Sergei Treil
In this paper we investigate the relations between (martingale) BMO spaces, paraproducts and commutators in non-homogeneous martingale settings. Some new, and one might add unexpected, results are obtained. Some alternative proof of known results are also presented.
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We address $L^p(μ)\rightarrow L^p(λ)$ bounds for paraproducts in the Bloom setting. We introduce certain "sparse BMO" functions associated with sparse collections with no infinitely increasing chains, and use these to express sparse operators as sums of paraproducts and martingale transforms -- essentially, as Haar multipliers -- as well as to obtain an equivalence of norms between sparse operators $\mathcal{A}_\mathcal{S}$ and compositions of paraproducts $Π^*_aΠ_b$.
We introduce multilinear analogues of dyadic paraproduct operators and Haar Multipliers, and study boundedness properties of these operators and their commutators. We also characterize dyadic BMO functions via the boundedness of certain paraproducts and also via the boundedness of the commutators of multilinear Haar Multipliers and paraproduct operators.
We study boundedness properties of a class of multiparameter paraproducts on the dual space of the dyadic Hardy space H_d^1(T^N), the dyadic product BMO space BMO_d(T^N). For this, we introduce a notion of logarithmic mean oscillation on the polydisc. We also obtain a result on the boundedness of iterated commutators on BMO([0,1]^2).
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…
Let $B$ be a locally integrable matrix function, $W$ a matrix A${}_p$ weight with $1 < p < \infty$, and $T$ be any of the Riesz transforms. We will characterize the boundedness of the commutator $[T, B]$ on $L^p(W)$ in terms of the membership of $B$ in a natural matrix weighted BMO space. To do this, we will characterize the boundedness of dyadic paraproducts on $L^p(W)$ via a new matrix weighted Carleson embedding theorem. Finally, we will use some of the ideas from these proofs to (among other things) obtain quantitative weighted norm inequalities for these operators and also use them to pro…
In this paper, we prove that the product (in the distribution sense) of two functions, which are respectively in $ \BMO(\bR^n)$ and $\H^1(\bR^n)$, may be written as the sum of two continuous bilinear operators, one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into $L^1(\bR^n)$, the other one from $\H^1(\bR^n)\times \BMO(\bR^n) $ into a new kind of Hardy-Orlicz space denoted by $\H^{\log}(\bR^n)$. More precisely, the space $\H^{\log}(\bR^n)$ is the set of distributions $f$ whose grand maximal function $\mathcal Mf$ satisfies $$\int_{\mathbb R^n} \frac {|\mathcal M f(x)|}{\log(e+|x|) +\log (e+ |\mathca…
We introduce another notion of bounded logarithmic mean oscillation in the N-torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little BMO of Cotlar-Sadosky to the product BMO of Chang-Fefferman. We also obtain a sufficient condition for the boundedness of iterated commutators with Hilbert transforms betweeen the strong notions of these two spaces.
In this paper, we study the product BMO space, little bmo space and their connections with the corresponding commutators associated with Bessel operators studied by Weinstein, Huber, and by Muckenhoupt-Stein. We first prove that the product BMO space in the Bessel setting can be used to prove the boundedness of the iterated commutators with the Bessel Riesz transforms. We next study the little $\rm bmo$ space in this Bessel setting and obtain the equivalent characterization of this space in terms of commutators. We further show that in analogy with the classical setting, the little $\rm bmo$ s…
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…
Characterization of BMO spaces via commutators of some maximal operators on the slice spaces
本文证明了切片空间中（分数）极大函数与尖锐极大函数交换子的有界性，其中交换子符号属于BMO空间，从而给出了BMO空间的新刻画。
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。
We consider iterated commutators of multiplication by a symbol function and tensor products of Hilbert or Riesz transforms. We establish mixed BMO classes of symbols that characterize boundedness of these objects in $L^p$. Little BMO and product BMO, big Hankel operators and iterated commutators are the base cases of our results. We use operator theoretical methods and existing profound results on iterated commutators for the Hilbert transform case, while the general result in several variables is obtained through the construction of a Journé operator that models the behavior of the multiple H…
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.
