Hardy spaces in the complex plane and in higher dimensions have natural finite-dimensional subspaces formed by polynomials or by linear maps. We use the restriction of Hardy norms to such subspaces to describe the set of possible derivatives of harmonic self-maps of a ball, providing a version of the Schwarz lemma for harmonic maps. These restricted Hardy norms display unexpected near-isometric duality between the exponents 1 and 4, which we use to give an explicit form of harmonic Schwarz lemma.
We obtain a necessary and sufficient condition for a weighted composition operator to be co-isometric on a general weighted Hardy space of analytic functions in the unit disk whose reproducing kernel has the usual natural form. This turns out to be equivalent to the property of being unitary. The result reveals a dichotomy identifying a specific family of weighted Hardy spaces as the only ones that support non-trivial operators of this kind.
In the first part of this study we consider the boundedness and compactness properties of Cauchy-Fantappie type operators on Poletsky-Stessin Hardy spaces $H^{p}_{u}(\mathbb{B}^{\textbf{p}})$ of complex ellipsoids. We show that boundedness and compactness criteria are given by the Carleson conditions. In addition we give a basic compactness property for the subsets of $H^{p}_{u}(\mathbb{B}^{\textbf{p}})$ spaces and the characterization of weakly convergent sequences in $H^{p}_{u}(\mathbb{B}^{\textbf{p}})$. In the second part we will discuss the dual complement of the complex ellipsoid and we w…
The authors study Hardy spaces, of arbitrary order, on a space of homogeneous type. This extends earlier work that treated only $H^p$ for $p$ near 1. Applications are given to the boundedness of certain singular integral operators, especially on domains in complex space.
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.
In \cite{CO-Tp-spaces}, the present authors initiated the study of composition operators on discrete analogue of generalized Hardy space $\mathbb{T}_{p}$ defined on a homogeneous rooted tree. In this article, we give equivalent conditions for the composition operator $C_ϕ$ to be bounded on $\mathbb{T}_{p}$ and on $\mathbb{T}_{p,0}$ spaces and compute their operator norm. We also characterize invertible composition operators as well as isometric composition operators on $\mathbb{T}_{p}$ and on $\mathbb{T}_{p,0}$ spaces. Also, we discuss the compactness of $C_ϕ$ on $\mathbb{T}_{p}$ and finally p…
We study the holomorphic tent spaces $\mathcal{HT}^p_{q,α}(\Bn)$, which are motivated by the area function description of the Hardy spaces on one hand, and the maximal function description of the Hardy spaces on the other. Characterizations for these spaces under general fractional differential operators are given. We describe the dual of $\mathcal{HT}^p_{q,α}(\Bn)$ for the full range $0<p,q<\infty$ and $α>-n-1$. Here the case $1<p<\infty$, $q\leq 1$ leads us to the Hardy-Bloch type spaces $\mathcal{BT}^{p}(\Bn)$ and $\mathcal{BT}^{p,0}(\Bn)$, the latter being the predual of $\mathcal{HT}^p_{1…
The definition of classical holomorphic function spaces such as the Hardy space or the Dirichlet space on the Hartogs triangle is not canonical. In this paper we introduce a natural family of holomorphic function spaces on the Hartogs triangle which includes some weighted Bergman spaces, a candidate Hardy space and a candidate Dirichlet space. For the weighted Bergman spaces and the Hardy space we study the $L^p$ mapping properties of Bergman and Szegő projection respectively, whereas for the Dirichlet space we prove it is isometric to the Dirichlet space on the bidisc.
We are concerned with Hardy and BMO spaces of operator-valued functions analytic in the unit disk of $\mathbb{C}.$ In the case of the Hardy space, we involve the atomic decomposition since the usual argument in the scalar setting is not suitable. Several properties (the Garsia-norm equivalent theorem, Carleson measure, and so on) of BMOA spaces are extended to the operator-valued setting. Then, the operator-valued $\mathrm{H}^1$-BMOA duality theorem is proved. Finally, by the $\mathrm{H}^1$-BMOA duality we present the Lusin area integral and Littlewood-Paley $g$-function characterizations of t…
We show that the efficiency of a natural pairing between certain projectively invariant Hardy spaces on dual strongly C-linearly convex real hypersurfaces in complex projective space is measured by the norm of the corresponding Leray transform.
Expected duality and approximation properties are shown to fail on Bergman spaces of domains in $\mathbb{C}^n$, via examples. When the domain admits an operator satisfying certain mapping properties, positive duality and approximation results are proved. Such operators are constructed on generalized Hartogs triangles. On a general bounded Reinhardt domain, norm convergence of Laurent series of Bergman functions is shown. This extends a classical result on Hardy spaces of the unit disc.
