Polynomial inequalities on the Hamming cube
Polynomial inequalities on the Hamming cube
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DOI：
10.1007/s00440-020-00973-y
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发表时间：
2020-06-04
影响因子：
2
通讯作者：
Ivanisvili, Paata
中科院分区：
文献类型：
--
作者：
Eskenazis, Alexandros;Ivanisvili, Paata
设（X，平行于。平行于（X））是Banach空间。本文系统地研究了Hamming立方上谱有上下界的向量值函数f：{-1，1}（n）-> X的维数无关性质.我们的证明利用了热流的收缩性质，由目标空间（X，平行于。平行于（X）），结合对偶参数和适当的工具，从近似理论和复分析。本文对谱有界函数的各种已有的估计进行了一系列改进，包括低次沃尔什多项式的矩比较结果和Bernstein-Markov型不等式，它们构成了Gauss空间中Freud不等式（1971）的离散向量值类似.这些不等式中的许多甚至是新的标量值函数。此外，我们提供了Mendel和Naor的热光滑定理（2014）的一个简短证明，用于尾空间中的函数，其值在非平凡型空间中，我们还证明了热半群作用于具有上界谱的函数的衰变的对偶下界。最后，我们改进了Meyer的反向Bernstein-Markov不等式（见：概率研讨会，XVIII，数学讲义。施普林格，柏林，1984年。org/10.1007/BFb0100043）和Mendel和Naor（Publ Math Inst Hautes Etudes Sci 119：1-95，2014. https：//doi.org/10.1007/s10240-013-0053-2）适用于频谱足够窄的函数，并改进了Filmus等人的边界。（Isr J Math 214（1）：167-192，2016. https://doi.org/ 10.1007/s11856-016-1355-0）关于有界函数的影响和的问题，其中p是（1，4/3）的元素.
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).