Kangwei Li, Henri Martikainen, Emil Vuorinen
We prove genuinely multilinear weighted estimates for singular integrals in product spaces. The estimates complete the qualitative weighted theory in this setting. Such estimates were previously known only in the one-parameter situation. Extrapolation gives powerful applications -- for example, a free access to mixed-norm estimates in the full range of exponents.
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In this paper, we will study the boundedness properties of multilinear Calderon--Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights.
In this note, we show that if $T$ is a multilinear singular integral operator associated with a kernel satisfies the so-called multilinear $L^{r}$-Hörmander condition, then $T$ can be dominated by multilinear sparse operators.
In this paper, we prove some BMO end-point estimates for some vector-valued multilinear operators related to certain singular integral operators.
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.
Multilinear Mikhlin Multipliers with Degenerate Singularities
本文证明了符号沿退化子空间奇点的多重线性 Mikhlin 乘子的 $L^p$ 有界性，通过将非退化理论的波包分析推广到温和退化情形，并指出指数范围由奇点几何决定。
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.
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.
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.
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.
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.
Multilinear estimates for maximal rough singular integrals
本文研究带粗糙齐次核的极大多（次）线性奇异积分，在核属于球面 L^q 且满足消失矩条件时证明多线性有界性，并推出双截断情形的几乎处处收敛。
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.
Multilinear singular integrals with homogeneous kernels near $L^1$
本文研究了具有近 $L^1$ 齐次核的多线性奇异积分算子，在核函数 $Ω$ 属于 $L^q$ 且积分消失的条件下，得到了其从乘积 Lebesgue 空间到 $L^p$ 空间有界性的最优开区间。
We study $m$-linear homogeneous rough singular integral operators $\mathcal{L}_Ω$ associated with integrable functions $Ω$ on $\mathbb{S}^{mn-1}$ with mean value zero. We prove boundedness for $\mathcal{L}_Ω$ from $L^{p_1}\times \cdots \times L^{p_m}$ to $L^p$ when $1<p_1,\dots, p_m<\infty$ and $1/p=1/p_1+\cdots +1/p_m$ in the largest possible open set of exponents when $Ω\in L^q(\mathbb S^{mn-1})$ and $q\ge 2$. This set can be described by a convex polyhedron in $\mathbb R^m$.
Non-Degenerate Multilinear Singular Multipliers with Fractional Rank
本研究针对满足m维线性子空间外符号估计的多线性乘子，建立了其作用于ℝᵈ上(n-1)元函数组的Lᵖ估计，重点分析了分数阶秩情形下的相关性质，为多线性奇异乘子的有界性研究提供了新结果。
Weighted estimates for Multilinear Singular Integrals with Rough Kernels
本文研究了具有粗糙核的多线性奇异积分算子的加权norm不等式，通过扩展Watson和Duoandikoetxea的经典结果，证明了在特定条件下该算子从多个加权Lebesgue空间到另一个加权Lebesgue空间的有界性。
We prove Bloom type two-weight inequalities for commutators of multilinear singular integral operators including Calderón-Zygmund operators and their dyadic counterparts. Such estimates are further extended to a general higher order multilinear setting. The proof involves a pointwise sparse domination of multilinear commutators.
We prove $L^{p_1}(\mathbb R^d)\times \dots \times L^{p_{n+2}}(\mathbb R^{d})$ polynomial growth estimates for the Christ-Journé multilinear singular integral forms and suitable generalizations.
We prove bounds for multilinear operators on $\R^d$ given by multipliers which are singular along a $k$ dimensional subspace. The new case of interest is when the rank $k/d$ is not an integer. Connections with the concept of {\em true complexity} from Additive Combinatorics are also investigated.
Let $T$ be a multilinear integral operator which is bounded on certain products of Lebesgue spaces on $\mathbb R^n$. We assume that its associated kernel 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. In this paper, given a suitable multiple weight $\vec{w}$, we obtain the bound for the weighted norm of multilinear operators $T$ in terms of $\vec{w}$. As applications, we exploit this result to obtain the weighted bounds for certain singular integral operators such as li…
We prove norm estimates for multilinear fractional integrals acting on weighted and variable Hardy spaces. In the weighted case we develop ideas we used for multilinear singular integrals [7]. For the variable exponent case, a key element of our proof is a new multilinear, off-diagonal version of the Rubio de Francia extrapolation theorem.
