E. Amar
We prove that the composition of a function in the Hardy class H^p of the unit ball B in C^n with an analytic disc is in the Bergman class of the unit disc. Then we use it to show that the natural "analysis by discs" fails in the case of H^p(B) interpolating sequences for finite p.
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We completely describe the boundedness of the Volterra type operator $J_ g$ between Hardy spaces in the unit ball of $\Cn$. The proof of the one dimensional case used tools, such as the strong factorization for Hardy spaces, that are not available in higher dimensions, and therefore new techniques are developed. In particular, a generalized version of the description of Hardy spaces in terms of the area function is needed.
We establish that the Volterra-type integral operator $J_b$ on the Hardy spaces $H^p$ of the unit ball $\mathbb{B}_n$ exhibits a rather strong rigid behavior. More precisely, we show that the compactness, strict singularity and $\ell^p$-singularity of $J_b$ are equivalent on $H^p$ for any $1 \le p < \infty$. Moreover, we show that the operator $J_b$ acting on $H^p$ cannot fix an isomorphic copy of $\ell^2$ when $p \ne 2.$
We study a Toeplitz type operator $Q_μ$ between the holomorphic Hardy spaces $H^p$ and $H^q$ of the unit ball. Here the generating symbol $μ$ is assumed to a positive Borel measure. This kind of operator is related to many classical mappings acting on Hardy spaces, such as composition operators, the Volterra type integration operators and Carleson embeddings. We completely characterize the boundedness and compactness of $Q_μ:H^p\to H^q$ for the full range $1<p,q<\infty$; and also describe the membership in the Schatten classes of $H^2$. In the last section of the paper, we demonstrate the usef…
In this paper, we investigate the spectra of invertible weighted composition operators with automorphism symbols, on Hardy space $H^2(\mathbb{B}_N)$ and weighted Bergman spaces $A_α^2(\mathbb{B}_N)$, where $\mathbb{B}_N$ is the unit ball of the $N$-dimensional complex space. By taking $N=1$, $\mathbb{B}_N=\mathbb{D}$ the unit disc, we also complete the discussion about the spectrum of a weighted composition operator when it is invertible on $H^2(\mathbb{D})$ or $A_α^2(\mathbb{D})$.
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.
Let $ϕ(z)=(ϕ_1(z),...,ϕ_n(z))$ be a holomorphic self-map of $B_n$ and $ψ(z)$ a holomorphic function on $B_n$, and $H(B_n)$ the class of all holomorphic functions on $B_n$, where $B_n$ is the unit ball of $C^n$, the weight composition operator $W_{ψ,ϕ}$ is defined by $W_{ψ,ϕ}=ψf(ϕ)$ for $f\in H(B_n)$. In this paper we estimate the essential norm for the weighted composition operator $W_{ψ,ϕ}$ acting from the Hardy space $H^p$ to $H^q$ ($0<p,q\leq \infty$). When $p=\infty$ and $q=2$, we give an exact formula for the essential norm. As their applications, we also obtain some sufficient and necess…
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.
We study the interpolation sets for the Hardy-Sobolev spaces defined on the unit ball of ${\bf C}^n$. We begin by giving a natural extension to ${\bf C}^n$ of the condition that is known to be necessay and sufficient for interpolation sets lying on the boundary of the unit disc. We show that under this condition the restriction of a function in the Hardy-Sobolev space to the set always exists, and lies in a Besov space. We then show that under the assumption that there is an holomorphic distance function for the set, there is an extension operator from these spaces to the Hardy-Sobolev ones.
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.
Random Carleson Sequences for the Hardy space on the Polydisc and the Unit Ball
本文研究多圆盘与单位球上Hardy空间随机Carleson序列的Kolmogorov 0-1律，通过随机Gramian矩阵和随机矩阵工具改进已有结果，并证明关于伪双曲距离划分的0-1律，应用于Bloch空间插值序列。
Riemannian metrics on the unit ball of the quaternions, which are naturally associated with the reproducing kernel Hilbert spaces. We study the metric arising from the Hardy space in detail. We show that, in contrast to the one-complex variable case, no Riemannian metric is invariant under regular self-maps of the quaternionic ball.
