Polynomial inequalities on the Hamming cube


Let \((X,\Vert \cdot \Vert _X)\) be a Banach space. The purpose of this article is to systematically investigate dimension independent properties of vector valued functions \(f:\{-1,1\}^n\rightarrow X\) on the Hamming cube whose spectrum is bounded above or below. Our proofs exploit contractivity properties of the heat flow, induced by the geometry of the target space \((X,\Vert \cdot \Vert _X)\), combined with duality arguments and suitable tools from approximation theory and complex analysis. We obtain a series of improvements of various well-studied estimates for functions with bounded spectrum, including moment comparison results for low degree Walsh polynomials and Bernstein–Markov type inequalities, which constitute discrete vector valued analogues of Freud’s inequality in Gauss space (1971). Many of these inequalities are new even for scalar valued functions. Furthermore, we provide a short proof of Mendel and Naor’s heat smoothing theorem (2014) for functions in tail spaces with values in spaces of nontrivial type and we also prove a dual lower bound on the decay of the heat semigroup acting on functions with spectrum bounded from above. Finally, we improve the reverse Bernstein–Markov inequalities of Meyer (in: Seminar on probability, XVIII, Lecture notes in mathematics. Springer, Berlin, 1984. https://doi.org/10.1007/BFb0100043) and Mendel and Naor (Publ Math Inst Hautes Études Sci 119:1–95, 2014. https://doi.org/10.1007/s10240-013-0053-2) for functions with narrow enough spectrum and improve the bounds of Filmus et al. (Isr J Math 214(1):167–192, 2016. https://doi.org/10.1007/s11856-016-1355-0) on the \(\ell _p\) sums of influences of bounded functions for \(p\in \big (1,\frac{4}{3}\big )\).

This is a preview of subscription content, log in to check access.


  1. 1.

    The bound (58) of Corollary 21 was pointed out to us by an anonymous referee, who improved a suboptimal \(O_q(d^2)\) estimate appearing in an earlier version of this manuscript. We are grateful to them for sharing their improvement with us and for their helpful comments.


  1. 1.

    Aaronson, S., Ambainis, A.: The need for structure in quantum speedups. Theory Comput. 10, 133–166 (2014). https://doi.org/10.4086/toc.2014.v010a006

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Bakry, D., Gentil, I., Ledoux, M.: Analysis and Geometry of Markov Diffusion Operators. Grundlehren der Mathematischen Wissenschaften. Springer, Cham (2014). https://doi.org/10.1007/978-3-319-00227-9

    Google Scholar 

  3. 3.

    Bačkurs, A., Bavarian, M.: On the sum of \(L_1\) influences. In: IEEE 29th Conference on Computational Complexity—CCC 2014, pp. 132–143. IEEE Computer Society, Los Alamitos, CA (2014). https://doi.org/10.1109/CCC.2014.21

  4. 4.

    Beckner, W.: Inequalities in Fourier analysis. Ann. Math. (2) 102(1), 159–182 (1975). https://doi.org/10.2307/1970980

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Ben Efraim, L., Lust-Piquard, F.: Poincaré type inequalities on the discrete cube and in the CAR algebra. Probab. Theory Relat. Fields 141(3–4), 569–602 (2008). https://doi.org/10.1007/s00440-007-0094-x

    Article  MATH  Google Scholar 

  6. 6.

    Bonami, A.: Étude des coefficients de Fourier des fonctions de \(L^{p}(G)\). Ann. Inst. Fourier (Grenoble) 20(fasc. 2), 335–402 (1970)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Borell, C.: On the integrability of Banach space valued Walsh polynomials. In: Séminaire de Probabilités, XIII, Lecture Notes in Math., vol. 721, pp. 1–3. Springer, Berlin (1979)

  8. 8.

    Borell, C.: On polynomial chaos and integrability. Probab. Math. Stat. 3(2), 191–203 (1984)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Texts in Mathematics, vol. 161. Springer, New York (1995). https://doi.org/10.1007/978-1-4612-0793-1

    Google Scholar 

  10. 10.

    Bourgain, J.: Walsh subspaces of \(L^{p}\)-product spaces. In: Seminar on Functional Analysis, 1979–1980 (French), pp. Exp. No. 4A, 9. École Polytech., Palaiseau (1980)

  11. 11.

    Epperson, J.B.: The hypercontractive approach to exactly bounding an operator with complex Gaussian kernel. J. Funct. Anal. 87(1), 1–30 (1989). https://doi.org/10.1016/0022-1236(89)90002-5

    MathSciNet  Article  Google Scholar 

  12. 12.

    Erdélyi, T.: Reverse Markov- and Bernstein-type inequalities for incomplete polynomials. J. Approx. Theory 251, 105341 (2020). https://doi.org/10.1016/j.jat.2019.105341

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Eskenazis, A., Ivanisvili, P.: Dimension independent Bernstein–Markov inequalities in Gauss space (2018). To appear in J. Approx. Theory. Preprint available at arXiv:1808.01273

  14. 14.

