Functiones et Approximatio Commentarii Mathematici

The Bohnenblust-Hille cycle of ideas from a modern point of view

Andreas Defant and Pablo Sevilla-Peris

Full-text: Open access

Abstract

In 1931 H.F. Bohnenblust and E. Hille published a very important paper in which not only did they solve a long standing problem on convergence of Dirichlet series, but also gave a~general version of a celebrated inequality of Littlewood. Although it is full of extremely valuable mathematical ideas, the paper has been overlooked for a~long time and even today we feel that it does not get the credit it deserves. This may be caused by the not always accessible style that makes that the ideas are sometimes hidden. It is our intention to try to study the paper from a~modern point of view and to bring to light the valuable aspects we believe it has.

Article information

Source
Funct. Approx. Comment. Math., Volume 50, Number 1 (2014), 55-127.

Dates
First available in Project Euclid: 27 March 2014

Permanent link to this document
https://projecteuclid.org/euclid.facm/1395924286

Digital Object Identifier
doi:10.7169/facm/2014.50.1.2

Mathematical Reviews number (MathSciNet)
MR3189502

Zentralblatt MATH identifier
1294.30009

Subjects
Primary: 30B50: Dirichlet series and other series expansions, exponential series [See also 11M41, 42-XX]
Secondary: 46G25: (Spaces of) multilinear mappings, polynomials [See also 46E50, 46G20, 47H60] 46E50: Spaces of differentiable or holomorphic functions on infinite- dimensional spaces [See also 46G20, 46G25, 47H60]

Keywords
Dirichlet series Bohnenblust-Hille inequality polynomials Bohr's problem

Citation

Defant, Andreas; Sevilla-Peris, Pablo. The Bohnenblust-Hille cycle of ideas from a modern point of view. Funct. Approx. Comment. Math. 50 (2014), no. 1, 55--127. doi:10.7169/facm/2014.50.1.2. https://projecteuclid.org/euclid.facm/1395924286


