Abstract
The Bohnenblust-Hille inequality says that the $\ell^{\frac{2m}{m+1}}$-norm of the coefficients of an $m$-homogeneous polynomial $P$ on $\Bbb{C}^n$ is bounded by $\| P \|_\infty$ times a constant independent of $n$, where $\|\cdot \|_\infty$ denotes the supremum norm on the polydisc $\mathbb{D}^n$. The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be $C^m$ for some $C>1$. Combining this improved version of the Bohnenblust-Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc $\mathbb{D}^n$ behaves asymptotically as $\sqrt{(\log n)/n}$ modulo a factor bounded away from $0$ and infinity, and the Sidon constant for the set of frequencies $\bigl\{ \log n: n \text{a positive integer} \le N\bigr\}$ is $\sqrt{N}\exp\{(-1/\sqrt{2}+o(1))\sqrt{\log N\log\log N}\}$ as $N\to \infty$.
-
[BCQ06]
R. Balasubramanian, B. Calado, and H. Queffélec, "The Bohr inequality for ordinary Dirichlet series," Studia Math., vol. 175, iss. 3, pp. 285-304, 2006.
@article {BCQ06, MRKEY = {2261747},
AUTHOR = {Balasubramanian, R. and Calado, B. and Queff{é}lec, H.},
TITLE = {The {B}ohr inequality for ordinary {D}irichlet series},
JOURNAL = {Studia Math.},
FJOURNAL = {Studia Mathematica},
VOLUME = {175},
YEAR = {2006},
NUMBER = {3},
PAGES = {285--304},
ISSN = {0039-3223},
MRCLASS = {30B50 (40A05 42A45 42B30 46E15)},
MRNUMBER = {2261747},
MRREVIEWER = {H. Turgay Kaptano{\u{g}}lu},
DOI = {10.4064/sm175-3-7},
ZBLNUMBER = {1110.30001},
} -
[Bayart02]
F. Bayart, "Hardy spaces of Dirichlet series and their composition operators," Monatsh. Math., vol. 136, iss. 3, pp. 203-236, 2002.
@article {Bayart02, MRKEY = {1919645},
AUTHOR = {Bayart, Fr{é}d{é}ric},
TITLE = {Hardy spaces of {D}irichlet series and their composition operators},
JOURNAL = {Monatsh. Math.},
FJOURNAL = {Monatshefte für Mathematik},
VOLUME = {136},
YEAR = {2002},
NUMBER = {3},
PAGES = {203--236},
ISSN = {0026-9255},
CODEN = {MNMTA2},
MRCLASS = {42B30 (30B50 32A55 46E15 47B33)},
MRNUMBER = {1919645},
MRREVIEWER = {Cristian E. Guti{é}rrez},
DOI = {10.1007/s00605-002-0470-7},
ZBLNUMBER = {1076.46017},
} -
[Blei79]
R. C. Blei, "Fractional Cartesian products of sets," Ann. Inst. Fourier $($Grenoble$)$, vol. 29, iss. 2, p. v, 79-105, 1979.
@article {Blei79, MRKEY = {0539694},
AUTHOR = {Blei, Ron C.},
TITLE = {Fractional {C}artesian products of sets},
JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
VOLUME = {29},
YEAR = {1979},
NUMBER = {2},
PAGES = {v, 79--105},
ISSN = {0373-0956},
CODEN = {AIFUA7},
MRCLASS = {43A46},
MRNUMBER = {0539694},
MRREVIEWER = {Gordon Woodward},
ZBLNUMBER = {0381.43003},
URL = {http://www.numdam.org/item?id=AIF_1979__29_2_79_0},
} -
[Boas00] H. P. Boas, "Majorant series," J. Korean Math. Soc., vol. 37, iss. 2, pp. 321-337, 2000.
@article {Boas00, MRKEY = {1775963},
AUTHOR = {Boas, Harold P.},
TITLE = {Majorant series},
JOURNAL = {J. Korean Math. Soc.},
FJOURNAL = {Journal of the Korean Mathematical Society},
VOLUME = {37},
YEAR = {2000},
NUMBER = {2},
PAGES = {321--337},
ISSN = {0304-9914},
CODEN = {JKMSDG},
MRCLASS = {32A05 (30B10)},
MRNUMBER = {1775963},
MRREVIEWER = {Rapha{ë}le Supper},
ZBLNUMBER = {0965.32001},
} -
[BoasKhav97]
H. P. Boas and D. Khavinson, "Bohr’s power series theorem in several variables," Proc. Amer. Math. Soc., vol. 125, iss. 10, pp. 2975-2979, 1997.