    Eskenazis, A., Naor, A.: Discrete Littlewood–Paley–Stein theory and Pisier’s inequality for superreflexive targets (2020) (Preprint)

  15. 15.

    Filmus, Y., Hatami, H., Heilman, S., Mossel, E., O’Donnell, R., Sachdeva, S., Wan, A., Wimmer, K.: Real Analysis in Computer Science: a collection of open problems (2014). Preprint available at https://simons.berkeley.edu/sites/default/files/openprobsmerged.pdf

  16. 16.

    Filmus, Y., Hatami, H., Keller, N., Lifshitz, N.: On the sum of the \(L_1\) influences of bounded functions. Isr. J. Math. 214(1), 167–192 (2016). https://doi.org/10.1007/s11856-016-1355-0

    Article  MATH  Google Scholar 

  17. 17.

    Freud, G.: A certain inequality of Markov type. Dokl. Akad. Nauk SSSR 197, 790–793 (1971)

    MathSciNet  Google Scholar 

  18. 18.

    Heilman, S., Mossel, E., Oleszkiewicz, K.: Strong contraction and influences in tail spaces. Trans. Am. Math. Soc. 369(7), 4843–4863 (2017). https://doi.org/10.1090/tran/6916

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Hytönen, T., Naor, A.: Pisier’s inequality revisited. Stud. Math. 215(3), 221–235 (2013). https://doi.org/10.4064/sm215-3-2

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Ivanisvili, P., Nazarov, F.: On Weissler’s conjecture on the Hamming cube I (2019). Preprint available at arXiv:1907.11359

  21. 21.

    Ivanisvili, P., Tkocz, T.: Comparison of moments of Rademacher chaoses. Ark. Mat. 57(1), 121–128 (2019). https://doi.org/10.4310/ARKIV.2019.v57.n1.a7

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Janson, S.: On complex hypercontractivity. J. Funct. Anal. 151(1), 270–280 (1997). https://doi.org/10.1006/jfan.1997.3144

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Kahane, J.P.: Sur les sommes vectorielles \(\sum \pm u_{n}\). C. R. Acad. Sci. Paris 259, 2577–2580 (1964)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Khintchine, A.: Über dyadische Brüche. Math. Z. 18(1), 109–116 (1923). https://doi.org/10.1007/BF01192399

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Krantz, S.G.: Geometric Function Theory. Cornerstones. Birkhäuser Boston, Inc., Boston, MA (2006). Explorations in complex analysis

    Google Scholar 

  26. 26.

    Kwapień, S.: A theorem on the Rademacher series with vector valued coefficients. In: Probability in Banach spaces, pp. 157–158. Lecture Notes in Math., Vol. 526. Springer, Berlin (1976)

  27. 27.

    Kwapień, S., Woyczyński, W.A.: Random Series and Stochastic Integrals: Single and Multiple. Probability and Its Applications. Birkhäuser Boston Inc, Boston, MA (1992). https://doi.org/10.1007/978-1-4612-0425-1

    Google Scholar 

  28. 28.

    Lust-Piquard, F.: Riesz transforms associated with the number operator on the Walsh system and the fermions. J. Funct. Anal. 155(1), 263–285 (1998). https://doi.org/10.1006/jfan.1997.3217

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Maurey, B.: Type, cotype and \(K\)-convexity. In: Handbook of the geometry of Banach spaces, Vol. 2, pp. 1299–1332. North-Holland, Amsterdam (2003). https://doi.org/10.1016/S1874-5849(03)80037-2

  30. 30.

    Maurey, B., Pisier, G.: Séries de variables aléatoires vectorielles indépendantes et propriétés géométriques des espaces de Banach. Stud. Math. 58(1), 45–90 (1976). https://doi.org/10.4064/sm-58-1-45-90

    Article  MATH  Google Scholar 

  31. 31.

    Mendel, M., Naor, A.: Nonlinear spectral calculus and super-expanders. Publ. Math. Inst. Hautes Études Sci. 119, 1–95 (2014). https://doi.org/10.1007/s10240-013-0053-2

    MathSciNet  Article  MATH  Google Scholar 

  32. 32.

    Meyer, P.A.: Transformations de Riesz pour les lois gaussiennes. In: Seminar on probability, XVIII, Lecture Notes in Math., vol. 1059, pp. 179–193. Springer, Berlin (1984). https://doi.org/10.1007/BFb0100043

  33. 33.

    Milman, V.D., Schechtman, G.: Asymptotic Theory of Finite-Dimensional Normed Spaces. Lecture Notes in Mathematics, vol. 1200. Springer, Berlin (1986)

    Google Scholar 

  34. 34.

    Naor, A.: An introduction to the Ribe program. Jpn. J. Math. 7(2), 167–233 (2012). https://doi.org/10.1007/s11537-012-1222-7

    MathSciNet  Article  MATH  Google Scholar 

  35. 35.

