The (weak-$L^2$) Boundedness of The Quadratic Carleson Operator
Victor Lie
证明了具有二次多项式相函数的广义Carleson算子是弱型（2，2）算子.为此，我们引入了一种新的方法的二次相位的时频分析。
We prove that the generalized Carleson operator with polynomial phase function of degree two is of weak type (2,2). For this, we introduce a new approach to the time-frequency analysis of the quadratic phase.
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On the Boundedness of the Carleson Operator near $L^1$
基于作者在“多项式Carleson算子”中所阐述的离散化方法，我们发展了Carleson算子的Calderon-Zygmund型分解。因此，通过一个单一的方法，不使用外推技术，我们恢复了以前已知的结果，关于最大的重排不变空间的功能与几乎处处收敛傅里叶级数。
We prove affirmatively the one dimensional case of a conjecture of Stein regarding the $L^p$-boundedness of the Polynomial Carleson operator, for $1<p<\infty$. The proof is based on two new ideas: i) developing a framework for \emph{higher-order wave-packet analysis} that is consistent with the time-frequency analysis of the (generalized) Carleson operator, and ii) a new tile discretization of the time-frequency plane that has the major consequence of \emph{eliminating the exceptional sets} from the analysis of the Carleson operator. As a further consequence, we are able to provide the full $L…
L^p bounds for a maximal dyadic sum operator
证明了$\rn$上极大并元和算子在值域$1<p<\infty$内的$L^p$界.这个极大算子提供了Carleson算子的离散多维模型。它的有界性是通过简单地扭曲莱西和蒂勒给出的Carleson定理的证明得到的，由Pramanik和Terwilleger在高维中改编。在一维，这个最大并元和的$\lp$有界性特别意味着另一种证明的亨特扩展Carleson的定理几乎处处收敛的傅里叶积分。
Carleson's theorem with quadratic phase
Carleson关于傅里叶级数逐点收敛的定理提供了极大算子的界限，其中最大值取相位参数的所有线性函数的选择。我们扩展到所有的二次选择相函数。具体来说，我们证明了下面的最大算子将$L^p$映射到$1<p<\infty$的自身。$$ \sup_a \sup_b |\int e^{i(ay^2+by)}f(x-y)dy/y| $$
We prove the $L^p$-boundedness, $1<p<\infty$, of the Polynomial Carleson operator in general dimension. This follows the author's resolution of the one dimensional case as well as the work of Zorin-Kranich on the higher dimensional case in the setting $2\leq p<\infty$. The techniques used in this paper are direct adaptations and natural extensions to the higher dimensional case of the one-dimensional methods developed by the author.
We make progress on an interesting problem on the boundedness of maximal modulations of the Hilbert transform along the parabola. Namely, if we consider the multiplier arising from it and restrict it to lines, we prove uniform $L^p$ bounds for maximal modulations of the associated operators. Our methods consist of identifying where to use effectively the polynomial Carleson theorem, and where we can take advantage of the presence of oscillation to obtain decay through the $TT^*$ method.
A Carleson type theorem for a Cantor group model of the scattering transform
我们考虑一个基本的$d$ -adic模型的散射线上的变换。我们证明了这种散射变换的$L^2$界和Carleson型极大算子的弱$L^2$界。后者暗示了具有$L^2(R)$中位势的Dirac型算子的广义本征函数的$d$-adic模型的有界性.我们表明，这一结果不能通过估计的散射变换的自然多线性展开的条款。
In this paper, we study the dyadic Carleson Embedding Theorem in the matrix weighted setting. We provide two new proofs of this theorem, which highlight connections between the matrix Carleson Embedding Theorem and both maximal functions and $H^1$-BMO duality. Along the way, we establish boundedness results about new maximal functions associated to matrix $A_2$ weights and duality results concerning $H^1$ and BMO sequence spaces in the matrix setting. As an application, we then use this Carleson Embedding Theorem to show that if $S$ is a sparse operator, then the operator norm of $S$ on $L^2(W…
We prove that a variety of oscillatory and polynomial Carleson operators are uniformly bounded on the family of parameters under considerations. As a particular application of our techniques, we prove uniform bounds for oscillatory Carleson operators near a single scale version of the quadratic Carleson operator.
Quantitative versions of weighted estimates obtained by F. Ruiz and J.L. Torrea in the 80's for the operator \[ \mathcal{M}f(x,t)=\sup_{x\in Q,\,l(Q)\geq t}\frac{1}{|Q|}\int_{Q}|f(x)|dx \qquad x\in\mathbb{R}^{n}, \, t \geq0 \] are obtained. As a consequence, some sufficient conditions for the boundedness of $\mathcal{M}$ in the two weight setting in the spirit of the results obtained by C. Pérez and E. Rela and very recently by M.T. Lacey and S. Spencer for the Hardy-Littlewood maximal operator are derived. As a byproduct some new quantitative estimates for the Poisson integral are obtained.
