The Bi-Carleson operator

Carleson's theorem with quadratic phase

Carleson's theorem on the pointwise convergence of Fourier series provides bounds for a maximal operator, with the maximum taken over all choices of linear functions of a phase argument. We extend this to all quadratic choices of phase functions. Specifically, we show that the maximal operator below maps $L^p$ into itself for $1<p<\infty$. $$ \sup_a \sup_b |\int e^{i(ay^2+by)}f(x-y)dy/y| $$

Uniform bounds for oscillatory and polynomial Carleson operators

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.

The Walsh model for $M_2^{*}$ Carleson

We study the Walsh model of a certain maximal truncation of Carleson's operator, related to the Return Times Theorem from Ergodic Theory.

Generalizations of the Carleson-Hunt theorem I. The classical singularity case

In this article we prove L^p estimates for a general maximal operator, which extend both the classical Coifman-Meyer and Carleson-Hunt theorems in harmonic analysis

L^p bounds for a maximal dyadic sum operator

We prove $L^p$ bounds in the range $1<p<\infty$ for a maximal dyadic sum operator on $\rn$. This maximal operator provides a discrete multidimensional model of Carleson's operator. Its boundedness is obtained by a simple twist of the proof of Carleson's theorem given by Lacey and Thiele, adapted in higher dimensions by Pramanik and Terwilleger. In dimension one, the $\lp$ boundedness of this maximal dyadic sum implies in particular an alternative proof of Hunt's extension of Carleson's theorem on almost everywhere convergence of Fourier integrals.

The Hilbert transform along the parabola, the polynomial Carleson theorem and oscillatory singular integrals

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.

Weak-L^p bounds for the Carleson and Walsh-Carleson operators

We prove a weak-$L^p$ bound for the Walsh-Carleson operator for $p $ near 1, improving on a theorem of Sjolin. We relate our result to the conjectures that the Walsh-Fourier and Fourier series of a function $f\in L\log L(\mathbb T)$ converge for almost every $x \in \mathbb T$.

A Fefferman-Stein inequality for the Carleson operator

We provide a Fefferman-Stein type weighted inequality for maximally modulated Calderón-Zygmund operators that satisfy \textit{a priori} weak type unweighted estimates. This inequality corresponds to a maximally modulated version of a result of Pérez. Applying it to the Hilbert transform we obtain the corresponding Fefferman-Stein inequality for the Carleson operator $\mathcal{C}$, that is $\mathcal{C}: L^p(M^{\lfloor p \rfloor +1}w) \to L^p(w)$ for any $1<p<\infty$ and any weight function $w$, with bound independent of $w$. We also provide a maximal-multiplier weighted theorem, a vector-valued…

Bounds for anisotropic Carleson operators

We prove weak $(2,2)$ bounds for maximally modulated anisotropically homogeneous smooth multipliers on $\mathbb{R}^n$. These can be understood as generalizing the classical one-dimensional Carleson operator. For the proof we extend the time-frequency method by Lacey and Thiele to the anistropic setting. We also discuss a related open problem concerning Carleson operators along monomial curves.

Maximal operators on variable Lebesgue spaces with weights related to oscillations of Carleson curves

We prove sufficient conditions for the boundedness of the maximal operator on variable Lebesgue spaces with weights $φ_{t,γ}(τ)=|(τ-t)^γ|$, where $γ$ is a complex number, over arbitrary Carleson curves. If the curve has different spirality indices at the point $t$ and $γ$ is not real, then $φ_{t,γ}$ is an oscillating weight lying beyond the class of radial oscillating weights considered recently by V. Kokilashvili, N. Samko, and S. Samko.

A Study of the Matrix Carleson Embedding Theorem with Applications to Sparse Operators

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…

The (weak-$L^2$) Boundedness of The Quadratic Carleson Operator

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.