For $1\leq p\leq \infty$ and $α>0$, Besov spaces $B^p_α$ play a key role in the theory of $α$-Möbius invariant function spaces. In some sense, $B^1_α$ is the minimal $α$-Möbius invariant function space, $B^2_α$ is the unique $α$-Möbius invariant Hilbert space, and $B^\infty_α$ is the maximal $α$-Möbius invariant function space. In this paper, under the $α$-Möbius invariant pairing and by the space $B^\infty_α$, we identify the predual and dual spaces of $B^1_α$. In particular, the corresponding identifications are isometric isomorphisms. The duality theorem via the $α$-Möbius invariant pairing…
Let $Ω$ be either the unit polydisc $\mathbb D^d$ or the unit ball $\mathbb B_d$ in $\mathbb C^d$ and $G$ be a finite pseudoreflection group which acts on $Ω.$ Associated to each one-dimensional representation $\varrho$ of $G,$ we provide a notion of the (weighted) Hardy space $H^2_\varrho(Ω/G)$ on $Ω/G.$ Subsequently, we show that each $H^2_\varrho(Ω/G)$ is isometrically isomorphic to the relative invariant subspace of $H^2(Ω)$ associated to the representation $\varrho.$ For $Ω=\mathbb D^d,$ $G=\mathfrak{S}_d,$ the permutation group on $d$ symbols and $\varrho = $ the sign representation of $…
We give various equivalent formulations to the (partially) open problem about $L^p$-boundedness of Bergman projections in tubes over cones. Namely, we show that such boundedness is equivalent to the duality identity between Bergman spaces, $A^{p'}=(A^p)^*$, and also to a Hardy type inequality related to the wave operator. We introduce analytic Besov spaces in tubes over cones, for which such Hardy inequalities play an important role. For $p\geq 2$ we identify as a Besov space the range of the Bergman projection acting on $L^p$, and also the dual of $A^{p'}$. For the Bloch space $\SB^\infty$ we…
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}:…
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…
We formalize the observation that the same summability methods converge in a Banach space $X$ and its dual $X^*$. At the same time we determine conditions under which these methods converge in the weak and weak*-topologies on $X$ and $X^*$ respectively. We also derive a general limitation theorem, which yields a necessary condition for the convergence of a summability method in $X$. These results are then illustrated by applications to a wide variety of function spaces, including spaces of continuous functions, Lebesgue spaces, the disk algebra, Hardy and Bergman spaces, the BMOA space, the Bl…
We study the multiplier algebras $A(\mathcal{H})$ obtained as the closure of the polynomials on certain reproducing kernel Hilbert spaces $\mathcal{H}$ on the ball $\mathbb{B}_d$ of $\mathbb{C}^d$. Our results apply, in particular, to the Drury-Arveson space, the Dirichlet space and the Hardy space on the ball. We first obtain a complete description of the dual and second dual spaces of $A(\mathcal H)$ in terms of the complementary bands of Henkin and totally singular measures for $\operatorname{Mult}(\mathcal{H})$. This is applied to obtain several definitive results in interpolation. In part…
We define by interpolation a scale analogous to the Hardy $H^p$ scale for complete Pick spaces, and establish some of the basic properties of the resulting spaces, which we call $\mathcal{H}^p$. In particular, we obtain an $\mathcal{H}^p-\mathcal{H}^q$ duality and establish sharp pointwise estimates for functions in $\mathcal{H}^p$.
Given an inner function $θ$ on the unit disk, let $K^p_θ:=H^p\capθ\bar z\bar{H^p}$ be the associated star-invariant subspace of the Hardy space $H^p$. Also, we put $K_{*θ}:=K^2_θ\cap{\rm BMO}$. Assuming that $B=B_{\mathcal Z}$ is an interpolating Blaschke product with zeros $\mathcal Z=\{z_j\}$, we characterize, for a number of smoothness classes $X$, the sequences of values $\mathcal W=\{w_j\}$ such that the interpolation problem $f\big|_{\mathcal Z}=\mathcal W$ has a solution $f$ in $K^2_B\cap X$. Turning to the case of a general inner function $θ$, we further establish a non-duality relatio…
Hardy Spaces and Toeplitz operators on the Pentablock
School of Mathematics and Statistics, Wuhan University of Technology（武汉理工大学数学与统计学院）；Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore（印度统计研究所班加罗尔统计与数学单元）