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References

  • A. Baernsteindee II and R.C. Culverhouse. Majorization of sequences, sharp vector Khinchin inequalities, and bisubharmonic functions, Studia Math. 152(3) (2002), 231–248.
  • R. Balasubramanian, B. Calado, and H. Queffélec, The Bohr inequality for ordinary Dirichlet series, Studia Math. 175(3) (2006), 285–304.
  • F. Bayart, Hardy spaces of Dirichlet series and their composition operators, Monatsh. Math. 136(3) (2002), 203–236.
  • F. Bayart, Maximum modulus of random polynomials, Q. J. Math. 63(1) (2012), 21–39.
  • G. Bennett, Inclusion mappings between $l\sp{p}$ spaces, J. Functional Analysis 13 (1973), 20–27.
  • J. Bergh and J. Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin, 1976, Grundlehren der Mathematischen Wissenschaften, No. 223.
  • R.. Blei, Fractional Cartesian products of sets, Ann. Inst. Fourier (Grenoble) 29(2) (1979), 79–105.
  • H.. Boas, The football player and the infinite series, Notices Amer. Math. Soc. 44(11) (1997), 1430–1435.
  • H.. Boas and D. Khavinson, Bohr's power series theorem in several variables, Proc. Amer. Math. Soc. 125(10) (1997), 2975–2979.
  • H.. Bohnenblust and E. Hille, On the absolute convergence of Dirichlet series, Ann. of Math. (2) 32(3) (1931), 600–622.
  • H. Bohr, Lösung des absoluten Konvergenzproblems einer allgemeinen Klasse dirichletscher Reihen, Acta Math. 36 (1913), 197–240.
  • H. Bohr, Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichlet–schen Reihen $\sum\,\frac{a_n}{n^s}$, Nachr. Ges. Wiss. Göttingen, Math. Phys. Kl., pages 441–488, 1913.
  • H. Bohr, Über die gleichmäßige Konvergenz Dirichletscher Reihen, J. Reine Angew. Math. 143 (1913), 203–211.
  • H. Bohr, A theorem concerning power series, Proc. London Math. Soc. (2) 13 (1914), 1–5.
  • P.. Boland and S. Dineen, Fonctions holomorphes sur des espaces pleinement nucléaires, C. R. Acad. Sci. Paris Sér. A-B 286(25) (1978), A1235–A1237.
  • F. Bombal, D. Pérez-García, and I. Villanueva, Multilinear extensions of Grothendieck's theorem, Q. J. Math. 55(4) (2004), 441–450.
  • A. Bonami, Étude des coefficients de Fourier des fonctions de $L^{p}(G)$, Ann. Inst. Fourier (Grenoble) 20(fasc. 2) (1970), 335–402 (1971).
  • J. Bonet and A. Peris, On the injective tensor product of quasinormable spaces, Results Math. 20(1-2) (1991), 431–443.
  • B. Carl, Absolut-$(p,\,1)$-summierende identische Operatoren von $l\sb{u}$ in $l\sb{v}$, Math. Nachr. 63 (1974), 353–360.
  • F. Carlson, Contributions à la théorie des séries de Dirichlet. Note i, Ark. för Mat., Astron. och Fys. 16(18) (1922), 1–19.
  • A.. Davie, Quotient algebras of uniform algebras, J. London Math. Soc. (2) 7 (1973), 31–40.
  • R. de la Bretèche, Sur l'ordre de grandeur des polynômes de Dirichlet, Acta Arith. 134(2) (2008), 141–148.
  • A. Defant, J.. Díaz, D. García, and M. Maestre, Unconditional basis and Gordon-Lewis constants for spaces of polynomials, J. Funct. Anal. 181(1) (2001), 119–145.
  • A. Defant and K. Floret, Tensor norms and operator ideals, volume 176 of North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam, 1993.
  • A. Defant and L. Frerick, A logarithmic lower bound for multi-dimensional Bohr radii, Israel J. Math. 152 (2006), 17–28.
  • A. Defant, L. Frerick, M. Maestre, and P. Sevilla-Peris, Monomial expansions of ${H}_{p}$–functions in infinitely many variables, preprint.
  • A. Defant, L. Frerick, J. Ortega-Cerdà M. Ounaïes, and K. Seip, The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive, Ann. of Math. (2) 174(1) (2011), 485–497.
  • A. Defant, D. García, and M. Maestre, Bohr's power series theorem and local Banach space theory, J. Reine Angew. Math. 557 (2003), 173–197.
  • A. Defant, D. García, M. Maestre, and D. Pérez-García, Bohr's strip for vector valued Dirichlet series, Math. Ann. 342(3) (2008), 533–555.
  • A. Defant and M. Maestre, Property (BB) and holomorphic functions on Fréchet-Montel spaces, Math. Proc. Cambridge Philos. Soc. 115(2) (1994), 305–313.
