Polynomial inequalities on the Hamming cube
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Polynomial inequalities on the Hamming cube Alexandros Eskenazis1,3 · Paata Ivanisvili2 Received: 21 February 2019 / Revised: 20 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Let (X , · 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 → 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 , · X ), combined with duality arguments and suitable tools from approximation theory and complex analysis. We obtain a series of improvements of various wellstudied 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 p sums of influences of bounded functions for p ∈ 1, 43 . Keywords Hamming cube · Heat semigroup · Hypercontractivity · Bernstein–Markov inequality · Moment comparison Mathematics Subject Classification Primary 42C10; Secondary 41A17 · 41A63 · 46B07
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. Extended author information available on the last page of the article
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A. Eskenazis, P. Ivanisvili
1 Introduction Fix n ∈ N and let (X , · X ) be a Banach space. If p ∈ [1, ∞), the vector valued L p norm of a function f : {−1, 1}n → X is defined as ⎛ f L p ({−1,1}n ;X )
def ⎝ 1 = 2n
⎞1/ p
f (ε) X ⎠ p
.
(1)
ε∈{−1,1}n
def
As usual, we denote f L ∞ ({−1,1}n ;X ) = maxε∈{−1,1}n f (ε) X . For a subset n Boolean funcA ⊆ {1, . . . , n} the Walsh function w A : {−1, 1} → {−1, 1} is the tion given by w A (ε) = i∈A εi , where ε = (ε1 , . . . , εn ) ∈ {−1, 1}n . Every function f : {−1, 1}n → X admits an expansion of the form f =
f (A)w A ,
(2)
f (δ)w A (δ) ∈ X .
(3)
A⊆{1,...,n}
where def 1
f (A) = n 2
δ∈{−1,1}n
For i ∈ {1, . . . , n} the i-th partial derivative of such a functi
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