[Paper Review] New atomic decompositons for Bergman spaces on the unit ball

Jens Gerlach Christensen, Karlheinz Gröchenig+1 more|arXiv (Cornell University)|Apr 1, 2015

Mathematical Analysis and Transform Methods24 references3 citations

TL;DR

This paper develops new atomic decompositions and frames for weighted Bergman spaces on the unit ball in several complex variables using group-theoretic methods, specifically the representation theory of the discrete series of SU(n,1) and its covering groups. By leveraging coorbit theory in this non-integrable setting, the authors extend atomic decompositions to a significantly larger class of admissible atoms than previously possible, offering a unified and more general framework for Bergman space analysis.

ABSTRACT

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

Motivation & Objective

• To extend coorbit theory to non-integrable representations of semisimple Lie groups, specifically the holomorphic discrete series of SU(n,1).
• To derive explicit atomic decompositions and frames for weighted Bergman spaces A^p_α on the unit ball in C^n.
• To overcome limitations of prior methods by using group-theoretic techniques to enlarge the class of admissible atoms.
• To establish the existence of Riesz sequences in Bergman spaces via K-separated sets and integrable representations.
• To unify and generalize atomic decomposition theory for Bergman spaces beyond the classical Coifman–Rochberg–Luecking framework.

Proposed method

• Utilizes coorbit theory with the holomorphic discrete series representation of SU(n,1) acting on Bergman spaces A^p_α.
• Applies the representation theory of SU(n,1) and its covering groups to construct coorbit spaces via representation coefficients.
• Employs the wavelet transform W^σ_ζ(f)(x) = ⟨f, π_σ(x)ζ⟩ to characterize functions in Bergman spaces.
• Introduces a new class of admissible atoms by extending beyond the standard L^1 or smooth vectors to intermediate function spaces.
• Applies duality and interpolation techniques to prove the existence of Riesz sequences in Bergman spaces for p ∈ (1, ∞).
• Uses the structure of the Iwasawa decomposition G = KAN to analyze the group action and derive frame properties.

Experimental results

Research questions

• RQ1 Can coorbit theory be extended to non-integrable representations of semisimple Lie groups such as SU(n,1)?
• RQ2 What is the class of admissible atoms that can be used in atomic decompositions for Bergman spaces on the unit ball in C^n?
• RQ3 How does the use of SU(n,1) representation theory improve the range of admissible atoms compared to classical methods?
• RQ4 Under what conditions do K-separated sets generate Riesz sequences in Bergman spaces A^p_α?
• RQ5 Can the coorbit framework be used to derive explicit and concrete atomic decompositions for Bergman spaces in several complex variables?

Key findings

• The paper establishes that Bergman spaces A^p_α for 1 ≤ p ≤ ∞ are coorbit spaces associated with the holomorphic discrete series of SU(n,1), with α = σp/2 − n − 1 and σ > 0.
• A significantly larger class of admissible atoms is obtained by using group-theoretic methods, extending beyond the standard smooth or L^1 atoms.
• For σ > 2n and σ ∈ ℚ, the representation π_σ is integrable, allowing the use of classical coorbit theory and enlarging the space of admissible atoms.
• The existence of K-separated sets X ⊆ G_σ that generate p-Riesz sequences in A^p_α is proven for 1 < p < ∞, with uniform bounds A, B > 0 in the Riesz sequence inequality.
• The wavelet transform W^σ_ζ is shown to be interpolating on A^q_σq/2−n−1 for conjugate exponents p, q, enabling the construction of frames via interpolation.
• The theory provides a unified framework that generalizes and extends the classical atomic decompositions of Coifman, Rochberg, and Luecking to higher-dimensional Bergman spaces.