非线性Carleson猜想的强版本失效

The strong version of nonlinear Carleson conjecture fails

研究Dirac方程与平方可和势的Jost解，证明传输系数幅值的最大函数无界，并展示非线性Carleson猜想的强版本在Dirac方程和Krein系统中失效。

On the Carleson duality

As a tool for solving the Neumann problem for divergence form equations, Kenig and Pipher introduced the space X of functions on the half space, such that the non-tangential maximal function of their L_2-Whitney averages belongs to L_2 on the boundary. In this paper, answering questions which arose from recent studies of boundary value problems by Auscher and the second author, we find the pre-dual of X, and characterize the pointwise multipliers from X to L_2 on the half space as the well-known Carleson-type space of functions introduced by Dahlberg. We also extend these results to L_p genera…

On weighted norm inequalities for the Carleson operator

We obtain $L^p(w)$ bounds for the Carleson operator ${\mathcal C}$ in terms of the $A_q$ constants $[w]_{A_q}$ for $1\le q\le p$. In particular, we show that, exactly as for the Hilbert transform, $\|{\mathcal C}\|_{L^p(w)}$ is bounded linearly by $[w]_{A_q}$ for $1\le q<p$. Our approach works in the general context of maximally modulated Calderón-Zygmung operators.

Polynomial Carleson operators along monomial curves in the plane

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…

A polynomial Carleson operator along the paraboloid

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…

On weighted norm inequalities for the Carleson and Walsh-Carleson operators

We prove $L^p(w)$ bounds for the Carleson operator ${\mathcal C}$, its lacunary version $\mathcal C_{lac}$, and its analogue for the Walsh series $\W$ in terms of the $A_q$ constants $[w]_{A_q}$ for $1\le q\le p$. In particular, we show that, exactly as for the Hilbert transform, $\|{\mathcal C}\|_{L^p(w)}$ is bounded linearly by $[w]_{A_q}$ for $1\le q<p$. We also obtain $L^p(w)$ bounds in terms of $[w]_{A_p}$, whose sharpness is related to certain conjectures (for instance, of Konyagin \cite{K2}) on pointwise convergence of Fourier series for functions near $L^1$. Our approach works in the g…

The Polynomial Carleson Operator

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…

Maximal polynomial modulations of singular integrals

Let $K$ be a standard Hölder continuous Calderón--Zygmund kernel on $\mathbb{R}^{\mathbf{d}}$ whose truncations define $L^2$ bounded operators. We show that the maximal operator obtained by modulating $K$ by polynomial phases of a fixed degree is bounded on $L^p(\mathbb{R}^{\mathbf{d}})$ for $1 < p < \infty$. This extends Sjölin's multidimensional Carleson theorem and Lie's polynomial Carleson theorem.

A Note on the Polynomial Carleson Operator in higher dimensions

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.

加倍度量测度空间上的Carleson算子

Carleson operators on doubling metric measure spaces

本文在加倍度量测度空间上提出调制函数的一般公理化方法，证明相应Carleson算子的L^p有界性，推广了经典结果，并用Lean验证。

On the Boundedness of the Carleson Operator near $L^1$

Based on the tile discretization elaborated by the author in "The Polynomial Carleson Operator", we develop a Calderon-Zygmund type decomposition of the Carleson operator. As a consequence, through a unitary method that makes no use of extrapolation techniques, we recover the previously known results regarding the largest rearrangement invariant space of functions with almost everywhere convergent Fourier series.

On Stein's Conjecture on the Polynomial Carleson Operator

We prove that the generalized Carleson operator $C_d$ with polynomial phase function is of strong type $(p,r)$, $1<r<p<\infty$; this yields a positive answer in the $1<p<2$ case to a conjecture of Stein which asserts that for $1<p<\infty$ we have that $C_d$ is of strong type $(p,p)$. A key ingredient in this proof is the further extension of the {\it relational} time-frequency perspective (introduced in \cite{q}) to the general polynomial phase.

