Frederic Bernicot
In this paper, we prove some uniform estimates between Lebesgue and Hardy spaces for operators closely related to the multilinear paraproducts on R^d. We are looking for uniformity with respect to parameters, which allow us to disturb the geometry and the metric on $R^d$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
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。
We prove L^p estimates for a class of two-dimensional multilinear forms that naturally generalize (dyadic variants of) both classical paraproducts and the twisted paraproduct introduced in [5] and studied in [1] and [6]. The method we use builds on the approach from [6] and we present it as a rather general technique for proving estimates on dyadic multilinear operators. In the particular application to "generalized paraproducts" this method is combined with combinatorics of integer partitions.
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.
Paraproducts for bilinear multipliers associated with convex sets
本文研究与凸集相关的双线性乘子，证明奇异副乘积在局部 $L^2$ 范围内的有界性，并分别在指数边界曲线和高阶空隙条件下给出结果。
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.
Multilinear Wavelet Compact T(1) Theorem
该文证明了多线性Calderón-Zygmund算子紧致性的小波T(1)定理，通过检验条件刻画紧致性，并给出紧致CZ形式的表示定理。
Multilinear extrapolation of compactness on mixed-norm spaces
本文发展了Rubio de Francia外推定理，用于混合范数Lebesgue空间上的多线性紧性，将紧性从无权重乘积空间外推到有权重的混合范数空间。
We study the natural resolution of the conjugated Haar multiplier $M_{w^{\frac{1}{2}}}T_σM_{w^{-\frac{1}{2}}},$ where the multiplication operators $M_{w^{\pm\frac{1}{2}}}$ are decomposed into their canonical paraproduct decompositions. We prove that each constituent operator obtained from this resolution has a linear bound on $L^2(\mathbb{R}^d;w)$ in terms of the $A_{2}$ characteristic of $w$. The main tools used are a product formula for Haar coefficients, the Carleson Embedding Theorem, the linear bound for the square function, and the well-known linear bound of $T_σ$ on $L^2(\mathbb{R}^d,w)…
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…
In the first part of the paper we prove a bi-parameter version of a well known multilinear theorem of Coifman and Meyer. As a consequence, we generalize the Kato-Ponce inequality in nonlinear PDE, obtaining a fractional Leibnitz rule for derivatives in the $x_1$ and $x_2$ directions simultaneously. Then, we show that the double bilinear Hilbert transform does not satisfy any $L^p$ estimates.
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…
Analogues of multi-paramter multiplier operators on R^d are defined on the torus T^d. It is shown that these operators satisfy the classical Coifman-Meyer theorem. In addition, L log L and L (log L)^n end-point estimates are proved.
The problem whether weighted estimates for multilinear Fourier multipliers with Sobolev regularity hold under weak condition on weights is considered.
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.
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 study properties of the bilinear multiplier space. We give a necessary condition for a continuous integrable function to be a bilinear multiplier on variable exponent Lebesgue spaces. And we prove the localization theorem of multipliers on variable exponent Lebesgue spaces. Moreover, we present a Mihlin-Hörmander type theorem for multilinear Fourier multipliers on weighted variable Lebesgue spaces and give some applications.
Boundedness of Multilinear Hilbert Transforms Along Moment Curves
本文通过建立多线性仿积的一致估计并应用索伯列夫平滑不等式，证明了沿矩曲线的多线性希尔伯特变换及其极大函数的有界性。
Characterizations for multi-sublinear operators and their commutators on three kinds of generalized weighted Morrey spaces and applications
本文研究多亚线性算子及其交换子在广义加权Morrey空间上有界的条件，给出充分条件及一类算子的充要条件，并应用于Calderón-Zygmund算子、Littlewood-Paley平方算子、伪微分算子和仿积。
Multilinear multiplier theorems and their applications to the Jacobian and the Hessian determinant
本文建立了Coifman和Meyer多线性乘子定理的几种变体，并提供了现有理论未涵盖的例子，旨在通过分布意义下雅可比行列式和海森矩阵行列式的定义进行应用。
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…
In this paper, we study multilinear Fourier multiplier operators on Hardy spaces. In particular, we prove that the multilinear Fourier multiplier operator of Hörmander type is bounded from $H^{p_1} \times \cdots \times H^{p_m}$ to $H^p$ for $0<p_1,\dots,p_m\le 1$ with $1/p_1 + \cdots 1/p_m = 1/p$, under suitable cancellation conditions. As a result, we extend the trilinear estimates of the authors(arXiv:2107.00225) to general multilinear ones and improve the boundedness result of the authors, Lee, Heo, Hong, Park, and Yang(Math. Ann. 381 : 499-555, 2021) in limiting situations.
