Multi-parameter paraproducts

Paraproducts with flag singularities I. A case study

We prove L^p estimates for a tri-linear operator, whose symbol is given by the product of two standard symbols, satisfying the well known Marcinkiewicz-Hormander-Mihlin condition. Our main result contains in particular the classical Coifman-Meyer theorem. This tri-linear operator is the simplest example of a large class of multi-linear operators, which we called "paraproducts with flag singularities".

Uniform estimates for paraproducts and related multilinear multipliers

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 $R^d$.

Logarithmic mean oscillation on the Polydisc, Multi-parameter paraproducts and iterated commutators

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.

Vector valued inequalities for families of bilinear Hilbert transforms and applications to bi-parameter problems

Muscalu, Pipher, Tao and Thiele \cite{MPTT} showed that the tensor product between two one dimensional paraproducts (also known as bi-parameter paraproduct) satisfies all the expected $L^p$ bounds. In the same paper they showed that the tensor product between two bilinear Hilbert transforms is unbounded in any range. They also raised the question about $L^p$ boundedness of the bilinear Hilbert transform tensor product with a paraproduct. We answer their question by obtaining a wide range of estimates for this hybrid bilinear operator. Our method relies on new vector valued estimates for a fami…

Logarithmic mean oscillation on the polydisc, endpoint results for multi-parameter paraproducts, and commutators on BMO

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).

旗多分辨率：T1、仿积与交换子

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。

Boundedness of some multi-parameter fiber-wise multiplier operators

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.

Multi-parameter estimates via operator-valued shifts

We prove new results for multi-parameter singular integrals. For example, we prove that bi-parameter singular integrals in $\mathbb{R}^{n+m}$ satisfying natural $T1$ type conditions map $L^q(\mathbb{R}^n; L^p(\mathbb{R}^m;E))$ to $L^q(\mathbb{R}^n; L^p(\mathbb{R}^m;E))$ for all $p,q \in (1,\infty)$ and UMD function lattices $E$. This result is shown to hold even in the $\mathcal{R}$-boundedness sense for all suitable families of bi-parameter singular integrals. On the technique side we demonstrate how many dyadic multi-parameter operators can be bounded by using, and further developing, the th…

混合范数空间上紧性的多线性外推

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。

Multi-linear operators given by singular multipliers

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.

A T(b) Theorem on Product Spaces

The main result of this paper is a bi-parameter T(b) theorem for the case that b is a tensor product of two pseudo-accretive functions. In the proof, we also discuss the L^2 boundedness of different types of the b-adapted bi-parameter paraproducts.

Upper Bound for Multi-parameter Iterated Commutators

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.

Bi-parameter Carleson embeddings with product weights

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…

Multi-parameter flag Leibniz rules of arbitrary complexity in mixed-norm spaces

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.

Zygmund 膨胀：双线性分析与交换子估计

Zygmund dilations: bilinear analysis and commutator estimates

本文研究了 $\mathbb{R}^3$ 中 Zygmund 膨胀下的双线性理论与交换子估计，构建了双线性二进多分辨率方法，证明了无抛积双线性 $T1$ 定理，并在不依赖加权估计的情况下建立了线性交换子估计。

Higher order Journe commutators and characterizations of multi-parameter BMO

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…

Commutators of bilinear bi-parameter singular integrals

We study the boundedness properties of commutators formed by $b$ and $T$, where $T$ is a bilinear bi-parameter singular integral satisfying natural $T1$ type conditions and $b$ is a little BMO function. For paraproduct free bilinear bi-parameter singular integrals $T$ we prove that $[b, T]_1 \colon L^p(\mathbb{R}^{n+m}) \times L^q(\mathbb{R}^{n+m}) \to L^r(\mathbb{R}^{n+m})$ in the full range $1 < p, q \le \infty$, $1/2 < r < \infty$ satisfying $1/p+1/q = 1/r$. A special case is when $T$ is a bilinear bi-parameter multiplier. We also prove the corresponding Banach range result for all singular…

多线性小波紧致T(1)定理

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。

Some obstacles in characterising the boundedness of bi-parameter singular integrals

The famous $T1$ theorem for classical Calderón-Zygmund operators is a characterisation for their boundedness in $L^{2}$. In the bi-parameter case, on the other hand, the current $T1$ theorem is merely a collection of sufficient conditions. This difference in mind, we study a particular dyadic bi-parameter singular integral operator, namely the full mixed bi-parameter paraproduct $P$, which is precisely the operator responsible for the outstanding problems in the bi-parameter theory. We make several remarks about $P$, the common theme of which is to demonstrate the delicacy of the problem of fi…