Sparse Bounds for Random Discrete Carleson Theorems

We study discrete random variants of the Carleson maximal operator. Intriguingly, these questions remain subtle and difficult, even in this setting. Let $\{X_m\}$ be an independent sequence of $\{0,1\}$ random variables with expectations \[ \mathbb E X_m = σ_m = m^{-α}, \ 0 < α< 1/2, \] and $ S_m = \sum_{k=1} ^{m} X_k$. Then the maximal operator below almost surely is bounded from $ \ell ^{p}$ to $ \ell ^{p}$, provided the Minkowski dimension of $ Λ\subset [-1/2, 1/2]$ is strictly less than $ 1- α$. \[ \sup_{λ\in Λ} \Bigl| \sum_{m\neq 0} X_{\lvert m\rvert } \frac{e( λm )}{ {\rm sgn} (m)S_{ |m|…

沿抛物面的一次Carleson算子

A degree one Carleson operator along the paraboloid

本研究证明了d≥2时沿抛物面的奇异积分的极大线性调制的L^p有界性，推广了Carleson定理并补充了Pierce与Yung的相关结果。

The non-resonant bilinear Hilbert--Carleson operator

In this paper we introduce the class of bilinear Hilbert--Carleson operators $\{BC^a\}_{a>0}$ defined by $$ BC^{a}(f,g)(x):= \sup_{λ\in {\mathbb R}} \Big|\int f(x-t)\, g(x+t)\, e^{iλt^a} \, \frac{dt}{t} \Big| $$ and show that in the non-resonant case $a\in (0,\infty)\setminus\{1,2\}$ the operator $BC^a$ extends continuously from $L^p({\mathbb R})\times L^q({\mathbb R})$ into $L^r({\mathbb R})$ whenever $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ with $1<p,\,q\leq\infty$ and $\frac{2}{3}<r<\infty$. A key novel feature of these operators is that -- in the non-resonant case -- $BC^{a}$ has a \emph{hybr…

A Carleson type theorem for a Cantor group model of the scattering transform

We consider a basic $d$-adic model for the scattering transform on the line. We prove $L^2$ bounds for this scattering transform and a weak $L^2$ bound for a Carleson type maximal operator. The latter implies boundedness of $d$-adic models of generalized eigenfunctions of Dirac type operators with potential in $L^2(R)$. We show that this result cannot be obtained by estimating the terms in the natural multilinear expansion of the scattering transform.

Positive sparse domination of variational Carleson operators

Due to its nonlocal nature, the $r$-variation norm Carleson operator $C_r$ does not yield to the sparse domination techniques of Lerner, Di Plinio and Lerner, Lacey. We overcome this difficulty and prove that the dual form to $C_r$ can be dominated by a positive sparse form involving $L^p$ averages. Our result strengthens the $L^p$ estimates by Oberlin et. al. As a corollary, we obtain quantitative weighted norm inequalities improving on previous results by Do and Lacey. Our proof relies on the localized outer $L^p$-embeddings of Di Plinio-Ou and Uraltsev.

Carleson--Dunkl 算子的振荡不等式

Oscillation inequalities for Carleson--Dunkl operator

本文在加权 L^p 空间上建立了 Carleson--Dunkl 算子的振荡半范数估计，由此导出了标准 Carleson 算子在径向 L^p 空间的振荡估计，并作为副产品提出了径向乘子的转移原理。

双侧马丁格尔变换与加权瓦尔什-卡拉森算子

Dyadic Martingale Transforms and Weighted Walsh-Carleson Operators

本文研究了由双侧马丁格尔变换产生的加权瓦尔什-卡拉森极大算子，证明了在满足均匀二进制变化条件和顶部二进制尺度上均匀界的情况下，这些算子在子序列上的弱型(1,1)估计，并给出了与权重在顶部二进制尺度附近行为相关的发散准则。

The pointwise convergence of Fourier Series (II). Strong $L^1$ case for the lacunary Carleson operator