In this paper, starting with a relatively simple observation that the variational estimates of the commutators of the standard Calderón-Zygmund operators with the BMO functions can be deduced from the weighted variational estimates of the standard Calderón-Zygmund operators themselves, we establish similar variational estimates for the commutators of the BMO functions with rough singular integrals which do not admit any weighted variational estimates. The proof involves many Littlewood-Paley type inequalities with commutators as well as Bony decomposition and related para-product estimates.
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
Paraproducts on local dyadic fractional Sobolev spaces
研究 dyadic paraproducts 在局部 dyadic 分数 Sobolev 空间上的有界性和紧性，提出新的 dyadic 分数 BMO 和 CMO 条件，并使用新的 dyadic 分数 Carleson 嵌入定理证明。
In this paper, we show that dyadic paraproducts $π_b$ with $b$ in dyadic BMO are bounded on matrix weighted $L^p(W)$ if $W$ is a matrix $\text{A}_p$ weight.
In this article, we investigate the boundedness properties of the multilinear dyadic paraproduct operators in the weighted setting. We also obtain weighted estimates for the multilinear Haar multipliers and their commutators with dyadic BMO functions.
Let $X$ be a space of homogeneous type and let $L$ be a sectorial operator with bounded holomorphic functional calculus on $L^2(X)$. We assume that the semigroup $\{e^{-tL}\}_{t>0}$ satisfies Davies-Gaffney estimates. In this paper, we introduce a new type of paraproduct operators that is constructed via certain approximations of the identity associated to $L$. We show various boundedness properties on $L^p(X)$ and the recently developed Hardy and BMO spaces $H^p_L(X)$ and $BMO_L(X)$. In generalization of standard paraproducts constructed via convolution operators, we show $L^2(X)$ off-diagona…
This paper is dedicated to study weighted $L^p$ inequalities for pseudo-differential operators with amplitudes and their commutators by using the new class of weights $A_p^\vc$ and the new BMO function space BMO$_\vc$ which are larger than the Muckenhoupt class of weights $A_p$ and classical BMO space BMO, respectively. The obtained results therefore improve substantially some well-known results.
We present a unified method to obtain weighted estimates of linear and multilinear commutators with BMO functions, that is amenable to a plethora of operators and functional settings. Our approach elaborates on a commonly used Cauchy integral trick, recovering many known results but yielding also numerous new ones.
In this paper we characterize the two matrix weighted boundedness of commutators with any of the Riesz transforms (when both are matrix A${}_p$ weights) in terms of a natural two matrix weighted BMO space. Furthermore, we identify this BMO space when $p = 2$ as the dual of a natural two matrix weighted H${}^1$ space, and use our commutator result to provide a converse to Bloom's matrix A${}_2$ theorem, which as a very special case proves Buckley's summation condition for matrix A${}_2$ weights. Finally, we use our results to prove a matrix weighted John-Nirenberg inequality, and we also briefl…
We show that the product BMO space can be characterized by iterated commutators of a large class of Calderón-Zygmund operators. This result follows from a new proof of boundedness of iterated commutators in terms of the BMO norm of their symbol functions, using Hytönen's representation theorem of Calderón-Zygmund operators as averages of dyadic shifts. The proof introduces some new paraproducts which have BMO estimates.
