://doi.org/10.4007/annals.2017.185.1.3},
zblnumber = {1393.58009},
} -
[Tolsa-llibre]
X. Tolsa, Analytic Capacity, the Cauchy Transform, and Non-Homogeneous Calderón-Zygmund Theory, Birkhäuser/Springer, Cham, 2014, vol. 307.
@BOOK{Tolsa-llibre,
author = {Tolsa, Xavier},
title = {Analytic Capacity, the {C}auchy Transform, and Non-Homogeneous {C}alderón-{Z}ygmund Theory},
series = {Progr. Math.},
volume = {307},
publisher = {Birkhäuser/Springer, Cham},
year = {2014},
pages = {xiv+396},
isbn = {978-3-319-00595-9; 978-3-319-00596-6},
mrclass = {42-02 (30C85 42B20 42B25 42B35)},
mrnumber = {3154530},
mrreviewer = {Tuomas P. Hytönen},
doi = {10.1007/978-3-319-00596-6},
url = {https://doi.org/10.1007/978-3-319-00596-6},
zblnumber = {1290.42002},
}