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 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…
Multilinear rough singular integral operators may not be of the weak type at the endpoint
该研究针对多线性粗糙奇异积分算子，证明其在端点处可能不具备从$L^1(\mathbb R^n)\times\cdots\times L^1(\mathbb R^n)$到$L^{\frac 1 m,\nf}(\mathbb R^n)$的有界性，仅在特定例外情况可能成立。
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。
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.
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 $i=1,…
Weighted weak-type bounds for multilinear singular integrals
本文为满足稀疏形式控制的多线性算子（如多线性Calderón-Zygmund奇异积分）建立了尖锐的加权弱型估计，适用于一般指数向量，并通过新的局部检验条件和好-$\lambda$不等式实现了定量改进。
Sparse domination for rough multilinear singular integrals
本文证明具有粗糙核的多线性奇异积分算子满足稀疏控制，并由此推导出定量加权范数不等式。
We prove $L^p$ bounds for the extensions of standard multilinear Calderón-Zygmund operators to tuples of UMD spaces tied by a natural product structure. This can, for instance, mean the pointwise product in UMD function lattices, or the composition of operators in the Schatten-von Neumann subclass of the algebra of bounded operators on a Hilbert space. We do not require additional assumptions beyond UMD on each space - in contrast to previous results, we e.g. show that the Rademacher maximal function property is not necessary. The obtained generality allows for novel applications. For instance…
Weighted Extensions of Stein's Theorem for Linear and Multilinear Operators
针对核函数仅满足尺寸条件的积分算子，在Herz和Cesàro型空间上建立加权估计，并推广到多线性强型和弱型情形，应用于粗糙奇异积分算子。
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 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…
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…
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…
Sharp weighted inequalities for iterated commutators of a class of multilinear operators
本文在点态稀疏控制估计的一般假设下，系统研究了一类多线性算子迭代交换子的精确量化加权界，建立了局部衰减、弱$A_\infty$衰减及相关不等式，并应用于多种多线性算子。
The Interplay of Shifted Square and Maximal Function Estimates in the Context of Multilinear Fourier Multipliers
本文通过移位正方形和最大函数的结合，改进了多线性傅里叶乘子的有界性定理，并应用于奇异积分在Orlicz空间中的有界性研究。
The $L^p$ boundedness theory of convolution operators is \linebreak based on an initial $L^2\to L^2$ estimate derived from the Fourier transform. The corresponding theory of multilinear operators lacks such a simple initial estimate in view of the unavailability of Plancherel's identity in this setting, and up to now it has not been clear what a natural initial estimate might be. In this work we achieve exactly this goal, i.e., obtain an initial $L^2\times\cdots\times L^2\to L^{2/m}$ estimate for general building blocks of $m$-linear multiplier operators. We apply this result to deduce analogo…
Dyadic analysis of compactness on product spaces
本研究采用现代视角建立乘积空间多线性奇异积分紧性理论，证明多线性紧性T1定理并给出交换子平均连续性结果，拓展了Journé定理的适用范围。
In this paper, the boundedness properties for some multilinear operators related to certain integral operators from Lebesgue spaces to Orlicz spaces are obtained. The operators include Calderón--Zygmund singular integral operator, fractional integral operator, Littlewood--Paley operator and Marcinkiewicz operator.
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…
Let $m(ξ,η)$ be a measurable locally bounded function defined in $\mathbb R^2$. Let $1\leq p_1,q_1,p_2,q_2<\infty $ such that $p_i=1$ implies $q_i=\infty $. Let also $0<p_3,q_3<\infty $ and $1/p=1/p_1+1/p_2-1/p_3$. We prove the following transference result: the operator $$ {\mathcal C}_m(f,g)(x)=\int_{\bbbr} \int_{\bbbr} \hat f(ξ) \hat g(η) m(ξ,η) e^{2πi x(ξ+η)}dξdη$$ initially defined for integrable functions with compact Fourier support, extends to a bounded bilinear operator from $L^{p_1,q_1}(\bbbr)\times L^{p_2,q_2}(\bbbr)$ into $L^{p_3,q_3}(\bbbr)$ if and only if the family of operators …
We study a multilinear singular integral obtained by taking averages of simplex Hilbert transforms. This multilinear form is also closely related to Calderón commutators and the twisted paraproduct. We prove $L^p$ bounds in dimensions two and three and give a conditional result valid in all dimensions.
