A problem for Hankel measures on Hardy space

加权 Hardy 空间上 Hankel 算子的再生核命题

Reproducing Kernel Thesis of Hankel Operators on Weighted Hardy Spaces

本文研究了带 Muckenhoupt 权的两加权空间间 Hankel 算子的有界性，提出双权版 Garsia 范数，证明在再生核上测试足以判定算子有界性，并证明了 Carleson 嵌入定理的双权版本。

Truncations of multilinear Hankel operators

We extend to multilinear Hankel operators the fact that some truncations of bounded Hankel operators are bounded. We prove and use a continuity property of bilinear Hilbert transforms on products of Lipschitz spaces and Hardy spaces.

Weighted model spaces and Schmidt subspaces of Hankel operators

For a bounded Hankel matrix $Γ$, we describe the structure of the Schmidt subspaces of $Γ$, namely the eigenspaces of $Γ^* Γ$ corresponding to non zero eigenvalues. We prove that these subspaces are in correspondence with weighted model spaces in the Hardy space on the unit circle. Here we use the term "weighted model space" to describe the range of an isometric multiplier acting on a model space. Further, we obtain similar results for Hankel operators acting in the Hardy space on the real line. Finally, we give a streamlined proof of the Adamyan-Arov-Krein theorem using the language of weight…

Hankel matrices acting on the Hardy space $H^1$ and on Dirichlet spaces

If $\,μ\,$ is a finite positive Borel measure on the interval $\,[0,1)$, we let $\,\mathcal H_μ\,$ be the Hankel matrix $\,(μ_{n, k})_{n,k\ge 0}\,$ with entries $\,μ_{n, k}=μ_{n+k}$, where, for $\,n\,=\,0, 1, 2, \dots $, $μ_n\,$ denotes the moment of order $\,n\,$ of $\,μ$. This matrix induces formally the operator $\,\mathcal{H}_μ(f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} μ_{n,k}{a_k}\right)z^n\,$ on the space of all analytic functions $\,f(z)=\sum_{k=0}^\infty a_kz^k\,$, in the unit disc $\,\mathbb D $. When $\,μ\,$ is the Lebesgue measure on $\,[0,1)\,$ the operator $\,\mathcal H_…

多圆盘上的配对算子与Toeplitz+Hankel算子的刻画

Generalization of Toeplitz + Hankel operators in the polydisc

本文完整分类了多圆盘上向量值Hardy空间中的Toeplitz+Hankel算子，并刻画了L^2(T^n)上的配对算子及内函数诱导的θ-配对算子。

A Hankel matrix acting on spaces of analytic functions

If $μ$ is a positive Borel measure on the interval $[0, 1)$ we let $\mathcal H_μ$ be the Hankel matrix $\mathcal H_μ=(μ_{n, k})_{n,k\ge 0}$ with entries $μ_{n, k}=μ_{n+k}$, where, for $n\,=\,0, 1, 2, \dots $, $μ_n$ denotes the moment of order $n$ of $μ$. This matrix induces formally the operator $$\mathcal{H}_μ(f)(z)= \sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty} μ_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb D $. This is a natural generalization of the classical Hilbert operator. In this paper we improve the results ob…

Toeplitz and Hankel operators and Dixmier traces on the unit ball of \mathbb C^n

We compute the Dixmier trace of pseudo-Toeplitz operators on the Fock space. As an application we find a formula for the Dixmier trace of the product of commutators of Toeplitz operators on the Hardy and weighted Bergman spaces on the unit ball of \mathbb C^d. This generalizes an earlier work of Helton-Howe for the usual trace of the anti-symmetrization of Toeplitz operators.

广义Foguel-汉克尔算子

Generalized Foguel-Hankel Operators

本文提出广义Foguel-汉克尔算子，探讨其幂有界性与收缩相似性条件的关系，揭示了希尔伯特矩阵下的等价条件。

环域上Hardy空间与Bergman空间上Toeplitz算子的零积问题

Zero Products of Toeplitz operators on the Hardy and Bergman spaces over an annulus

本文研究环域上Hardy与Bergman空间中Toeplitz算子的零积问题，在符号傅里叶展开条件下证明无零因子，并借助Abrahamse约化定理刻画紧Hankel算子，得出零积结果。

有界对称域商空间上的 Hankel 算子与投射 Hilbert 模

Hankel operators and Projective Hilbert modules on quotients of bounded symmetric domains

本文研究有限伪反射群作用下有界对称域商空间的 Hardy 空间，证明小 Hankel 算子满足 Hartman 型定理，而大 Hankel 算子不满足 Nehari 定理，并据此说明多圆盘商域上的 Hardy 空间非投射 Hilbert 模。

