Liu Lanzhe
In this paper, we prove some BMO end-point estimates for some vector-valued multilinear operators related to certain singular integral operators.
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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。
Sharp bounds for the multilinear integral operators on Heisenberg group BMO space
本文研究了 Heisenberg 群 BMO 空间上的 m-线性 n-维 Hardy-Littlewood-Pólya 算子与 Hilbert 算子，并证明了这两个算子在该空间中的有界性。
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 $T$ be a multilinear operator which is bounded on certain products of unweighted Lebesgue spaces of $\mathbb R^n$. We assume that the associated kernel of $T$ satisfies some mild regularity condition which is weaker than the usual Hölder continuity of those in the class of multilinear Calderón-Zygmund singular integral operators. We then show the boundedness for $T$ and the boundedness of the commutator of $T$ with BMO functions on products of weighted Lebesgue spaces of $\mathbb R^n$. As an application, we obtain the weighted norm inequalities of multilinear Fourier multipliers and of the…
We prove a Hörmander type multiplier theorem for multilinear Fourier multipiers with multiple weights. We also give weighted estimates for their commutators with vector $BMO$ functions.
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.
In the present work we extend a local Tb theorem for square functions of Christ and Hofmann to the multilinear setting. We also present new BMO type interpolation result for square functions associated to multilinear operators. These square function bounds are applied to prove a multilinear local Tb theorem for singular integral operators.
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.
A general class of weighted multilinear Hardy-Cesàro operators that acts on the product of Lebesgue spaces and central Morrey spaces. Their sharp bounds are also obtained. In addition, we obtain sufficient and necessary conditions on weight functions so that the commutators of these weighted multilinear Hardy-Cesàro operators (with symbols in central BMO space) are bounded on the product of central Morrey spaces. These results extends known results on multilinear Hardy operators.
In this paper, we first introduce several new classes of weighted amalgam spaces. Then we discuss both strong type and weak type estimates for certain multilinear $θ$-type Calderón--Zygmund operators $T_θ$ recently introduced in the literature on products of these spaces with multiple weights. Furthermore, the strong type and weak end-point estimates for both multilinear commutators and iterated commutators of $T_θ$ and pointwise multiplication with BMO functions are established as well.
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.
Let $T$ be a multilinear Calderón-Zygmund operator of type $ω$ with $ω(t)$ being nondecreasing and satisfying a kind of Dini's type condition. Let $T_{Π\vec{b}}$ be the iterated commutators of $T$ with $BMO$ functions. The weighted strong and weak $L(\log{L})$-type endpoint estimates for $T_{Π\vec{b}}$ with multiple weights are established. Some boundedness properties on weighted variable exponent Lebesgue spaces are also obtained. As applications, multiple weighted estimates for iterated commutators of paraproducts and bilinear pseudo-differential operators with mild regularity are given.
In this paper, we gave a weighted compactness theory for the generalized commutators of vecotor-valued multilinear Calderón-Zygmund operators. This was done by establishing a weighted Fréchet-Kolmogorov theorem, which holds for weights not merely in $A_\infty$. This weighted theory also extends the previous known unsatisfactory results in the terms of relaxing the index to the natural range. As consequences, we not only obtained the weighted compactness theory for the commutators of multilinear Calderón-Zygmund operators, but also extended the same results to the commutators of multilinear Lit…
In this paper we present (quasi-)norm equivalence on a vector-valued function space $L^p_A(l^q)$ and extend the equivalence to $p=\infty$ and $0<q<\infty$ in the scale of Triebel-Lizorkin space, motivated by Fraizer-Jawerth. By applying the results, we improve the multilinear Hormander's multiplier theorem of Tomita, that of Grafakos-Si, and the boundedness results for bilinear pseudo-differential operators, given by Koezuka-Tomita.
Characterization of Banach valued BMO functions and UMD Banach spaces by using Bessel convolutions
本文利用Bessel卷积与$\gamma$-radonifying范数刻画取值于UMD Banach空间的BMO奇函数空间，并证明UMD Banach空间是使Bessel-Poisson积分满足特定Carleson不等式的唯一空间。
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.
In this paper, we study a class of multilinear fractional integral operators which have correlation kernels $\prod_{1\leq i<j \leq k}|x_i-x_j|^{-α_{ij}}$. The necessary and sufficient conditions are obtained under which these oprators are bounded from $L^{p_1}\times \cdots \times L^{p_k}$ into $L^q$. As a consequence, we also get the endpoint estimates from $L^{p_1}\times \cdots \times L^{p_k}$ to $BMO$ of these operators.
