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A Counterexample to the Liu--Lou--Zhu $\mathcal Q_p$--Carleson Embedding Conjecture
Bingyang Hu, Jie Xiao, Xiaojing Zhou
https://arxiv.org/abs/2608.04981 https://arxiv.org/pdf/2608.04981 https://arxiv.org/html/2608.04981
arXiv:2608.04981v1 Announce Type: new
Abstract: In this paper, we disprove a conjecture of Liu, Lou, and Zhu concerning Carleson embeddings of $\mathcal Q_p$ spaces into tent spaces for $0<p><1$. More precisely, we construct a finite positive $p$-Carleson measure $\mu$ on the unit disc $\mathbb D$ such that the canonical embedding $$ \operatorname{id}:\mathcal Q_p \longrightarrow \mathcal T_{p,2}^2(\mu) $$ is not bounded. The main ingredient is a new family of $\mathcal Q_p$ test functions that encodes Cantor-type structures on the unit circle $\mathbb T$ into the analytic behavior of functions in $\mathcal Q_p$. This construction is inspired by ideas developed in a recent work of the first and third authors on composition operators on $\mathcal Q_p$ spaces.
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