Irina Holmes, Sergei Treil, Alexander Volberg
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.
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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…
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…
We investigate a scale of dyadic operator-valued BMO spaces, corresponding to the different yet equivalent characterizations of dyadic BMO in the scalar case. In the language of operator spaces, we investigate different operator space structures on the scalar dyadic BMO space which arise naturally from the different characterisations of scalar BMO. We also give sharp dimensional growth estimates for the sweep of functions and its bilinear extension in some of those different dyadic BMO spaces.
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$(R)$.
Parabolic Extrapolation and Its Applications to Characterizing Parabolic BMO Spaces via Parabolic Fractional Commutators
本文建立抛物版Rubio de Francia外推定理，利用带时间滞后的抛物分数次算子交换子，首次在高维情形下刻画抛物BMO型空间。
We first consider two types of localizations of singular integral operators of convolution type, and show, under mild decay and smoothness conditions on the auxiliary functions, that their boundedness on the local Hardy space $h^1(\mathbb{R}^n)$ is equivalent. We then study the boundedness on $h^1(\mathbb{R}^n)$ of the commutator $[b,T]$ of an inhomogeneous singular integral operator with $b$ in $bmo(\mathbb{R}^n)$, the nonhomogeneous space of functions of bounded mean oscillation. We define local analogues of the atomic space $H^1_b(\mathbb{R}^n)$ introduced by Pérez in the case of the homoge…
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).
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。
In this paper, we establish multilinear BMO estimates for commutators of multilinear fractional maximal and integral operators both on product generalized Morrey spaces and product generalized vanishing Morrey spaces, respectively. Similar results are still valid for commutators of multilinear maximal and singular integral operators.
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 $p\in(1, 1/β)$.
We consider BMO spaces of operator-valued functions, among them the space of operator-valued functions $B$ which define a bounded paraproduct on $L^2(H)$. We obtain several equivalent formulations of $\|π_B\|$ in terms of the norm of the "sweep" function of $B$ or of averages of the norms of martingales transforms of $B$ in related spaces. Furthermore, we investigate a connection between John-Nirenberg type inequalities and Carleson-type inequalities via a product formula for paraproducts and deduce sharp dimensional estimates for John-Nirenberg type inequalities.
Let $p\in(1, \infty)$, $κ\in(0, 1)$ and $w\in A_p(\mathbb C).$ In this article, the authors obtain a boundedness (resp., compactness) characterization of the Buerling-Ahlfors commutator $[\mathcal B, b]$ on the weighted Morrey space $L_w^{p,\,κ}(\mathbb C)$ via $\mathrm{BMO}(\mathbb C)$ [resp., $\mathrm{CMO}(\mathbb C)$], where $\mathcal B$ denotes the Buerling-Ahlfors transform and $b\in \mathrm{BMO}(\mathbb C)$ [resp., $\mathrm{CMO}(\mathbb C)$]. Moreover, an application to the Beltrami equation is also given.
Multilinear Commutators of Multilinear Square Operators Associated with New $BMO$ Functions and New Weight Functions
本文利用新权与新BMO函数定义了一类具有广义核的多线性平方算子，并证明了其生成的多线性交换子及迭代交换子在加权Lebesgue空间和加权Morrey空间上的有界性，推广了若干已知结论。
Characterization of BMO spaces via commutators of fractional maximal function on Lorentz spaces
本文通过分数阶极大函数的极大交换子和非线性交换子在Lorentz空间上的有界性，给出了BMO空间子类的新的充分必要条件刻画。
Let $δ\in(0,1]$ and $T$ be a $δ$-Calderón-Zygmund operator. Let $w$ be in the Muckenhoupt class $A_{1+δ/n}({\mathbb R}^n)$ satisfying $\int_{{\mathbb R}^n}\frac {w(x)}{1+|x|^n}\,dx<\infty$. When $b\in{\rm BMO}(\mathbb R^n)$, it is well known that the commutator $[b, T]$ is not bounded from $H^1(\mathbb R^n)$ to $L^1(\mathbb R^n)$ if $b$ is not a constant function. In this article, the authors find out a proper subspace ${\mathop\mathcal{BMO}_w({\mathbb R}^n)}$ of $\mathop\mathrm{BMO}(\mathbb R^n)$ such that, if $b\in {\mathop\mathcal{BMO}_w({\mathbb R}^n)}$, then $[b,T]$ is bounded from the we…
We prove that the weak Morrey space $WM^{p}_{q}$ is contained in the Morrey space $M^{p}_{q_{1}}$ for $1\leq q_{1}< q\leq p<\infty$. As applications, we show that if the commutator $[b,T]$ is bounded from $L^p$ to $L^{p,\infty}$ for some $p\in (1,\infty)$, then $b\in \mathrm{BMO}$, where $T$ is a Calderón-Zygmund operator. Also, for $1<p\leq q<\infty$, $b\in \mathrm{BMO}$ if and only if $[b,T]$ is bounded from $M^{p}_{q}$ to $WM_{q}^{p}$. For $b$ belonging to Lipschitz class, we obtain similar results.