Necessary conditions for weighted estimates of Multilinear Multipliers and Pseudo-Differential Operators
本文研究了多线性傅里叶乘法算子和伪微分算子加权估计的必要条件，证明了多重权重假设的最佳性，并改进了经典线性情况下的结果。
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。
We prove L^p estimates for a large class of multi-linear operators, which includes the multi-linear paraproducts studied by Coifman and Meyer, as well as the bilinear Hilbert transform.
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。
We prove $L^p$ estimates for various multi-parameter bi- and trilinear operators with symbols acting on fibers of the two-dimensional functions. In particular, this yields estimates for the general bi-parameter form of the twisted paraproduct studied in arXiv:1011.6140.
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…
We provide necessary and sufficient conditions for multilinear multiplier operators with symbols in $L^r$-based product-type Sobolev spaces uniformly over all annuli to be bounded from products of Hardy spaces to a Lebesgue space. We consider the case $1<r\leq 2$ and we characterize boundedness in terms of inequalities relating the Lebesgue indices (or Hardy indices), the dimension, and the regularity and integrability indices of the Sobolev space. The case $r>2$ cannot be handled by known techniques and remains open. Our result not only extends but also establishes the sharpness of previous r…
We study a multilinear version of Hörmander multiplier theorem, namely \begin{equation*} \Vert T_σ(f_1,\dots,f_n)\Vert_{L^p}\lesssim \sup_{k\in\mathbb{Z}}{\Vert σ(2^k\cdot,\dots,2^k\cdot)\widehat{ϕ^{(n)}}\Vert_{L^{2}_{(s_1,\dots,s_n)}}}\Vert f_1\Vert_{H^{p_1}}\cdots\Vert f_n\Vert_{H^{p_n}}. \end{equation*} We show that the estimate does not hold in the limiting case $\min{(s_1,\dots,s_n)}=d/2$ or $\sum_{k\in J}{({s_k}/{d}-{1}/{p_k})}=-{1}/{2}$ for some $J \subset \{1,\dots,n\}$. This provides the necessary and sufficient condition on $(s_1,\dots,s_n)$ for the boundedness of $T_σ$.
