Alexei Yu. Karlovich, Andrei K. Lerner
A classical theorem of Coifman, Rochberg, and Weiss on commutators of singular integrals is extended to the case of generalized $L^p$ spaces with variable exponent.
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In this paper, the behavior for commutators of a class of bilinear singular integral operator associated with non-smooth kernels on the products of weighted Lebesgue spaces is considered. By some new maximal functions to control the commutators of bilinear singular integral operators and CMO functions, compactness of the commutators is proved.
We obtain a Bloom-type characterization of the two-weighted boundedness of iterated commutators of singular integrals. The necessity is established for a rather wide class of operators, providing a new result even in the unweighted setting for the first order commutators.
A homage to Guido Weiss and his leadership of the Saint Louis team: Commutators of Singular Integrals and Sobolev inequalities
本文扩展了线性与非线性交换子的经典Sobolev型不等式。
In this paper we characterize BMO in terms of the boundedness of commutators of various bilinear singular integral operators with pointwise multiplication. In particular, we study commutators of a wide class of bilinear operators of convolution type, including bilinear Calderón-Zygmund operators and the bilinear fractional integral operators.
We provide a characterization of $\mathrm{BMO}$ in terms of endpoint boundedness of commutators of singular integrals. In particular, in one dimension, we show that $\|b\|_{\mathrm{BMO}}\eqsim B$, where $B$ is the best constant in the endpoint $L\log L$ modular estimate for the commutator $[H,b]$. We provide a similar characterization of the space $\mathrm{BMO}$ in terms of endpoint boundedness of higher order commutators of the Hilbert transform. In higher dimension we give the corresponding characterization of $\mathrm{BMO}$ in terms of the first order commutators of the Riesz transforms. We…
The commutators of bilinear Calderón-Zygmund operators and point-wise multiplication with a symbol in $cmo$ are bilinear compact operators on product of Lebesgue spaces. This work shows that, for certain non-degenerate Calderón-Zygmund operators, the symbol being in $cmo$ is not only sufficient but actually necessary for the compactness of the commutators.
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…
For commutators of the form [b,T] where T is any Calderon--Zygmund operator and b is any BMO function we derive weighted quadratic type estimates in term of the A1 constant of the weight both in the Lp context or of LlogL type at the endpoint
In this paper, the main aim is to consider the boundedness of commutators of multilinear Calderón-Zygmund operators with Lipschitz functions in the context of the variable exponent Lebesgue spaces. Furthermore, the variable versions of the Lipschitz spaces are also discussed, and with the help of a key tool of a pointwise estimate involving the sharp maximal operator of the multilinear fractional commutator and certain associated maximal operators.
Let $L=-Δ+V$ with non-negative potential $V$ satisfying some appropriate reverse Hölder inequality. In this paper, we study the boundedness of the commutators of some singular integrals associated to $L$ such as Riesz transforms and fractional integrals with the new BMO functions introduced in \cite{BHS1} on the weighted spaces $L^p(w)$ where $w$ belongs to the new classes of weights introduced by \cite{BHS2}.
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 prove a wide range of L^p estimates for a trilinear singular integral operator motivated by dropping one average in Calderón's second commutator. For comparison by dropping two averages in Calderón's second commutator one faces the trilinear Hilbert transform. The novelty in this paper is that in order to avoid difficulty of the level of the trilinear Hilbert transform, we choose to view the symbol of the operator as a non-standard symbol. The methods used come from time-frequency analysis but must be adapted to the fact that our symbol is non-standard.