  • A. Defant, M. Maestre, and C. Prengel, Domains of convergence for monomial expansions of holomorphic functions in infinitely many variables, J. Reine Angew. Math. 634 (2009), 13–49.
  • A. Defant, M. Maestre, and U. Schwarting, Bohr radii of vector valued holomorphic functions, Adv. Math. 231(5) (2012), 2837–2857.
  • A. Defant, M. Maestre, and P. Sevilla-Peris, Cotype 2 estimates for spaces of polynomials on sequence spaces, Israel J. Math. 129 (2002), 291–315.
  • A. Defant, D. Popa, and U. Schwarting, Coordinatewise multiple summing operators in Banach spaces, J. Funct. Anal. 259(1) (2010), 220–242.
  • A. Defant and C. Prengel, Volume estimates in spaces of homogeneous polynomials, Math. Zeit. 261(4) (2009), 909–932.
  • A. Defant, U. Schwarting, and P. Sevilla-Peris, Bohr's absolute convergence problem for $\mathcal{H}_p$-dirichlet series in banach spaces, preprint.
  • A. Defant, U. Schwarting, and P. Sevilla-Peris, Estimates for vector valued dirichlet polynomials, preprint.
  • A. Defant and P. Sevilla-Peris, A new multilinear insight on Littlewood's 4/3-inequality, J. Funct. Anal. 256(5) (2009), 1642–1664.
  • A. Defant and P. Sevilla-Peris, Convergence of Dirichlet polynomials in Banach spaces, Trans. Amer. Math. Soc. 363(2) (2011), 681–697.
  • A. Defant and P. Sevilla-Peris, Convergence of monomial expansions in Banach spaces, Q. J. Math. 63(3) (2012), 569–584.
  • J. Diestel, H. Jarchow, and A. Tonge, Absolutely summing operators, volume 43 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1995.
  • S. Dineen, Complex analysis on infinite-dimensional spaces, Springer Monographs in Mathematics. Springer-Verlag London Ltd., London, 1999.
  • D. Diniz, G.A. Muñoz-Fernández, D. Pellegrino, and J.B. Seoane-Sepúlveda, The asymptotic growth of the constants in the Bohnenblust-Hille inequality is optimal, J. Funct. Anal. 263(2) (2012), 415–428.
  • N. Dunford, Einar Hille (June 28, 1894–February 12, 1980), Bull. Amer. Math. Soc. (N.S.) 4(3) (1981), 303–319 (1 plate).
  • K. Floret, Natural norms on symmetric tensor products of normed spaces, Note Mat. 17 (1997), 153–188 (1999).
  • K. Floret, The extension theorem for norms on symmetric tensor products of normed spaces, in Recent progress in functional analysis (Valencia, 2000), volume 189 of North-Holland Math. Stud., pages 225–237, North-Holland, Amsterdam, 2001.
  • A. Grothendieck, Résumé de la théorie métrique des produits tensoriels topologiques, Bol. Soc. Mat. São Paulo 8 (1953), 1–79.
  • U. Haagerup, The best constants in the Khintchine inequality, Studia Math. 70(3) (1981), 231–283 (1982).
  • G. Hardy and J. Littlewood, Bilinear forms bounded in space $[p,q]$, Q. J. Math. 5 (1934), 241–54.
  • L.A. Harris, Bounds on the derivatives of holomorphic functions of vectors, in Analyse fonctionnelle et applications (Comptes Rendus Colloq, Analyse, Inst. Mat., Univ. Federal Rio de Janeiro, Rio de Janeiro, 1972), pages 145–163. Actualités Aci. Indust., No. 1367. Hermann, Paris, 1975.
  • H. Hedenmalm, P. Lindqvist, and K. Seip, A Hilbert space of Dirichlet series and systems of dilated functions in $L^2(0,1)$, Duke Math. J. 86(1) (1997), 1–37.
  • D. Hilbert, Wesen und Ziele einer Analysis der unendlichvielen unabhängigen Variablen, Rend. del Circolo mat. di Palermo 27 (1909), 59–74.
  • N. Jacobson, Einar Hille, his Yale years, a personal recollection, Integral Equations Operator Theory 4(3) (1981), 307–310.
  • J.-P. Kahane, Some random series of functions, volume 5 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, second edition, 1985.
  • S. Kaijser, Some results in the metric theory of tensor products, Studia Math. 63(2) (1978), 157–170.
  • H. König and S. Kwapień, Best Khintchine type inequalities for sums of independent, rotationally invariant random vectors, Positivity 5(2) (2001), 115–152.
