This paper gives new two-weight bump conditions for the sparse operators related to iterated commutators of fractional integrals. As applications, the two-weight bounds for iterated commutators of fractional integrals under more general bump conditions are obtained. Meanwhile, the necessity of two-weight bump conditions as well as the converse of Bloom type estimates for iterated commutators of fractional integrals are also given.
We prove new sufficient bump conditions for general iterated commutators of Calderón-Zygmund operators and fractional integral operators. As an application of our results we derive two weight compactness theorems for higher order iterated commutators with a $CMO$ function.
We prove certain two weight bump conditions are sufficient for the compactness of the commutator $[b,T]$ where $b\in CMO$ and $T$ is a Calderón- Zygmund operator. This is the first result for compactness in the two weight setting without additional assumptions on the individual weights.
In this paper we extend the bump conjecture and a particular case of the separated bump conjecture with logarithmic bumps to iterated commutators $T_b^m$. Our results are new even for the first order commutator $T_b^1$. A new bump type necessary condition for the two-weighted boundedness of $T_b^m$ is obtained as well. We also provide some results related to a converse to Bloom's theorem.
In this paper we consider two weight bump conditions for higher order commutators. Given $b$ and a Calderón-Zygmund operator $T$, define the commutator $T^1_bf=[T,b]f= bTf-T(bf)$, and for $m\geq 2$ define the iterated commutator $T^m_b f = [b,T_b^{m-1}]f$. Traditionally, commutators are defined for functions $b\in BMO$, but we show that if we replace $BMO$ by an oscillation class first introduced by Pérez [31], we can give a range of sufficient conditions on a pair of weights $(u,v)$ for $T^m_b : L^p(v)\rightarrow L^p(u)$ to be bounded. Our results generalize work of the first two authors in […
In this paper we prove two matrix weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a matrix symbol. More precisely, we extend the recent results of the second author, Pott, and Treil on two matrix weighted norm inequalities for commutators of Calderon-Zygmund operators and multiplication by a matrix symbol to the fractional integral operator setting. In particular, we completely extend the fractional Bloom theory of Holmes, Rahm, and Spencer to the two matrix weighted setting with a matrix
This note is devoted to establishing two-weight estimates for commutators of singular integrals. We combine multilinearity with product spaces. A new type of two-weight extrapolation result is used to yield the quasi-Banach range of estimates.
This paper gives the pointwise sparse dominations for variation operators of singular integrals and commutators with kernels satisfying the $L^r$-Hörmander conditions. As applications, we obtain the strong type quantitative weighted bounds for such variation operators as well as the weak-type quantitative weighted bounds for the variation operators of singular integrals and the quantitative weighted weak-type endpoint estimates for variation operators of commutators, which are completely new even in the unweighted case. In addition, we also obtain the local exponential decay estimates for such…
We prove mixed inequalities for commutators of Calderón-Zygmund operators (CZO) with multilinear symbols. Concretely, let $m\in\mathbb{N}$ and $\mathbf{b}=(b_1,b_2,\dots, b_m)$ be a vectorial symbol such that each component $b_i\in \mathrm{Osc}_{\mathrm{exp}\, L^{r_i}}$, with $r_i\geq 1$. If $u\in A_1$ and $v\in A_\infty(u)$ we prove that the inequality \[uv\left(\left\{x\in \mathbb{R}^n: \frac{|T_\mathbf{b}(fv)(x)|}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\|\mathbf{b}\|\frac{|f(x)|}{t}\right)u(x)v(x)\,dx\] holds for every $t>0$, where $Φ(t)=t(1+\log^+t)^r$, with $1/r=\sum_{i=1}^…
Hörmander's conditions for vector-valued kernels of singular integrals and its commutators
本文研究了奇异积分算子$T$及其分解算子与向量值核$K$的卷积的Coifman型估计和加权模不等式.我们定义了一个较弱的Hörmander型条件与杨函数的向量值内核。在这个一般框架下，作为例子，我们得到了[ M]中关于平方算子及其交换子的结果。Lorente，M.S. Riveros，A. de la Torre \emph{On the Coifman type inequality for the oscillation of the one-sided averages }，Journal of Mathematical Analysis and Applications，2007年，第336期，第1期，第577-592页。]
We prove mixed inequalities for the generalized maximal operator $M_Φ$ when the function $v$ is a radial power function that fails to be locally integrable. Concretely, let $u$ be a weight, $v(x)=|x|^β$ with $β<-n$ and $r\geq 1$. If $Φ$ is a Young function with certain properties, then the inequality \[uv^r\left(\left\{x\in\mathbb{R}^n: \frac{M_Φ(fv)(x)}{v(x)}>t\right\}\right)\leq C\int_{\mathbb{R}^n}Φ\left(\frac{|f(x)|}{t}\right)v^r(x)Mu(x)\,dx\] holds for every $t>0$ and every bounded function. This improves a similar mixed estimate proved in \cite{BCP-M}. As an application, we give mixed es…
Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: General operator theory and weights
这是一系列四篇文章的第一部分。在这项工作中，我们感兴趣的加权范数估计。我们把重点放在两个不同性质的结果：一个是基于一个好的$λ$不等式与两个参数和其他使用Calderón-Zygmund分解。这些结果很好地适用于奇异的“非积分”算子及其具有有界平均振荡函数的算子。奇异意味着它们是0阶的，“非整数”意味着它们没有一个带有大小估计的核的整表示，甚至是粗略的，所以它们可能不在$1 < p < \infty$的所有$L^p$空间上有界。然后用适当的局部$L^p-L^q$估计值替换逐点估计值。我们获得加权$L^p$估计的$p$的范围是不同的$(1,\infty)$和隔离的权利类的权重。特别是，我们证明了这样一个类的外推定理' à la Rubio de Francia'，从而向量值估计。