We prove weighted estimates for rough bilinear singular integral operators with kernel $$K(y_1, y_2) = \frac{Ω((y_1,y_2)/|(y_1,y_2)|)}{|(y_1, y_2)|^{2d}},$$ where $y_i \in \mathbb{R}^{d}$ and $Ω\in L^{\infty}(S^{2d-1})$ with $\int_{S^{2d-1}}Ωdσ= 0.$ The argument is by sparse domination of rough bilinear operators, via an abstract theorem that is a multilinear generalization of recent work by Conde-Alonso, Culiuc, Di Plinio and Ou. We also use recent results due to Grafakos, He, and Honz\'ık for the application to rough bilinear operators. In particular, since the weighted estimates are proved …
Extrapolation for multilinear compact operators and applications
本文研究多重线性紧算子的 Rubio de Francia 外推，通过建立加权 Fréchet-Kolmogorov 定理及混合范数 Lebesgue 空间中的加权插值定理，将紧性从一个空间外推至全范围加权空间，并应用于证明多种多重线性算子交换子的加权紧性。
This paper studies dyadic singular integral forms associated with $r$-partite $r$-uniform hypergraphs such that all their connected components are complete. We characterize their $L^p$ boundedness by T(1)-type conditions in two different ways. We also dominate these forms by positive sparse forms and prove weighted estimates with multilinear Muckenhoupt weights.
The multilinear fractional sparse operator theory I: pointwise domination and weighted estimate
本文针对多线性分数阶奇异积分算子的广义交换子，通过引入高阶多符号稀疏算子并利用极大权与迭代权方法，建立了Bloom型定量加权估计及端点估计，改进了现有方法并推广了结论。
The multilinear fractional sparse operator theory II: refining weighted estimates via multilinear fractional sparse forms
本文通过引入新的多线性分数阶稀疏形式和权类，改进多线性分数阶奇异积分算子广义交换子的加权估计，获得尖锐定量界并应用于分数阶Laplacian方程解的正则性。
We obtain sharp estimates for certain trilinear oscillatory integrals. In particular, we extend Phong and Stein's seminal result to a trilinear setting. This result partially answers a question raised by Christ, Li, Tao and Thiele concerning the sharp estimates for certain multilinear oscillatory integrals. The method in this paper relies on a self-contained algorithm of resolution of singularities in $\mathbb R^2$, which may be of independent interest.
Weighted Bilinear Multiplier Theorems in Dunkl Setting via Singular Integrals
本文在Dunkl背景下，通过证明Littlewood-Paley型定理和多线性Calderón-Zygmund算子的加权不等式新结果，建立了具有多个Muckenhoupt权的双线性乘子算子的单权与双权不等式。
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。
We prove uniform $L^p$ bounds for multilinear operators which are given by multipliers whose symbols are singular on a one dimensional subspace. The novelty is that these bounds are uniform in the choice of the subspace.
Limited range extrapolation with quantitative bounds and applications
本文发展了具有定量界的 Rubio de Francia 外推方法，建立了定量多重线性有界范围外推定理，并将其应用于研究超越 Calderón-Zygmund 理论的算子的定量加权不等式。
Knapp-type obstructions in multilinear fractal Fourier extension
本文研究了多线性分形傅里叶扩展估计，提出了一种适用于卷积属于$ L^p $空间的测度的估计方法，并获得了Knapp型例子及两个必要条件。
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.
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…
The reappearance of a sometimes called exotic behavior for linear and multilinear pseudodifferential operators is investigated. The phenomenon is shown to be present in a recently introduced class of bilinear pseudodifferential operators which can be seen as more general variable coefficient counterparts of the bilinear Hilbert transform and other singular bilinear multipliers operators. The unboundedness on product of Lebesgue spaces but the boundedness on spaces of smooth functions (which is the exotic behavior referred to) of such operators is obtained. In addition, by introducing a new way…
The following subexponential estimate for commutators is proved |[|\{x\in Q: |[b,T]f(x)|>tM^2f(x)\}|\leq c\,e^{-\sqrt{α\, t\|b\|_{BMO}}}\, |Q|, \qquad t>0.\] where $c$ and $α$ are absolute constants, $T$ is a Calderón--Zygmund operator, $M$ is the Hardy Littlewood maximal function and $f$ is any function supported on the cube $Q$. It is also obtained \[|\{x\in Q: |f(x)-m_f(Q)|>tM_{1/4;Q}^#(f)(x) \}|\le c\, e^{-α\,t}|Q|,\qquad t>0,\] where $m_f(Q)$ is the median value of $f$ on the cube $Q$ and $M_{1/4;Q}^#$ is Strömberg's local sharp maximal function. As a consequence it is derived Karagulyan'…
Weighted Norm Inequalities for Maximal Multi-sublinear Rough Singular Integrals
针对多线性粗糙齐次奇异积分的极大截断，建立其加权范数不等式，证明该估计满足与对应主值估计相同的多重权条件。
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