We study the boundedness properties of commutators formed by $b$ and $T$, where $T$ is a bilinear bi-parameter singular integral satisfying natural $T1$ type conditions and $b$ is a little BMO function. For paraproduct free bilinear bi-parameter singular integrals $T$ we prove that $[b, T]_1 \colon L^p(\mathbb{R}^{n+m}) \times L^q(\mathbb{R}^{n+m}) \to L^r(\mathbb{R}^{n+m})$ in the full range $1 < p, q \le \infty$, $1/2 < r < \infty$ satisfying $1/p+1/q = 1/r$. A special case is when $T$ is a bilinear bi-parameter multiplier. We also prove the corresponding Banach range result for all singular…
Let $Ω$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$, $T_Ω$ be the homogeneous singular integral operator with kernel $\frac{Ω(x)}{|x|^d}$ and $T_{Ω,\,b}$ be the commutator of $T_Ω$ with symbol $b$. In this paper, we prove that if $Ω\in L(\log L)^2(S^{d-1})$, then for $b\in {\rm BMO}(\mathbb{R}^d)$, $T_{Ω,\,b}$ satisfies an endpoint estimate of $L\log L$ type.
In this paper we study Coifman type estimates and weighted norm inequalities for singular integral operators $T$ and its commutators, given by the convolution with a vector valued kernel $K$. We define a weaker Hörmander type condition associated with Young functions for the vector valued kernels. With this general framework we obtain as an example the result for the square operator and its commutator given in [ M. 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, Vol 33…
In this paper, a weak type (1,1) bound criterion is established for singular integral operator with rough kernel. As some applications of this criterion, we prove some important operators with rough kernel in harmonic analysis, such as Calderón commutator, higher order Calderón commutator, general Calderón commutator, Calderón commutator of Bajsanski-Coifman type and general singular integral of Muckenhoupt type, are all of weak type (1,1).
In this article, we show that multilinear fractional type operators are bounded from product Hardy spaces with variable exponents into Lebesgue spaces with variable exponents via the atomic decomposition theory. We also study continuity properties of commutators of multilinear fractional type operators on product of certain Hardy spaces with variable exponents.
Let $Ω$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{n-1}$, $T_Ω$ be the convolution singular integral operator with kernel $\frac{Ω(x)}{|x|^n}$. For $b\in{\rm BMO}(\mathbb{R}^n)$, let $T_{Ω,\,b}$ be the commutator of $T_Ω$. In this paper, by establishing suitable sparse dominations, the authors establish some weak type endpoint estimates of $L\log L$ type for $T_{Ω,\,b}$ when $Ω\in L^q(S^{n-1})$ for some $q\in (1,\,\infty]$.
Weighted mixed endpoint estimates of Fefferman-Stein type for commutators of singular integral operators
本文研究了具有 BMO 符号的 Calderón-Zygmund 算子高阶交换子及较不光滑核算子交换子的 Fefferman-Stein 型混合弱估计，推广了先前关于奇异积分算子和交换子的弱端点估计结果。
In this paper we prove two-weighted norm estimates for higher order commutator of singular integral and fractional type operators between weighted $L^p$ and certain spaces that include Lipschitz, BMO and Morrey spaces. We also give the optimal parameters involved with these results, where the optimality is understood in the sense that the parameters defining the corresponding spaces belong to certain region out of which the classes of weights are satisfied by trivial weights. We also exhibit pairs of non-trivial weights in the optimal region satisfying the conditions required.
Quantitative $A_1-A_\infty$ estimates for rough homogeneous singular integrals $T_Ω$ and commutators of $BMO$ symbols and $T_Ω$ are obtained. In particular the following estimates are proved: % \[ \|T_Ω\|_{L^p(w)}\le c_{n,p}\|Ω\|_{L^\infty} [w]_{A_1}^{\frac{1}{p}}\,[w]_{A_{\infty}}^{1+\frac{1}{p'}}\|f\|_{L^p(w)} \] % and % \[ \| [b,T_Ω]f\|_{L^{p}(w)}\leq c_{n,p}\|b\|_{BMO}\|Ω\|_{L^{\infty}} [w]_{A_1}^{\frac{1}{p}}[w]_{A_{\infty}}^{2+\frac{1}{p'}}\|f\|_{L^{p}\left(w\right)}, \] % for $1<p<\infty$ and $1/p+1/p'=1$.