The Hardy space $H^1$ consists of the integrable functions $f$ on the unit circle whose Fourier coefficients $\widehat f(k)$ vanish for $k<0$. We are concerned with $H^1$ functions that have some additional (finitely many) holes in the spectrum, so we fix a finite set $\mathcal K$ of positive integers and consider the "punctured" Hardy space $$H^1_{\mathcal K}:=\{f\in H^1:\,\widehat f(k)=0\,\,\,\text{for all }\, k\in\mathcal K\}.$$ We then investigate the geometry of the unit ball in $H^1_{\mathcal K}$. In particular, the extreme points of the ball are identified as those unit-norm functions i…
Suppose $n\geq 3$ and let $B$ be the open unit ball in $\mathbb{R}^n$. Let $φ: B\to B$ be a $C^2$ map whose Jacobian does not change sign, and let $ψ$ be a $C^2$ function on $B$. We characterize bounded weighted composition operators $W_{φ,ψ}$ acting on harmonic Hardy spaces $h^p(B)$. In addition, we compute the operator norm of $W_{φ,ψ}$ on $h^p(B)$ when $φ$ is a Möbius transformation of $B$.
In this work, we prove that if S is a dual bounded sequence in the unit ball B of C^n for the Hardy class H^p(B), then S is H^s(B) interpolating with the linear extension property.
Reverse Carleson measures for Hardy spaces in the unit ball
本文刻画了单位球Hardy空间的反向Carleson测度，并推导了非内全纯函数对应的de Branges-Rovnyak空间的反向Carleson测度性质。
A note on extreme points of the unit of Hardy-Lorentz spaces
本文研究 Hardy-Lorentz 空间单位球的极点问题，证明在 $φ$ 严格递增且严格凹时，内函数是 $H(Λ(φ))$ 单位球的极点。
We prove that the multiplier algebra of the Drury-Arveson Hardy space $H_{n}^{2}$ on the unit ball in $\mathbb{C}^{n}$ has no corona in its maximal ideal space, thus generalizing the famous Corona Theorem of L. Carleson to higher dimensions. This result is obtained as a corollary of the Toeplitz corona theorem and a new Banach space result: the Besov-Sobolev space $B_{p}^σ$ has the "baby corona property" for all $σ\geq 0$ and $1<p<\infty $. In addition we obtain infinite generator and semi-infinite matrix versions of these theorems.
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.
Let $\mathbb B_n$ be the open unit ball in $\mathbb C^n$. We characterize the spectra of pointwise multipliers $M_u$ acting on Banach spaces of analytic functions on $\mathbb B_n$ satisfying some general conditions. These spaces include Bergman-Sobolev spaces $A^p_{α,β}$, Bloch-type spaces $\mathcal B_α$, weighted Hardy spaces $H^p_w$ with Muckenhoupt weights and Hardy-Sobolev Hilbert spaces $H^2_β$. Moreover, we describe the essential spectra of multipliers in most of the aforementioned spaces, in particular, in those spaces for which the set of multipliers is a subset of the ball algebra.
A slice regular analogue of the Malmquist-Takenaka system is investigated. It is proved that they form a complete orthonormal system in the quaternionic Hardy spaces of the unit ball. The properties of associated projection operator are studied. Key Words and Phrases: Functions of hypercomplex variables, quaternionic Hardy spaces, series expansions, interpolation, approximation by rational functions, quaternionic Malmquist-Takenaka system
Characterizations of composition operators on Bloch and Hardy type spaces
本文利用一般倍增权重和新方法刻画不同区域上Bloch型、Hardy型及Lipschitz型空间之间的复合算子，并给出有界性、紧性刻画及对已有结果的推广。
We study several aspects concerning slice regular functions mapping the quaternionic open unit ball into itself. We characterize these functions in terms of their Taylor coefficients at the origin and identify them as contractive multipliers of the Hardy space. In addition, we formulate and solve the Nevanlinna-Pick interpolation problem in the class of such functions presenting necessary and sufficient conditions for the existence and for the uniqueness of a solution. Finally, we describe all solutions to the problem in the indeterminate case.
A description of the Bloch functions that can be approximated in the Bloch norm by functions in the Hardy space $H^p$ of the unit ball of $\Cn$ for $0<p<\infty$ is given. When $0<p\leq1$, the result is new even in the case of the unit disk.
Recovering Hardy spaces from optimal domains of integration operators
研究了Hardy空间H^p到H^q的有界Volterra积分算子T_g的最佳积分域，证明其严格包含H^p，并讨论了不同p和q情况下最优域的交集性质。
We show that the weighted Bergman-Orlicz space $A\_α^ψ$ coincides with some weighted Banach space of holomorphic functions if and only if the Orlicz function $ψ$ satisfies the so-called $Δ^{2}$--condition. In addition we prove that this condition characterizes those $A\_α^ψ$ on which every composition operator is bounded or order bounded into the Orlicz space $L\_α^ψ$. This provides us with estimates of the norm and the essential norm of composition operators on such spaces. We also prove that when $ψ$ satisfies the $Δ^{2}$--condition, a composition operator is compact on $A\_α^ψ$ if and only …
In this article, we initiate the study of Bloch type spaces on the unit ball of a Hilbert space. As applications, the Hardy-Littlewood theorem in infinite-dimensional Hilbert spaces and characterizations of some holomorphic function spaces related to the Bloch type space are presented.