We prove that the lacunary Carleson operator is bounded from $L \log L$ to $L^{1}$. This result is sharp. The proof is based on two newly introduced concepts: 1) the \emph{time-frequency regularization of a measurable set} and 2) the \emph{set-resolution of the time-frequency plane at $0-$frequency}. These two concepts will play the central role in providing a special tile decomposition adapted to the interaction between the \emph{structure} of the lacunary Carleson operator and the corresponding \emph{structure} of a fix measurable set. Another key insight of our paper is that it provides for…

高维下的变分范数Carleson定理

A variation norm Carleson theorem in higher dimensions

本文将Fefferman的多边形傅里叶部分和极大函数界推广到高维下的强$r$-变分范数界与2-振荡界，并针对嵌套矩形的部分傅里叶和建立了相关变分与振荡界，完善了高维Carleson定理的研究。

Oscillatory integrals related to Carleson's theorem: fractional monomials

Stein and Wainger proved the $L^p$ bounds of the polynomial Carleson operator for all integer-power polynomials without linear term. In the present paper, we partially generalise this result to all fractional monomials in dimension one. Moreover, the connections with Carleson's theorem and the Hilbert transform along vector fields or (variable) curves are also discussed in details.

A note on two weight bounds for the generalized Hardy-Littlewood Maximal operator

We give a straighforward proof of the two weight estimates of the generalized maximal operator under Sawyer type testing conditions. The proof relies on the Martingale Carleson Embedding Theorem.

Algebras of singular integral operators on Nakano spaces with Khvedelidze weights over Carleson curves with logarithmic whirl points

We establish a Fredholm criterion for an arbitrary operator in the Banach algebra of singular integral operators with piecewise continuous coefficients on Nakano spaces (generalized Lebesgue spaces with variable exponent) with Khvedelidze weights over Carleson curves with logarithmic whirl points.

A Discrete Carleson Theorem Along the Primes with a Restricted Supremum

Consider the discrete maximal function acting on finitely supported functions on the integers, \[ \mathcal{C}_Λf(n) := \sup_{λ\in Λ} | \sum_{p \in \pm \mathbb{P}} f(n-p) \log |p| \frac{e^{2πi λp}}{p} |,\] where $\pm \mathbb{P} := \{ \pm p : p \text{ is a prime} \}$, and $Λ\subset [0,1]$. We give sufficient conditions on $Λ$, met by (finite unions of) lacunary sets, for this to be a bounded sublinear operator on $\ell^p(\mathbb{Z})$ for $\frac{3}{2} < p < 4$.

Carleson embeddings and two operators on Bergman spaces of tube domains over symmetric cones

We prove Carleson embeddings for Bergman spaces of tube domains over symmetric cones, we apply them to characterize symbols of bounded Cesàro-type operators from weighted Bergman spaces to weighted Besov spaces. We also obtain Schatten class criteria of Toeplitz operators and Cesàro-type operators on weighted Hilbert-Bergman spaces.

A Discrete Quadratic Carleson Theorem on $ \ell ^2 $ with a Restricted Supremum

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…

On two-weight norm estimates for multilinear fractional maximal function

We prove some Sawyer-type characterizations for multilinear fractional maximal function for the upper triangle case. We also provide some two-weight norm estimates for this operator. As one of the main tools, we use an extension of the usual Carleson Embedding that is an analogue of the P. L. Duren extension of the Carleson Embedding for measures.

A variation norm Carleson theorem

We strengthen the Carleson-Hunt theorem by proving $L^p$ estimates for the $r$-variation of the partial sum operators for Fourier series and integrals, for $p>\max\{r',2\}$. Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

A Carleson problem for the Boussinesq operator

In this paper, Theorems 1.1- 1.2 show that the Boussinesq operator $\mathcal{B}_tf$ converges pointwise to its initial data $f\in H^s(\mathbb{R})$ as $t\to 0$ provided $s\geq\frac{1}{4}$ -- more precisely -- on the one hand, by constructing a counterexample in $\mathbb{R}$ we discover that the optimal convergence index $s_{c,1}=\frac14$; on the other hand, we find that the Hausdorff dimension of the disconvergence set for $\mathcal{B}_tf$ is \begin{align*} α_{1,\mathcal{B}}(s)&=\begin{cases} 1-2s&\ \ \text{as}\ \ \frac{1}{4}\leq s\leq\frac{1}{2};\\ 1 &\ \ \text{as}\ \ 0<s<\frac{1}{4}. \end{cas…

二次超曲面上的多项式Carleson算子研究

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。

A quantitative approach to weighted Carleson Condition

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.