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…
Weighted $L^p\to L^q$-boundedness of commutators and paraproducts in the Bloom setting
本文在 Bloom 背景下，针对缺失指数范围 $1<q<p<\infty$，通过引入新的具有抵消性的条件，补充刻画了 Calderón--Zygmund 算子交换子与二进仿积的加权 $L^p\to L^q$ 有界性，从而完成了对所有指数的有界性刻画。
In this paper, we introduce a new weighted Hardy-Cesàro operator defined by $U_{ψ,s}f(x)=\int\limits_0^1 f(s(t)\cdot x) ψ(t)dt$, which is associated to the parameter curve $s(t,x)=s(t)x$. Under certain conditions on $s(t)$ and on an absolutely homogeneous weight function $ω$, we characterize the weight function $ψ$ such that $U_{ψ,s}$ is bounded on $L^p(ω)$, $BMO(ω)$. The corresponding operator norms are worked out too. These results extend the ones of Jie Xiao \cite{xiao}. We also give a sufficient and a necessary condition on the weight function $ψ$, which ensure the boundedness of the commu…
Commutators of Hilbert transforms along monomial curves
本文研究沿单项式曲线的 Hilbert 变换与符号函数的交换子，利用 Cauchy 积分技巧建立非各向同性 BMO 上界，并通过测试 BMO 空间给出下界及包含关系。
We characterize the boundedness of the commutators $[b, T]$ with biparameter Journé operators $T$ in the two-weight, Bloom-type setting, and express the norms of these commutators in terms of a weighted little $bmo$ norm of the symbol $b$. Specifically, if $μ$ and $λ$ are biparameter $A_p$ weights, $ν:= μ^{1/p}λ^{-1/p}$ is the Bloom weight, and $b$ is in $bmo(ν)$, then we prove a lower bound and testing condition $\|b\|_{bmo(ν)} \lesssim \sup \| [b, R_k^1 R_l^2]: L^p(μ) \rightarrow L^p(λ)\|$, where $R_k^1$ and $R_l^2$ are Riesz transforms acting in each variable. Further, we prove that for suc…
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。
Ó. Blasco and S. Pott showed that the supremum of operator norms over $L^2$ of all bicommutators (with the same symbol) of one-parameter Haar multipliers dominates the biparameter dyadic product BMO norm of the symbol itself. In the present work we extend this result to the Bloom setting, and to any exponent $1<p<\infty$. The main tool is a new characterization in terms of paraproducts and two-weight John--Nirenberg inequalities for dyadic product BMO in the Bloom setting. We also extend our results to the whole scale of indexed spaces between little bmo and product BMO in the general multipar…
This article is devoted to a study of the Hardy space $H^{\log} (\mathbb{R}^d)$ introduced by Bonami, Grellier, and Ky. We present an alternative approach to their result relating the product of a function in the real Hardy space $H^1$ and a function in $BMO$ to distributions that belong to $H^{\log}$ based on dyadic paraproducts. We also point out analogues of classical results of Hardy-Littlewood, Zygmund, and Stein for $H^{\log}$ and related Musielak-Orlicz spaces.
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…
We establish the equivalent characterisation of the weighted BMO space on the complex plane $\mathbb{C}$ via the two weight commutator of the Beurling--Ahlfors operator with a BMO function. Our method of proofs relies on the explicit kernel of the Beurling--Ahlfors operator and the properties of Muckenhoupt weight class.
Characterizations of a class of Musielak--Orlicz BMO spaces via commutators of Riesz potential operators
本文研究了通过Riesz势算子的交换子将Musielak-Orlicz Hardy空间映射到Musielak-Orlicz空间的条件，提出了新的BMO子空间定义并验证了其有界性。
In this article we extend recent results by the first author on the necessity of $BMO$ for the boundedness of commutators on the classical Lebesgue spaces. We generalize these results to a large class of Banach function spaces. We show that with modest assumptions on the underlying spaces and on the operator $T$, if the commutator $[b,T]$ is bounded, then the function $b$ is in $BMO$.
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…
In this paper, we provide a direct and constructive proof of weak factorization of $h^1(\mathbb{R})$ (the predual of little BMO space bmo$(\mathbb{R}\times\mathbb{R})$ studied by Cotlar-Sadosky and Ferguson-Sadosky), i.e., for every $f\in h^1(\mathbb{R}\times\mathbb{R})$ there exist sequences $\{α_j^k\}\in\ell^1$ and functions $g_j^k,h^k_j\in L^2(\mathbb{R}^2)$ such that \begin{align*} f=\sum_{k=1}^\infty\sum_{j=1}^\inftyα^k_j\Big(\, h^k_j H_1H_2 g^k_j - g^k_j H_1H_2 h^k_j\Big) \end{align*} in the sense of $h^1(\mathbb{R})$, where $H_1$ and $H_2$ are the Hilbert transforms on the first and sec…
BMO-Type characterizations and endpoint estimates for commutators of intrinsic square functions on Orlicz-Hardy spaces
本文研究内蕴平方函数交换子，证明其有界性等价于BMO型空间成员资格，并将加权结果推广至Musielak-Orlicz与Orlicz-Hardy空间，给出端点估计。
We provide an explicit technical framework for proving very general two-weight commutator estimates in arbitrary parameters. The aim is to both clarify existing literature, which often explicitly focuses on two parameters only, and to extend very recent results to the full generality of arbitrary parameters. More specifically, we study two-weight commutator estimates -- Bloom type estimates -- in the multi-parameter setting involving weighted product BMO and little BMO spaces, and their combinations.
