Near-endpoints Carleson Embedding of $\mathcal Q_s$ and $F(p, q, s)$ into tent spaces

TL;DR

This paper investigates Carleson embedding problems for $\

Contribution

It introduces a new approach to near-endpoints Carleson embeddings for $\\mathcal Q_s$ and $F(p, q, s)$ spaces, extending previous results and confirming a conjecture.

Findings

Established equivalence of $s$-Carleson measures and bounded embeddings for $\\mathcal Q_t$ spaces.

Proved near-endpoints Carleson embedding results for $F(p, p\alpha-2, s)$ spaces.

Achieved compactness results and confirmed a conjecture on $\\mathcal Q_s$ embeddings.

Abstract

This paper aims to study the $\mathcal Q_s$ and $F(p, q, s)$ Carleson embedding problems near endpoints. We first show that for $0<t<s \le 1$, $\mu$ is an $s$-Carleson measure if and only if $id: \mathcal Q_t \mapsto \mathcal T_{s, 2}^2(\mu)$ is bounded. Using the same idea, we also prove a near-endpoints Carleson embedding for $F(p, p\alpha-2, s)$ for $\alpha>1$. Our method is different from the previously known approach which involves a delicate study of Carleson measures (or logarithmic Carleson measures) on weighted Dirichlet spaces. As some byproducts, the corresponding compactness results are also achieved. Finally, we compare our approach with the existing solutions of Carleson embedding problems proposed by Xiao, Pau, Zhao, Zhu, etc. Our results assert that a "tiny-perturbed" version of a conjecture on the $\mathcal Q_s$ Carleson embedding problem due to Liu, Lou, and Zhu is true. Moreover, we answer an open question by Pau and Zhao on the $F(p, q, s)$ Carleson embedding near endpoints.