Polynomial inequalities on the Hamming cube

Abstract

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 \(p\in \big (1,\frac{4}{3}\big )\).

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Notes

  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.

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Acknowledgements

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.

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Correspondence to Alexandros Eskenazis.

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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.

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

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Keywords

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

Mathematics Subject Classification

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