Sparse domination for singular integral operators and their commutators in Dunkl setting with applications
本文在 Dunkl 框架下建立奇异积分算子及其交换子的稀疏控制，定义 Dunkl-Muckenhoupt 权，获得加权界与双权界，并证明算子在插值空间上的有界性。
We introduce the iterated commutator for the Riesz transforms in the multi-parameter flag setting, and prove the upper bound of this commutator with respect to the symbol $b$ in the flag BMO space. Our methods require the techniques of semigroups, harmonic functions and multi-parameter flag Littlewood-Paley analysis. We also introduce the big commutator in this multi-parameter flag setting and prove the upper bound with symbol $b$ in the flag little-bmo space by establishing the "exponential-logarithmic" bridge between this flag little bmo space and the Muckenhoupt $A_p$ weights with flag stru…
This paper contains three observations on commutators of Singular Integral Operators with BMO functions: 1) The subgaussian local decay for the commutator, namely \[\frac{1}{|Q|}\left|\left\{x\in Q\, : \, |[b,T](fχ_Q)(x)|>M^2f(x)t\right\}\right|\leq c e^{-\sqrt{ct\|b\|_{BMO}}} \] is sharp, that is, it is subgaussian and not better. 2) It is not possible to obtain a pointwise control of the commutator by a finite sum of sparse operators defined with $L\log L$ averages. 3) If $w\in A_p\setminus A_1$ then $\left\| wM\left(\frac{f}{w}\right)\right\|_{L^1(\mathbb{R}^n)\rightarrow L^{1,\infty}(\math…
We complete our boundedness theory of commutators of bilinear bi-parameter singular integrals by establishing the following result. If $T$ is a bilinear bi-parameter singular integral satisfying suitable $T1$ type assumptions, $\|b\|_{\operatorname{bmo}(\mathbb{R}^{n+m})} = 1$ and $1 < p, q \le \infty$ and $1/2 < r < \infty$ satisfy $1/p+1/q = 1/r$, then we have $$ \|[b, T]_1(f_1, f_2)\|_{L^r(\mathbb{R}^{n+m})} \lesssim \|f_1\|_{L^p(\mathbb{R}^{n+m})} \|f_2\|_{L^q(\mathbb{R}^{n+m})}. $$ Previously the range $r \le 1$ was proved only in the paraproduct free situation. The main novelty lies in t…
In this paper, we establish the boundedness of the commutators of multilinear Hausdorff operators on the product of some weighted Morrey-Herz type spaces with variable exponent with their symbols belong to both Lipschitz space and central BMO space. By these, we generalize and strengthen some previous known results.
In this paper, the weighted Lp boundedness of multilinear commutators and iterated commutators of multilinear singular integral operators with generalized kernels is established, where the weight is multiple weight. Our results are generalizations of the corresponding results for multilinear singular integral operators with standard kernels and Dini kernels.