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.
Let $b$ be a $BMO$-function. It is well-known that the linear commutator $[b, T]$ of a Calderón-Zygmund operator $T$ does not, in general, map continuously $H^1(\mathbb R^n)$ into $L^1(\mathbb R^n)$. However, Pérez showed that if $H^1(\mathbb R^n)$ is replaced by a suitable atomic subspace $\mathcal H^1_b(\mathbb R^n)$ then the commutator is continuous from $\mathcal H^1_b(\mathbb R^n)$ into $L^1(\mathbb R^n)$. In this paper, we find the largest subspace $H^1_b(\mathbb R^n)$ such that all commutators of Calderón-Zygmund operators are continuous from $H^1_b(\mathbb R^n)$ into $L^1(\mathbb R^n)$…
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。
We characterize the Hilbert--Schmidt class membership of commutator with the Hilbert transform in the two weight setting. The characterization depends upon the symbol of the commutator being in a new weighted Besov space. This follows from a Schatten class $S_p$ result for dyadic paraproducts, where $1< p < \infty $. We discuss the difficulties in extending the dyadic result to the full range of Schatten classes for the Hilbert transform.
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.
In this paper, the central BMO spaces with variable exponent are introduced. As an application, we characterize these spaces by the boundedness of commutators of Hardy operator and its dual operator on variable Lebesgue spaces. The boundedness of vector-valued commutators on Herz spaces with variable exponent are also considered.
Boundedness of commutator generated by fractional integral operator and Orlicz-BMO function
研究了分数积分算子与Orlicz-BMO函数生成的交换子从Orlicz-Hardy空间到Lebesgue空间的有界性，以及其在弱Lebesgue空间中的有界性。
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.
Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2(R^n)$ with Gaussican kernel bounds, and let $L^{-α/2}$ be the fractional integrals of $L$ for $0<α<n.$ For any locally integrable function $b$, The commutators associated with $L^{-α/2}$ are defined by $[b,L^{-α/2}](f)(x)=b(x)L^{-α/2}(f)(x)-L^{-α/2}(bf)(x)$. When $b\in BMO(ω)$(weighted $BMO$ space) or $b\in BMO$, the author obtain the necessary and sufficient conditions for the boundedness of $[b,L^{-α/2}]$ on weighted Morrey spaces respectively.
