Komla Domelevo, Spyridon Kakaroumpas, Stefanie Petermichl, Odí Soler i Gibert
We characterize dyadic little BMO via the boundedness of the tensor commutator with a single well chosen dyadic shift. It is shown that several proof strategies work for this problem, both in the unweighted case as well as with Bloom weights. Moreover, we address the flexibility of one of our methods.
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Let T be the unite circle on $R^2$. Denote by BMO(T) the classical BMO space and denote by BMO_D(T) the usual dyadic BMO space on T. We prove that, BMO(T) is the intersction of BMO_D(T) and a translate of BMO_D(T).
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.
We introduce the space of dyadic bounded mean oscillation functions $f$ defined on $[0,1]^n$ and study the behavior of the nonincreasing rearrangement of $f$, as an element of the space $BMO((0,1])$. We also study the analogous class of functions that satisfy the dyadic Gurov-Reshetnyak condition and look upon their integrability properties.
The usual one third trick allows to reduce problems involving general cubes to a countable family. Moreover, this covering lemma uses only dyadic cubes, which allows to use nice martingale properties in harmonic analysis problems. We consider alternatives to this technique in spaces equipped with nonhomogeneous measures. This entails additional difficulties which forces us to consider martingale filtrations that are not regular. The dyadic covering that we find can be used to clarify the relationship between martingale BMO spaces and the most natural BMO space in this setting, which is the spa…
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.
A Bellman function approach to Fefferman's $H^1-BMO$ duality theorem is presented. One Bellman-type argument is used to handle two different one-dimensional cases, dyadic and continuous. An explicit estimate for the constant of embedding $BMO\subset (H^1)^*$ is given in the dyadic case. The same Bellman function is then used to establish a multi-dimensional analog.
We develop technical tools that enable the use of Bellman functions for BMO defined on $α$-trees, which are structures that generalize dyadic lattices. As applications, we prove the integral John--Nirenberg inequality and an inequality relating $L^1$- and $L^2$-oscillations for BMO on $α$-trees, with explicit constants. When the tree in question is the collection of all dyadic cubes in $\mathbb{R}^n,$ the inequalities proved are sharp. We also reformulate the John--Nirenberg inequality for the continuous BMO in terms of special martingales generated by BMO functions. The tools presented can be…
We obtain the explicit upper Bellman function for the natural dyadic maximal operator acting from ${\rm BMO}(\mathbb{R}^n)$ into ${\rm BLO}(\mathbb{R}^n).$ As a consequence, we show that the ${\rm BMO}\to{\rm BLO}$ norm of the natural operator equals 1 for all $n,$ and so does the norm of the classical dyadic maximal operator. The main result is a partial corollary of a theorem for the so-called $α$-trees, which generalize dyadic lattices. The Bellman function in this setting exhibits an interesting quasi-periodic structure depending on $α,$ but also allows a majorant independent of $α,$ hence…
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.
We describe the spaces $H^1(R)$ and BMO$(R)$ in terms of their closely related, simpler dyadic and two-sided counterparts. As a result of these characterizations we establish when a bounded linear operator defined on dyadic or two-sided $H^1(R)$ into a Banach space has a continuous extension to $H^1(R)$ and when a bounded linear operator that maps a Banach space into dyadic or two-sided BMO$(R)$ actually maps continuously into BMO$(R)$.
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.
Consider a tensor product of simple dyadic shifts defined below. We prove here that for dyadic bi-parameter repeated commutator its norm can be estimated from below by Chang-Fefferman $BMO$ norm pertinent to its symbol. See Theorems in Section 8 at the end of this article. But this is done below under an extra assumption on the Haar--Fourier side of the symbol. In Section 7 we carefully analyze what goes wrong in the absence of this extra assumption. At the end of this note we also list a counterexample to the existing proof of characterization of bi-parameter repeated commutator with the Hilb…
In this paper we give a simple proof of the fact that the average over all dyadic lattices of the dyadic $H^1$-norm of a function gives an equivalent $H^1$-norm. The proof we present works for both one-parameter and multi-parameter Hardy spaces. The results of such type are known. The first result (for one-parameter Hardy spces) belongs to Burgess Davis (1980). Also, by duality, such results are equivalent to the "BMO from dyadic BMO" statements proved by Garnett-Jones(1982} for one parameter case, and by Pipher-Ward (2008) for two-parameter case. While the paper generalizes these results to t…
We extend the definitions of dyadic paraproduct and t-Haar multipliers to dyadic operators that depend on the complexity (m,n), for m and n positive integers. We will use the ideas developed by Nazarov and Volberg to prove that the weighted L^2(w)-norm of a paraproduct with complexity (m,n) associated to a function b\in BMO, depends linearly on the A_2-characteristic of the weight w, linearly on the BMO-norm of b, and polynomially in the complexity. This argument provides a new proof of the linear bound for the dyadic paraproduct (the one with complexity (0,0)). Also we prove that the L^2-norm…
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).