In this paper we show that that on a strongly pseudoconvex domain $D$ the projective limit of all Poletsky--Stessin Hardy spaces $H^p_u(D)$, introduced in \cite{PS}, is isomorphic to the space $H^\infty(D)$ of bounded holomorphic functions on $D$ endowed with a special topology. To prove this we show that Carathéodory balls lie in approach regions, establish a sharp inequality for the Monge--Ampére mass of the envelope of plurisubharmonic exhaustion functions and use these facts to demonstrate that the intersection of all Poletsky--Stessin Hardy spaces $H^p_u(D)$ is $H^\infty(D)$.
In this paper we prove generic results concerning Hardy spaces in one or several complex variables. More precisely, we show that the generic function in certain Hardy type spaces is totally unbounded and hence non-extentable, despite the fact that these functions have non tangential limits at the boundary of the domain. We also consider local Hardy spaces and show that generically these functions do not belong, not even locally, to Hardy spaces of higher order. We work first in the case of the unit ball of Cn where the calculations are easier and the results are somehow better, and then we ext…
Contraction property on complex hyperbolic ball
本文在特定假设下证明了复双曲球上的等周不等式，并由此推导出Hardy和加权Bergman空间中全纯函数的收缩性质，部分推广了Kulikov在复双曲平面上的结果。
As outlined below, this paper is devoted to a Carleson-type-measure-based study of the holomorphic Campanato $2$-space on the open unit ball $\mathbb B_n$ of $\mathbb C^n$, comprising all Hardy $2$-functions whose oscillations in non-isotropic metric balls on the compact unit sphere $\mathbb S_n$ are proportional to some power of the radius other than the dimension $n\ge 1$.
In this paper, a family of holomorphic spaces of tent type in the unit ball of $\mathbb{C}^n$ is introduced, which is closely related to maximal and area integral functions in terms of the Bergman metric. It is shown that these spaces coincide with Bergman spaces. Furthermore, Littlewood-Paley type $g$-functions for the Bergman spaces are introduced in terms of the radial derivative, the complex gradient, and the invariant gradient. The corresponding characterizations for Bergman spaces are obtained as well. As an application, we obtain new maximal and area integral characterizations for Hardy…
Regularly oscillating mappings between metric spaces and a theorem of Hardy and Littlewood
受Hardy-Littlewood关于单位圆盘解析映射导数增长与Hölder连续性关联的经典定理启发，本文将该结果推广到带权重度量空间上的正则振荡映射情形，并推导出赋范空间单位球上解析映射的对应定理。
We prove that there is a continuous non-negative function $g$ on the unit sphere in $\cd$, $d \geq 2$, whose logarithm is integrable with respect to Lebesgue measure, and which vanishes at only one point, but such that no non-zero bounded analytic function $m$ in the unit ball, with boundary values $m^\star$, has $|m^\star| \leq g$ almost everywhere. The proof analyzes the common range of co-analytic Toeplitz operators in the Hardy space of the ball.
We characterize the symbols of Hankel operators that ex- tend into bounded operators from the Hardy-Orlicz $H^{Φ_1} (\mathbb B^n)$ into $H^{Φ_2} (\mathbb B^n)$ in the unit ball of Cn, in the case where the growth functions $?Φ_1$ and $Φ_2$ are either concave or convex. The case where the growth functions are both concave has been studied by Bonami and Sehba. We also obtain several weak factorization theorems for functions in $H^Φ(\mathbb B^n)$, with concave growth function, in terms of products of Hardy-Orlicz functions with convex growth functions.
We study the holomorphic Hardy-Orlicz spaces H^Φ(Ω), where Ωis the unit ball or, more generally, a convex domain of finite type or a strictly pseudoconvex domain in Cn . The function Φis in particular such that H ^1(Ω) \subset H^Φ(Ω) \subset H ^p (Ω) for some p > 0. We develop for them maximal characterizations, atomic and molecular decompositions. We then prove weak factorization theorems involving the space BMOA(Omega). As a consequence, we characterize those Hankel operators which are bounded from H ^Φ(Ω) into H^1 (Ω).