On $\mathbb R^N$ equipped with a root system $R$, multiplicity function $k \geq 0$, and the associated measure $dw(\mathbf{x})=\prod_{α\in R}|\langle \mathbf{x},α\rangle|^{k(α)}\,d\mathbf{x}$, we consider a (non-radial) kernel ${K}(\mathbf{x})$ which has properties similar to those from the classical theory of singular integrals and the Dunkl convolution operator $\mathbf{T}f=f*K$ associated with ${K}$. Assuming that $b$ belongs to the ${\rm BMO}$ space on the space of homogeneous type $X=(\mathbb{R}^N,\|\cdot\|,dw)$, we prove that the commutator $[b,\mathbf{T}]f(\mathbf{x})=b(\mathbf{x})\math…
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_{-…
Endpoint boundedness of Orlicz-BMO commutators on Orlicz-Hardy type spaces
本文研究次线性算子与Orlicz-BMO函数生成的交换子在Orlicz-Hardy型空间上的端点有界性，证明了两类有界性结果，指出仅Bochner-Riesz平均算子对应的交换子满足特定端点有界性并给出反例。
A characterization of mixed $λ$-central $\mathrm{BMO}$ space via the commutators of Hardy type operators
本文利用 Hardy 算子及其对偶算子的交换子在混合 Lebesgue 空间上的有界性刻画了混合 $\lambda$-中心 BMO 空间，并建立了相应交换子在混合 $\lambda$-中心 Morrey 空间上的有界性。
Dyadic fractional integral operators are shown to be bounded on Morrey spaces and their preduals. It seems that the proof of the boundedness by means of dyadic fractional integral operators is effective particularly on the preduals. In the present paper the commutators are proved to be bounded as well.
The Commutators of $n$-dimensional Rough Fractional Hardy Operators on Two Weighted Grand Herz-Morrey Spaces with Variable Exponents
研究证明了带粗糙核的 n 维分数 Hardy 算子及其伴随算子与 BMO 函数生成的 m 阶交换子在两类变指数加权大 Herz-Morrey 空间上的有界性。
We prove that for any operator $T$ on bi--parameter BMO the identity factors through $T$ or $I - T$. Bourgain's localization method provides the conceptual framework of our proof. It consists in replacing the factorization problem on the non--separable bi--parameter BMO by its localized, finite dimensional counterpart. We solve the resulting finite dimensional factorization problems by exploiting the geometry and combinatorics of colored dyadic rectangles.
Let $μ_{Ω,\vec{b}}$ be the multilinear commutator generalized by $μ_Ω$, the $n$-dimensional Marcinkiewicz integral, with $\Osc_{\exp L^{^τ}}(\R^{n})$ functions for $τ\ge 1$, where $\Osc_{\exp L^{^τ}}(\R^{n})$ is a space of Orlicz type satisfying that $\Osc_{\exp L^{^τ}}(\R^{n})=\BMO(\R^{n})$ if $τ=1$ and $\Osc_{\exp L^{^τ}}(\R^{n})\subset\BMO(\R^{n})$ if $τ>1$. The authors establish the weighted weak $L\log L$-type estimates for $μ_{Ω,\vec{b}}$ when $Ω$ satisfies a kind of Dini conditions.
In this paper, we establish BMO estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively.
Boundedness of operator-valued commutators involving martingale paraproducts
本文证明了涉及鞅仿积与左乘算子的算子值交换子在非交换$L_p$空间上的有界性，并探讨了半交换鞅仿积基于$BMO$空间的外推性质。
In this paper we give BMO (bounded mean oscillation) space estimates for commutators of fractional type sublinear operators in generalized Morrey spaces on Heisenberg groups. The boundedness conditions are also formulated in terms of Zygmund type integral inequalities.
The aim of this paper is to get the boundedness of certain sublinear operators with rough kernel generated by Calderón-Zygmund operators on the generalized weighted Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. We also prove that the commutator operators formed by BMO functions and certain sublinear operators with rough kernel are also bounded on the generalized weighted Morrey spaces. Marcinkiewicz operator which satisfies the conditions of these theorems can be considered as an example.
Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2(\mathbb{R}^n)$ with Gaussian kernel bounds, and let $L^{-α/2}$ be the fractional integrals of $L$ for $0<α<n$. Assume that $\vec{b}=(b_1,b_2,\cdots,b_m)$ is a finite family of locally integrable functions, then the multilinear commutators generated by $\vec{b}$ and $L^{-α/2}$ is defined by \begin{equation*} L_{\vec{b}}^{-α/2}f=[b_m,\cdots,[b_2,[b_1,L^{-α/2}]],\cdots,]f \end{equation*} when $b_j\in BMO(w)$, $j=1,2,\cdots,m$, the authors obtain the boundedness of $L_{\vec{b}}^{-α/2}$ on weighted Morrey spaces.
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