@article {BoasKhav97, MRKEY = {1443371},
AUTHOR = {Boas, Harold P. and Khavinson, Dmitry},
TITLE = {Bohr's power series theorem in several variables},
JOURNAL = {Proc. Amer. Math. Soc.},
FJOURNAL = {Proceedings of the American Mathematical Society},
VOLUME = {125},
YEAR = {1997},
NUMBER = {10},
PAGES = {2975--2979},
ISSN = {0002-9939},
CODEN = {PAMYAR},
MRCLASS = {32A05},
MRNUMBER = {1443371},
MRREVIEWER = {Richard M. Timoney},
DOI = {10.1090/S0002-9939-97-04270-6},
ZBLNUMBER = {0888.32001},
} -
[BH31]
H. F. Bohnenblust and E. Hille, "On the absolute convergence of Dirichlet series," Ann. of Math., vol. 32, iss. 3, pp. 600-622, 1931.
@article {BH31, MRKEY = {1503020},
AUTHOR = {Bohnenblust, H. F. and Hille, Einar},
TITLE = {On the absolute convergence of {D}irichlet series},
JOURNAL = {Ann. of Math.},
FJOURNAL = {Annals of Mathematics. Second Series},
VOLUME = {32},
YEAR = {1931},
NUMBER = {3},
PAGES = {600--622},
ISSN = {0003-486X},
CODEN = {ANMAAH},
MRCLASS = {Contributed Item},
MRNUMBER = {1503020},
DOI = {10.2307/1968255},
ZBLNUMBER = {0001.26901},
} -
[B13a] H. Bohr, "Über die gleichmässige Konvergenz Dirichletscher Reihen," J. Reine Angew. Math., vol. 143, pp. 203-211, 1913.
@article{B13a,
author={Bohr, H.},
TITLE={Über die gleichmässige {K}onvergenz {D}irichletscher {R}eihen},
JOURNAL={J. Reine Angew. Math.},
VOLUME={143},
YEAR={1913},
PAGES={203--211},
JFMNUMBER={44.0307.01},
} -
[B13b] H. Bohr, "Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reihen," Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl., pp. 441-488, 1913.
@article{B13b,
author={Bohr, H.},
TITLE={Über die {B}edeutung der {P}otenzreihen unendlich vieler {V}ariabeln in der {T}heorie der {D}irichletschen {R}eihen},
JOURNAL={ Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl.},
YEAR={1913},
PAGES={441--488},
JFMNUMBER={44.0306.01},
} -
[Bohr14] H. Bohr, "A theorem concerning power series," Proc. London Math. Soc., vol. 13, pp. 1-5, 1914.
@article{Bohr14,
author={Bohr, H.},
TITLE={A theorem concerning power series},
JOURNAL={Proc. London Math. Soc.},
VOLUME={13},
YEAR={1914},
PAGES={1--5},
JFMNUMBER={44.0289.01},
} -
[Bo]
A. Bonami, "Étude des coefficients de Fourier des fonctions de $L^{p}(G)$," Ann. Inst. Fourier $($Grenoble$)$, vol. 20, iss. 2, pp. 335-402 (1971), 1970.
@article {Bo, MRKEY = {0283496},
AUTHOR = {Bonami, Aline},
TITLE = {\'{E}tude des coefficients de {F}ourier des fonctions de {$L\sp{p}(G)$}},
JOURNAL = {Ann. Inst. Fourier $($Grenoble$)$},
FJOURNAL = {Université de Grenoble. Annales de l'Institut Fourier},
VOLUME = {20},
YEAR = {1970},
NUMBER = {2},
PAGES = {335--402 (1971)},
ISSN = {0373-0956},
MRCLASS = {42.50},
MRNUMBER = {0283496},
MRREVIEWER = {S. Izumi},
URL = {http://www.numdam.org/item?id=AIF_1970__20_2_335_0},
ZBLNUMBER = {0195.42501},
} -
[Br08]
R. de la Bretèche, "Sur l’ordre de grandeur des polynômes de Dirichlet," Acta Arith., vol. 134, iss. 2, pp. 141-148, 2008.
