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