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A Bloch-type space and the predual of $A_λ^1$ on the Siegel upper half-space

Congwen Liu, Jiajia Si

Abstract

This paper aims to determine the predual of the Bergman space $A_λ^1$ on the Siegel upper half-space. To achieve this, a Bloch-type space $\widetilde{\calB}$ is introduced and studied, and some of its essential properties are established. We identify the little Bloch-type space $\widetilde{\calB}_0$ with the predual of $A_λ^1$ via a duality pairing.

A Bloch-type space and the predual of $A_λ^1$ on the Siegel upper half-space

Abstract

This paper aims to determine the predual of the Bergman space on the Siegel upper half-space. To achieve this, a Bloch-type space is introduced and studied, and some of its essential properties are established. We identify the little Bloch-type space with the predual of via a duality pairing.
Paper Structure (9 sections, 19 theorems, 96 equations)

This paper contains 9 sections, 19 theorems, 96 equations.

Table of Contents

  1. introduction
  2. Preliminaries
  3. Elementary results
  4. Heisenberg Group
  5. Cancellation property of $A_{\lambda}^1(\mathcal{U})$
  6. The projection $\widetilde{\mathcal{P}_{\lambda}}$ from $L^{\infty}(\mathcal{U})$ into $\widetilde{\mathcal{B}}$
  7. Integral representation
  8. Norm equivalence
  9. $\widetilde{\mathcal{B}}_0$ as the predual of $A_{\lambda}^1(\mathcal{U})$

Key Result

Theorem 1.1

If $f\in \widetilde{\mathcal{B}}$, then for any $N\in\mathbb{N}_0$ and $\lambda>-1$.

Theorems & Definitions (34)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 2.1: DK93
  • Lemma 2.2: Liu18
  • Lemma 2.3: LS20
  • Lemma 2.4
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 24 more