Weak $L^2$ bound of the lacunary Carleson operator for the non-linear Fourier transform
本文证明了在位势属于$L^1$的条件下，与$SU(1,1)$值非线性傅里叶变换相关的缺项Carleson极大函数具有弱$L^2$有界性。
We prove $L^p$-boundedness of variational Carleson operators for functions valued in intermediate UMD spaces. This provides quantitative information on the rate of convergence of partial Fourier integrals of vector-valued functions. Our proof relies on bounds on wave packet embeddings into outer Lebesgue spaces on the time-frequency-scale space $\mathbb{R}^3_+$, which are the focus of this paper.
On Polynomial Carleson operators along quadratic hypersurfaces
本文研究二次超曲面上的多项式Carleson算子，证明其对任意非退化二次型和所有1<p<∞的Lp空间存在先验界，方法适用于任意符号二次型，突破了此前仅正定情形的限制。
Maximal operators on variable Lebesgue spaces with weights related to oscillations of Carleson curves
本文证明了极大算子在带权的变Lebesgue空间$φ_{t,γ}(τ)=|(τ-t)^γ|$（其中$γ$为复数）上在任意Carleson曲线上有界的充分条件.如果曲线在点$t$处具有不同的螺旋指数，并且$γ$不是真实的，则$φ_{t,γ}$是一个摆锤，其位于最近由V.Kokilashvili，N. Samko和S.三哥
Consider the discrete maximal function acting on $\ell^2(\mathbb Z)$ functions \[ \mathcal{C}_Λ f( n ) := \sup_{ λ\in Λ} \left| \sum_{m \neq 0} f(n-m) \frac{e^{2 πiλm^2}} {m} \right| \] where $Λ\subset [0,1]$. We give sufficient conditions on $Λ$, met by certain kinds of Cantor sets, for this to be a bounded sublinear operator. This result is a discrete analogue of E. M. Stein's integral result, that the maximal operator below is bounded on $L^2(\mathbb R)$. \[ \mathcal{C}_2 f(x):= \sup_{λ\in \mathbb R} \left| \int f(x-y) \frac{e^{2πi λy^2}}{y} \ dy \right|.\] The proof of our result relies he…
Characterization of infinitesimal boundedness of Schrödinger operator
本文利用 Carleson 条件、局部化估计和容量理论，刻画了薛定谔算子势函数 V 的加权无穷小有界性，推广了 Maz'ya 与 Verbitsky 的经典结果。
On the Boundedness in $H^{1/4}$ of the Maximal Square Function Associated with the Schroedinger Equation
线性薛定谔方程的一个长期存在的猜想指出，在索伯列夫空间的意义下，$L^2$中的1/4导数足以使该方程的解在任何维度上几乎处处收敛到初始数据。这是唯一已知的是真实的一维工作的Carleson。在本文中，我们证明了该猜想是正确的球平均。更精确地说，我们证明了Sobolev类$H^{1/4}(R^n)$上相关极大平方函数在任意维数n上的$L^2$有界性.
Remark on the Boundedness of the Cauchy Singular Integral Operator on Variable Lebesgue Spaces with Radial Oscillating Weights
最近，V. Kokilashvili，N. Samko和S. Samko证明了Carleson曲线上Cauchy奇异积分算子在带径向振动权的变Lebesgue空间上有界的一个充分条件。这个条件是制定的Matuszewska-Orlicz指数的权重。我们证明了他们的结果的部分逆。
Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves
1968年，Israel Gohberg和Naum Krupnik发现了李雅普诺夫曲线上Lebesgue空间$L^p(Γ)$上具有分段连续系数的奇异积分算子的局部谱具有圆弧形状.大约25年后，Albrecht Böttcher和Yuri Karlovich意识到，当李雅普诺夫曲线变形为任意Carleson曲线时，这些圆弧变形为具有中值分离点的所谓对数叶。我们表明，这一结果仍然有效的可变Lebesgue空间$L^{p(\cdot)}(Γ)$，其中$p:Γ\to(1,\infty)$满足Dini-利普希茨条件的更一般的设置。证明的主要内容之一是一个新的充分条件的Cauchy奇异积分算子的有界性的可变Lebesgue空间的重量有关的Carleson曲线的振动。
Let $p\in (1,\infty)$. In this paper, for any given measurable function $u:\ \mathbb{R}\rightarrow \mathbb{R}$ and a generalized plane curve $γ$ satisfying some conditions, the $L^p(\mathbb{R}^2)$ boundedness of the Hilbert transform along the variable plane curve $u(x_1)γ$ $$H_{u,γ}f(x_1,x_2):=\mathrm{p.\,v.}\int_{-\infty}^{\infty}f(x_1-t,x_2-u(x_1)γ(t)) \,\frac{\textrm{d}t}{t}, \quad \forall\, (x_1,x_2)\in\mathbb{R}^2, $$ is obtained. At the same time, the $L^p(\mathbb{R})$ boundedness of the corresponding Carleson operator along the general curve $γ$ $$\mathcal{C}_{u,γ}f(x):=\mathrm{p.\,v.}…
We prove $L^p$ bounds for partial polynomial Carleson operators along monomial curves $(t,t^m)$ in the plane $\mathbb{R}^2$ with a phase polynomial consisting of a single monomial. These operators are "partial" in the sense that we consider linearizing stopping-time functions that depend on only one of the two ambient variables. A motivation for studying these partial operators is the curious feature that, despite their apparent limitations, for certain combinations of curve and phase, $L^2$ bounds for partial operators along curves imply the full strength of the $L^2$ bound for a one-dimensio…