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.

Variational estimates for the bilinear iterated Fourier integral

We prove pointwise variational Lp bounds for a bilinear Fourier integral operator in a large but not necessarily sharp range of exponents. This result is a joint strengthening of the corresponding bounds for the classical Carleson operator, the bilinear Hilbert transform, the variation norm Carleson operator, and the bi-Carleson operator. Terry Lyon's rough path theory allows for extension of our result to multilinear estimates. We consider our result a proof of concept for a wider array of similar estimates with possible applications to ordinary differential equations.

Generalized Carleson perturbations of elliptic operators and applications

We extend in two directions the notion of perturbations of Carleson type for the Dirichlet problem associated to an elliptic real second-order divergence-form (possibly degenerate, not necessarily symmetric) elliptic operator. First, in addition to the classical perturbations of Carleson type, that we call additive Carleson perturbations, we introduce scalar-multiplicative and antisymmetric Carleson perturbations, which both allow non-trivial differences at the boundary. Second, we consider domains which admit an elliptic PDE in a broad sense: we count as examples the 1-sided NTA (a.k.a. unifo…

Variational Carleson operators in UMD spaces

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.

Weighted inequlities for a generalized dyadic maximal operator involving the infinite product

We define a generalized dyadic maximal operator involving the infinite product and discuss weighted inequalities for the operator. A formulation of the Carleson embedding theorem is proved. Our results depend heavily on a generalized Hölder's inequalities.

非线性傅里叶变换的缺项Carleson算子的弱$L^2$有界性

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。

Singular Integral Operators on Variable Lebesgue Spaces over Arbitrary Carleson Curves

In 1968, Israel Gohberg and Naum Krupnik discovered that local spectra of singular integral operators with piecewise continuous coefficients on Lebesgue spaces $L^p(Γ)$ over Lyapunov curves have the shape of circular arcs. About 25 years later, Albrecht Böttcher and Yuri Karlovich realized that these circular arcs metamorphose to so-called logarithmic leaves with a median separating point when Lyapunov curves metamorphose to arbitrary Carleson curves. We show that this result remains valid in a more general setting of variable Lebesgue spaces $L^{p(\cdot)}(Γ)$ where $p:Γ\to(1,\infty)$ satisfie…

On estimate of operator for $0<p<\infty $

Operators such as Carleson operator are known to be bounded on $L^p$ for all $1<p<\infty$, but not from $L^1$ to weak-$L^1$ and from $H^p$ to $L^p$ for each $0<p\leq 1$, the object of this article is to give a estimate for all $0<p<\infty$. For the weights $w$ satisfying the doubling condition of order $q$ with $0<q<p$ and the reverse Hölder condition, by using some new functions spaces, we prove that: $\bullet$ some sublinear operators are bounded from some subspaces of $L^p_w$ to $L^p_w$ and to themselves for all $0<p< \infty$; in particular, these imply the endpoint estimates from $H^p_w$ t…

Dyadic Carleson embedding and sparse domination of weighted composition operators on strictly pseudoconvex domains

In this paper, we study the behavior of the weighted composition operators acting on Bergman spaces defined on strictly pseudoconvex domains via the sparse domination technique from harmonic analysis. As a byproduct, we also prove a weighted type estimate for the weighted composition operators which is adapted to Sawyer-testing conditions. Our results extend the work by the first author, Li, Shi and Wick under a much more general setting.

Maximal Operators Associated to Multiplicative Characters

We show that the natural analogue of the Carleson-Hunt inequality fails for multiplicative characters.