In this paper, we establish sufficient conditions for a singular integral $T$ to be bounded from certain Hardy spaces $H^p_L$ to Lebesgue spaces $L^p$, $0< p \le 1$, and for the commutator of $T$ and a BMO function to be weak-type bounded on Hardy space $H_L^1$. We then show that our sufficient conditions are applicable to the following cases: (i) $T$ is the Riesz transform or a square function associated with the Laplace-Beltrami operator on a doubling Riemannian manifold, (ii) $T$ is the Riesz transform associated with the magnetic Schrödinger operator on an Euclidean space, and (iii) $T = g…
Two inequalities for commutators of singular integral operators satisfying Hörmander conditions of Young type
本文针对满足 Young 型 Hörmander 条件的奇异积分算子一般交换子，建立逐点稀疏控制并证明两类加权不等式，解耦交换子数量与 ε 指标，改进已有结果并适用于多类算子。
Let $T$ be a bilinear Calderón-Zygmund singular integral operator and $T_*$ be its corresponding truncated maximal operator. The commutators in the $i$-$th$ entry and the iterated commutators of $T_*$ are defined by $$ T_{\ast,b,1}(f,g)(x)=\sup_{δ>0}\bigg|\iint_{|x-y|+|x-z|>δ}K(x,y,z)(b(y)-b(x))f(y)g(z)dydz\bigg|, $$ $$T_{\ast,b,2}(f,g)(x)=\sup_{δ>0}\bigg|\iint_{|x-y|+|x-z|>δ}K(x,y,z)(b(z)-b(x))f(y)g(z)dydz\bigg|,$$ \begin{align*} T_{\ast,(b_1,b_2)}(f,g)(x)=\sup\limits_{δ>0}\bigg|\iint_{|x-y|+|x-z|>δ} K(x,y,z)(b_1(y)-b_1(x))(b_2(z)-b_2(x))f(y)g(z)dydz\bigg|. \end{align*} In this paper, the com…
Weighted Estimates of Singular Integrals and Commutators in the Zygmund Dilation Setting
本文通过离散Littlewood-Paley理论建立Zygmund伸缩相关奇异积分的加权估计，得到其与小bmo空间函数交换子范数上界，还证明相关奇异积分族不包含Nagel-Wainger算子且该算子具理论意义。
We study mixed weak type inequalities for the commutator $[b,T]$, where $b$ is a BMO function and $T$ is a Calderón-Zygmund operator. More precisely, we prove that for every $t>0$ \begin{equation*}%\label{tesis_teo2.2} uv(\{x\in\R^n: |\frac{[b,T](fv)(x)}{v(x)}|>t\})\leq C\int_{\R^n}ϕ(\frac{|f(x)|}{t})u(x)v(x)\,dx, \end{equation*} where $ϕ(t)=t(1+\log^{+}{t})$, $u\in A_1$ and $v\in A_{\infty}(u)$. Our technique involves the classical Calderón-Zygmund decomposition, which allow us to give a direct proof. We use this result to prove an analogous inequality for higher order commutators. We also ob…
Iterated commutators of multilinear Calderon-Zygmund operators and pointwise multiplication with functions in $BMO$ are studied in products of Lebesgue spaces. Both strong type and weak end-point estimates are obtained, including weighted results involving the vectors weights of the multilinear Calderon-Zygmund theory recently introduced in the literature. Some better than expected estimates for certain multilinear operators are presented too.
This paper has been withdrawn by the author due to crucial sign error in Lemma 8.1.
Utilising some recent ideas from our bilinear bi-parameter theory, we give an efficient proof of a two-weight Bloom type inequality for iterated commutators of linear bi-parameter singular integrals. We prove that if $T$ is a bi-parameter singular integral satisfying the assumptions of the bi-parameter representation theorem, then $$ \| [b_k,\cdots[b_2, [b_1, T]]\cdots]\|_{L^p(μ) \to L^p(λ)} \lesssim_{[μ]_{A_p}, [λ]_{A_p}} \prod_{i=1}^k\|b_i\|_{\operatorname{bmo}(ν^{θ_i})} , $$ where $p \in (1,\infty)$, $θ_i \in [0,1]$, $\sum_{i=1}^kθ_i=1$, $μ, λ\in A_p$, $ν:= μ^{1/p}λ^{-1/p}$. Here $A_p$ stan…
An $A_1-A_\infty$ estimate improving a previous result in arXiv:1607.06432 is obtained. Also new a result in terms of the ${A_\infty}$ constant and the one supremum $A_q-A_\infty^{\exp}$ constant, is proved, providing a counterpart for the result obained in arXiv:1705.08364. Both of the preceding results rely upon a sparse domination in terms of bilinear forms for $[b,T_Ω]$ with $Ω\in L^\infty(\mathbb{S}^{n-1})$ and $b\in BMO$ which is established relying upon techniques from arXiv:1705.07397.
