On the Hilbert Transform and $C^{1+\ze}$ Families of Lines

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.

Bilinear Hilbert transforms associated to plane curves

We prove that the bilinear Hilbert transforms and maximal functions along certain general plane curves are bounded from $L^2(\mathbb{R})\times L^2(\mathbb{R})$ to $L^1(\mathbb{R})$.

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.

Bilinear Hilbert transforms along curves I. The monomial case

We establish an L^2 \times L^2 to L^1 estimate for the bilinear Hilbert transform along a curve defined by a monomial. Our proof is closely related to multi-linear oscillatory integrals.

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

We show sufficient conditions on matrix weights $U$ and $V$ for the martingale transforms to be uniformly bounded from $L^2(V)$ to $L^2(U)$. We also show that these conditions imply the uniform boundedness of the dyadic shifts as well as the boundedness of the Hilbert transform between these spaces.

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.

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

We study the bilinear Hilbert transform and bilinear maximal functions associated to polynomial curves and obtain uniform $L^r$ estimates for $r>\frac{d-1}{d}$ and this index is sharp up to the end point.

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

We prove $L^p$, $p\in (1,\infty)$ estimates on the Hilbert transform along a one variable vector field acting on functions with frequency support in an annulus. Estimates when $p>2$ were proved by Lacey and Li in \cite{LL1}. This paper also contains key technical ingredients for a companion paper \cite{BT} with Christoph Thiele in which $L^p$ estimates are established for the full Hilbert transform.

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.

On the Hilbert transform and lacunary directions in the plane

Let $H_k$ be the one dimensional Hilbert transform computed in the direction $(1,2^k)$ in the plane. We show that the maximal operator $\sup_k |H_kf|$ maps $L^p$ of the plane into itself for $1<p<\infty$. The same result with the Hilbert transform replaced by the one dimensional maximal function was proved by Nagel, Stein and Wainger in 1978.

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

We prove old and new $L^p$ bounds for the quartile operator, a Walsh model of the bilinear Hilbert transform, uniformly in the parameter that models degeneration of the bilinear Hilbert transform. We obtain the full range of exponents that can be expected from known bounds in the degenerate and non-degenerate cases. For the new estimates with exponents p close to 1 the argument relies on a multi-frequency Calderon-Zygmund decomposition.

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

While exploiting the generalized Parseval equality for the Mellin transform, we derive the reciprocal inverse operator in the weighted L_2-space related to the Hilbert transform on the nonnegative half-axis. Moreover, employing the convolution method, which is based on the Mellin-Barnes integrals, we prove the corresponding convolution and Titchmarsh's theorems for the half-Hilbert transform. Some applications to the solvability of a new class of singular integral equations are demonstrated. Our technique does not require the use of methods of the Riemann-Hilbert boundary value problems for an…

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.

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

We prove that the analogue for the Hilbert transform of a classical weighted inequality by Fefferman and Stein for the Hardy Littlewood maximal operator does not hold. This is a sequel to paper arXiv:1008.3943 by the first author, which shows the same for the general Haar multiplier operator.

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.

Triple Hilbert transforms along polynomial surfaces

This paper has been withdrawn since it contains some discrepancy with othe authers's recent result. We will not post this until this discrepancy is resolved.

《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.

Variational bounds for a dyadic model of the bilinear Hilbert transform

We prove variation-norm estimates for the Walsh model of the truncated bilinear Hilbert transform, extending related results of Lacey, Thiele, and Demeter. The proof uses analysis on the Walsh phase plane and two new ingredients: (i) a variational extension of a lemma of Bourgain by Nazarov-Oberlin-Thiele, and (ii) a variation-norm Rademacher-Menshov theorem of Lewko-Lewko.

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

We consider the two weight problem for the Hilbert transform, namely the question of finding real-variable characterization of those pair of weights for which the Hilbert transform acts boundedly on $ L ^2 $ of the weights. Such a characterization is known subject to certain side conditions. We give a new proof, simpler in many details, of the best such result. In addition, we analyze underlying assumptions in the proof, especially in terms of two alternate side conditions. A new characterization in the case of one doubling weight is given.

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.

Sampling and interpolation for the discrete Hilbert and Kak-Hilbert transforms

The goal of the paper is to obtain analogs of the sampling theorems and of the Riesz-Boas interpolation formulas which are relevant to the Discrete Hilbert and Kak-Hilbert transforms in $l^{2}$.

On maximal function of discrete rough truncated Hilbert transforms

We prove the weak type (1,1) estimate for maximal function of the truncated rough Hilbert transform considered in [9] and [10]

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。

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.

On the two dimensional Bilinear Hilbert Transform

We investigate the Bilinear Hilbert Transform in the plane and the pointwise convergence of bilinear averages in Ergodic theory, arising from $\Z^2$ actions. Our techniques combine novel one and a half dimensional phase-space analysis with more standard one dimensional theory.

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

This paper has been withdrawn by the author due to some error in the result

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

We consider multilinear averages in ergodic theory and harmonic analysis and prove their divergence in some range of $L^p$ spaces, with $p$ close enough to 1. We also prove that the trilinear Hilbert transform is unbounded in a similar range of $L^p$ spaces.

The Hilbert transform, rearrangements, and logarithmic determinants

This is an extended version of notes prepared for the talk at the conference "Rajchman-Zygmund-Marcinkiewicz 2000" based on recent works of the authors.

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。

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。

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.

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.

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.