关于希尔伯特变换与C^2直线族

Schatten classes and commutator in the two weight setting, I. Hilbert transform

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.

加权Lebesgue空间上希尔伯特变换的交换子的精确估计

Sharp estimates for the commutator of the Hilbert transform on weighted Lebesgue spaces

由于结果中的一些错误，本文已被作者撤回

Hilbert变换的Rellich恒等式

Rellich identities for the Hilbert transform

本文利用Rellich恒等式和图Lipschitz区域上的Neumann问题证明涉及共形映射的Hilbert变换恒等式，并由此得到新的加权L^2估计及与Helson-Szegö定理相关的精确范数界。

与多项式相关的多重希尔伯特变换

Multiple Hilbert transform associated with polynomials

研究了与多项式相关的多重希尔伯特变换的$L^p$有界性的条件。

Quasi pieces of the bilinear Hilbert transform incorporated into a paraproduct

We prove the boundedness of a class of tri-linear operators consisting of a quasi piece of bilinear Hilbert transform whose scale equals to or dominates the scale of its linear counter part. Such type of operators is motivated by the tri-linear Hilbert transform and its curved versions.

Remarks on the maximal operator and Hilbert transform along variable parabolas

We prove the boundedness of the maximal operator and Hilbert transform along certain variable parabolas in $L^p$ for $p>p_0$ with some $p_0\in (1, 2)$. Connections with the Hilbert transform along vector fields and the polynomial Carleson's maximal operator are also discussed.

Cancellation for the simplex Hilbert transform

We show that the truncated simplex Hilbert transform enjoys some cancellation in the sense that its norm grows sublinearly in the number of scales retained in the truncation. This extends the recent result by Tao on cancellation for the multilinear Hilbert transform. Our main tool is the Hilbert space regularity lemma due to Gowers, which enables a very short proof.

Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^2$ boundedness

We prove the $L^2$ boundedness of the directional Hilbert transform in the plane relative to measurable vector fields which are constant on suitable Lipschitz curves.

与平面曲线相关的双线性希尔伯特变换

Bilinear Hilbert transforms associated to plane curves

证明了双线性希尔伯特变换和极大函数沿某些一般平面曲线从$L^2(\mathbb{R})\times L^2(\mathbb{R})$到$L^1(\mathbb{R})$是有界的.

Hilbert transform along measurable vector fields constant on Lipschitz curves: $L^p$ boundedness

We prove the $L^p (p > 3/2)$ boundedness of the directional Hilbert transform in the plane relative to measurable vector fields which are constant on suitable Lipschitz curves.

沿曲线I的双线性希尔伯特变换。单项的情况

Bilinear Hilbert transforms along curves I. The monomial case

我们建立了一个L^2 \times L^2到L^1估计的双线性希尔伯特变换沿曲线定义的单项式。我们的证明是密切相关的多线性振荡积分。

鞅变换、并元移位和希尔伯特变换：矩阵加权空间有界的一个充分条件

Martingale transforms, the dyadic shift and the Hilbert transform: a sufficient condition for boundedness between matrix weighted spaces

给出了鞅变换从$L^2(V)$到$L^2(U)$是一致有界的矩阵权$U$和$V$的充分条件.我们还表明，这些条件意味着一致有界的并矢位移以及这些空间之间的希尔伯特变换的有界性。

Sparse domination of Hilbert transforms along curves

We obtain sharp sparse bounds for Hilbert transforms along curves in $\mathbb{R}^n$, and derive as corollaries weighted norm inequalities for such operators. The curves that we consider include monomial curves and arbitrary $C^n$ curves with nonvanishing torsion.

双线性希尔伯特变换和多项式双线性极大函数的一致估计

Uniform estimates for bilinear Hilbert transform and bilinear maximal functions associated to polynomials

本文研究了多项式曲线上的双线性希尔伯特变换和双线性极大函数，得到了$r>\frac{d-1}{d}$的一致$L^r$估计，并且该估计到终点是尖锐的.

On the $l^p$-norm of the discrete Hilbert transform

Using a representation of the discrete Hilbert transform in terms of martingales arising from Doob $h$-processes, we prove that its $l^p$-norm, $1<p<\infty$, is bounded above by the $L^p$-norm of the continuous Hilbert transform. Together with the already known lower bound, this resolves the long-standing conjecture that the norms of these operators are equal.

沿向量场的希尔伯特变换的单环$L^p$估计

Single annulus $L^p$ estimates for Hilbert transforms along vector fields

我们证明了$L^p$，$p\in (1,\infty)$估计的希尔伯特变换沿一个变量的向量场作用于函数的频率支持在一个环。当Lacey和Li在\cite{LL1}中证明$p>2$时的估计。本文还包含关键技术成分的配套文件\cite{BT}与克里斯托夫Thiele在其中$L^p$估计建立了完整的希尔伯特变换。

On exceptional sets of Hilbert transform

We prove several theorems concerning the exceptional sets of Hilbert transform on the real line. In particular, it is proved that any null set is exceptional set for the Hibert transform of an indicator function. The paper also provides a real variable approach to the Kahane-Katsnelson theorem on divergence of Fourier series.

