Embedding theorems and integration operators on Hardy--Carleson type tent spaces induced by doubling weights

TL;DR

The paper develops a comprehensive theory for Hardy–Carleson–type analytic tent spaces $AT_q^\infty(\omega)$ induced by radial doubling weights in $\mathcal{D}$. It establishes sharp Carleson-type embedding theorems into weighted tent spaces, derives a Littlewood–Paley formula for these spaces, and completely characterizes the boundedness/compactness of Volterra-type integration operators between $AT_p^\infty(\omega)$ and $AT_q^\infty(\omega)$. The results rely on atomic decompositions, Carleson-measure criteria, and Forelli–Rudin-type weight estimates, providing a unified framework for analytic tent spaces with doubling weights and extending previous work to the general $\mathcal{D}$ setting. Collectively, this work advances the functional-analytic understanding of analytic tent spaces and their operator theory, with potential applications in harmonic analysis and complex function theory.

Abstract

This paper develops the function and operator theory of Hardy--Carleson--type analytic tent spaces $AT_q^\infty(ω)$ induced by radial weights $ω$ satisfying a two-sided doubling condition. We first characterize the positive Borel measures $μ$ for which the embedding from $AT_p^\infty(ω)$ into the tent space $T_q^\infty(μ)$ is bounded for all $0 < p, q < \infty$. A Littlewood--Paley formula for $AT_q^\infty(ω)$ is then established. Using these results, we give a complete characterization of the boundedness (compactness) of Volterra-type integration operators between $AT_p^\infty(ω)$ and $AT_q^\infty(ω)$.

Theorems & Definitions (33)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1: Zh07
• Lemma 2.2: Pe16
• Lemma 2.3: PR21
• Lemma 2.4: PPR20
• Lemma 2.5
• proof
• Theorem 3.1