Boundedness of Forelli-Rudin Type Operators on Tubular Domains over The Generalized Light Cones

TL;DR

The paper addresses the boundedness of Forelli-Rudin type operators on weighted Lebesgue spaces over tubular domains associated with the generalized light cone. It develops explicit kernel estimates and a Schur-type framework to obtain a complete characterization: Theorem 1 provides necessary parameter relations, including $c_j = a_j + b_j + n + 1 + \frac{\beta_j + n + 1}{q} - \frac{\alpha_j + n + 1}{p}$ and the inequalities $-a_j q < \beta_j + \frac{n+1}{2}$ (and $-a_n q < \beta_n+1$), along with duality-implied conditions on $\alpha_j$ and $b_j$. Theorem 2 gives sufficient conditions, requiring $c_j>n$ and a set of linear bounds on $\alpha$, $\beta$, $p$, and $q$ that also enforce the same $c_j$ relation. Together, these results yield a complete boundedness characterization for two Forelli-Rudin type operator classes in this tubular-light-cone setting, advancing Bergman-projection analysis on these domains.

Abstract

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

Theorems & Definitions (7)

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