Power-type cancellation for the simplex Hilbert transform

We prove $\mathrm{L}^p$ bounds for the truncated simplex Hilbert transform which grow with a power less than one of the truncation range in the logarithmic scale.

平面上的希尔伯特变换与缺项方向

On the Hilbert transform and lacunary directions in the plane

令$H_k$为在平面中的方向$(1,2^k)$上计算的一维希尔伯特变换。我们证明了极大算子$\sup_k |H_kf|$将平面的$L^p$映射到自身的$1<p<\infty$。Nagel，Stein和Wainger在1978年用一维极大函数代替希尔伯特变换证明了同样的结果。

双线性希尔伯特变换Walsh模型的新的一致界

New uniform bounds for a Walsh model of the bilinear Hilbert transform

我们证明了旧的和新的$L^p$界的四分位数算子，沃尔什模型的双线性希尔伯特变换，统一的参数模型退化的双线性希尔伯特变换。我们得到了完整的范围内的指数，可以预期从已知的退化和非退化的情况下的界限。对于指数p接近1的新估计，参数依赖于多频Calderon-Zygmund分解。

半希尔伯特变换的新逆、卷积和Titchmarsh定理

New inversion, convolution and Titchmarsh's theorems for the half-Hilbert transform

利用Mellin变换的广义Parseval等式，在非负半轴上导出了与希尔伯特变换相关的加权L_2-空间中的互逆算子.此外，利用基于Mellin-Barnes积分的卷积方法，证明了半希尔伯特变换的相应卷积定理和Titchmarsh定理.给出了一类新的奇异积分方程可解性的一些应用。我们的技术不需要使用的方法的Riemann-Hilbert边值问题的解析函数。同样的方法将被应用在即将到来的研究中，以反转半哈特莱变换，并建立其卷积定理。

Hilbert transforms along variable planar curves: Lipschitz regularity

In this paper, for $1<p<\infty$, we obtain the $L^p$-boundedness of the Hilbert transform $H^γ$ along a variable plane curve $(t,u(x_1, x_2)γ(t))$, where $u$ is a Lipschitz function with small Lipschitz norm, and $γ$ is a general curve satisfying some suitable smoothness and curvature conditions.

On the Boundedness of The Bilinear Hilbert Transform along "non-flat" smooth curves

We are proving $L^2(\R)\times L^2(\R)\,\rightarrow\,L^1(\R)$ bounds for the bilinear Hilbert transform $H_Γ$ along curves $Γ=(t,-γ(t))$ with $γ$ being a smooth "non-flat" curve near zero and infinity.

希尔伯特变换不将L ^1（Mw）映射到L^{1，\infty }（w）

The Hilbert transform does not map L^1(Mw) to L^{1,\infty}(w)

本文证明了Festivalman和Stein关于Hardy Littlewood极大算子的一个经典加权不等式的希尔伯特变换的类似不成立.这是第一作者的论文arXiv：1008.3943的续篇，该论文对一般的Haar乘子算子也给出了同样的结果。

Averages of simplex Hilbert transforms

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.

论希尔伯特变换的局部正则性

On the Local Regularity of the Hilbert Transform

本文研究希尔伯特变换的局部正则性，证明了其局部光滑性与实解析性。

沿多项式曲面的三重希尔伯特变换

Triple Hilbert transforms along polynomial surfaces

这篇论文因与其他作者最近的结果有出入而被撤回。我们不会发布这个，直到这个矛盾得到解决。

《三线性平滑不等式与三角Hilbert变换的一个变体》的学习指南

A study guide for "Trilinear smoothing inequalities and a variant of the triangular Hilbert transform"

本文是一篇学习指南，介绍了带曲率的三角Hilbert变换$L^p$有界性证明中的核心创新——三线性平滑不等式，并对比了传统振荡积分技术的局限性。

Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann-Hilbert approach

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms $\mathcal{H}_L:L^2([b_L,0])\to L^2([0,b_R])$ and $\mathcal{H}_R:L^2([0,b_R])\to L^2([b_L,0])$. These operators arise when one studies the interior problem of tomography. The diagonalization of $\mathcal{H}_R,\mathcal{H}_L$ has been previously obtained, but only asymptotically when $b_L\neq-b_R$. We implement a novel approach based on the method of matrix Riemann-Hilbert problems (RHP) which diagonalizes $\mathcal{H}_R,\mathcal{H}_L$ explicitly. We also find the asymptotics …

Vector-valued Hilbert transforms along curves

In this paper, we show that Hilbert transforms along some curves are bounded on $L^p({\mathbb R}^n;X)$ for some $1<p<\infty$ and some UMD spaces $X$. In particular, we prove that the Hilbert transform along some curves are completely $L^p$-bounded in the terminology from operator space theory. Moreover, we obtain the $L^p(\mathbb{R}^n;X)$-boundedness of anisotropic singular integrals by using the "method of rotations" of Calderón-Zygmund. All these results extend the existing related ones.