@article {Br08, MRKEY = {2429643},
AUTHOR = {de la Bret{è}che, R{é}gis},
TITLE = {Sur l'ordre de grandeur des polynômes de {D}irichlet},
JOURNAL = {Acta Arith.},
FJOURNAL = {Acta Arithmetica},
VOLUME = {134},
YEAR = {2008},
NUMBER = {2},
PAGES = {141--148},
ISSN = {0065-1036},
CODEN = {AARIA9},
MRCLASS = {11M41 (11N37)},
MRNUMBER = {2429643},
MRREVIEWER = {Wolfgang Schwarz},
DOI = {10.4064/aa134-2-5},
ZBLNUMBER = {05306447},
} -
[Da73]
A. M. Davie, "Quotient algebras of uniform algebras," J. London Math. Soc., vol. 7, pp. 31-40, 1973.
@article {Da73, MRKEY = {0324427},
AUTHOR = {Davie, A. M.},
TITLE = {Quotient algebras of uniform algebras},
JOURNAL = {J. London Math. Soc.},
FJOURNAL = {Journal of the London Mathematical Society. Second Series},
VOLUME = {7},
YEAR = {1973},
PAGES = {31--40},
ISSN = {0024-6107},
MRCLASS = {46J35},
MRNUMBER = {0324427},
MRREVIEWER = {D. R. Wilken},
DOI = {10.1112/jlms/s2-7.1.31},
ZBLNUMBER = {0264.46055},
} -
[DeDiGaMa]
A. Defant, J. C. D’iaz, D. Garc’ia, and M. Maestre, "Unconditional basis and Gordon-Lewis constants for spaces of polynomials," J. Funct. Anal., vol. 181, iss. 1, pp. 119-145, 2001.
@article {DeDiGaMa, MRKEY = {1818112},
AUTHOR = {Defant, Andreas and D{\'ı}az, Juan Carlos and Garc{\'ı}a, Domingo and Maestre, Manuel},
TITLE = {Unconditional basis and {G}ordon-{L}ewis constants for spaces of polynomials},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {181},
YEAR = {2001},
NUMBER = {1},
PAGES = {119--145},
ISSN = {0022-1236},
CODEN = {JFUAAW},
MRCLASS = {46B28 (46G25)},
MRNUMBER = {1818112},
MRREVIEWER = {Ignacio Zalduendo},
DOI = {10.1006/jfan.2000.3702},
ZBLNUMBER = {0986.46031},
} -
[DeGarMa_BorhLoc]
A. Defant, D. Garc’ia, and M. Maestre, "Bohr’s power series theorem and local Banach space theory," J. Reine Angew. Math., vol. 557, pp. 173-197, 2003.
@article {DeGarMa_BorhLoc, MRKEY = {1978407},
AUTHOR = {Defant, Andreas and Garc{\'ı}a, Domingo and Maestre, Manuel},
TITLE = {Bohr's power series theorem and local {B}anach space theory},
JOURNAL = {J. Reine Angew. Math.},
FJOURNAL = {Journal für die Reine und Angewandte Mathematik},
VOLUME = {557},
YEAR = {2003},
PAGES = {173--197},
ISSN = {0075-4102},
CODEN = {JRMAA8},
MRCLASS = {46B07 (46G25)},
MRNUMBER = {1978407},
MRREVIEWER = {G. Schechtman},
DOI = {10.1515/crll.2003.030},
ZBLNUMBER = {1031.46014},
} -
[DefFrer06]
A. Defant and L. Frerick, "A logarithmic lower bound for multi-dimensional Bohr radii," Israel J. Math., vol. 152, pp. 17-28, 2006.
@article {DefFrer06, MRKEY = {2214450},
AUTHOR = {Defant, Andreas and Frerick, Leonhard},
TITLE = {A logarithmic lower bound for multi-dimensional {B}ohr radii},
JOURNAL = {Israel J. Math.},
FJOURNAL = {Israel Journal of Mathematics},
VOLUME = {152},
YEAR = {2006},
PAGES = {17--28},
ISSN = {0021-2172},
CODEN = {ISJMAP},
MRCLASS = {32A07 (46B07 46G20 46G25)},
MRNUMBER = {2214450},
MRREVIEWER = {Richard M. Timoney},
DOI = {10.1007/BF02771973},
ZBLNUMBER = {1124.32003},
} -
[DefFrer] A. Defant and L. Frerick, The Bohr radius of the unit ball of $\ell_n^p$.