Fix $λ>0$. Consider the Hardy space $H^1(\mathbb{R}_+,dm_λ)$ in the sense of Coifman and Weiss, where $\mathbb{R_+}:=(0,\infty)$ and $dm_λ:=x^{2λ}dx$ with $dx$ the Lebesgue measure. Also consider the Bessel operators $Δ_λ:=-\frac{d^2}{dx^2}-\frac{2λ}{x} \frac d{dx}$, and $S_λ:=-\frac{d^2}{dx^2}+\frac{λ^2-λ}{x^2}$ on $\mathbb{R_+}$. The Hardy spaces $H^1_{Δ_λ}$ and $H^1_{S_λ}$ associated with $Δ_λ$ and $S_λ$ are defined via the Riesz transforms $R_{Δ_λ}:=\partial_x (Δ_λ)^{-1/2}$ and $R_{S_λ}:= x^λ\partial_x x^{-λ} (S_λ)^{-1/2}$, respectively. It is known that $H^1_{Δ_λ}$ and $H^1(\mathbb{R}_+,d…
Characterization of boundedness of some commutators of fractional maximal functions in terms of $p$-adic vector spaces
本文在p-adic域上刻画了分数次极大算子交换子在变指标Lebesgue空间和Morrey空间中的有界性，并据此得到BMO函数的新刻画及BMO范数与相关范数的等价关系。
Optimal Sparse Bounds and Commutator Characterizations Without Doubling
本文在非加倍测度的非齐次背景下研究二进仿积与交换子，建立了BMO符号下二进仿积的逐点稀疏控制，改进了已有结果，并刻画了二进Hilbert变换交换子的有界符号类及其嵌套性质。
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…
Multilinear paraproducts on Sobolev spaces
该研究利用符号的Triebel-Lizorkin型范数刻画多线性仿积的Sobolev空间有界性，结合小波表示定理导出多线性Calderón-Zygmund算子的新Sobolev空间T(1)型定理，完善了相关算子的空间估计理论。
This paper obtains new characterizations of weighted Hardy spaces and certain weighted $BMO$ type spaces via the boundedness of variation operators associated with approximate identities and their commutators, respectively.
Let $\bfT$ is a certain tensor product of simple dyadic shifts defined below. We prove here that for dyadic bi-parameter commutator the following equivalence holds $ \|\bfT b-b \bfT \| \asymp \|b\|_{bmo^d}$. This result is well-known for many types of bi-parameter commutators, see \cite{FS}, \cite{DLWY} and \cite{DPSK} for more details.
In this paper, the boundedness properties of commutators generated by $b$ and intrinsic square functions in the endpoint case are discussed, where $b\in BMO(\mathbb R^n)$. We first establish the weighted weak $L\log L$-type estimates for these commutator operators. Furthermore, we will prove endpoint estimates of commutators generated by $BMO(\mathbb R^n)$functions and intrinsic square functions in the weighted Morrey spaces $L^{1,κ}(w)$ for $0<κ<1$ and $w\in A_1$, and in the generalized Morrey spaces $L^{1,Θ}$, where $Θ$ is a growth function on $(0,+\infty)$ satisfying the doubling condition.