作用于Dirichlet空间的Hankel矩阵

Hankel matrices acting on the Dirichlet space

本文利用Carleson测度、Widom型条件和再生核论点完整解决了Dirichlet空间上Hankel矩阵诱导算子的有界性问题，并进一步刻画了Cesàro型矩阵诱导算子的有界性与紧性，同时证明随机Dirichlet函数几乎必然诱导紧Hankel型算子。

Hankel算子的强再生核定理与Garsia范数

The strong reproducing kernel thesis for Hankel operators and the Garsia norm

本文针对再生核希尔伯特空间上的有界算子提出强再生核定理（SRKT）的判定问题，并具体研究Hardy空间上Hankel算子满足SRKT的条件，同时探讨BMO单位球在Garsia范数下的几何性质。

加权 Bergman 空间上的 Hankel 形式与测度

Hankel forms and measures on weighted Bergman spaces

本文刻画了上加倍权条件下加权 Bergman 空间上由测度诱导的 Hankel 形式与算子的有界性及 $A^p_\omega$ Hankel 测度，并利用双权分数导数理论简化了小 Bergman 空间的 $A^1$ 对偶结果。

A generalized Hilbert matrix acting on Hardy spaces

If $μ$ is a positive Borel measure on the interval $[0, 1)$, the Hankel matrix $\mathcal H_μ=(μ_{n,k})_{n,k\ge 0}$ with entries $μ_{n,k}=\int_{[0,1)}t^{n+k}\,dμ(t)$ induces formally the operator $$\mathcal{H}_μ(f)(z)=\sum_{n=0}^{\infty}\left(\sum_{k=0}^{\infty}μ_{n,k}{a_k}\right)z^n$$ on the space of all analytic functions $f(z)=\sum_{k=0}^\infty a_kz^k$, in the unit disc $\mathbb{D} $. In this paper we describe those measures $μ$ for which $\mathcal{H}_μ$ is a bounded (compact) operator from $H^p$ into $H^q$, $0<p,q<\infty $. We also characterize the measures $μ$ for which $\mathcal H_μ$ lies…

A Derivative-Hilbert operator acting on Hardy spaces

Let $μ$ be a positive Borel measure on the interval [0,1). The Hankel matrix $\mathcal{H}_μ= (μ_{n,k})_{n,k\geq0}$ with entries $μ_{n,k}= μ_{n+k}$, where $μ_n=\int_{ [0,1)}t^ndμ(t)$, induces formally the operator $$\mathcal{DH}_μ(f)(z)=\sum_{n=0}^\infty (\sum_{k=0}^\infty μ_{n,k}a_k)(n+1)z^n$$ on the space of all analytic function $f(z)=\sum_{k=0}^ \infty a_k z^n$ in the unit disc $\mathbb{D}$. We characterize those positive Borel measures on $[0,1)$ such that $\mathcal{DH}_μ(f)(z)= \int_{[0,1)} \frac{f(t)}{(1-tz)^2} dμ(t)$ for all in Hardy spaces $H^p(0<p<\infty)$, and among them we describe …

On singular values of Hankel operators on Bergman spaces

In this paper, we study the behavior of the singular values of Hankel operators on weighted Bergman spaces $A^2_{ω_φ}$, where $ω_φ= e^{-φ}$ and $φ$ is a subharmonic function. We consider compact Hankel operators $H_{\overline ϕ}$, with anti-analytic symbols ${\overline ϕ}$, and give estimates of the trace of $h(|H_{\overline ϕ}|)$ for any convex function $h$. This allows us to give asymptotic estimates of the singular values $(s_n(H_{\overline ϕ}))_n$ in terms of decreasing rearrangement of $|ϕ'|/\sqrt{Δφ}$. For the radial weights, we first prove that the critical decay of $(s_n(H_{\overline ϕ…

Interpolating sequences and Carleson measures in the Hardy-Sobolev spaces of the ball in $C^n$

In this work we study Hardy Sobolev spaces in the ball of $C^n$ with respect to interpolating sequences and Carleson measures. We compare them with the classical Hardy spaces of the ball and we stress analogies and differences.

Carleson measures for weighted Hardy-Sobolev spaces

We obtain characterizations of positive Borel measures $μ$ on $\B^n$ so that some weighted Hardy-Sobolev are imbedded in $L^p(dμ)$, where $w$ is an $A_p$ weight in the unit sphere of $\C^n$.