@misc{DefFrer,
author={Defant, Andreas and Frerick, Leonhard},
TITLE={The {B}ohr radius of the unit ball of $\ell_n^p$},
NOTE={to appear in {\it J. {R}eine {A}ngew. {M}ath.}},
} -
[DeSe]
A. Defant and P. Sevilla-Peris, "A new multilinear insight on Littlewood’s 4/3-inequality," J. Funct. Anal., vol. 256, iss. 5, pp. 1642-1664, 2009.
@article {DeSe, MRKEY = {2490233},
AUTHOR = {Defant, Andreas and Sevilla-Peris, Pablo},
TITLE = {A new multilinear insight on {L}ittlewood's 4/3-inequality},
JOURNAL = {J. Funct. Anal.},
FJOURNAL = {Journal of Functional Analysis},
VOLUME = {256},
YEAR = {2009},
NUMBER = {5},
PAGES = {1642--1664},
ISSN = {0022-1236},
CODEN = {JFUAAW},
MRCLASS = {46G25 (47B10)},
MRNUMBER = {2490233},
MRREVIEWER = {Geraldo Botelho},
DOI = {10.1016/j.jfa.2008.07.005},
ZBLNUMBER = {1171.46034},
} -
[Harris75] L. A. Harris, "Bounds on the derivatives of holomorphic functions of vectors," in Analyse Fonctionnelle et Applications, Paris, 1975, pp. 145-163.
@inproceedings {Harris75, MRKEY = {0477773},
AUTHOR = {Harris, Lawrence A.},
TITLE = {Bounds on the derivatives of holomorphic functions of vectors},
BOOKTITLE = {Analyse Fonctionnelle et Applications},
VENUE={{C}omptes {R}endus {C}olloq. {A}nalyse, {I}nst. {M}at., {U}niv. {F}ederal {R}io de {J}aneiro, {R}io de {J}aneiro, 1972},
PAGES = {145--163},
SERIES={Actualités Aci. Indust.},
VOLUME={1367},
PUBLISHER = {Hermann},
ADDRESS = {Paris},
YEAR = {1975},
MRCLASS = {46G20 (32A10)},
MRNUMBER = {0477773},
MRREVIEWER = {R. M. Aron},
ZBLNUMBER = {0315.46040},
} -
[Helson]
H. Helson, "Hankel forms and sums of random variables," Studia Math., vol. 176, iss. 1, pp. 85-92, 2006.
@article {Helson, MRKEY = {2263964},
AUTHOR = {Helson, Henry},
TITLE = {Hankel forms and sums of random variables},
JOURNAL = {Studia Math.},
FJOURNAL = {Studia Mathematica},
VOLUME = {176},
YEAR = {2006},
NUMBER = {1},
PAGES = {85--92},
ISSN = {0039-3223},
MRCLASS = {43A15 (15A63 30B50 30D55 47B35)},
MRNUMBER = {2263964},
MRREVIEWER = {V. V. Peller},
DOI = {10.4064/sm176-1-6},
ZBLNUMBER = {1108.43003},
} -
[Kahane] J. Kahane, Some random series of functions, Second ed., Cambridge: Cambridge Univ. Press, 1985.
@book {Kahane, MRKEY = {0833073},
AUTHOR = {Kahane, Jean-Pierre},
TITLE = {Some random series of functions},
SERIES = {Cambridge Studies in Advanced Mathematics},
NUMBER = {5},
EDITION = {Second},
PUBLISHER = {Cambridge Univ. Press},
ADDRESS = {Cambridge},
YEAR = {1985},
PAGES = {xiv+305},
ISBN = {0-521-24966-X; 0-521-45602-9},
MRCLASS = {60G99 (42A61 60G15 60G17)},
MRNUMBER = {0833073},
MRREVIEWER = {Michael Marcus},
ZBLNUMBER = {0571.60002},
} -
[Ka78] S. Kaijser, "Some results in the metric theory of tensor products," Studia Math., vol. 63, iss. 2, pp. 157-170, 1978.