In this work we extend consideration of the polynomial Carleson operator to the setting of a Radon transform acting along the paraboloid in $\mathbb{R}^{n+1}$ for $n \geq 2$. Inspired by work of Stein and Wainger on the original polynomial Carleson operator, we develop a method to treat polynomial Carleson operators along the paraboloid via van der Corput estimates. A key new step in the approach of this paper is to approximate a related maximal oscillatory integral operator along the paraboloid by a smoother operator, which we accomplish via a Littlewood-Paley decomposition and the use of a s…
We prove the boundedness of the maximal operator and Hilbert transform along certain variable parabolas in $L^p$ for $p>p_0$ with some $p_0\in (1, 2)$. Connections with the Hilbert transform along vector fields and the polynomial Carleson's maximal operator are also discussed.
On the boundedness for the bilinear quadratic functional given by arbitrary strips
研究由任意不相交条带定义的光滑双线性二次泛函的有界性，该泛函结合了线性Rubio de Francia平方函数与双线性Hilbert变换的调制不变奇异结构。
We consider boundedness of a certain positive dyadic operator $$ T^σ\colon L^p(σ; \ \! \ell^2) \to L^p(ω), $$ that arose during our attempts to develop a two-weight theory for the Hilbert transform in $L^p$. Boundedness of $T^σ$ is characterized when $p \in [2, \infty)$ in terms of certain testing conditions. This requires a new Carleson-type embedding theorem that is also proved.
Endpoint Estimates for Stein's Purely Quadratic Carleson Operator
该研究针对Stein纯二次Carleson算子，证明其间隙与全版本均非弱型$(1,1)$，并分别建立了二者对应的$L\bmod$估计。
A stochastic Carleson embedding theorem with constant $e$ and the vector of Riesz transforms
本文证明带常数 $e$ 的随机 Carleson 嵌入定理，适用于满足广义 Cauchy--Riemann 关系的连续鞅系统，并通过 Gundy--Varopoulos 表示推广至任意维数函数的 Riesz 变换向量。
We prove boundedness of Calderón-Zygmund operators acting in Banach functions spaces on domains, defined by the $L_1$ Carleson functional and $L_q$ ($1<q<\infty$) Whitney averages. For such bounds to hold, we assume that the operator maps towards the boundary of the domain. We obtain the Carleson estimates by proving a pointwise domination of the operator, by sparse operators with a causal structure. The work is motivated by maximal regularity estimates for elliptic PDEs and is related to one-sided weighted estimates for singular integrals.
Let $B$ be a locally integrable matrix function, $W$ a matrix A${}_p$ weight with $1 < p < \infty$, and $T$ be any of the Riesz transforms. We will characterize the boundedness of the commutator $[T, B]$ on $L^p(W)$ in terms of the membership of $B$ in a natural matrix weighted BMO space. To do this, we will characterize the boundedness of dyadic paraproducts on $L^p(W)$ via a new matrix weighted Carleson embedding theorem. Finally, we will use some of the ideas from these proofs to (among other things) obtain quantitative weighted norm inequalities for these operators and also use them to pro…
In this paper, we consider $L^p$- estimate for a class of oscillatory integral operators satisfying the Carleson-Sjölin conditions with further convex and straight assumptions. As applications, the multiplier problem related to a general class of hypersurfaces with nonvanishing Gaussian curvature, local smoothing estimates for the fractional Schrödinger equation and the sharp resolvent estimates outside of the uniform boundedness range are discussed.
