Polynomial inequalities on the Hamming cube

• 成果类型: Article; Early Access
• 署名作者: Eskenazis, Alexandros; Ivanisvili, Paata
• 署名单位: Princeton University; University of California System; University of California Irvine; Sorbonne Universite; Universite Paris Cite
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-020-00973-y
• 发表日期: 2020
• 关键词: operator; geometry; riesz

摘要

Let (X, parallel to . parallel to(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, parallel to . parallel to (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 Etudes 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 l(p) sums of influences of bounded functions for p is an element of (1, 4/3).