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…
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。
Matrix-weighted little BMO spaces in two parameters
本文建立了双参数双矩阵加权小BMO理论，证明其等价于在各变量中一致属于单参数BMO，并据此推导了Journé算子交换子范数的上下界。
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.
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.
From little BMO to product BMO through multiplication operator
该研究刻画了多圆盘上Cotlar-Sadosky小BMO空间到Chang-Fefferman乘积BMO空间的乘法算子特征，丰富了多圆盘调和分析中BMO空间乘子理论。
Haar bases on quasi-metric measure spaces, and dyadic structure theorems for function spaces on product spaces of homogeneous type
本文研究了在几何上双重的准度量空间中，利用二进制立方体系统显式构造Haar函数，并证明这些Haar函数构成L^p基。接着，针对同质类型空间X，利用这些Haar函数定义离散平方函数，从而定义H^1(X)和BMO(X)的二进制版本。在同质类型乘积空间X~上，证明BMO(X~)可以表示为有限多个二进制BMO空间的交集，同样适用于A_p(X~)、反Hölder权和双重权。此外，证明Hardy空间H^1(X~)是有限多个二进制Hardy空间之和，且X~上的强极大函数与有限多个二进制强极大函数点wise可比较。这些二进制结构定理将Mei和Li、Pipher和Ward的欧几里得类似结果推广到同质类型乘积空间。
Hardy-Littlewood maximal, generalized Bessel-Riesz and generalized fractional integral operators in generalized Morrey and $BMO_ϕ$ spaces associated with Dunkl operator on the real line
本文研究了与Dunkl算子相关的广义Morrey和$BMO_ϕ$空间中Hardy-Littlewood极大函数、Bessel-Riesz算子和广义分数积分算子的有界性问题。
Boundedness of multilinear Littlewood--Paley operators with convolution type kernels on products of BMO spaces
该研究证明了卷积型多线性 Littlewood–Paley 算子在 BMO 空间乘积上的存在性、几乎处处有限性及到 BLO 空间的有界性，并讨论了非卷积型核情形。
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。
We consider the dyadic paraproducts $π_\f$ on $\T$ associated with an $\M$-valued function $\f.$ Here $\T$ is the unit circle and $\M$ is a tracial von Neumann algebra. We prove that their boundedness on $L^p(\T,L^p(\M))$ for some $1<p<\infty $ implies their boundedness on $L^p(\T,L^p(\M))$ for all $1<p<\infty$ provided $\f$ is in an operator-valued BMO space. We also consider a modified version of dyadic paraproducts and their boundedness on $L^p(\T,L^p(\M)).
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…
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…
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.
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…
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…
Remark on dimension-free estimates for discrete maximal functions over $\ell^q$ balls: small dyadic scales
本文证明小二进半径下 ℓ^q 球离散 Hardy–Littlewood 极大算子在 ℓ^p(Z^d) 上的无维数界，并与已有结果结合得到 q≥2 时二进情形的完整无维数估计。
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 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.
We give a simple proof of L^p boundedness of iterated commutators of Riesz transforms and a product BMO function. We use a representation of the Riesz transforms by means of simple dyadic operators - dyadic shifts - which in turn reduces the estimate quickly to paraproduct estimates.
We give a leisurely proof of a result of Ferguson--Lacey (math.CA/0104144) and Lacey--Terwelleger (math.CA/0601192) on a Nehari theorem for "little" Hankel operators on a polydisk. If H_b is a little Hankel operator with symbol b on product Hardy space we have || H_b || \simeq || b ||_{BMO} where BMO is the product BMO space identified by Chang and Fefferman. This article begins with the classical Nehari theorem, and presents the necessary background for the proof of the extension above. The proof of the extension is an induction on parameters, with a bootstrapping argument. Some of the more t…
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.
Weighted BMO-BLO estimates for Littlewood--Paley square operators
本文研究A₁权下Littlewood-Paley平方算子的加权估计，引入加权BLO空间，证明算子从BMO(ω)到BLO(ω)有界，给出对应John-Nirenberg不等式与BLO(ω)等价刻画。
In the two-parameter setting, we say a function belongs to the mean little $BMO$, if its mean over any interval and with respect to any of the two variables has uniformly bounded mean oscillation. This space has been recently introduced by S. Pott and the author in relation with the multiplier algebra of the product $BMO$ of Chang-Fefferman. We prove that the Cotlar-Sadosky space of functions of bounded mean oscillation $bmo(\mathbb{T}^N)$ is a strict subspace of the mean little $BMO$.
