Related papers: Estimating Norms of Commutators

We find the optimal constant $C$ such that \begin{equation*} \|f_1*f_2*\dots*f_{k}\|_{\infty}\geq C\prod_{i=1}^{k}\|f_i\|_1 \end{equation*} for functions $f_i:\{0,1\}^d\to\mathbb{R}$. As applications, we derive bounds for Sidon sets on…

Let $f \geq 0$ be operator monotone on $[0, \infty)$. In this paper we prove that for any unitarily-invariant norm $|||-|||$ on $M_n(\mathbb{C})$ and matrices $A, B, X \in M_n(\mathbb{C})$ with $A, B \geq 0$ and $|||X||| \leq 1$,…

We consider the problem of estimating small ball probabilities $\mathbb P\{f(G) \leqslant \delta \mathbb Ef(G)\}$ for sub-additive,positively homogeneous functions $f$ with respect to the Gaussian measure. We establish estimates that depend…

In this paper, we establish a determinantal formula for 2 x 2 matrix commutators [X,Y] = XY - YX over a commutative ring, using (among other invariants) the quantum traces of X and Y. Special forms of this determinantal formula include a…

Given a Calder\'on-Zygmund operator $T$, a classic result of Coifman-Rochberg-Weiss relates the norm of the commutator $[b, T]$ with the BMO norm of $b$. We focus on a weighted version of this result, obtained by Bloom and later generalized…

We study Whitney-type estimates for approximation of convex functions in the uniform norm on various convex multivariate domains while paying a particular attention to the dependence of the involved constants on the dimension and the…

Let $A$ be a $C^*$-algebra and $I$ be a closed ideal in $A$. For $x\in A$, its image under the canonical surjection $A\to A/I$ is denoted by $\dot x$, and the spectral radius of $x$ is denoted by $r(x)$. We prove that $$\max\{r(x), \|\dot…

We propose to extend ``invertibility'' to ``regularity'' for categories in general abstract algebraic manner. Higher regularity conditions and ``semicommutative'' diagrams are introduced. Distinction between commutative and…

A convex function $f:[a,b]\to\mathbb{R}$ satisfies the so-called Hermite-Hadamard inequality $$ f\left(\frac{a+b}{2}\right)\leq \frac{1}{b-a}\int_a^{b}f(t)dt\leq \frac{f(a)+f(b)}{2}. $$ Motivated by the above estimates, in this paper we…

For commutators of the form [b,T] where T is any Calderon--Zygmund operator and b is any BMO function we derive weighted quadratic type estimates in term of the A1 constant of the weight both in the Lp context or of LlogL type at the…

Let $S(x,t)$ denote the Weyl sum with associated polynomial $xn + tn^2$. Suppose that $|S(x,t)|$ attains its maximum for given $x$ at $t = t(x)$. We give upper and lower bounds of the same order of magnitude for the $L^p$ norm of…

Let $A\in\mathbb{C}^{n\times n}$ and $\widetilde{A}\in\mathbb{C}^{n\times n}$ be two normal matrices with spectra $\{\lambda_{i}\}_{i=1}^{n}$ and $\{\widetilde{\lambda}_{i}\}_{i=1}^{n}$, respectively. The celebrated Hoffman--Wielandt…

This paper is devoted to the estimators of the mean that provide strong non-asymptotic guarantees under minimal assumptions on the underlying distribution. The main ideas behind proposed techniques are based on bridging the notions of…

We provide a general approach to obtain upper bounds for small deviations $ \mathbb{P}(\Vert y \Vert \le \epsilon)$ in different norms, namely the supremum and $\beta$- H\"older norms. The large class of processes $y$ under consideration…

We develop an $\e$-regularity theory at the boundary for a general class of Monge-Amp\`ere type equations arising in optimal transportation. As a corollary we deduce that optimal transport maps between H\"older densities supported on $C^2$…

Let $f$ be a symmetric norm on ${\mathbb R}^n$ and let ${\mathcal B}({\mathcal H})$ be the set of all bounded linear operators on a Hilbert space ${\mathcal H}$ of dimension at least $n$. Define a norm on ${\mathcal B}({\mathcal H})$ by…

We investigate the representation of homogeneous, symmetric means in the form M(x,y)=\frac{x-y}{2f((x-y)/(x+y))}. This allows for a new approach to comparing means. As an example, we provide optimal estimate of the form (1-\mu)min(x,y)+ \mu…

The size estimation problem in electrical impedance tomography is considered when the conductivity is a complex number and the body is two-dimensional. Upper and lower bounds on the volume fraction of the unknown inclusion embedded in the…

We give a new fractal Weyl upper bound for resonances of convex co-compact hyperbolic manifolds in terms of the dimension $n$ of the manifold and the dimension $\delta$ of its limit set. More precisely, we show that as $R\to\infty$, the…

We derive an upper bound for the mean of the supremum of the empirical process indexed by a class of functions that are known to have variance bounded by a small constant $\delta$. The bound is expressed in the uniform entropy integral of…