Let $T$ be a bilinear Calderón-Zygmund singular integral operator and $T^*$ be its corresponding truncated maximal operator. For any $b\in\text{BMO}(\mathbb {R}^n)$ and $\vec{b}=(b_1,\ b_2)\in\text{BMO}(\mathbb {R}^n)\times\text {BMO}(\mathbb{R}^n)$, let $T^*_{b,j}$ (j=1,2), $T^*_{\vec{b}}\ $ be the commutators in the j-th entry and the iterated commutators of $T^*$, respectively. In this paper, for all $1<p_1,p_2<\infty$, $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}$, we show that $T^*_{b,j}$ and $T^*_{\vec{b}}$ are compact operators from $L^{p_1}(w_1)\times L^{p_2}(w_2)$ to $L^p(v_{\vec{w}})$, i…
Sparse bounds and joint oscillation for bilinear rough commutators
本文研究双线性粗糙奇异积分交换子的联合振荡条件，证明二阶混合与重复位置交换子的下界，并建立稀疏控制，得到Banach与拟Banach范围内的强、弱界。
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.
$L^p$ boundness of Oscillatory singular integral with Calderón Type Commutators
本文研究了具有Calderón型交换子的振荡奇异积分算子$T_{P,K,A}$的$L^p$有界性，通过vanishing moment和CZ(δ)条件证明其在$R^n$上的统一有界性，扩展了先前结果。
Endpoint estimates for commutators of singular integral operators associated with admissible functions
本文针对容许函数相关的奇异积分算子，在齐型空间上建立了交换子从 Hardy 空间到弱 L¹、L¹ 及 Hardy 空间的有界性，并给出充要刻画，结果适用于 Schrödinger 与 Laguerre 算子等新情形。
In this paper we establish the following estimate \[ w\left(\left\{ x\in\mathbb{R}^{n}\,:\,\left|[b,T]f(x)\right| > λ\right\} \right)\leq \frac{c_{T}}{\varepsilon^{2}}\int_{\mathbb{R}^{n}}Φ\left(\|b\|_{BMO}\frac{|f(x)|}λ\right)M_{L(\log L)^{1+\varepsilon}}w(x)dx \] where $w\geq0, \, 0<\varepsilon<1$ and $Φ(t)=t(t+\log^+(t))$. This inequality relies upon the following sharp $L^p$ estimate \[ \|[b,T]f\|_{L^{p}(w)}\leq c_{T}\left(p'\right)^{2}p^{2}\left(\frac{p-1}δ\right)^{\frac{1}{p'}} \|b\|_{BMO} \, \|f \|_{L^{p}(M_{L(\log L)^{2p-1+δ}}w)} \]where $1<p<\infty, w\geq0 \text{ and } 0<δ<1.$ As a co…
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。
Let $T$ be a Calderón-Zygmund singular integral operator. In this paper, we will show some weighted boundedness properties of commutator $[b,T]$ on the weighted Morrey spaces $L^{p,κ}(w)$ under appropriate conditions on the weight $w$, where the symbol $b$ belongs to weighted $BMO$ or Lipschitz space or weighted Lipschitz space.