Dimension-free estimates on $l^2 (\mathbb{Z} ^d)$ for discrete dyadic maximal function over $l^1$ balls: small scales
本文针对$\mathbb{Z}^d$上$l^1$球的离散Hardy-Littlewood算子，在二进小半径条件下，给出了$l^p(\mathbb{Z}^d)$（$p \in [2, \infty]$）的无维数界估计。
Quantitative BMO-BLO Estimates for the Hardy-Littlewood Maximal Function
本文研究了Hardy-Littlewood极大函数在BMO-BLO空间中的量化估计，扩展了经典结果并给出了A_∞类的新刻画。
In this paper we establish a T1 criterion for the boundedness of Hermite-Calderon-Zygmund operators on the BMO_H(R^n) space naturally associated to the Hermite operator H. We apply this criterion in a systematic way to prove the boundedness on BMO_H(R^n) of certain harmonic analysis operators related to H (Riesz transforms, maximal operators, Littlewood-Paley g-functions and variation operators).
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.
Vector-valued estimates for shifted operators
该研究针对移位算子估计的对数增长最优性未完全解决的问题，给出广义Peetre极大函数的精确向量值移位极大不等式，以新方式推导出多类移位算子带最优对数增长的改进估计，还得到移位二进Hardy-Littlewood极大算子的向量值极大不等式。
Commutator estimates for Haar shifts with general measures
本文研究一般非倍测度下Haar位移算子与BMO函数的交换子，证明其L^p有界性并建立加权不等式及适配权类的John-Nirenberg不等式，为非齐次情形下的算子理论提供新工具。
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。
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…
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。
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 establish the product Hardy spaces associated with the Bessel Schrödinger operator introduced by Muckenhoupt and Stein, and provide equivalent characterizations in terms of the Bessel Riesz transforms, non-tangential and radial maximal functions, and Littlewood--Paley theory, which are consistent with the classical product Hardy space theory developed by Chang and Fefferman. Moreover, in this specific setting, we also provide another characterization via the Telyakovskií transform, which further implies that the product Hardy space associated with this Bessel Schrödinger opera…
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. Associated to $L$ are certain approximations of the identity. We call an operator $T$ a non-integral operator if compositions involving $T$ and these approximations satisfy certain weighted norm estimates. The Davies-Gaffney and the weighted norm estimates are together a substitute for the usual kernel estimates on $T$ in Calderón-Zygmund theory. In this paper, we show, under the…
Let $f:=(f_n)_{n\in \mathbb{Z}_+}$ and $g:=(g_n)_{n\in \mathbb{Z}_+}$ be two martingales related to the probability space $(Ω,\mathcal F,\mathbb P)$ equipped with the filtration $(\mathcal F_n)_{n\in \mathbb{Z}_+}.$ Assume that $f$ is in the martingale Hardy space $H_1$ and $g$ is in its dual space, namely the martingale $\rm BMO.$ Then the semi-martingale $f\cdot g:=(f_ng_n)_{n\in \mathbb{Z}_+}$ may be written as the sum $$f\cdot g=G(f, g)+L( f,g).$$ Here $L( f,g):=(L( f,g)_n)_{n\in\mathbb{Z}_+}$ with $L( f,g)_n:=\sum_{k=0}^n(f_k-f_{k-1})(g_k-g_{k-1)})$ for any $n\in\mathbb{Z}_+$, where $f_{-…
We develop a wide general theory of bilinear bi-parameter singular integrals $T$. First, we prove a dyadic representation theorem starting from $T1$ assumptions and apply it to show many estimates, including $L^p \times L^q \to L^r$ estimates in the full natural range together with weighted estimates and mixed-norm estimates. Second, we develop commutator decompositions and show estimates in the full range for commutators and iterated commutators, like $[b_1,T]_1$ and $[b_2, [b_1, T]_1]_2$, where $b_1$ and $b_2$ are little BMO functions. Our proof method can be used to simplify and improve lin…
New characterizations of BLO spaces by heat semigroups and applications
本文通过热半群对BLO空间进行新刻画，并应用该结果证明热方程解的正则性及Littlewood-Paley $g$-函数的BMO-BLO有界性。
In this paper we give a criterion to prove boundedness results for several operators from $H^1((0,\infty),γ_α)$ to $L^1((0,\infty),γ_α)$ and also from $L^\infty((0,\infty),γ_α)$ to $\BMO((0,\infty),γ_α)$, with respect to the probability measure $dγ_α(x)=\frac{2}{Γ(α+1)} x^{2α+1} e^{-x^2} dx$ on $(0,\infty)$ when ${α>-\frac12}$. We shall apply it to establish endpoint estimates for Riesz transforms, maximal operators, Littlewood-Paley functions, multipliers of Laplace transform type, fractional integrals and variation operators in the Laguerre setting.
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