Let $$(X,\Vert \cdot \Vert _X)$$ ( X , ‖ · ‖ X ) be a Banach space. The purpose of this article is to systematically investigate dimension independent properties of vector valued functions $$f:\{-1,1\}^n\rightarrow X$$ 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 $$(X,\Vert \cdot \Vert _X)$$ ( X , ‖ · ‖ 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 $$\ell _p$$ ℓ p sums of influences of bounded functions for $$p\in \big (1,\frac{4}{3}\big )$$ p ∈ ( 1 , 4 3 ) .

中文翻译：

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汉明立方体上的多项式不等式

令 $$(X,\Vert \cdot \Vert _X)$$ ( X , ‖ · ‖ X ) 是一个 Banach 空间。本文的目的是系统地研究向量值函数 $$f:\{-1,1\}^n\rightarrow X$$ f : { - 1 , 1 } n → X 在汉明立方体上的维数无关性质其频谱上界或下界。我们的证明利用了热流的收缩特性，由目标空间 $$(X,\Vert \cdot \Vert _X)$$ ( X , ‖ · ‖ X ) 的几何结构引起，结合对偶参数和来自近似理论和复分析。我们获得了对有界函数的各种经过充分研究的估计的一系列改进，包括低阶 Walsh 多项式和 Bernstein-Markov 型不等式的矩比较结果，它们构成了高斯空间中弗洛伊德不等式的离散向量值类似物（1971）。即使对于标量值函数，其中许多不等式也是新的。此外，我们为尾部空间中具有非平凡类型空间中的值的函数提供了孟德尔和 Naor 的热平滑定理 (2014) 的简短证明，我们还证明了热半群衰减的双重下界作用于具有谱有界的函数从上面。最后，我们改进了 Meyer 的反向 Bernstein-Markov 不等式（在：概率研讨会，XVIII，数学讲义。Springer，柏林，1984。https://doi.org/10.1007/BFb0100043）和 Mendel 和 Naor（Publ Math Inst Hautes Études Sci 119:1–95, 2014. https://doi.org/10.1007/s10240-013-0053-2 ) 用于具有足够窄谱的函数并改进 Filmus 等人的界限。