Emil Airta, Tuomas Hytönen, Kangwei Li, Henri Martikainen, Tuomas Oikari
We study the bi-commutators $[T_1, [b, T_2]]$ of pointwise multiplication and Calderón-Zygmund operators, and characterize their $L^{p_1}L^{p_2} \to L^{q_1}L^{q_2}$ boundedness for several off-diagonal regimes of the mixed-norm integrability exponents $(p_1,p_2)\neq(q_1,q_2)$. The strategy is based on a bi-parameter version of the recent approximate weak factorization method.
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In this paper, we establish a new characterization of weighted $\rm{VMO}$ spaces, which are essential different from the classical $\rm{VMO}$ spaces, via the compactness of sparse operators, commutators of Riesz potentials and fractional maximal operators. Some of our results are new even in unweighted setting.
We study the boundedness of commutators of bi-parameter singular integrals between mixed spaces $$ [b,T]: L^{p_1}L^{p_2} \to L^{q_1}L^{q_2} $$ in the off-diagonal situation $q_i,p_i\in(1,\infty)$ where we also allow $q_i\not= p_i.$ Boundedness is fully characterized for several arrangements of the integrability exponents with some open problems presented.
We find a minimal notion of non-degeneracy for bilinear singular integral operators $T$ and identify testing conditions on the multiplying function $b$ that characterize the $L^p\times L^q\to L^r,$ $1<p,q<\infty$ and $r>\frac{1}{2},$ boundedness of the bilinear commutator $[b,T]_1(f,g) = bT(f,g) - T(bf,g).$ Our arguments cover almost all arrangements of the integrability exponents $p,q,r,$ with a single open problem presented in the end. Additionally, the arguments extend to the multilinear setting.
Off-diagonal bloom weighted estimates for bilinear commutators
本文通过稀疏控制和中位数方法，证明了双权设置下双线性迭代交换子的非对角估计，并指出方法可推广至多线性情形。
Bloom weighted bounds for sparse forms associated to commutators
本文研究了一般算子类迭代交换子相关的双线性稀疏形式，在指数全范围内建立了对角与非对角的 Bloom 加权估计，并应用于粗糙齐次奇异积分与 Bochner-Riesz 算子，揭示了高阶交换子已有估计的不尖锐性。
Note on the compactness of commutators
本文研究了Calderón-Zygmund和粗糙型奇异积分算子交换子的紧致性条件，并提出了一种新的外推原理。
On the compactness of the bi-commutator
本文研究双交换子 [T_1,[b,T_2]] 的紧性，利用混合范数非对角估计、新外推方案、双参数空间逼近及矩形近似弱分解，给出乘积 VMO 充分条件与完整刻画。
On off-diagonal operators in matrix-weighted spaces
研究矩阵加权空间中非对角算子，通过凸体控制理论证明分数阶算子及其交换子的矩阵加权不等式，给出相关定量估计及向量值函数的矩阵加权加利亚尔多 - 尼伦伯格 - 索伯列夫不等式。
The multilinear fractional bounded mean oscillation operator theory I: sparse domination, sparse $T1$ theorem, off-diagonal extrapolation, quantitative weighted estimate -- for generalized commutators
本文在球基测度空间上引入并研究了多线性分数次有界平均振荡算子及其广义交换子，建立了稀疏控制与稀疏 $T1$ 定理，并在多线性分数次权下获得了多种加权不等式与非对角外推框架。
Compactness of commutators of rough singular integrals
本文刻画粗糙奇异积分算子 T_Ω 交换子在双权非对角情形下的紧性，并利用矩阵权 Kolmogorov-Riesz 定理得到矩阵权紧性结果。
This is the first part of a series of four articles. In this work, we are interested in weighted norm estimates. We put the emphasis on two results of different nature: one is based on a good-$λ$ inequality with two-parameters and the other uses Calderón-Zygmund decomposition. These results apply well to singular 'non-integral' operators and their commutators with bounded mean oscillation functions. Singular means that they are of order 0, 'non-integral' that they do not have an integral representation by a kernel with size estimates, even rough, so that they may not be bounded on all $L^p$ sp…
We prove several off-diagonal and pointwise estimates for singular integral operators that extend compactly on $L^{p}(\mathbb R^{n})$.