Sharp weighted weak-type bounds for commutators of singular integrals
本文建立了Calderón--Zygmund算子与BMO符号交换子的尖锐加权弱型界，并证明了对数符号情形的最优指数依赖，方法涉及点态控制和奇异积分算子的一般结果。
In this paper, the following iterated commutators $T_{*,Πb}$ of maximal operator for multilinear singular integral operators and $I_{α, Πb}$ of multilinear fractional integral operator are introduced and studied $$\aligned T_{*,Πb}(\vec{f})(x)&=\sup_{δ>0}\bigg|[b_1,[b_2,...[b_{m-1},[b_m,T_δ]_m]_{m-1}...]_2]_1 (\vec{f})(x)\bigg|,$$ $$\aligned I_{α, Πb}(\vec{f})(x)&=[b_1,[b_2,...[b_{m-1},[b_m,I_α]_m]_{m-1}...]_2]_1 (\vec{f})(x),$$ where $T_δ$ are the smooth truncations of the multilinear singular integral operators and $I_α$ is the multilinear fractional integral operator, $b_i\in BMO$ for $i=1,…
On entangled and multi-parameter commutators
该研究补充了Zygmund膨胀不变奇异积分理论，证明交换子[b,T]从L^p到L^q有界性和紧性的充要条件，澄清下界混淆并给出p≤q时的完整刻画，发现紧性迫使b为常数的意外结果。
Zygmund dilations: bilinear analysis and commutator estimates
本文研究了 $\mathbb{R}^3$ 中 Zygmund 膨胀下的双线性理论与交换子估计，构建了双线性二进多分辨率方法，证明了无抛积双线性 $T1$ 定理，并在不依赖加权估计的情况下建立了线性交换子估计。
Characterization of Lipschitz Spaces via Commutators of Fractional Maximal Function on the $p$-Adic Variable Exponent Lebesgue Spaces
本文在 $p$-adic 变指数 Lebesgue 空间上，通过分数次极大算子交换子的有界性刻画了 Lipschitz 空间，并给出了 Lipschitz 范数与变指数 Lebesgue 范数的等价关系。
Convex body domination for rough singular integrals
该研究将粗糙奇异积分的稀疏控制结果推广至凸体情形，方法适用于角部无界的齐次粗糙奇异积分，且凸体控制可推导新的双权交换子界。
Boundedness of a class of multilinear operators and their iterated commutators on Morrey-Banach function spaces
本文在广义Morrey-Banach函数空间框架下，针对含多线性ω-Calderón-Zygmund算子等的算子类，通过逐点稀疏控制假设或Wᵣ性质建立其及交换子的有界性，并将迭代交换子有界性推广到Morrey-Lorentz空间和变指数Morrey空间。
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.
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…
Weak Endpoint Estimate for Commutators of Rough Maximal Singular Integral operators
本文证明粗糙极大奇异积分交换子在BMO乘子与L(logL)^2核条件下满足弱型端点估计，通过Calderón-Zygmund分解、二进线性化、核分解及解析族插值实现。
On extension of Calderón-Zygmund type singular integrals and their commutators
本文研究了Calderón-Zygmund型奇异积分及其交换子的延拓，建立了在Lipschitz、Hardy和加权空间上的一致估计，并通过取极限推导出经典情形的类似估计。
Bounds for $p$-adic Hardy-type Operators and Commutator On $p$-adic Variable Herz-Morrey Spaces
本文研究分数p-adic Hardy算子在变指数p-adic Herz-Morrey空间中的有界性，并进一步估计了其与p-adic变指数CBMO函数生成的交换子算子的有界性。
Let $L= -Δ+ V$ be a Schrödinger operator on $\mathbb R^d$, $d\geq 3$, where $V$ is a nonnegative potential, $V\ne 0$, and belongs to the reverse Hölder class $RH_{d/2}$. In this paper, we study the commutators $[b,T]$ for $T$ in a class $\mathcal K_L$ of sublinear operators containing the fundamental operators in harmonic analysis related to $L$. More precisely, when $T\in \mathcal K_L$, we prove that there exists a bounded subbilinear operator $\mathfrak R= \mathfrak R_T: H^1_L(\mathbb R^d)\times BMO(\mathbb R^d)\to L^1(\mathbb R^d)$ such that $|T(\mathfrak S(f,b))|- \mathfrak R(f,b)\leq |[b,…
Schatten Properties of Calderón--Zygmund Singular Integral Commutator on stratified Lie groups
本文完整刻画了分层李群上Calderón--Zygmund奇异积分交换子的Schatten性质，证明了在临界指标处范数等价性，并提出了可推广的新方法。
A weak type $(1,1)$ estimate is established for the first order $d$-commutator introduced by Christ and Journé, in dimension $d\ge 2$.
Flag multiresolutions: T1, paraproducts and commutators
本文发展二进-概率多分辨率方法，证明双参数旗奇异积分完整T1表示定理，并推广至多参数旗设置，蕴含旗A_p加权估计及交换子估计。
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