We give a constructive proof of the factorization theorem for the classical Hardy space in terms of fractional integral operator. Moreover, the result is extended to the multilinear case and weighted case. As an application, we obtain the characterization of $BMO$ via the weighted boundedness of commutators of the multilinear fractional integral operator, without individual conditions on the weights class.
In this paper, we investigate the weighted vector-valued bounds for a class of multilinear singular integral operators, and its commutators, from $L^{p_1}(l^{q_1};\,\mathbb{R}^n,w_1)\times\dots\times L^{p_m}(l^{q_m};\,\mathbb{R}^n,w_m)$ to $L^{p}(l^q;\,\mathbb{R}^n,ν_{\vec{w}})$, with $p_1,\dots,p_m\in (1,\,\infty)$ and $1/p=1/p_1+\dots+1/p_m$ and $\vec{w}$ is a multiple $A_{\vec{P}}$ weights. Our argument also leads to the weighted weak type endpoint estimates for the commutators.
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,…
We extend the recently much-studied Hardy factorization theorems to the weight case. The key point of this paper is to establish the factorization theorems without individual condition on the weight functions. As a direct application, we obtain the characterizations of $BMO(\mathbb{R}^n)$ space and Lipschitz spaces via the weighted boundedness of commutators of multilinear Calderón-Zygmund operators with the genuinely multilinear weights.
Let $m\in \mathbb{N}$ and $\vec{b}=(b_{1},\cdots,b_{m})$ be a collection of locally integrable functions. It is proved that $b_{1},b_{2},\cdots, b_{m}\in BMO$ if and only if $$\sup_{Q}\frac{1}{|Q|^{m}}\int_{Q^{m}}\Big|\sum_{i=1}^{m}\big(b_{i}(x_{i})-(b_{i})_{Q}\big)\Big|d\vec{x}<\infty,$$ where $(b_{i})_{Q}=\frac{1}{|Q|}\int_{Q}b_{i}(x)dx$. As an application, we show that if the linear commutator of certain multilinear Calderón-Zygmund operator $[Σ\vec{b},T]$ is bounded from $L^{p_{1}}\times\cdots\times L^{p_{m}}$ to $L^{p}$ with $\sum_{i=1}^{m}1/p_{i}=1/p$ and $1<p,p_{1},\cdots,p_{m}<\infty$,…
We develop a general theory of multilinear singular integrals with operator-valued kernels, acting on tuples of UMD Banach spaces. This, in particular, involves investigating multilinear variants of the $\mathcal R$-boundedness condition naturally arising in operator-valued theory. We proceed by establishing a suitable representation of multilinear, operator-valued singular integrals in terms of operator-valued dyadic shifts and paraproducts, and studying the boundedness of these model operators via dyadic-probabilistic Banach space-valued analysis. In the bilinear case, we obtain a $T(1)$-typ…
Vector-valued extensions of operators through multilinear limited range extrapolation
本文将Rubio de Francia外推定理延拓至多重线性有限范围设定，证明了m-（次）线性算子向Bochner空间的延拓，并应用于双线性Hilbert变换等算子获得新的向量值界。
A Representation of Matrix-Valued Harmonic Functions by the Poisson Integral of Non-commutative BMO Functions
本文研究矩阵值调和函数，通过非交换BMO空间函数的Poisson积分对其进行刻画，从而为Fabes等人的经典结果提供了非交换类比。
In this paper we solve a long standing problem about the multivariable Rubio de Francia extrapolation theorem for the multilinear Muckenhoupt classes $A_{\vec{p}}$, which were extensively studied by Lerner et al. and which are the natural ones for the class of multilinear Calderón-Zygmund operators. Furthermore, we go beyond the classes $A_{\vec{p}}$ and extrapolate within the classes $A_{\vec{p},\vec{r}}$ which appear naturally associated to the weighted norm inequalities for multilinear sparse forms which control fundamental operators such as the bilinear Hilbert transform. We give several a…
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。
Vector valued estimates for matrix weighted maximal operators and product $\mathrm{BMO}$
本文研究了矩阵加权极大算子在复数空间上的向量值估计，利用Bownik和Cruz-Uribe的凸体值函数外推结果，并应用到双参数paraproducts的上界估计中。
UMD-valued square functions associated with Bessel operators in Hardy and BMO spaces
本文研究 Bessel 背景下 Banach 取值 Hardy 和 BMO 空间，利用分数阶导数定义平方函数，证明了 UMD 空间中涉及 $γ$-radonifying 算子的等价范数，并基于 g-函数的有界性刻画了 UMD Banach 空间。
$γ$-radonifying operators and UMD-valued Littlewood-Paley-Stein functions in the Hermite setting on BMO and Hardy spaces
本文研究了 Hermite 算子下取值于 UMD Banach 空间的 Littlewood-Paley-Stein 函数，证明了 Hermite 平方函数在 BMO 和 Hardy 空间上的有界性，并给出了这些空间的等价范数及 UMD Banach 空间的新刻画。
Basic questions concerning nonsingular multilinear operators with oscillatory factors are posed and partially answered. Lebesgue space norm inequalities are established for multilinear integral operators of Calderon-Zygmund type which incorporate oscillatory factors exp(iP), where P is a real-valued polynomial with large coefficients. A related problem concerning upper bounds for measures of sublevel sets is solved.
