Related papers: Weighted bound for commutators

In this paper we prove that if $\Lambda\in M_p(\mathbb R^N)$ and has compact support then $\Lambda$ is a weak summability kernel for $1<p<\infty$, where $M_p(\mathbb R^N)$ is the space of multipliers of $L^p(\mathbb R^N)$.

For polynomial $ P (x,y)$, and any Calder\'{o}n-Zygmund kernel, $K$, the operator below satisfies a $ (1,r)$ sparse bound, for $ 1< r \leq 2$. $$ \sup _{\epsilon >0} \Bigl\lvert \int_{|y| > \epsilon} f (x-y) e ^{2 \pi i P (x,y) } K(y) \; dy…

We establish weighted weak-type bounds for the Bergman projection with respect to Bekoll\'e-Bonami characteristics. We present two proofs of an improved quantitative weak-type $(1,1)$ estimate, as well as sharp weak-type $(p,p)$ bounds for…

Let T be the singular integral operator with variable kernel defined by $Tf(x)= p.v. \int_{\mathbb{R}^{n}}K(x,x-y)f(y)\mathrm{d}y$ and $D^{\gamma}(0\leq\gamma\leq1)$ be the fractional differentiation operator, where…

Calder\'on-Zygmund operators with noncommuting kernels may fail to be Lp-bounded for $p \neq 2$, even for kernels with good size and smoothness properties. Matrix-valued paraproducts, Fourier multipliers on group vNa's or noncommutative…

Let $\mu_{\Omega,\vec{b}}$ be the multilinear commutator generalized by $\mu_{\Omega}$, the $n$-dimensional Marcinkiewicz integral with the bounded kernel, and $b_{j}\in \Osc_{\exp L^{r_{j}}}(1\le j\le m)$. In this paper, the following…

We give a weak factorization proof of the Hardy space $H^{p}(\mathbb{R}^{n})$ in the multilinear setting, for $\frac{n}{n+1} < p <1$. As a consequence, we obtain a characterization of the boundedness of the commutator $[b,T]$ from…

In this paper we study the boundedness and compactness characterizations of the commutator of Cauchy type integrals $\mathcal C$ on a bounded strongly pseudoconvex domain $D$ in $C^n$ with boundary $bD$ satisfying the minimum regularity…

This paper continues the investigation begun in arXiv:1906.05602 of extending the T1 theorem of David and Journ\'e, and optimal cancellation conditions, to more general weight pairs. The main additional tool developed here is a two weight…

A variant of the global $T(1)$ criterion to characterize the bounded Calder\'{o}n--Zygmund operators on BMO($\mathbb{R}^d$) is proved. We apply it to the certain Calder\'on commutators.

Let $T_a$ be a pseudo-differential operator defined by exotic symbol $a$ in H\"{o}rmander class $S^m_{0,\delta}$ with $m \in \mathbb{R} $ and $0 \leq \delta \leq 1 $. It is well-known that the weak type (1,1) behavior of $T_a $ is not fully…

Let $p\in(1,\infty)$, $\rho\in (2, \infty)$ and $W$ be a matrix $A_p$ weight. In this article, we introduce a version of variation $\mathcal{V}_{\rho}({\mathcal T_n}_{\,,\,\ast})$ for matrix Calder\'on--Zygmund operators with modulus of…

The weak boundedness property associated with a standard alpha-fractional Calderon-Zygmund operator and a weight pair is good-lambda controlled by the testing conditions and the Muckenhoupt and energy side conditions. As a consequence,…

Given a bilinear (or sub-bilinear) operator $B$, we prove restricted weighted weak type inequalities of the form $$ ||B(f_1, f_2)||_{L^{p, \infty}(w_1^{p/p_1}w_2^{p/p_2})}\lesssim ||f_1||_{L^{p_1, 1}(w_1)}||f_2||_{L^{p_2, 1}(w_2)}, $$…

Let $T_m$ be the $m$-th Calder\'on-Zygmund type singular integral. In the paper, we consider the boundedness of $T_m$ on the generalized product local Morrey spaces $LM_{p_1,\varphi_1}^{\{x_0\}}\times…

We present a new proof of the classical weak-type $(1,1)$ estimate for Calder\'on-Zygmund operators. This proof is inspired by ideas of Nazarov, Treil, and Volberg that address the non-doubling setting. An application to a weighted…

In this work we obtain weighted boundedness results for singular integral operators with kernels exhibiting exponential decay. We also show that the classes of weights are characterized by a suitable maximal operator. Additionally, we study…

Let $p\in[1,\infty]$, $q\in(1,\infty)$, $s\in\mathbb{Z}_+:=\mathbb{N}\cup\{0\}$, and $\alpha\in\mathbb{R}$. In this article, the authors introduce a reasonable version $\widetilde T$ of the Calder\'on--Zygmund operator $T$ on…

Let $\mathcal T_\alpha~(0\leq\alpha<n)$ be a class of sublinear operators satisfying certain size conditions introduced by Soria and Weiss, and let $[b,\mathcal T_\alpha]~(0\leq\alpha<n)$ be the commutators generated by…

The purpose of this article is to provide an alternative proof of the weak-type $\left(1,\ldots,1;\frac{1}{m}\right)$ estimate for $m$-multilinear Calder\'on-Zygmund operators on $\mathbb{R}^n$ first proved by Grafakos and Torres.…