Let $(X,\|\cdot\|_X)$ be a Banach space. The purpose of this article is to systematically investigate dimension independent properties of vector valued functions $f:\{-1,1\}^n\to X$ 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 $(X,\|\cdot\|_X)$, 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 on 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 (1984) and Mendel and Naor (2014) for functions with narrow enough spectrum and improve the bounds of Filmus, Hatami, Keller and Lifshitz (2016) on the $\ell_p$ sums of influences of bounded functions for $p\in\big(1,\frac{4}{3}\big)$.
This note contains some estimates for tail spaces on Hamming cube. We use analytic paraproduct operator for that purpose. We also show several types of Bernstein--Markov inequalities for Banach space valued functions on Hamming cube. Here the novelty is in getting rid of some irritating logarithms and in proving Bernstein--Markov inequalities for $|\nabla f|_X$ rather than for $Δ^{1/2} f$ for $X$ valued polynomials on Hamming cube.
The Lipschitz geometry of segments of the infinite Hamming cube is studied. Tight estimates on the distortion necessary to embed the segments into spaces of continuous functions on countable compact metric spaces are given. As an application, the first nontrivial lower bounds on the $C(K)$-distortion of important classes of separable Banach spaces, where $K$ is a countable compact space in the family $ \{ [0,ω],[0,ω\cdot 2],\dots, [0,ω^2], \dots, [0,ω^k\cdot n],\dots,[0,ω^ω]\}\ ,$ are obtained.
Let $1\leq p \leq q <\infty$, and let $w \in \mathbb{C}$. Weissler conjectured that the Hermite operator $e^{wΔ}$ is bounded as an operator from $L^{p}$ to $L^{q}$ on the Hamming cube $\{-1,1\}^{n}$ with the norm bound independent of $n$ if and only if \begin{align*} |p-2-e^{2w}(q-2)|\leq p-|e^{2w}|q. \end{align*} It was proved by Bonami (1970), Beckner (1975), and Weissler (1979) in all cases except $2<p\leq q <3$ and $3/2<p\leq q <2$, which stood open until now. The goal of this paper is to give a full proof of Weissler's conjecture in the case $p=q$. Several applications will be presented.
This is an attempt to build Banach space valued theory for certain singular integrals on Hamming cube. Of course all estimates below are dimension independent, and we tried to find ultimate sharp assumptions on the Banach space for a corresponding operators to be bounded. In certain cases we succeeded, although there are still many open questions, some of them are listed in the last Section. Using the approach of \cite{IVHV} and also quantum random variables approach of \cite{ELP} we generalize several theorems of Pisier \cite{P} and Hytönen-Naor \cite{HN}. We also improve the constant in $L^1…
Graham and Winkler derived a formula for the determinant of the distance matrix of a full-dimensional set of $n + 1$ points $\{ x_{0}, x_{1}, \ldots , x_{n} \}$ in the Hamming cube $H_{n} = ( \{ 0,1 \}^{n}, \ell_{1} )$. In this article we derive a formula for the determinant of the distance matrix $D$ of an arbitrary set of $m + 1$ points $\{ x_{0}, x_{1}, \ldots , x_{m} \}$ in $H_{n}$. It follows from this more general formula that $\det (D) \not= 0$ if and only if the vectors $x_{0}, x_{1}, \ldots , x_{m}$ are affinely independent. Specializing to the case $m = n$ provides new insights into …
We establish a sharp moment comparison inequality between an arbitrary negative moment and the second moment for sums of independent uniform random variables, which extends Ball's cube slicing inequality.
We prove Lp Poincare inequalities for functions on the discrete cube and their discrete gradient. We thus recover an exponential inequality and the concentration phenomenon for the uniform probability on the cube first obtained by Bobkov and Gotze. Inequalities involving the discrete gradient and powers of the discrete Laplacian are also considered, for the Lp norm or more general ones. Similar results hold true, replacing functions on the cube by elements of the CAR algebra and considering the annihilation operators and the number operator.