We prove a sparse bound for the $m$-sublinear form associated to vector-valued maximal functions of Fefferman-Stein type. As a consequence, we show that the sparse bounds of multisublinear operators are preserved via $\ell^r$-valued extension. This observation is in turn used to deduce vector-valued, multilinear weighted norm inequalities for multisublinear operators obeying sparse bounds, which are out of reach for the extrapolation theory recently developed by Cruz-Uribe and Martell. As an example, vector-valued multilinear weighted inequalities for bilinear Hilbert transforms are deduced fr…
We develop a general framework for the analysis of operator-valued multilinear multipliers acting on Banach-valued functions. Our main result is a Coifman-Meyer type theorem for operator-valued multilinear multipliers acting on suitable tuples of UMD spaces. A concrete case of our theorem is a multilinear generalization of Weis' operator-valued Hörmander-Mihlin linear multiplier theorem. Furthermore, we derive from our main result a wide range of mixed $L^p$-norm estimates for multi-parameter multilinear paraproducts, leading to a novel mixed norm version of the partial fractional Leibniz rule…
In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing so, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding ``finite'' case obtained by Duoandikoetxea, which wa…
We prove that the class of trilinear multiplier forms with singularity over a one dimensional subspace, including the bilinear Hilbert transform, admit bounded $L^p$-extension to triples of intermediate $\mathrm{UMD}$ spaces. No other assumption, for instance of Rademacher maximal function type, is made on the triple of $\mathrm{UMD}$ spaces. Among the novelties in our analysis is an extension of the phase-space projection technique to the $\mathrm{UMD}$-valued setting. This is then employed to obtain appropriate single tree estimates by appealing to the $\mathrm{UMD}$-valued bound for bilinea…
In this work we extend Lacey's domination theorem to prove the pointwise control of bilinear Calderón--Zygmund operators with Dini--continuous kernel by sparse operators. The precise bounds are carefully tracked following the spirit in a recent work of Hytönen, Roncal and Tapiola. We also derive new mixed weighted estimates for a general class of bilinear dyadic positive operators using multiple $A_{\infty}$ constants inspired in the Fujii-Wilson and Hrusčěv classical constants. These estimates have many new applications including mixed bounds for multilinear Calderón--Zygmund operators and th…
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。
The aim of this paper is to obtain mixed weak-type inequalities for multilinear fractional operators, extending results by F. Berra, M. Carena and G. Pradolini \cite{BCP}. We prove that, under certain conditions on the weights, there exists a constant $C$ such that $$\Bigg\| \frac{\mathcal G_α(\vec f \,)}{v}\Bigg\|_{L^{q, \infty}(νv^q)} \leq C \ \prod_{i=1}^m{\|f_i\|_{L^1(u_i)}},$$ where $\mathcal G_α(\vec f \,)$ is the multilinear maximal function $\mathcal M_α(\vec f\,)$ that was introduced by K. Moen in \cite{M} or the multilineal fractional integral $\mathcal I_α(\vec f \,)$. As an applica…
Quantitative estimates and extrapolation for multilinear weight classes
该研究提出定量多线性有限范围外推定理，拓展已有外推结果，发展适配多子线性哈代-李特尔伍德极大算子的多线性Rubio de Francia算法，得到多类空间相关估计及推论。
In this paper we present a theorem that generalizes Sawyer's classic result about mixed weighted inequalities to the multilinear context. Let $\vec{w}=(w_1,...,w_m)$ and $ν= w_1^\frac{1}{m}...w_m^\frac{1}{m}$, the main result of the paper sentences that under different conditions on the weights we can obtain $$\Bigg\| \frac{T(\vec f\,)(x)}{v}\Bigg\|_{L^{\frac{1}{m}, \infty}(νv^\frac{1}{m})} \leq C \ \prod_{i=1}^m{\|f_i\|_{L^1(w_i)}}, $$ where $T$ is a multilinear Calderón-Zygmund operator. To obtain this result we first prove it for the $m$-fold product of the Hardy-Littlewood maximal operator…