Analog version of Hausdorff--Young's theorem for quadratic Fourier transforms and boundedness of oscillatory integral operator
本文基于 Riesz--Thorin 插值定理证明了二次 Fourier 变换和线性正则变换的 Hausdorff--Young 型不等式，并研究了多项式相位振荡积分算子的有界性。
On the uniform rectifiability of AD regular measures with bounded Riesz transform operator: the case of codimension 1
证明了如果$μ$是$\R^{d+1}$中的d维Ahlfors-David正则测度，则$d$ -维里斯变换在$L^2(μ)$中的有界性意味着非BAUP的David-Semmes胞腔构成Carleson族.结合David和Semmes的早期结果，这产生了$μ$的一致可求直性。
The first three results in this thesis are motivated by a far-reaching conjecture on boundedness of singular Brascamp-Lieb forms. Firstly, we improve over the trivial estimate for their truncations, thus excluding potential trivial counterexamples. Secondly, in a dyadic model we give a partial result in the trilinear, 2-dimensional case when one of the functions depends only on one variable. Thirdly, we estimate the multidimensional version of the polynomial Carleson operator, whose boundedness would also be a consequence of the conejcture. The final pair of results concerns directional square…
Pseudodifferential operators with rough symbols
在这项工作中，我们开发了$L^p$有界性理论的伪微分算子粗糙（甚至不连续的一般）符号的$x$变量。此外，$B(L^p)$运营商的范数估计明确的尺度不变量涉及的符号。相对于$ξ$变量中所需的平滑度，所有估计值均显示为尖锐。作为推论，我们获得了最大方向希尔伯特变换和Carleson算子（平滑版本）的$L^p$界。
A non-testing characterization of bounded and compact composition operators on $\mathcal Q_p$ spaces
本文通过广义p-Nevanlinna计数函数的二进迹公式，给出了Q_p空间上有界和紧复合算子的非测试刻画，解决了Xiao和Zhao提出的开放问题。
The Bilinear Hilbert-Carleson operator along curves. The purely non-zero curvature case
本文研究纯非零曲率情形下沿曲线的双线性Hilbert-Carleson算子，利用Rank II LGC方法确定了其最大有界性范围（直至端点），并给出了完整的Hölder型L^p估计。
We improve the range of $\ell^p(\mathbb Z^d)$-boundedness of the integral $k$-spherical maximal functions introduced by Magyar. The previously best known bounds for the full $k$-spherical maximal function require the dimension $d$ to grow at least cubicly with the degree $k$. Combining ideas from our prior work with recent advances in the theory of Weyl sums by Bourgain, Demeter, and Guth and by Wooley, we reduce this cubic bound to a quadratic one. As an application, we deduce improved bounds in the ergodic Waring--Goldbach problem.
Recently in joint work with E. Sert, we proved sharp boundedness results on discrete fractional integral operators along binary quadratic forms. Present work vastly enhances the scope of those results by extending boundedness to bivariate quadratic polynomials. We achieve this in part by establishing connections to problems on concentration of lattice points on short arcs of conics, whence we study discrete fractional integrals and lattice point concentration from a unified perspective via tools of sieving and diophantine approximation, and prove theorems that are of interest to researchers in…
Quadratic estimates and functional calculi of perturbed Dirac operators
我们证明了Dirac型算子的复扰动的二次估计，从而证明了这种算子具有有界的谓词演算。作为一个应用程序，我们表明，谱投影的霍奇-狄拉克算子紧流形上的解析依赖于$L_\infty$变化的度量。我们还恢复了Calderón程序中的许多结果的统一证明，包括Kato平方根问题和Cauchy算子在Lipschitz曲线和曲面上的有界性。
Logarithmic bump conditions and the two weight boundedness of Calderón-Zygmund operators
我们证明了如果一对权重$(u,v)$在对数凸块和某些对数凸块的尺度上满足一个尖锐的$A_p$ -凸块条件，那么Haar将映射$L^p(v)$变换为$L^p(u)$，且变换的复杂度为常数平方.这反过来又意味着所有Calderón-Zygmund算子的双权有界性。这对一个长期存在的猜想给出了部分答案。我们也给出了弱型不等式的一个相关猜想的部分结果。为了证明我们的主要结果，我们结合了几种不同的方法来解决这些问题，特别是我们使用的许多想法来证明$A_2$猜想。作为我们工作的副产品，我们还反驳了Muckenhoupt和Wheeden关于希尔伯特变换的弱型不等式的猜想。这与最近Reguera、Scurry和Thiele的反例密切相关。
Classical Proofs Of Kato Type Smoothing Estimates for The Schrödinger Equation with Quadratic Potential in R^n+1 with application
利用Hermite函数技巧给出了R^n+1中具有二次位势的薛定谔方程的Kato型光滑估计的初等证明.这等价于证明一个奇异化Hermite投影核族的一致L^2（R ^n）到L^2（R^n）有界性。作为上述估计的一个应用，我们还证明了R^9塌缩变量型的Lahartz估计。
Non-zero to zero curvature transition: Operators along hybrid curves with no quadratic (quasi-)resonances
本文基于LGC方法，统一研究沿无二次共振混合曲线的Carleson型、Hilbert变换及双线性Hilbert变换等算子的L^p有界性，并给出平滑不等式的新证明。
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