Intrinsic dimensional functional inequalities on model spaces
CNRS, Institut de Mathématiques de Jussieu, Sorbonne Université（法国国家科学研究中心，朱西厄数学研究所，索邦大学）；Trinity College, University of Cambridge（剑桥大学三一学院）；Division of Applied Mathematics, Brown University（布朗大学应用数学系）
Suppose that the $d$-dimensional unit cube $Q$ is the union of three disjoint "simple" sets $E$, $F$ and $G$ and that the volumes of $E$ and $F$ are both greater than half the volume of $G$. Does this imply that, for some cube $W$ contained in $Q$. the volumes of $E\cap W$ and $F\cap W$ both exceed $s$ times the volume of $W$ for some absolute positive constant $s$? Here, by "simple" we mean a set which is a union of finitely many dyadic cubes. We prove that an affirmative answer to this question would have deep consequences for the important space $BMO$ of functions of bounded mean oscillatio…
Support-Sensitive Bohnenblust-Hille Inequalities and Local Invariants on Hamming Schemes
Institut für Mathematik, Carl von Ossietzky Universität（卡尔·冯·奥西茨基大学数学研究所）；Universidad Torcuato Di Tella. Departamento de Matemáticas y Estadística. IMAS-CONICET（托尔夸托·迪泰拉大学数学与统计系 IMAS-CONICET）；Departamento de Matemática, Facultad de Cs. Exactas y Naturales, Universidad de Buenos Aires and IAM-CONICET（布宜诺斯艾利斯大学理学院数学系 IAM-CONICET）；Faculty of Mathematics and Computer Science, Adam Mickiewicz University（亚当·密茨凯维奇大学数学与计算机科学学院）；FCEIA, Universidad Nacional de Rosario and CIFASIS, CONICET（罗萨里奥国立大学经济与应用科学学院 CIFASIS-CONICET）
We study the generalized roundness of finite metric spaces whose distance matrix $D$ has the property that every row of $D$ is a permutation of the first row. The analysis provides a way to characterize subsets of the Hamming cube $\{ 0, 1 \}^{n} \subset \ell_{1}^{(n)}$ ($n \geq 1$) that have strict $1$-negative type. The result can be stated in two ways: a subset $S = \{ \vc{x}_0,\vc{x}_1,\ldots,\vc{x}_k \}$ of the Hamming cube $\{ 0, 1 \}^{n} \subset \ell_{1}^{(n)}$ has generalized roundness one if and only if the vectors $\{ \vc{x}_1 - \vc{x}_0,\vc{x}_2 - \vc{x}_0,\ldots,\vc{x}_k - \vc{x}_0…
A rather tricky question is the construction of wavelet bases on domains for suitable function spaces (Sobolev, Besov, Triebel-Lizorkin type). In his monograph from 2008, Triebel presented an approach how to construct wavelet (Riesz) bases in function spaces of Besov and Triebel-Lizorkin type on cellular domains, in particular on the cube. However, he had to exclude essential exceptional values of the smoothness parameter $s$, for instance the theorems do not cover the Sobolev space W_2^1(Q) on the n-dimensional cube Q for n at least 2. Triebel also gave an idea how to deal with those exceptio…
We obtain an optimal deviation from the mean upper bound \begin{equation} D(x)\=\sup_{f\in \F}μ\{f-\E_μ f\geq x\},\qquad\ \text{for}\ x\in\R\label{abstr} \end{equation} where $\F$ is the class of the integrable, Lipschitz functions on probability metric (product) spaces. As corollaries we get exact solutions of $\eqref{abstr}$ for Euclidean unit sphere $S^{n-1}$ with a geodesic distance and a normalized Haar measure, for $\R^n$ equipped with a Gaussian measure and for the multidimensional cube, rectangle, torus or Diamond graph equipped with uniform measure and Hamming distance. We also prove …
An operator C on a Hilbert space H dilates to an operator T on a Hilbert space K if there is an isometry V from H to K such that C=V^*TV. A main result of this paper is, for a positive integer d, the simultaneous dilation, up to a sharp factor $\vartheta(d)$, of all d-by-d symmetric matrices of operator norm at most one to a collection of commuting self-adjoint contraction operators on a Hilbert space. An analytic formula for $\vartheta(d)$ is derived, which as a by-product gives new probabilistic results for the binomial and beta distributions. Dilating to commuting operators has consequences…
We prove structure theorems for measures on the discrete cube and on Gaussian space, which provide sufficient conditions for mean-field behavior. These conditions rely on a new notion of complexity for such measures, namely the Gaussian-width of the gradient of the log-density. On the cube $\{-1,1\}^n$, we show that a measure $ν$ which exhibits low complexity can be written as a mixture of measures $\{ν_θ\}_{θ\in \mathcal{I}}$ such that: i. for each $θ$, the measure $ν_θ$ is a small perturbation of $ν$ such that $\log \tfrac{d ν_θ}{d ν}$ is a linear function whose gradient is small and, ii. $ν…
We study Hausdorff-Young type inequalities for vector-valued Dirichlet series which allow to compare the norm of a Dirichlet series in the Hardy space $\mathcal{H}_{p} (X)$ with the $q$-norm of its coefficients. In order to obtain inequalities completely analogous to the scalar case, a Banach space must satisfy the restrictive notion of Fourier type/cotype. We show that variants of these inequalities hold for the much broader range of spaces enjoying type/cotype. We also consider Hausdorff-Young type inequalities for functions defined on the infinite torus $\mathbb{T}^{\infty}$ or the boolean …
Log-Sobolev inequalities (LSIs) upper-bound entropy via a multiple of the Dirichlet form (i.e. norm of a gradient). In this paper we prove a family of entropy-energy inequalities for the binary hypercube which provide a non-linear comparison between the entropy and the Dirichlet form and improve on the usual LSIs for functions with small support. These non-linear LSIs, in turn, imply a new version of the hypercontractivity for such functions. As another consequence, we derive a sharp form of the uncertainty principle for the hypercube: a function whose energy is concentrated on a set of small …
On cost-induced Santaló-type inequalities in Polish measure spaces
Carnegie Mellon University（卡内基梅隆大学）；University of Toulouse & CNRS（图卢兹大学及法国国家科学研究中心）；University of Cincinnati（辛辛那提大学）；The Ohio State University（俄亥俄州立大学）
