We provide a large family of atoms for Bergman spaces on irreducible bounded symmetric domains. This vastly generalizes results by Coifman and Rochberg from 1980. The atomic decompositions are derived using the holomorphic discrete series representations for the domain, and the approach is inspired by recent advances in wavelet and coorbit theory. This approach also settles the relation between atomic decompositions for the bounded and unbounded realizations of the domain.
We use the author's previous work on atomic decompositions of Besov spaces with spectrum on symmetric cones, to derive new atomic decompositions for Bergman spaces on tube type domains. It is related to work by Ricci and Taibleson who derived decompositions for classical Besov spaces from atomic decompositions of Bergman spaces on the upper half plane. Moreover, for this class of domains our method is an alternative to classical results by Coifman and Rochberg, and it works for a larger range of Bergman spaces.
For $α>-1$ and $0<p<\infty$, we study weighted Bergman spaces $\mathcal B^p_α$ of harmonic functions on the real hyperbolic ball and obtain an atomic decomposition of these spaces in terms of reproducing kernels. We show that an $r$-separated sequence $\{a_m\}$ with sufficiently large $r$ is an interpolating sequence for $\mathcal B^p_α$. Using these we determine precisely when a Bergman space $\mathcal B^p_α$ is included in another Bergman space $\mathcal B^q_β$.
In this paper, we show that every (weighted) Bergman space $\mathcal{A}^p_α (\mathbb{B}_n)$ in the complex ball admits an atomic decomposition of real-variable type for any $0 < p \le 1$ and $α> -1.$ More precisely, for each $f \in \mathcal{A}^p_α (\mathbb{B}_n)$ there exist a sequence of real-variable $(p, \8)_α$-atoms $a_k$ and a scalar sequence $\{λ_k \}$ with $\sum_k | λ_k |^p < \8$ such that $f = \sum_k λ_k P_α (a_k),$ where $P_α$ is the Bergman projection from $L^2_α (\mathbb{B}_n)$ onto $\mathcal{A}^2_α (\mathbb{B}_n).$ The proof is constructive, and our construction is based on some sh…
We derive atomic decompositions and frames for weighted Bergman spaces of several complex variables on the unit ball in the spirit of Coifman, Rochberg, and Luecking. In contrast to our predecessors, we use group theoretic methods, in particular the representation theory of the discrete series of $SU(n,1)$ and its covering groups. One of the benefits is a much larger class of admissible
In this paper, we present maximal and area integral characterizations of Bergman spaces in the unit ball of $\mathbb{C}^n.$ The characterizations are in terms of maximal functions and area integral functions on Bergman balls involving the radial derivative, the complex gradient, and the invariant gradient. As an application, we obtain new maximal and area integral characterizations of Besov spaces. Moreover, we give an atomic decomposition of real-variable type with respect to Carleson tubes for Bergman spaces.
For appropriate domains $Ω_{1}, Ω_{2}$ we consider mappings $Φ_{\mathbf A}:Ω_{1}\toΩ_{2}$ of monomial type. We obtain an orthogonal decomposition of the Bergman space $\mathcal A^{2}(Ω_{1})$ into finitely many closed subspaces indexed by characters of a finite Abelian group associated to the mapping $Φ_{\mathbf A}$. We then show that each subspace is isomorphic to a weighted Bergman space on $Ω_{2}$. This leads to a formula for the Bergman kernel on $Ω_{1}$ as a sum of weighted Bergman kernels on $Ω_{2}$
Let $\mathcal{W}$ be the corresponding wandering subspace of an invariant subspace of the Bergman shift. By identifying the Bergman space with $H^2(\mathbb{D}^2)\ominus[z-w]$, a sufficient and necessary conditions of a closed subspace of $H^2(\mathbb{D}^2)\ominus[z-w]$ to be a wandering subspace of an invariant subspace is given also, and a functional charaterization and a coefficient characterization for a function in a wandering subspace are given. As a byproduct, we proved that for two invariant subspaces $\mathcal{M}$, $\mathcal{N}$ with $\mathcal{M}\supsetneq\mathcal{N}$ and $dim(\mathcal…
We characterize the space of restrictions of real rational functions to certain algebraic Jordan curves in the plane via the Dirichlet-to-Neumann map associated to the domain in the complex plane bounded by the curve and its Bergman kernel. The characterization leads to a partial fractions-like decomposition for such rational functions and new ways to describe such Jordan curves. The multiply connected case is also explored.
In this paper, we introduce a dyadic structure on convex domains of finite type via the so-called dyadic flow tents. This dyadic structure allows us to establish weighted norm estimates for the Bergman projection $P$ on such domains with respect to Muckenhoupt weights. In particular, this result gives an alternative proof of the $L^p$ boundedness of $P$. Moreover, using extrapolation, we are also able to derive weighted vector-valued estimates and weighted modular inequalities for the Bergman projection.