In this paper, the objects of our investigation are some dyadic operators, including dyadic shifts, multilinear paraproducts and multilinear Haar multipliers. We mainly focus on the continuity and compactness of these operators. First, we consider the continuity properties of these operators. Then, by the Fréchet-Kolmogorov-Riesz-Tsuji theorem, the non-compactness properties of these dyadic operators will be studied. Moreover, we show that their commutators are compact with \textit{CMO} functions, which is quite different from the non-compaceness properties of these dyadic operators. These res…
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…
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。
In this article, we provide a multilinear version of the Hörmander multiplier theorem with a Lorentz-Sobolev space condition. The work is motivated by the recent result of the first author and Slavíková where an analogous version of classical Hörmander multiplier theorem was obtained; this version is sharp in many ways and reduces the number of indices that appear in the statement of the theorem. As a natural extension of the linear case, in this work, we prove that if $mn/2<s<mn$, then $$ \big\Vert T_σ(f_1,\dots,f_m)\big\Vert_{L^p((\mathbb{R})^n)}\lesssim \sup_{k\in\mathbb{Z}}\big\Vert σ(2^k\…
We investigate pointwise multipliers on vector-valued function spaces over $\mathbb{R}^d$, equipped with Muckenhoupt weights. The main result is that in the natural parameter range, the characteristic function of the half-space is a pointwise multiplier on Bessel-potential spaces with values in a UMD Banach space. This is proved for a class of power weights, including the unweighted case, and extends the classical result of Shamir and Strichartz. The multiplication estimate is based on the paraproduct technique and a randomized Littlewood-Paley decomposition. An analogous result is obtained fo…
We consider bilinear multipliers that appeared as a distinguished particular case in the classification of two-dimensional bilinear Hilbert transforms by Demeter and Thiele [9]. In this note we investigate their boundedness on Sobolev spaces. Furthermore, we study structurally similar operators with symbols that also depend on the spatial variables. The new results build on the existing L^p estimates for a paraproduct-like operator previously studied by the authors in [5] and [10]. Our primary intention is to emphasize the analogies with Coifman-Meyer multipliers and with bilinear pseudodiffer…
Multilinear transference on product spaces and applications
本文建立了从欧几里得空间到乘积空间的多重线性傅里叶乘子算子的强、弱型转移原理，并应用于获得多重线性及PDE相关乘子算子的界。
We prove multi-parameter Leibniz rules corresponding to flag paraproducts of arbitrary complexity in mixed-norm spaces, including endpoint estimates. The proof relies on multi-linear harmonic analysis techniques and a quantitative treatment of the commutators introduced by Bourgain and Li. The argument is robust and applicable to a generic class of multipliers, including (symmetric) Mikhlin multipliers of positive order and asymmetric variants of partial differential operators and Mikhlin multipliers of positive order.
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 paper, we study the $A_p$-$A_\infty$ estimates for a class of multilinear dyadic positive operators. As applications, the $A_p$-$A_\infty$ estimates for different operators e.g. multilinear square functions and multilinear Fourier multipliers can be deduced very easily.
Let $n\ge 1$ and $\mathfrak{T}_{m}$ be the bilinear square Fourier multiplier operator associated with a symbol $m$, which is defined by $$ \mathfrak{T}_{m}(f_1,f_2)(x) = \biggl( \int_{0}^\infty\Big|\int_{(\mathbb{R}^n)^2} e^{2πix\cdot (ξ_1 +ξ_2) }m(tξ_1,tξ_2) \hat{f}_{1}(ξ_1)\hat{f}_{2}(ξ_2)dξ_1 dξ_2\Big|^2\frac{dt}{t } \biggr)^{\frac 12}. $$ Let $s$ be an integer with $s\in[n+1,2n]$ and $p_0$ be a number satisfying $2n/s\le p_0\le 2$. Suppose that $ν_{\vecω}=\prod_{i=1}^2ω_i^{p/ p_i}$ and each $ω_i$ is a nonnegative function on $\mathbb{R}^n$. In this paper, we show that $\mathfrak{T}_{m}$ i…
Compact sets in quasi-Banach function spaces and multilinear extrapolation of compactness on $L^{p(\cdot)}(w)$
本文通过建立加权Riesz-Kolmogorov定理，在变指数Lebesgue空间上实现了多线性紧算子的外推，并得到了交换子、分数次积分和Fourier乘子的新加权紧性估计。
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.