Analysis of Toeplitz Operators with $BMO^1_α$ operator-valued symbols on $\ell^2-$Valued Bergman Spaces
研究单位球上ℓ²值Bergman空间上具有BMO¹_α算子值符号的Toeplitz算子，通过有限维限制方法及充分条件刻画其紧性，并探讨逆蕴含问题。
We prove a limited range, off-diagonal extrapolation theorem that generalizes a number of results in the theory of Rubio de Francia extrapolation, and use this to prove a limited range, multilinear extrapolation theorem. We give two applications of this result to the bilinear Hilbert transform. First, we give sufficient conditions on a pair of weights $w_1,\,w_2$ for the bilinear Hilbert transform to satisfy weighted norm inequalities of the form \[ BH : L^{p_1}(w_1^{p_1}) \times L^{p_2}(w_2^{p_2}) \longrightarrow L^p(w^p), \] where $w=w_1w_2$ and $\frac{1}{p}=\frac{1}{p_1}+\frac{1}{p_2}<\frac…
Sparse domination implies vector-valued sparse domination
本文证明多线性算子的标量值稀疏控制蕴含向量值稀疏控制，引入UMD条件的多线性类比，从而直接获得精确的向量值加权界，并应用于改进Calderón-Zygmund算子与双线性Hilbert变换的界。
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…
We provide a natural BMO-criterion for the $L_2$-boundedness of Calderón-Zygmund operators with operator-valued kernels satisfying a symmetric property. Our arguments involve both classical and quantum probability theory. In the appendix, we give a proof of the $L_2$-boundedness of the commutators $[R_j,b]$ whenever $b$ belongs to the Bourgain's vector-valued BMO space, where $R_j$ is the $j$-th Riesz transform. A common ingredient is the operator-valued Haar multiplier studied by Blasco and Pott.
We present a framework based on modified dyadic shifts to prove multiple results of modern singular integral theory under mild kernel regularity. Using new optimized representation theorems we first revisit a result of Figiel concerning the UMD-extensions of linear Calderón-Zygmund operators with mild kernel regularity and extend our new proof to the multilinear setting improving recent UMD-valued estimates of multilinear singular integrals. Next, we develop the product space theory of the multilinear singular integrals with modified Dini-type assumptions, and use this theory to prove bi-param…
On regularity of solutions to the Navier--Stokes equation with initial data in $\mathrm{BMO}^{-1}$
本文研究了Navier-Stokes方程初值在BMO^{-1}中的解的正则性，证明了其在时间上的弱*连续性及在无穷远处的消失性。
We establish a uniform domination of the family of trilinear multiplier forms with singularity over a one-dimensional subspace by positive sparse forms involving $L^p$-averages. This class includes the adjoint forms to the bilinear Hilbert transforms. Our result strengthens the $L^p$-boundedness proved in \cite{MTT} and entails as a corollary a rich multilinear weighted theory. In particular, we obtain $L^{q_1}(v_1) \times L^{q_2}(v_2)$-boundedness of the bilinear Hilbert transform when the weights $v_j$ belong to the class $A_{\frac{q+1}{2}}\cap RH_2$. Our proof relies on a stopping time cons…
We prove multiple vector-valued and mixed-norm estimates for multilinear operators in $\rr R^d$, more precisely for multilinear operators $T_k$ associated to a symbol singular along a $k$-dimensional space and for multilinear variants of the Hardy-Littlewood maximal function. When the dimension $d \geq 2$, the input functions are not necessarily in $L^p(\rr R^d)$ and can instead be elements of mixed-norm spaces $L^{p_1}_{x_1} \ldots L^{p_d}_{x_d}$. Such a result has interesting consequences especially when $L^\infty$ spaces are involved. Among these, we mention mixed-norm Loomis-Whitney-type i…
In this paper, the boundedness properties of vector-valued intrinsic square functions and their vector-valued commutators with $BMO(\mathbb R^n)$ functions are discussed. We first show the weighted strong type and weak type estimates of vector-valued intrinsic square functions in the Morrey type spaces. Then we obtain weighted strong type estimates of vector-valued analogues of commutators in Morrey type spaces. In the endpoint case, we establish the weighted weak $L\log L$-type estimates for these vector-valued commutators in the setting of weighted Lebesgue spaces. Furthermore, we prove weig…