In this paper we study the boundedness of Bergman projectors on weighted Bergman spaces on homogeneous Siegel domains of Type II. As it appeared to be a natural approach in the special case of tube domains over irreducible symmetric cones, we study such boundedness on the scale of mixed-norm weighted Lebesgue spaces. The sharp range for the boundedness of such operators is essentially known only in the case of tube domains over Lorentz cones. In this paper we prove that the boundedness of such Bergman projectors is equivalent to variuos notions of atomic decomposition, duality, and characteriz…
We describe the range of of weighted Cauchy transform and its $k$-Bergman projection when action on weighted true poly-Bargmann spaces constituting an orthogonal Hilbertian decomposition of the Hilbert space of Gaussian functions on the complex plane.
The purpose of this paper is to establish an atomic decomposition for functions in the weighted mixed norm space $A^{p,q}_ω$ induced by a radial weight $ω$ in the unit disc admitting a two-sided doubling condition. The obtained decomposition is further applied to characterize Carleson measures for $A^{p,q}_ω$, and bounded differentiation operators $D^{(n)}(f)=f^{(n)}$ acting from $A^{p,q}_ω$ to $L^p_μ$, induced by a positive Borel measure $μ$, on the full range of parameters $0<p,q,s<\infty$.
In this paper we study spaces of holomorphic functions on the right half-plane $\cal R$, that we denote by $\cal M^p_ω$, whose growth conditions are given in terms of a translation invariant measure $ω$ on the closed half-plane $\overline\cal R$. Such a measure has the form $ω=ν\otimes m$, where $m$ is the Lebesgue measure on $\mathbb R$ and $ν$ is a regular Borel measure on $[0,+\infty)$. We call these spaces generalized Hardy-Bergman spaces on the half-plane $\cal R$. We study in particular the case of $ν$ purely atomic, with point masses on an arithmetic progression on $[0,+\infty)$. We obt…
We consider a compact Riemann surface $R$ of arbitrary genus, with a finite number of non-overlapping quasicircles, which separate $R$ into two subsets: a connected Riemann surface $Σ$, and the union $\mathcal{O}$ of a finite collection of simply-connected regions. We prove that the Schiffer integral operator mapping the Bergman space of anti-holomorphic one-forms on $\mathcal{O}$ to the Bergman space of holomorphic forms on $Σ$ is an isomorphism. We then apply this to prove versions of the Plemelj-Sokhotski isomorphism and jump decomposition for such a configuration. Finally we obtain some ap…
The main purpose of this paper is to study the generalized Hilbert operator {equation*} \mathcal{H}_g(f)(z)=\int_0^1f(t)g'(tz)\,dt {equation*} acting on the weighted Bergman space $A^p_\om$, where the weight function $\om$ belongs to the class $\R$ of regular radial weights and satisfies the Muckenhoupt type condition {equation}\label{Mpconditionaabstract} \sup_{0\le r<1}\bigg(\int_{r}^1(\int_t^1\om(s)ds)^{-\frac{p'}{p}}\,dt\bigg)^\frac{p}{p'} \int_{0}^r(1-t)^{-p}(\int_t^1\om(s)ds)\,dt<\infty. \tag† {equation} If $q=p$, the condition on $g$ that characterizes the boundedness (or the compactnes…
We study several connected problems of holomorphic function spaces on homogeneous Siegel domains. The main object of our study concerns weighted mixed norm Bergman spaces on homogeneous Siegel domains of type II. These problems include: sampling, atomic decomposition, duality, boundary values, boundedness of the Bergman projectors. Our analysis include the Hardy spaces, and suitable generalizations of the classical Bloch and Dirichlet spaces. One of the main novelties in this work is the generality of the domains under consideration, that is, homogeneous Siegel domains, extending many results …
We give an atomic decomposition of closed forms on R n , the coefficients of which belong to some Hardy space of Musielak-Orlicz type. These spaces are natural generalizations of weighted Hardy-Orlicz spaces, when the Orlicz function depends on the space variable. One of them, called H log , appears naturally when considering products of functions in the Hardy space H 1 and in BM O. As a main consequence of the atomic decomposition, we obtain a weak factorization of closed forms whose coefficients are in H log. Namely, a closed form in H log is the infinite sum of the wedge product between an …
In this paper, we develop via real variable methods various characterisations of the Hardy spaces in the multi-parameter flag setting. These characterisations include those via, the non-tangential and radial maximal function, the Littlewood--Paley square function and area integral, Riesz transforms and the atomic decomposition in the multi-parameter flag setting. The novel ingredients in this paper include (1) establishing appropriate discrete Calderón reproducing formulae in the flag setting and a version of the Plancherel--Pólya inequalities for flag quadratic forms; (2) introducing the maxi…
Recently, several papers have considered a nonlinear analogue of Fourier series in signal analysis, referred to as either nonlinear phase unwinding or adaptive Fourier decomposition. In these processes, a signal is represented as the real component of a complex function $F: \partial\mathbb{D}\to \mathbb{C}$, and by performing an iterative method to obtain a sequence of Blaschke decompositions, the signal can be approximated using only a few terms. To better understand the convergence of these methods, the study of Blaschke decompositions on weighted Hardy spaces was studied by Coifman and Stei…