多参数多线性Fourier乘子算子的一个精确Hörmander估计

A sharp Hörmander estimate for multi-parameter and multi-linear Fourier multiplier operators

本文研究多线性多参数Fourier乘子的Hörmander型定理，给出了基于L^u的Sobolev范数刻画下的光滑性假设结果，在多参数双线性情形下精确、多线性情形下几乎精确，且改进了单参数多线性情形的已有结论。

Multi-parameter singular integral operators and representation theorem

We formulate a class of singular integral operators in arbitrarily many parameters using mixed type characterizing conditions. We also prove a multi-parameter representation theorem saying that a general operator in our class can be represented as an average of sums of dyadic shifts, which implies a new multi-parameter $T1$ theorem as a byproduct. Furthermore, an equivalence result between ours and Journé's class of multi-parameter operators is established, whose proof requires the multi-parameter $T1$ theorem. These results generalize to arbitrarily many parameters recent results of Hytönen, …

多参数电影曲率

A multi-parameter cinematic curvature

本文提出多参数电影曲率条件，证明相关极大算子的$L^p$有界性，并验证了Zahl的局部光滑猜想。

Regularity of multi-parameter Fourier integral operator

We study a family of Fourier integral operators, by allowing their symbols to satisfy a multi-parameter differential inequality. We extend the sharp L^p-result obtained by Seeger, Sogge and Stein to product spaces.

Two-weight commutator estimates: general multi-parameter framework

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.

Commutators of multi-parameter flag singular integrals and applications

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…

多参数分数次积分的加权$L^p$范数不等式

Weighted $L^p$-norm inequality of multi-parameter fractional integration

本文研究核在每个坐标子空间上具有奇异性的强分数次积分算子，证明了当多参数theta-bump特征有限时，相应的双权$L^p$范数不等式成立。

The multi-parameter Hausdorff operators on $H^1$ and $L^p$

In the present paper, we characterize the nonnegative functions $φ$ for which the multi-parameter Hausdorff operator $\mathcal H_φ$ generated by $φ$ is bounded on the multi-parameter Hardy space $H^1(\mathbb R\times\cdots\times\mathbb R)$ or $L^p(\mathbb R^n)$, $p\in [1,\infty]$. The corresponding operator norms are also obtained. Our results improve some recent results in \cite{FZ, LM2, LM3, We} and give an answer to an open question posted by Liflyand \cite{Li}.

Uniform multi-parameter limit transitions in the Askey tableau

Extended abstract for the Proceedings of the Conference ``Modern developments in complex analysis and related topics'' (on the occasion of the 70th birthday of prof.\ dr.\ J. Korevaar), University of Amsterdam, January 27--29, 1993.

关于 Bellow-Furstenberg 问题的一个多参数变体

On a multi-parameter variant of the Bellow-Furstenberg problem

本文通过建立多参数极大与振荡不等式，证明了多参数多项式遍历平均在 $L^p$ 上的范数收敛和几乎处处逐点收敛，肯定回答了 Bellow-Furstenberg 问题的多参数变体，并首次系统引入多参数振荡半范数及圆法新思想。

Oscillation inequalities in ergodic theory and analysis: one-parameter and multi-parameter perspectives

In this survey we review useful tools that naturally arise in the study of pointwise convergence problems in analysis, ergodic theory and probability. We will pay special attention to quantitative aspects of pointwise convergence phenomena from the point of view of oscillation estimates in both the single and several parameter settings. We establish a number of new oscillation inequalities and give new proofs for known results with elementary arguments.

Some properties related to trace inequalities for the multi-parameter Hardy operators on poly-trees

In this note we investigate the multi-parameter Potential Theory on the weighted $d$-tree (Cartesian product of several copies of uniform dyadic tree), which is connected to the discrete models of weighted Dirichlet spaces on the polydisc. We establish some basic properties of the respective potentials, capacities and equilibrium measures (in particular in the case of product polynomial weights). We explore multi-parameter Hardy inequality and its trace measures, and discuss some open problems of potential-theoretic and combinatorial nature.

Multi-Parameter Div-Curl Lemmas

We study the possible analogous of the Div-Curl Lemma in classical harmonic analysis and partial differential equations, but from the point of view of the multi-parameter setting. In this context we see two possible Div-Curl lemmas that arise. Extensions to differential forms are also given.

Multi-parameter Carnot-Caratheodory balls and the theorem of Frobenius

We study multi-parameter Carnot-Caratheodory balls, generalizing results due to Nagel, Stein, and Wainger in the single parameter setting. The main technical result is seen as a uniform version of the theorem of Frobenius. In addition, we study maximal functions associated to certain multi-parameter families of Carnot-Caratheodory balls.