Commutators with Reisz Potentials in One and Several Parameters

Sharp commutator estimates via harmonic extensions

We give an alternative proof of several sharp commutator estimates involving Riesz transforms, Riesz potentials, and fractional Laplacians. Our methods only involve harmonic extensions to the upper half-space, integration by parts, and trace space characterizations. The commutators we investigate are Jacobians, more generally Coifman-Rochberg-Weiss commutators, Chanillo's commutator with the Riesz potential, Coifman-Meyer or Kato-Ponce-Vega type commutators, and the Da Lio-Rivière three-term commutators. We also give a new limiting $L^1$-estimate for a double commutator of Coifman-Rochberg-Wei…

Endpoint estimates for the commutators of multilinear Calderón-Zygmund operators with Dini type kernels

Let $T_{\vec{b}}$ and $T_{Πb}$ be the commutators in the $j$-th entry and iterated commutators of the multilinear Calderón-Zygmund operators, respectively. It was well-known that $T_{\vec{b}}$ and $T_{Πb}$ were not of weak type $(1,1)$ and $(H^1, L^1)$, but they did satisfy certain endpoint $L\log L$ type estimates. In this paper, our aim is to give more natural sharp endpoint results. We show that $T_{\vec{b}}$ and $T_{Πb}$ are bounded from product Hardy space $H^1\times\cdot\cdot\cdot\times H^1$ to weak $L^{\frac{1}{m},\infty}$ space, whenever the kernel satisfies a class of Dini type condit…

On pointwise and weighted estimates for commutators of Calderón-Zygmund operators

In recent years, it has been well understood that a Calderón-Zygmund operator $T$ is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise estimate for the commutator $[b,T]$ with a locally integrable function $b$. This result is applied into two directions. If $b\in BMO$, we improve several weighted weak type bounds for $[b,T]$. If $b$ belongs to the weighted $BMO$, we obtain a quantitative form of the two-weighted bound for $[b,T]$ due to Bloom-Holmes-Lacey-Wick.

Sharp norm inequalities for commutators of classical operators

We prove several sharp weighted norm inequalities for commutators of classical operators in harmonic analysis. We find sufficient $A_p$-bump conditions on pairs of weights $(u,v)$ such that $[b,T]$, $b\in BMO$ and $T$ a singular integral operator (such as the Hilbert or Riesz transforms), maps $L^p(v)$ into $L^p(u)$. Because of the added degree of singularity, the commutators require a "double log bump" as opposed to that of singular integrals, which only require single log bumps. For the fractional integral operator $I_\al$ we find the sharp one-weight bound on $[b,I_\al]$, $b\in BMO$, in ter…

Two-weight, weak type norm inequalities for a class of sublinear operators on weighted Morrey and amalgam spaces

Let $\mathcal T_α~(0\leqα<n)$ be a class of sublinear operators satisfying certain size conditions introduced by Soria and Weiss, and let $[b,\mathcal T_α]~(0\leqα<n)$ be the commutators generated by $\mathrm{BMO}(\mathbb R^n)$ functions and $\mathcal T_α$. This paper is concerned with two-weight, weak type norm estimates for these sublinear operators and their commutators on the weighted Morrey and amalgam spaces. Some boundedness criterions for such operators are given, under the assumptions that weak-type norm inequalities on weighted Lebesgue spaces are satisfied. As applications of our ma…

Extrapolation of compactness on weighted spaces

Let $T$ be a linear operator that, for some $p_1\in(1,\infty)$, is bounded on $L^{p_1}(\tilde w)$ for all $\tilde w\in A_{p_1}(\mathbb R^d)$ and in addition compact on $L^{p_1}(w_1)$ for some $w_1\in A_{p_1}(\mathbb R^d)$. Then $T$ is bounded and compact on $L^p(w)$ for all $p\in(1,\infty)$ and all $w\in A_p(\mathbb R^d)$. This "compact version" of Rubio de Francia's celebrated weighted extrapolation theorem follows from a combination of results in the interpolation and extrapolation theory of weighted spaces on the one hand, and of compact operators on abstract spaces on the other hand. Moreo…

Hankel Operators in Several Complex Variables and Product $BMO\zProd$

$H^2\zProd$ denotes the Hardy space of square integrable functions analytic in each variable separately. Let $P^{\ominus}$ be the natural projection of $L^2\zProd$ onto $\z8{H^2\zProd}$. A Hankel operator with symbol $b$ is the linear operator from $H^2\zProd$ to $\z8{H^2\zProd}$ given by $H_b \zvf=P^{\ominus}\bar b \zvf$. We show that \md0 \norm H_b ..\simeq \norm P^{\oplus}b.BMO\zProd., \emd where the right hand norm is S.-Y. Chang and R. Fefferman product $BMO$. This fact has well known equivalences in terms of commutators and the weak factorization of $H^1\zProd$. In the case of two comple…

多线性分数阶稀疏算子理论I：逐点控制与加权估计

The multilinear fractional sparse operator theory I: pointwise domination and weighted estimate

本文针对多线性分数阶奇异积分算子的广义交换子，通过引入高阶多符号稀疏算子并利用极大权与迭代权方法，建立了Bloom型定量加权估计及端点估计，改进了现有方法并推广了结论。