We provide off-diagonal estimates for a maximal operator arising from a geometric problem of estimating the size of certain geometric configuration of k- skeletons in $\mathbb{R}^n$. This is achieved by interpolating a weak-type endpoint estimate with the known diagonal bounds. The endpoint estimate is proved by combining a geometric result about k-skeletons and adapting an argument used to prove off-diagonal estimates for the circular maximal function in the plane.
This is the third part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. For $L$ in some class of elliptic operators, we study weighted norm $L^p$ inequalities for singular 'non-integral' operators arising from $L$ ; those are the operators $ϕ(L)$ for bounded holomorphic functions $ϕ$, the Riesz transforms $\nabla L^{-1/2}$ (or $(-Δ)^{1/2}L^{-1/2}$) and its inverse $L^{1/2}(-Δ)^{-1/2}$, some quadratic functionals $g\_{L}$ and $G\_{L}$ of Littlewood-Paley-Stein type and also some vector-valued inequalities such as the ones involved for ma…
We give an overview of the generalized Calderón-Zygmund theory for "non-integral" singular operators, that is, operators without kernels bounds but appropriate off-diagonal estimates. This theory is powerful enough to obtain weighted estimates for such operators and their commutators with $\BMO$ functions. $L^p-L^q$ off-diagonal estimates when $p\le q$ play an important role and we present them. They are particularly well suited to the semigroups generated by second order elliptic operators and the range of exponents $(p,q)$ rules the $L^p$ theory for many operators constructed from the semigr…
In a previous paper by one of us, a "compact version" of Rubio de Francia's weighted extrapolation theorem was proved, which allows one to extrapolate the compactness of an operator from just one space to the full range of weighted spaces, where this operator is bounded. In this paper, we obtain generalizations of this extrapolation of compactness for operators that are bounded from one space to a different one ("off-diagonal estimates") or only in a limited range of the $L^p$ scale. As applications, we easily recover recent results on the weighted compactness of commutators of fractional inte…
In this paper, we continue some recent work on two weight boundedness of sparse operators to the "off-diagonal" setting. We use the new "entropy bumps" introduced in by Treil-Volberg ([21]) and improved by Lacey-Spencer ([9]) and the "direct comparison bumps" introduced by Rahm-Spencer ([19]) and improved by Lerner ([10]). Our results are "sharp" in the sense that they are sharp in various particular cases. A feature is that given the current machinery, the proofs are almost trivial.
This is the second part of a series of four articles on weighted norm inequalities, off-diagonal estimates and elliptic operators. We consider a substitute to the notion of pointwise bounds for kernels of operators which usually is a measure of decay. This substitute is that of off-diagonal estimates expressed in terms of local and scale invariant $L^p-L^q$ estimates. We propose a definition in spaces of homogeneous type that is stable under composition. It is particularly well suited to semigroups. We study the case of semigroups generated by elliptic operators.
We obtain some necessary and sufficient conditions for the boundedness of a family of positive operators defined on symmetric cones, we then deduce off-diagonal boundedness of associated Bergman-type operators in tube domains over symmetric cones.
We investigate $L^p(γ)$-$L^q(γ)$ off-diagonal estimates for the Ornstein-Uhlenbeck semigroup $(e^{tL})_{t > 0}$. For sufficiently large $t$ (quantified in terms of $p$ and $q$) these estimates hold in an unrestricted sense, while for sufficiently small $t$ they fail when restricted to maximal admissible balls and sufficiently small annuli. Our counterexample uses Mehler kernel estimates.
Another look at regularity in transport-commutator estimates
研究探讨了运输-交换算子估计中速度场正则性的必要条件，发现通常无法放松到BMO假设，但在一维端点可实现，并揭示了相互作用势奇异性与速度正则性的权衡关系。
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