We prove a global well-posedness result for the Landau-Lifshitz equation with Gilbert damping provided that the BMO semi-norm of the initial data is small. As a consequence, we deduce the existence of self-similar solutions in any dimension. In the one-dimensional case, we characterize the self-similar solutions associated with an initial data given by some ($\mathbb{S}^2$-valued) step function and establish their stability. We also show the existence of multiple solutions if the damping is strong enough. Our arguments rely on the study of a dissipative quasilinear Schrödinger obtained via the…
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。
We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on $\ell^r$-valued extensions of linear operators. We show that for certain $1 \leq p, q_1, \dots, q_m, r \leq \infty$, there is a constant $C\geq 0$ such that for every bounded multilinear operator $T\colon L^{q_1}(μ_1) \times \cdots \times L^{q_m}(μ_m) \to L^p(ν)$ and functions $\{f_{k_1}^1\}_{k_1=1}^{n_1} \subset L^{q_1}(μ_1), \dots, \{f_{k_m}^m\}_{k_m=1}^{n_m} \subset L^{q_m}(μ_m)$, the following inequality holds \begin{equation}\label{MZ ineq abstract} (1) \quad \quad \left\Vert \left(\sum_{k_1, \dots…
Convex body domination for the commutator of vector valued operators with multiple matrix-valued symbols
研究凸体支配性质在向量值算子交换子中的应用，证明强类型估计及BMO空间的相关性质。
In this paper, the author considers the weighted vector-valued estimate for the operator defined by $$T_Af(x)={\rm p.\,v.}\int_{\mathbb{R}^n}\frac{Ω(x-y)}{|x-y|^{n+1}}\big(A(x)-A(y)-\nabla A(y)\big)f(y){\rm d}y,$$ and the corresponding maximal operator $T_A^*$, where $Ω$ is homogeneous of degree zero, has vanishing moment of order one, $A$ is a function in $\mathbb{R}^n$ such that $\nabla A\in {\rm BMO}(\mathbb{R}^n)$. By a pointwise estimate for $\|\{T_Af_k(x)\}\|_{l^q}$ and the weighted $L^p$ estimates for the sparse operator $$\mathcal{A}_{\mathcal{S},\,L(\log L)^β}f(x)=\sum_{Q\in\mathcal{S…
A nonlinear superposition operator $T_g$ related to a Borel measurable function $g:\ {\mathbb C}\to {\mathbb C}$ is defined via $T_g(f):=g\circ f$ for any complex-valued function $f$ on ${\mathbb R^n}$. This article is devoted to investigating the mapping properties of $T_g$ on a new BMO type space recently introduced by Bourgain, Brezis and Mironescu [J. Eur. Math. Soc. (JEMS) 17 (2015), 2083-2101], as well as its VMO and CMO type subspaces. Some sufficient and necessary conditions for the inclusion result and the continuity property of $T_g$ on these spaces are obtained.
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)).
A class of multilinear bounded oscillation operators on measure spaces and applications
本文为测度空间上取值于Banach空间的多线性有界振荡算子发展了全面的加权理论，证明了其被稀疏二进算子逐点控制，建立了三类典型估计，并通过外推法获得了交换子的加权紧性。
Multilinear matrix weights
本文全面刻画了使多线性 Calderón-Zygmund 算子延拓为 matrix weighted Lebesgue 空间有界算子的 matrix weights 类，发展了张量积取值的多线性奇异积分理论，建立了多种算子的定量界并提出了方向非退化新概念。
In this paper, we first introduce some new kinds of weighted amalgam spaces. Then we deal with the vector-valued intrinsic square functions, which are given by \begin{equation*} \mathcal S_γ(\vec{f})(x) = \Bigg(\sum_{j=1}^\infty \big|\mathcal S_γ(f_j)(x) \big|^2\Bigg)^{1/2}, \end{equation*} where $0<γ\leq1$ and \begin{equation*} \mathcal S_γ(f_j)(x) = \left(\iint_{Γ(x)} \Big[\sup_{φ\in{\mathcal C}_γ} \big|φ_t*f_j(y) \big|\Big]^2 \frac{dydt}{t^{n+1}}\right)^{1/2}, \quad j=1,2,\dots. \end{equation*} In his fundamental work, Wilson established strong-type and weak-type estimates for vector-valued…
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