Dyadic triangular Hilbert transform of two general and one not too general function

The so-called triangular Hilbert transform is an elegant trilinear singular integral form which specializes to many well studied objects of harmonic analysis. We investigate $L^p$ bounds for a dyadic model of this form in the particular case when one of the functions on which it acts is essentially one-dimensional. This special case still implies dyadic analogues of boundedness of the Carleson maximal operator and of the uniform estimates for the one-dimensional bilinear Hilbert transform.

双线性希尔伯特变换的并元模型的变分界

Variational bounds for a dyadic model of the bilinear Hilbert transform

证明了截断双线性希尔伯特变换的Walsh模型的变分范数估计，推广了Lacey，Thiele和Demeter的相关结果.证明使用沃尔什相平面上的分析和两个新的成分：（i）Nazarov-Oberlin-Thiele的Bourgain引理的变分扩展，以及（ii）Lewko-Lewko的变分范数Rademacher-Menshov定理。

希尔伯特变换、日冕和能量条件的双权不等式

The Two Weight Inequality for Hilbert Transform, Coronas, and Energy Conditions

我们考虑的两个重量问题的希尔伯特变换，即问题的实变量特征的那些对的重量，希尔伯特变换有界作用于$ L ^2 $的重量。这样的表征是已知的，受制于某些附带条件。我们给出了一个新的证明，在许多细节上更简单，最好的这样的结果。此外，我们分析了证明中的基本假设，特别是在两个交替的边条件。在一个二重权的情况下，给出了一个新的刻画。

Sparse Bounds for Discrete Quadratic Phase Hilbert Transform

Consider the discrete quadratic phase Hilbert Transform acting on $\ell^{2}$ finitely supported functions $$ H^α f(n) : = \sum_{m \neq 0} \frac{e^{2 πiαm^2} f(n - m)}{m}. $$ We prove that, uniformly in $α\in \mathbb{T}$, there is a sparse bound for the bilinear form $\langle H^α f , g \rangle$. The sparse bound implies several mapping properties such as weighted inequalities in an intersection of Muckenhoupt and reverse Hölder classes.

关于二维双线性希尔伯特变换

On the two dimensional Bilinear Hilbert Transform

本文研究了平面上的双线性希尔伯特变换和遍历论中双线性平均的逐点收敛性，这些双线性平均是由$\Z^2$作用引起的.我们的技术结合了新颖的一维半相空间分析与更标准的一维理论。

组合平均的发散性与三线性希尔伯特变换的无界性

Divergence of combinatorial averages and the unboundedness of the trilinear Hilbert transform

我们考虑遍历论和调和分析中的多重线性平均，并证明了它们在$L^p$空间的某些范围内的发散性，其中$p$足够接近1。我们还证明了三线性希尔伯特变换在类似的$L^p$空间范围内是无界的。

希尔伯特变换、重排和对数行列式

The Hilbert transform, rearrangements, and logarithmic determinants

这是根据作者最近的作品为“Rajchman-Zygmund-Marcinkiewicz 2000”会议上的演讲准备的笔记的扩展版本。

Krein condition and the Hilbert transform

Krein condition have been used as a qualitative result to show the M-indeterminacy of some kind of densities. In this work we use results from the theory of the Hilbert transform to construct families of densities having all the same finite moment sequence as a density $f$ with finite logarithmic integral. Actually, our approach explicitly gives Stieltjes classes with center at $f$ and perturbations involving the Hilbert transform of $\ln f.$ We consider densities supported on the whole real line or the positive half line.

Hilbert Transforms and Sum Rules of Bessel Moments

Using Hilbert transforms, we establish two families of sum rules involving Bessel moments, which are integrals associated with Feynman diagrams in two-dimensional quantum field theory. With these linear relations among Bessel moments, we verify and generalize two conjectures by Bailey-Borwein-Broadhurst-Glasser and Broadhurst-Mellit.

变换对计算的Riemann-Hilbert方法

A Riemann-Hilbert approach to the computation of transform pairs

提出一种统一方法，结合谱理论、Riemann-Hilbert问题和逆散射理论，构建并数值评估线性变系数偏微分方程的变换对，通过混合解析-数值策略实现精确计算，尤其适用于间断系数情形。

The Fourier-Like and Hartley-Like Wavelet Analysis Based on Hilbert Transforms

In continuous-time wavelet analysis, most wavelet present some kind of symmetry. Based on the Fourier and Hartley transform kernels, a new wavelet multiresolution analysis is proposed. This approach is based on a pair of orthogonal wavelet functions and is named as the Fourier-Like and Hartley-Like wavelet analysis. A Hilbert transform analysis on the wavelet theory is also included.