Analysis & PDE

  • Anal. PDE
  • Volume 9, Number 5 (2016), 1235-1258.

Bohnenblust–Hille inequalities for Lorentz spaces via interpolation

Andreas Defant and Mieczysław Mastyło

Full-text: Access denied (no subscription detected)

However, an active subscription may be available with MSP at

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We prove that the Lorentz sequence space 2m(m+1),1 is, in a precise sense, optimal among all symmetric Banach sequence spaces satisfying a Bohnenblust–Hille-type inequality for m-linear forms or m-homogeneous polynomials on n . Motivated by this result we develop methods for dealing with subtle Bohnenblust–Hille-type inequalities in the setting of Lorentz spaces. Based on an interpolation approach and the Blei–Fournier inequalities involving mixed-type spaces, we prove multilinear and polynomial Bohnenblust–Hille-type inequalities in Lorentz spaces with subpolynomial and subexponential constants. An application to the theory of Dirichlet series improves a deep result of Balasubramanian, Calado and Queffélec.

Article information

Anal. PDE, Volume 9, Number 5 (2016), 1235-1258.

Received: 14 January 2016
Revised: 12 February 2016
Accepted: 30 March 2016
First available in Project Euclid: 12 December 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B70: Interpolation between normed linear spaces [See also 46M35] 47A53: (Semi-) Fredholm operators; index theories [See also 58B15, 58J20]

Bohnenblust–Hille inequality Dirichlet polynomials Dirichlet series homogeneous polynomials interpolation spaces Lorentz spaces


Defant, Andreas; Mastyło, Mieczysław. Bohnenblust–Hille inequalities for Lorentz spaces via interpolation. Anal. PDE 9 (2016), no. 5, 1235--1258. doi:10.2140/apde.2016.9.1235.

Export citation


  • N. Albuquerque, F. Bayart, D. Pellegrino, and J. B. Seoane-Sepúlveda, “Sharp generalizations of the multilinear Bohnenblust–Hille inequality”, J. Funct. Anal. 266:6 (2014), 3726–3740.
  • 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, A. Defant, L. Frerick, M. Maestre, and P. Sevilla-Peris, “Multipliers of Dirichlet series and monomial series expansions of holomorphic functions in infinitely many variables”, preprint, 2014.
  • F. Bayart, D. Pellegrino, and J. B. Seoane-Sepúlveda, “The Bohr radius of the $n$-dimensional polydisk is equivalent to $\sqrt{(\log n)/n}$”, Adv. Math. 264 (2014), 726–746.
  • C. Bennett and R. Sharpley, Interpolation of operators, Pure and Applied Mathematics 129, Academic Press, Boston, MA, 1988.
  • R. C. Blei and J. J. F. Fournier, “Mixed-norm conditions and Lorentz norms”, pp. 57–78 in Commutative harmonic analysis (Canton, NY, 1987), edited by D. Colella, Contemp. Math. 91, Amer. Math. Soc., Providence, RI, 1989.
  • H. F. Bohnenblust and E. Hille, “On the absolute convergence of Dirichlet series”, Ann. of Math. $(2)$ 32:3 (1931), 600–622.
  • H. Bohr, “Über die Bedeutung der Potenzreihen unendlich vieler Variabeln in der Theorie der Dirichletschen Reihen $\sum{a_n}/{n^2}$”, Nachr. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1913 (1913), 441–488.
  • A.-P. Calderón, “Intermediate spaces and interpolation, the complex method”, Studia Math. 24 (1964), 113–190.
  • A. Defant and M. Mastyło, “$L_p$-norms and Mahler's measure of polynomials on the $n$-dimensional torus”, Constr. Approx. (online publication December 2015).
  • A. Defant and P. Sevilla-Peris, “The Bohnenblust–Hille cycle of ideas from a modern point of view”, Funct. Approx. Comment. Math. 50:1 (2014), 55–127.
  • 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, M. Maestre, and U. Schwarting, “Bohr radii of vector valued holomorphic functions”, Adv. Math. 231:5 (2012), 2837–2857.
  • A. Defant, D. Garcia, M. Maestre, and P. Sevilla-Peris, book manuscript, 2016.
  • V. Dimant and P. Sevilla-Peris, “Summation of coefficients of polynomials on $\ell_p$-spaces”, preprint, 2013. To appear in Publ. Mat.
  • J. J. F. Fournier, “Mixed norms and rearrangements: Sobolev's inequality and Littlewood's inequality”, Ann. Mat. Pura Appl. $(4)$ 148 (1987), 51–76.
  • L. Grafakos and M. Mastyło, “Analytic families of multilinear operators”, Nonlinear Anal. 107 (2014), 47–62.
  • T. Holmstedt, “Interpolation of quasi-normed spaces”, Math. Scand. 26 (1970), 177–199.
  • J.-P. Kahane, Some random series of functions, 2nd ed., Cambridge Studies in Advanced Mathematics 5, Cambridge University Press, 1985.
  • H. K önig, “On the best constants in the Khintchine inequality for Steinhaus variables”, Israel J. Math. 203:1 (2014), 23–57.
  • S. G. Kreĭn, Y. \=I. Petunīn, and E. M. Semënov, Interpolation of linear operators, Translations of Mathematical Monographs 54, American Mathematical Society, Providence, R.I., 1982.
  • J. E. Littlewood, “On bounded bilinear forms in an infinite number of variables”, Q. J. Math., Oxf. Ser. 1 (1930), 164–174.
  • A. Montanaro, “Some applications of hypercontractive inequalities in quantum information theory”, J. Math. Phys. 53:12 (2012), 122206, 15.
  • R. O'Donnell, Analysis of Boolean functions, Cambridge University Press, 2014.
  • H. Queffélec and M. Queffélec, Diophantine approximation and Dirichlet series, Harish-Chandra Research Institute Lecture Notes 2, Hindustan, New Delhi, 2013.
  • J. Sawa, “The best constant in the Khintchine inequality for complex Steinhaus variables, the case $p=1$”, Studia Math. 81:1 (1985), 107–126.