Universal multipliers for Sub-Hardy Hilbert spaces
本文研究 de Branges-Rovnyak 空间的通用乘子，建立一般框架，重新证明 Davis-McCarthy 定理，并用商 b/a 所属的函数类刻画 Lipschitz 类与 Gevrey 类乘子。
Weighted norm inequalities of various square functions and Volterra integral operators on the unit ball
本文研究复单位球上的平方函数与Volterra积分算子，证明了泊松积分相关Lusin面积积分的A_p权加权不等式（解答公开问题），得到全纯函数加权Hardy空间的等价刻画及Volterra算子的加权不等式。
In this paper, we introduce a discrete analogue of weighted Hardy spaces on rooted trees and study weighted composition operators between them in detail. In particular, we characterize bounded and compact weighted composition operators between discrete Hardy spaces. We also consider isometric weighted composition operators between these spaces.
We define Hardy spaces $H^p$, $0<p<\infty$, for quasiconformal mappings on the Korányi unit ball $B$ in the first Heisenberg group $\mathbb{H}^1$. Our definition is stated in terms of the Heisenberg polar coordinates introduced by Korányi and Reimann, and Balogh and Tyson. First, we prove the existence of $p_0(K)>0$ such that every $K$-quasiconformal map $f:B \to f(B) \subset \mathbb{H}^1$ belongs to $H^p$ for all $0<p<p_0(K)$. Second, we give two equivalent conditions for the $H^p$ membership of a quasiconformal map $f$, one in terms of the radial limits of $f$, and one using a nontangential …
There has been a great deal of work done in recent years on weighted Bergman spaces $\apa$ on the unit ball $\bn$ of $\cn$, where $0<p<\infty$ and $α>-1$. We extend this study in a very natural way to the case where $α$ is {\em any} real number and $0<p\le\infty$. This unified treatment covers all classical Bergman spaces, Besov spaces, Lipschitz spaces, the Bloch space, the Hardy space $H^2$, and the so-called Arveson space. Some of our results about integral representations, complex interpolation, coefficient multipliers, and Carleson measures are new even for the ordinary (unweighted) Bergm…
We study properties of inner and outer functions in the Hardy space of the quaternionic unit ball. In particular, we give sufficient conditions as well as necessary ones for functions to be inner or outer.
In this paper, we investigate the properties of hyperbolic harmonic mappings in the unit ball $\mathbb{B}^{n}$ in $\IR^n$ $(n\geq 2)$. Firstly, we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the Bloch space $\mathcal{B}(\mathbb{B}^{n})$ and the generalized Bloch space $\mathcal{L}_{\infty,ω}\mathcal{B}_{α,\mathrm{a}}^{0}(\mathbb{B}^{n})$, respectively. Secondly, we discuss the relationship between the integral means of hyperbolic harmonic mappings and that of their gradients. The obtained results are the generalizations of Hardy and Littlewood's rel…
Composition operators between de Branges-Rovnyak and Hardy spaces
研究从de Branges-Rovnyak空间到Hardy空间的复合算子的紧性，通过切片函数的Nevanlinna计数函数给出自然限制条件。
It is a classical theorem that if a function is integrable along the boundary of the unit circle, then the function is the nontangential limit of a holomorphic function on the open disc if and only if its Fourier coefficients for nonnegative integers are zero. In this article we generalize this result to higher complex dimensions by proving that for an integrable function on the unit sphere, it is ``boundary trace'' of a holomorphic function on the open unit ball if and only if two particular families of integral equations are satisfied. To do this, we use the theory of Hardy spaces as well as…
Cyclicity of stable matrix free polynomials over non-commutative operator unit balls
研究非交换算子单位球上矩阵自由多项式的循环性，通过稳定性和弱*拓扑特性，结合实化理论，探讨稳定非交换有理函数的循环性及非交换自由 Hardy 空间中结果的恢复。
We study almost sure separating and interpolating properties of random sequences in the polydisc and the unit ball. In the unit ball, we obtain the 0-1 Komolgorov law for a sequence to be interpolating almost surely for all the Besov-Sobolev spaces $B_{2}^σ\left(\mathbb{B}_{d}\right)$, in the range $0 < σ\leq1 / 2$. For those spaces, such interpolating sequences coincide with interpolating sequences for their multiplier algebras, thanks to the Pick property. This is not the case for the Hardy space $\mathrm{H}^2(\mathbb{D}^d)$ and its multiplier algebra $\mathrm{H}^\infty(\mathbb{D}^d)$: in th…
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