  • S.V. Konyagin and H. Queffélec, The translation $\frac12$ in the theory of Dirichlet series, Real Anal. Exchange 27(1) (2001/02), 155–175.
  • L. Lempert, The Dolbeault complex in infinite dimensions. II, J. Amer. Math. Soc. 12(3) (1999), 775–793.
  • J. Lindenstrauss and L. Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin, 1977, Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92.
  • J.E. Littlewood, On bounded bilinear forms in an infinite number of variables, Quarterly Journ. (Oxford Series) 1 (1930), 164–174.
  • M.C. Matos, Fully absolutely summing and Hilbert-Schmidt multilinear mappings, Collect. Math. 54(2) (2003), 111–136.
  • B. Maurizi and H. Queffélec, Some remarks on the algebra of bounded Dirichlet series, J. Fourier Anal Appl (2010), DOI 10.1007/s00041-009-9112-y.
  • P. Mellon, The polarisation constant for ${\rm JB}\sp \ast$-triples, Extracta Math. 9(3) (1994), 160–163.
  • A. Montanaro, Some applications of hypercontractive inequalities in quantum information theory, arXiv, 1208.0161v2, 2012.
  • G.A. Muñoz-Fernández, D. Pellegrino, and J.B. Seoane-Sepúlveda, Estimates for the asymptotic behaviour of the constants in the Bohnenblust-Hille inequality, Linear Multilinear Algebra 60(5) (2012), 573–582.
  • D. Nuñez-Alarcón, D. Pellegrino, J. Seoane-Sepúlveda, and D. Serrano-Rodríguez, There exist multilinear Bohnenblust–Hille constants $(C_n)_{n=1}^\infty$ with $\lim_{n\rightarrow\infty}(C_{n+1}-C_n)=0$, J. Funct. Anal. 264(2) (2013), 429–463.
  • D. Nuñez-Alarcón, D. Pellegrino, and J.B. Seoane-Sepúlveda, On the Bohnenblust-Hille inequality and a variant of Littlewood's 4/3 inequality, J. Funct. Anal. 264(1) (2013), 326–336.
  • W. Orlicz, Über unbedingte Konvergenz in Funktionenräumen i, Studia Math. 4 (1933), 33–37.
  • D. Pellegrino and J.B. Seoane-Sepúlveda, New upper bounds for the constants in the Bohnenblust-Hille inequality, J. Math. Anal. Appl. 386(1) (2012), 300–307.
  • R. Phillips, Einar Hille, a biographical Memoir, Biographical Memoirs of the National Academy of Sciences, pages 217–244, 1994.
  • A. Pietsch, Operator ideals, volume 20 of North-Holland Mathematical Library, North-Holland Publishing Co., Amsterdam, 1980 (translated from German by the author).
  • D. Popa and G. Sinnamon, Blei's inequality and coordinatewise multiple summing operators, preprint.
  • C. Prengel, Domains of convergence in infinite dimensional holomorphy, PhD thesis, Universität Oldenburg, Oldenburg, Germany, 2005.
  • H. Queffélec, H. Bohr's vision of ordinary Dirichlet series; old and new results, J. Anal. 3 (1995), 43–60.
  • W. Rudin, Some theorems on Fourier coefficients, Proc. Amer. Math. Soc. 10 (1959), 855–859.
  • R.A. Ryan, Holomorphic mappings on $l\sb 1$, Trans. Amer. Math. Soc. 302(2) (1987), 797–811.
  • J. Sawa, The best constant in the Khintchine inequality for complex Steinhaus variables, the case $p=1$, Studia Math. 81(1) (1985), 107–126.
  • U. Schwarting, Vector valued Bohnenblust–Hille inequalities, PhD thesis, Universität Oldenburg, Oldenburg, Germany, 2013.
  • H. Shapiro, Extremal problems for polynomials and power series, Master's thesis, Massachusetts Institute of Technology, 1951.
  • O. Toeplitz, Über eine bei den Dirichletschen Reihen auftretende Aufgabe aus der Theorie der Potenzreihen von unendlichvielen Veränderlichen, Nachrichten von der Königlichen Gesellschaft der Wissenschaften zu Göttingen, pages 417–432, 1913.
  • N. Tomczak-Jaegermann, Banach-Mazur distances and finite-dimensional operator ideals, volume 38 of Pitman Monographs and Surveys in Pure and Applied Mathematics, Longman Scientific & Technical, Harlow, 1989.
  • A. Tonge, Polarization and the two-dimensional Grothendieck inequality, Math. Proc. Cambridge Philos. Soc. 95(2) (1984), 313–318.
  • M. Vilela de Souza, PhD thesis, IUMPA, Campinas, Brazil, 2003.
  • F.B. Weissler, Logarithmic Sobolev inequalities and hypercontractive estimates on the circle, J. Funct. Anal. 37(2) (1980), 218–234.