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$L^{\vec{p}}-L^{\vec{q}}$ Boundedness of Multiparameter Forelli-Rudin Type Operators on Tube Domains Over The Forward Light Cones

Xin Xia, Guan Tie Deng

TL;DR

This paper analyzes when multiparameter Forelli-Rudin type operators on tubular domains $T_{\Lambda_n}$ over the forward light cone are bounded between weighted mixed-norm Lebesgue spaces. Using Schur-type testing for product kernels and adjoint-duality arguments, it derives sharp parameter relations and explicit coupling conditions, notably $c_i = n + a_i + b_i + \frac{n + \beta_i}{q_i} - \frac{n + \alpha_i}{p_i}$ and the bounds $-q_i a_i < \beta_i + 1$, $\alpha_i + 1 < p_i(b_i + 1)$. The main results (Theorems 1–5) provide complete necessary and sufficient characterizations for two operator classes, covering general and endpoint mixed-norm regimes, with important implications for Bergman projections in this geometry. The work extends classical Forelli-Rudin theory to a multiparameter, tube-domain setting, offering a rigorous framework for weighted Bergman-type operators on domains tied to the forward light cone.

Abstract

This study investigates necessary and sufficient conditions for the boundedness of Forelli-Rudin type operators on weighted Lebesgue spaces associated with tubular domains over the forward light cone. We establish a complete characterization of the boundedness for two classes of multiparameter Forelli-Rudin type operators from the mixed-norm Lebesgue space $L^{\vec{p}}$ to $L^{\vec{q}}$, in the range $1 \leq \vec{p} \leq \vec{q} < \infty$. The findings contribute significantly to the analysis of Bergman projection operators in this setting.

$L^{\vec{p}}-L^{\vec{q}}$ Boundedness of Multiparameter Forelli-Rudin Type Operators on Tube Domains Over The Forward Light Cones

TL;DR

This paper analyzes when multiparameter Forelli-Rudin type operators on tubular domains over the forward light cone are bounded between weighted mixed-norm Lebesgue spaces. Using Schur-type testing for product kernels and adjoint-duality arguments, it derives sharp parameter relations and explicit coupling conditions, notably and the bounds , . The main results (Theorems 1–5) provide complete necessary and sufficient characterizations for two operator classes, covering general and endpoint mixed-norm regimes, with important implications for Bergman projections in this geometry. The work extends classical Forelli-Rudin theory to a multiparameter, tube-domain setting, offering a rigorous framework for weighted Bergman-type operators on domains tied to the forward light cone.

Abstract

This study investigates necessary and sufficient conditions for the boundedness of Forelli-Rudin type operators on weighted Lebesgue spaces associated with tubular domains over the forward light cone. We establish a complete characterization of the boundedness for two classes of multiparameter Forelli-Rudin type operators from the mixed-norm Lebesgue space to , in the range . The findings contribute significantly to the analysis of Bergman projection operators in this setting.

Paper Structure

This paper contains 7 sections, 7 theorems, 105 equations.

Table of Contents

  1. Introduction and main results
  2. Preliminaries
  3. Proof of Theorem 1
  4. Proof of Theorem 2
  5. Proof of Theorem 3
  6. Proof of Theorem 4
  7. Proof of Theorem 5

Key Result

Lemma 2.1

Suppose that $r, s > \frac{n - 1}{2}$, $l> -1$, and $r + s - l > \frac{3}{2}n - 1$. Then for all $z$, $\xi$ in $T_{\Lambda_n}$, where $C_{1,l,r,s}$ is a constant in terms of n, r, s, l.

Theorems & Definitions (8)

  • Lemma 2.1
  • Remark 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Lemma 2.7