@article {Ka78, MRKEY = {0511301},
AUTHOR = {Kaijser, Sten},
TITLE = {Some results in the metric theory of tensor products},
JOURNAL = {Studia Math.},
FJOURNAL = {Polska Akademia Nauk. Institut Matematyczny. Studia Mathematica},
VOLUME = {63},
YEAR = {1978},
NUMBER = {2},
PAGES = {157--170},
ISSN = {0039-3223},
CODEN = {SMATAZ},
MRCLASS = {46M05 (43A15)},
MRNUMBER = {0511301},
MRREVIEWER = {Gilles Pisier},
ZBLNUMBER = {0392.46047},
} -
[KQ01] S. V. Konyagin and H. Queffélec, "The translation $\frac12$ in the theory of Dirichlet series," Real Anal. Exchange, vol. 27, iss. 1, pp. 155-175, 2001/02.
@article {KQ01, MRKEY = {1887689},
AUTHOR = {Konyagin, S. V. and Queff{é}lec, H.},
TITLE = {The translation {$\frac12$} in the theory of {D}irichlet series},
JOURNAL = {Real Anal. Exchange},
FJOURNAL = {Real Analysis Exchange},
VOLUME = {27},
YEAR = {2001/02},
NUMBER = {1},
PAGES = {155--175},
ISSN = {0147-1937},
MRCLASS = {42A32},
MRNUMBER = {1887689},
MRREVIEWER = {Kazaros Kazarian},
ZBLCOMMENT = {BIBPROC: YEAR doesn't match found ZBLNUMBER},
ZBLNUMBER = {1026.42011},
} -
[Li30] J. E. Littlewood, "On bounded bilinear forms," Quart. J. Math., Oxford Ser., vol. 1, pp. 164-174, 1930.
@article{Li30,
author={Littlewood, J. E.},
TITLE={On bounded bilinear forms},
JOURNAL={Quart. J. Math., Oxford Ser.},
VOLUME={1},
YEAR={1930},
PAGES={164--174},
JFMNUMBER = {56.0335.01},
} -
[Ma80] M. Mateljević, "The isoperimetric inequality and some extremal problems in $H^{1}$," in Analytic Functions, Kozubnik 1979, New York, 1980, pp. 364-369.
@inproceedings {Ma80, MRKEY = {0577467},
AUTHOR = {Mateljevi{ć},
Miodrag},
TITLE = {The isoperimetric inequality and some extremal problems in {$H\sp{1}$}},
BOOKTITLE = {Analytic Functions, {K}ozubnik 1979},
VENUE={{P}roc. {S}eventh {C}onf., {K}ozubnik, 1979},
SERIES = {Lecture Notes in Math.},
VOLUME = {798},
PAGES = {364--369},
PUBLISHER = {Springer-Verlag},
ADDRESS = {New York},
YEAR = {1980},
MRCLASS = {30D55},
MRNUMBER = {0577467},
MRREVIEWER = {Kenneth Stephenson},
ZBLNUMBER = {0446.30020},
} -
[Queffelec] H. Queffélec, "H. Bohr’s vision of ordinary Dirichlet series; old and new results," J. Anal., vol. 3, pp. 43-60, 1995.
@article {Queffelec, MRKEY = {1340125},
AUTHOR = {Queff{é}lec, H.},
TITLE = {H. {B}ohr's vision of ordinary {D}irichlet series; old and new results},
JOURNAL = {J. Anal.},
FJOURNAL = {Journal of Analysis},
VOLUME = {3},
YEAR = {1995},
PAGES = {43--60},
ISSN = {0971-3611},
MRCLASS = {11M41 (11K99)},
MRNUMBER = {1340125},
MRREVIEWER = {Douglas Hensley},
ZBLNUMBER = {0881.11068},
} -
[Vukotic]
D. Vukotić, "The isoperimetric inequality and a theorem of Hardy and Littlewood," Amer. Math. Monthly, vol. 110, iss. 6, pp. 532-536, 2003.
@article {Vukotic, MRKEY = {1984405},
AUTHOR = {Vukoti{ć},
Dragan},
TITLE = {The isoperimetric inequality and a theorem of {H}ardy and {L}ittlewood},
JOURNAL = {Amer. Math. Monthly},
FJOURNAL = {The American Mathematical Monthly},
VOLUME = {110},
YEAR = {2003},
NUMBER = {6},
PAGES = {532--536},
ISSN = {0002-9890},
CODEN = {AMMYAE},
MRCLASS = {30D55 (46E15 51M16)},
MRNUMBER = {1984405},
DOI = {10.2307/3647909},
ZBLNUMBER = {1061.30027},
}