Duality theorems for coinvariant subspaces of $H^1$

Let $θ$ be an inner function satisfying the connected level set condition of B. Cohn, and let $K^{1}_θ$ be the shift-coinvariant subspace of the Hardy space $H^1$ generated by $θ$. We describe the dual space to $K^{1}_θ$ in terms of a bounded mean oscillation with respect to the Clark measure $σ_α$ of $θ$. Namely, we prove that $(K^{1}_θ \cap zH^1)^* = {\rm BMO}(σ_α)$. The result implies a two-sided estimate for the operator norm of a finite Hankel matrix of size $n\times n$ via ${\rm BMO}(μ_{2n})$-norm of its standard symbol, where $μ_{2n}$ is the Haar measure on the group $\{ξ\in \mathbb{C}:…

Reverse Carleson measures in Hardy spaces

We give a necessary and sufficient condition for a measure $μ$ in the closed unit disk to be a reverse Carleson measure for Hardy spaces. This extends a previous result of Lefévre, Li, Queffélec and Rodr\'ıguez-Piazza \cite{LLQR}. We also provide a simple example showing that the analogue for the Paley-Wiener space does not hold. This example can be generalised to model spaces associated to one-component inner functions.

Carleson measures for Hardy and Bergman spaces in the quaternionic unit ball

We study a characterization of slice Carleson measures and of Carleson measures for the both the Hardy spaces $H^p(\mathbb B)$ and the Bergman spaces $\mathcal A^p(\mathbb B)$ of the quaternionic unit ball $\mathbb B$. In the case of Bergman spaces, the characterization is done in terms of the axially symmetric completion of a pseudohyperbolic disc in a complex plane. We also show that a characterization in terms of pseudohyperbolic balls is not possible.

Equivalence among various variable exponent Hardy or Bergman spaces

We study the question of when two weighted variable exponent Bergman spaces or Hardy spaces are equivalent. As an application, we show that variable exponent Hardy spaces have a close relation to classical Hardy spaces when the exponent is log-Hölder continuous and has bounded harmonic conjugate (when extended from its boundary values to be harmonic in the disc). We use this to characterize Carleson measures for these variable exponent Hardy spaces. We also prove under certain conditions an analogue of Littlewood subordination and a result on the boundedness of composition operators.

An extremal problem for characteristic functions

Suppose $E$ is a subset of the unit circle $\mathbb{T}$ and $H^\infty\subset L^\infty$ is the Hardy subalgebra. We examine the problem of finding the distance from the characteristic function of $E$ to $z^nH^\infty$. This admits an alternate description as a dual extremal problem. Precise solutions are given in several important cases. The techniques used involve the theory of Toeplitz and Hankel operators as well as the construction of certain conformal mappings.

Volterra operators on Hardy spaces of Dirichlet series

For a Dirichlet series symbol $g(s) = \sum_{n \geq 1} b_n n^{-s}$, the associated Volterra operator $\mathbf{T}_g$ acting on a Dirichlet series $f(s)=\sum_{n\ge 1} a_n n^{-s}$ is defined by the integral $f\mapsto -\int_{s}^{+\infty} f(w)g'(w)\,dw$. We show that $\mathbf{T}_g$ is a bounded operator on the Hardy space $\mathcal{H}^p$ of Dirichlet series with $0 < p < \infty$ if and only if the symbol $g$ satisfies a Carleson measure condition. When appropriately restricted to one complex variable, our condition coincides with the standard Carleson measure characterization of ${\operatorname{BMOA…

Hankel and Toeplitz-Schur Multipliers

We study the problem of characterizing Hankel-Schur multipliers and Toeplitz-Schur multipliers of Schatten-von Neumann class $\bS_p$ for $0<p<1$. We obtain various sharp necessary conditions and sufficient conditions for a Hankel matrix to be a Schur multiplier of $\bS_p$. We also give a characterization of the Hankel-Schur multipliers of $\bS_p$ whose symbols have lacunary power series. Then the results on Hankel-Schur multipliers are used to obtain a characterization of the Toeplitz-Schur multipliers of $\bS_p$. Finally, we return to Hankel-Schur multipliers and obtain new results in the cas…

函数空间中的Riesz-Kolmogorov型紧性准则及其应用

Riesz-Kolmogorov type compactness criteria in function spaces with applications

本文提出适用于多种函数空间的Riesz-Kolmogorov型紧性准则，并用于刻画Bergman空间上的紧Toeplitz算子及推导Hardy空间上Hankel算子的紧性。

Carleson Measures for the Drury-Arveson Hardy space and other Besov-Sobolev spaces on Complex Balls

We characterize the Carleson measures for the Drury-Arveson Hardy space and other Hilbert spaces of analytic functions of several complex variables. This provides sharp estimates for Drury's generalization of Von Neumann's inequality. The characterization is in terms of a geometric condition, the "split tree condition", which reflects the nonisotropic geometry underlying the Drury-Arveson Hardy space.