The fractional Riesz transform and their commutator in Dunkl setting
本文研究 Dunkl 设置下分数阶 Riesz 变换的有界性，并给出其交换子在中心 BMO 与 Dunkl BMO 空间有界性的充要条件，进而用 VMO 空间刻画交换子的紧性。
Given a Calderón-Zygmund operator $T$, a classic result of Coifman-Rochberg-Weiss relates the norm of the commutator $[b, T]$ with the BMO norm of $b$. We focus on a weighted version of this result, obtained by Bloom and later generalized by Lacey and the authors, which relates $\| [b, T] : L^p(\mathbb{R}^n; μ) \to L^p(\mathbb{R}^n; λ) \|$ to the norm of $b$ in a certain weighted BMO space determined by $A_p$ weights $μ$ and $λ$. We extend this result to higher iterates of the commutator and recover a one-weight result of Chung-Pereyra-Perez in the process.
Let $0<α<n$ and $M_α$ be the fractional maximal function. The nonlinear commutator of $M_α$ and a locally integrable function $b$ is given by $[b,M_α](f)=bM_α(f)-M_α(bf)$. In this paper, we mainly give some necessary and sufficient conditions for the boundedness of $[b,M_α]$ on variable Lebesgue spaces when $b$ belongs to Lipschitz or $BMO(\rn)$ spaces, by which some new characterizations for certain subclasses of Lipschitz and $BMO(\rn)$ spaces are obtained.
In this paper, we study commutator of generalized Hausdorff operator on function spaces. We mainly discuss the continuity criteria for such commutator operator when the symbol functions are either from central-$BMO$ or Lipschitz class of functions.
Commutators of Fractional Integrals with $\operatorname{BMO}^β$ Functions
本文研究了分数积分算子与$\operatorname{BMO}^β$函数的交换子，证明了其在容积Lebesgue空间上的有界性，并获得了端点估计。
Let $dγ(x)\equivπ^{-n/2}e^{-|x|^2}dx$ for all $x\in{\mathbb R}^n$ be the Gauss measure on ${\mathbb R}^n$. In this paper, the authors establish the characterizations of the space BMO$(γ)$ of Mauceri and Meda via commutators of either local fractional integral operators or local fractional maximal operators. To this end, the authors first prove that such a local fractional integral operator of order $β$ is bounded from $L^p(γ)$ to $L^{p/(1-pβ)}(γ)$, or from the Hardy space $H^1(γ)$ of Mauceri and Meda to $L^{1/(1-β)}(γ)$ or from $L^{1/β}(γ)$ to BMO$(γ)$, where $β\in(0, 1)$ and $p\in(1, 1/β)$.
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 $1/p+1/p'=1$.
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 […
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…
This paper provides a deeper study of the Hardy and $\rm BMO$ spaces associated to the Neumann Laplacian $Δ_N$. For the Hardy space $H^1_{Δ_N}(\mathbb{R}^n)$ (which is a proper subspace of the classical Hardy space $H^1(\mathbb{R}^n)$) we demonstrate that the space has equivalent norms in terms of Riesz transforms, maximal functions, atomic decompositions, and weak factorizations. While for the space ${\rm BMO}_{Δ_N}(\mathbb{R}^n)$ (which contains the classical $\rm BMO(\mathbb{R}^n)$) we prove that it can be characterized in terms of the action of the Riesz transforms associated to the Neuman…
Let b be a function on the plane. Let H_j, j=1,2, be the Hilbert transform acting on the j-th coordinate on the plane. We show that the operator norm of the double commutator [[ M_b, H_1], H_2] is equivalent to the Chang-Fefferman BMO norm of b. Here, M_b denotes the operator which is multiplication by b. This result extends a well known theorem of Nehari on weak factorization in the Hardy space H^1 to the same theorem on H^1 of a product domain. The product setting is more delicate because of the presence of a two parameter family of dilations. The method of proof depends upon (a) A dyadic de…
In this paper we investigate weighted norm inequalities for the commutator of a fractional integral operator and multiplication by a function. In particular, we show that, for $μ,λ\in A_{p,q}$ and $α/n+1/q=1/p$, the norm $\| [b,I_α]:L^p(μ^p)\to L^q(λ^q) \|$ is equivalent to the norm of $b$ in the weighted BMO space $BMO(ν)$, where $ν=μλ^{-1}$. This work extends some of the results on this topic existing in the literature, and continues a line of investigation which was initiated by Bloom in 1985 and was recently developed further by the first author, Lacey, and Wick.
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.
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