Calderón-Zygmund operators with noncommuting kernels may fail to be Lp-bounded for $p \neq 2$, even for kernels with good size and smoothness properties. Matrix-valued paraproducts, Fourier multipliers on group vNa's or noncommutative martingale transforms are frameworks where we find such difficulties. We obtain weak type estimates for perfect dyadic CZO's and cancellative Haar shifts associated to noncommuting kernels in terms of a row/column decomposition of the function. Arbitrary CZO's satisfy $H_1 \to L_1$ type estimates. In conjunction with $L_\infty \to BMO$, we get certain row/column …
Quantitative weighted estimates for the multilinear pseudo-differential operators in function spaces
该研究系统推导了重排不变Banach与拟Banach空间中多线性伪微分算子及其交换子的加权有界性和模估计，并将结果推广至多线性Fourier乘子等三类算子。
For each $1 \leq p \leq \infty$, let $W_{p}(\mathbb{R}) = \left\{ f \in L^p(\mathbb{R}): \hat{f} \in L^{p^\prime}(\mathbb{R}) \right\}$ with norm $||f||_{W_{p}(\mathbb{R})} = ||\hat{f}||_{L^{p^\prime}(\mathbb{R})}$. Moreover, let $ Γ= \left\{ ξ_1 + ξ_2 =0\right\} \subset \mathbb{R}^2$ and $a_1,a_2 : \mathbb{R}^2 \rightarrow \mathbb{C}$ satisfy the Hörmander-Mikhlin condition \begin{eqnarray*} \left| \partial^{\vecα} a_j \left(\vecξ\right) \right| \lesssim_{\vecα} \frac{1}{dist(\vecξ, Γ)^{|\vecα|}}~~~\forall \vecξ \in \mathbb{R}^2, j \in \{1, 2\} \end{eqnarray*} for sufficiently many multi-indi…
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.
Coifman--Meyer multipliers represent a very important class of bi-linear singular operators, which were extensively studied and generalized. They have a natural multi-parameter counterpart. Decomposition of those operators into paraproducts, and, more generally to multi-parameter paraproducts is a staple of the theory. In this paper we consider weighted estimates for bi-parameter paraproducts that appear from such multipliers. Then we apply our harmonic analysis results to several complex variables. Namely, we show that a (weighted) Carleson embedding for a scale of Dirichlet spaces from the b…
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。
We develop a special multilinear complex interpolation theorem that allows us to prove an optimal version of the bilinear Hörmander multiplier theorem concerning symbols that lie in the Sobolev space $L^r_s(\mathbb R^{2n})$, $2\le r<\infty$, $rs>2n$, uniformly over all annuli. More precisely, given a smoothness index $s$, we find the largest open set of indices $(1/p_1,1/p_2 )$ for which we have boundedness for the associated bilinear multiplier operator from $L^{p_1}(\mathbb R^{ n})\times L^{p_2} (\mathbb R^{ n})$ to $ L^p(\mathbb R^{ n})$ when $1/p=1/p_1+1/p_2$, $1<p_1,p_2<\infty$.
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。
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.
We dominate non-integral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces. Our assumptions on the operators are minimal and our result applies to an array of situations, whose prototype are Riesz transforms / multipliers or paraproducts associated with a second order elliptic operator. It also applies to such operators whose unweighted continuity is restricted to Lebesgue spaces with certain ranges of exponents $(p_0,q_0)$ where $1\le p_0<2<q_0\le \infty$. The norm estimates obtained are powers $α$ of the characteristic used by Auscher and Mar…
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 main purpose of this paper is to study $L^r$ Hölder type estimates for a bi-parameter trilinear Fourier multiplier with flag singularity, and the analogous pseudo-differential operator, when the symbols are in a certain product form. More precisely, for $f,g,h\in \mathcal{S}(\mathbb{R}^{2})$, the bi-parameter trilinear flag Fourier multiplier operators we consider are defined by $$ T_{m_1,m_2}(f,g,h)(x):=\int_{\mathbb{R}^{6}}m_1(ξ,η,ζ)m_2(η,ζ)\hat f(ξ) \hat g(η)\hat h(ζ)e^{2πi(ξ+η+ζ)\cdot x}dξdηdζ, $$ when $m_1,m_2$ are two bi-parameter symbols. We will show that our problem can be reduced…
Recent progress of Littlewood-paley Theory with chirp function
本文基于啁啾函数发展了分数傅里叶变换下的小沃德-帕利框架，提出了多种估计和理论，包括乘法恒等式、平方函数估计、Marcinkiewicz和Mihlin-Hörmander乘法结果等。
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…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。