Invariant tori for the cubic Szegö equation

We continue the study of the following Hamiltonian equation on the Hardy space of the circle, $$i\partial _tu=Π(|u|^2u)\ ,$$ where $Π$ denotes the Szegö projector. This equation can be seen as a toy model for totally non dispersive evolution equations. In a previous work, we proved that this equation admits a Lax pair, and that it is completely integrable. In this paper, we construct the action-angle variables, which reduces the explicit resolution of the equation to a diagonalisation problem. As a consequence, we solve an inverse spectral problem for Hankel operators. Moreover, we establish t…

广义Hilbert算子从Hardy空间到加权Bergman空间的作用

Generalized Hilbert operators acting from Hardy spaces to weighted Bergman spaces

本文刻画了使广义Hilbert算子$\mathcal{H}_{\mu,\alpha}$在Hardy空间上有定义并满足积分表示，且从$H^p$到加权Bergman空间$A_{\alpha-2}^q$有界或紧的测度条件。

Siegel上半空间中Hardy-Sobolev空间的Carleson测度

Carleson measures for Hardy-Sobolev spaces in the Siegel upper half-space

针对Siegel上半空间中Arcozzi等人提出的Hardy-Sobolev空间的Carleson测度刻画问题，给出了容型刻画，部分解答了原作者提出的相关问题。

Fock型空间上有界、紧与Schatten类Hankel算子

Bounded, compact and Schatten class Hankel operators on Fock-type spaces

本文研究了Fock型空间上具有局部可积符号的Hankel算子，利用BDA条件刻画了其有界性、紧性与Schatten类成员资格，并给出了小指数下正测度符号Toeplitz算子的Schatten类刻画。

Some Carleson measures for the Hilbert-Hardy space of tube domains over symmetric cones

In this note, we obtain a full characterization of radial Carleson measures for the Hilbert-Hardy space on tube domains over symmetric cones. For large derivatives, we also obtain a full characterization of the measures for which the corresponding embedding operator is continuous. Restricting to the case of light cones of dimension three, we prove that by freezing one or two variables, the problem of embedding derivatives of the Hilbert-Hardy space into Lebesgue spaces reduces to the characterization of Carleson measures for Hilbert-Bergman spaces of the upper-half plane or the product of two …

狄利克雷级数哈代空间上有限素数复合算子的临界范数轮廓

Critical Norm Profiles for Finite-Prime Composition Operators on the Hardy Space of Dirichlet Series

本文研究狄利克雷级数哈代空间上有限素数复合算子，确定其临界边界范数轮廓，推导相关算子的收敛关系、结构性质、集中不等式及有限维近似，揭示单一系数算子结构的支配作用。

A matrix characterization of boundary representations of positive matrices in the Hardy space

Spectral measures give rise to a natural harmonic analysis on the unit disc via a boundary representation of a positive matrix arising from a spectrum of the measure. We consider in this paper the reverse: for a positive matrix in the Hardy space of the unit disc we consider which measures, if any, yield a boundary representation of the positive matrix. We introduce a potential characterization of those measures via a matrix identity and show that the characterization holds in several important special cases.

A Characterization of Boundary Representations of Positive Matrices in the Hardy Space via the Abel Product

Spectral measures give rise to a natural harmonic analysis on the unit disc via a boundary representation of a positive matrix arising from a spectrum of the measure. We consider in this paper the reverse: for a positive matrix in the Hardy space of the unit disc we consider which measures, if any, yield a boundary representation of the positive matrix. We prove a characterization of those representing measures via a matrix identity by introducing a new operator product called the Abel Product.

Compact embedding derivatives of Hardy spaces into Lebesgue spaces

We characterize the positive Borel measures such that the differentiation operator of order $n\in\mathbb{N}\cup\{0\}$ is compact from the Hardy space $H^p$ into $L^q(μ)$, $0<p,q<\infty$.

加权 Bergman 空间上 Hankel 算子奇异值的 IDA 函数与渐近行为

IDA function and asymptotic behavior of singular values of Hankel operators on weighted Bergman spaces

本文以 IDA 函数的不增重排刻画加权 Bergman 空间中 Hankel 算子奇异值的渐近行为，并推广至共轭符号联合刻画及加权 Fock 空间中的 Berger-Coburn 现象。

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.