    Naor, A., Schechtman, G.: Remarks on non linear type and Pisier’s inequality. J. Reine Angew. Math. 552, 213–236 (2002). https://doi.org/10.1515/crll.2002.092

    MathSciNet  Article  MATH  Google Scholar 

  36. 36.

    O’Donnell, R.: Analysis of Boolean Functions. Cambridge University Press, New York (2014). https://doi.org/10.1017/CBO9781139814782

    Google Scholar 

  37. 37.

    Oleszkiewicz, K.: On a nonsymmetric version of the Khinchine-Kahane inequality. In: Stochastic Inequalities and Applications, Progr. Probab., vol. 56, pp. 157–168. Birkhäuser, Basel (2003)

  38. 38.

    Pisier, G.: Sur les espaces de Banach qui ne contiennent pas uniformément de \(l^{1}_{n}\). C. R. Acad. Sci. Paris Sér. A-B 277, A991–A994 (1973)

    MATH  Google Scholar 

  39. 39.

    Pisier, G.: Les inégalités de Khintchine-Kahane, d’après C. Borell. In: Séminaire sur la Géométrie des Espaces de Banach (1977–1978), pp. Exp. No. 7, 14. École Polytech., Palaiseau (1978)

  40. 40.

    Pisier, G.: Holomorphic semigroups and the geometry of Banach spaces. Ann. Math. (2) 115(2), 375–392 (1982). https://doi.org/10.2307/1971396

    MathSciNet  Article  MATH  Google Scholar 

  41. 41.

    Pisier, G.: Probabilistic methods in the geometry of Banach spaces. In: Probability and Analysis (Varenna, 1985), Lecture Notes in Math., vol. 1206, pp. 167–241. Springer, Berlin (1986). https://doi.org/10.1007/BFb0076302

  42. 42.

    Pisier, G.: Riesz transforms: a simpler analytic proof of P.-A. Meyer’s inequality. In: Séminaire de Probabilités, XXII, Lecture Notes in Math., vol. 1321, pp. 485–501. Springer, Berlin (1988). https://doi.org/10.1007/BFb0084154

  43. 43.

    Pisier, G.: A remark on hypercontractive semigroups and operator ideals (2007). Preprint available at arXiv:0708.3423

  44. 44.

    Sarantopoulos, Y.: Bounds on the derivatives of polynomials on Banach spaces. Math. Proc. Camb. Philos. Soc. 110(2), 307–312 (1991). https://doi.org/10.1017/S0305004100070389

    MathSciNet  Article  MATH  Google Scholar 

  45. 45.

    Stein, E.M.: Topics in Harmonic Analysis Related to the Littlewood–Paley Theory. Annals of Mathematics Studies, No. 63. Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo (1970)

  46. 46.

    Szegö, G.: Über einen Satz von A. Markoff. Math. Z. 23(1), 45–61 (1925). https://doi.org/10.1007/BF01506220

    Article  MATH  Google Scholar 

  47. 47.

    Talagrand, M.: Isoperimetry, logarithmic Sobolev inequalities on the discrete cube, and Margulis’ graph connectivity theorem. Geom. Funct. Anal. 3(3), 295–314 (1993). https://doi.org/10.1007/BF01895691

    MathSciNet  Article  MATH  Google Scholar 

  48. 48.

    Wagner, R.: Notes on an inequality by Pisier for functions on the discrete cube. In: Geometric Aspects of Functional analysis. Lecture Notes in Math., vol. 1745, pp. 263–268. Springer, Berlin (2000). https://doi.org/10.1007/BFb0107220

  49. 49.

    Weissler, F.B.: Two-point inequalities, the Hermite semigroup, and the Gauss–Weierstrass semigroup. J. Funct. Anal. 32(1), 102–121 (1979). https://doi.org/10.1016/0022-1236(79)90080-6

    MathSciNet  Article  MATH  Google Scholar 

  50. 50.

    Wolff, P.: Hypercontractivity of simple random variables. Stud. Math. 180(3), 219–236 (2007). https://doi.org/10.4064/sm180-3-3

    MathSciNet  Article  MATH  Google Scholar 

Download references


We are indebted to Assaf Naor for many helpful discussions. We are also very grateful to Tamás Erdélyi for proving the main result of [12] upon our request and to an anonymous referee for sharing with us an argument which improved Corollary 21. Finally, we would like to thank Françoise Lust-Piquard for valuable feedback.

Author information



Corresponding author

Correspondence to Alexandros Eskenazis.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

P. I.  was partially supported by NSF DMS-1856486 and NSF CAREER-1945102. This work was carried out under the auspices of the Simons Algorithms and Geometry (A&G) Think Tank.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Eskenazis, A., Ivanisvili, P. Polynomial inequalities on the Hamming cube. Probab. Theory Relat. Fields (2020). https://doi.org/10.1007/s00440-020-00973-y

Download citation


  • Hamming cube
  • Heat semigroup
  • Hypercontractivity
  • Bernstein–Markov inequality
  • Moment comparison

Mathematics Subject Classification

  • Primary 42C10
  • Secondary 41A17
  • 41A63
  • 46B07