Related papers: The A_2 theorem with the Dini condition

We construct a slightly new noncommutative Calder\'on-Zygmund decomposition by further splitting the bad function. Using this tool, we prove the weak type (1,1) boundedness of noncommutative Calder\'on-Zygmund operators under a class of…

We develop the compactness theory of multilinear singular integrals on product spaces using a modern point of view. The first main result is a compact $T1$ theorem for multilinear Calder\'{o}n--Zygmund operators on product spaces. More…

The Coifman-Fefferman inequality implies quite easily that a Calderon-Zygmund operator $T$ acts boundedly in a Banach lattice $X$ on $\mathbb R^n$ if the Hardy-Littlewood maximal operator $M$ is bounded in both $X$ and $X'$. We discuss this…

In this paper we consider nonlinear problems with an operator depending only on the deformation tensor. We consider the class of operators derived from a potential and with $(p,\delta)$ structure, for $1<p\leq 2$ and for all $\delta\geq0$.…

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group $G$. In particular, we show that the operators $T_\alpha: f \mapsto |.|^{-\alpha} L^{-\alpha/2} f$, where $|.|$ is a homogeneous norm, $0 < \alpha < Q/p$,…

We investigate certain singular integral operators with Riesz-type kernels on s-dimensional Ahlfors-David regular subsets of Heisenberg groups. We show that $L^2$-boundedness, and even a little less, implies that $s$ must be an integer and…

We prove a necessary and sufficient condition for embeddability of an operator system into $\mathcal{O}_2$. Using Kirchberg's theorems on a tensor product of $\mathcal{O}_2$ and $\mathcal{O}_{\infty}$, we establish results on their operator…

In this expository article, we briefly survey the main known schemes of proof of sparse domination principles within harmonic analysis. We then use the one based on the Calder\'on-Zygmund decomposition to prove a dual sparse domination…

We establish a new $T1$ theorem for the compactness of bi-parameter Calder\'on-Zygmund singular integral operators. Namely, we show that if a bi-parameter CZO $T$ satisfies the product weak compactness property, the mixed weak…

We give explicit necessary and sufficient conditions for the boundedness of the general second order differential operator L with real- or complex-valued distributional coefficients acting from the Sobolev space W^{1,2}(R^n) to its dual…

A martingale transform $ T$, applied to an integrable locally supported function $ f$, is pointwise dominated by a positive sparse operator applied to $ \lvert f\rvert $, the choice of sparse operator being a function of $ T$ and $ f$. As a…

We study the positive-definiteness of a family of $L^2(\mathbf{R})$ integral operators with kernel $K_{t, a}(x, y) = (1 + (x - y)^2 + a(x^2 + y^2)^t)^{-1}$, with $t > 0$ and $a > 0$. When $0 < t \le 1$, the known theory of positive-definite…

Let T : X --> X$ be a power bounded operator on Banach space. An operator C : X --> Y$ is called admissible for T if it satisfies an estimate $\sum_k\norm{CT^k(x)}^2\,\leq M^2\norm{x}^2$. Following Harper and Wynn, we study the validity of…

We give a characterization of the continuity properties of a Luxemburg maximal type operator associated to a critical radius function $\rho$ between Orlicz spaces. This goal is achieved by means of a Dini type condition that includes…

On the twisted Fock spaces $ \mathcal{F}^\lambda(\mathbb{C}^{2n}) $ we consider two types of convolution operators $ S_\varphi $ and $ \widetilde{S}_\varphi $ associated to an element $ \varphi \in \mathcal{F}^\lambda(\mathbb{C}^{2n}) .$ We…

Let $\mu$ be a finite Radon measure in $\mathbb{R}^d$ with polynomial growth of degree $n$, although not necessarily $n$-AD-regular. We prove that under some geometric conditions on $\mu$ that are closely related to rectifiability and…

Given a Calder\'{o}n--Zygmund (C--Z for short) operator $T$, which satisfies H\"ormander condition, we prove that: if $T$ maps all the characteristic atoms to $WL^{1}$, then $T$ is continuous from $L^{p}$ to $L^{p}(1<p<\infty)$. So the…

In the present paper, we consider elliptic operators $L=-\textrm{div}(A\nabla)$ in a domain bounded by a chord-arc surface $\Gamma$ with small enough constant, and whose coefficients $A$ satisfy a weak form of the Dahlberg-Kenig-Pipher…

We study orthogonally additive operators between Riesz spaces without the Dedekind completeness assumption on the range space. Our first result gives necessary and sufficient conditions on a pair of Riesz spaces $(E,F)$ for which every…

Motivated by the study of kernels of bilinear pseudodifferential operators with symbols in a H\"ormander class of critical order, we investigate boundedness properties of strongly singular Calder\'on--Zygmund operators in the bilinear…