A Counterexample to the Liu--Lou--Zhu ${\mathcal{Q}}_p$--Carleson Embedding Conjecture

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 $$\mathrm{id}:{\mathcal{Q}}_p⟶{\mathcal{T}}_{p,2}^2(\mu )$$ 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.

18 pages, 3 figures. Comments welcome!