A non-testing characterization of bounded and compact composition operators on $\mathcal Q_p$ spaces

Abstract

In this paper, we give symbol-only, non-testing characterizations of bounded and compact composition operators on $\mathcal Q_p$, $0<p\le 1$, via a novel dyadic trace formulation over Carleson tents for generalized $p$-Nevanlinna counting functions. Our results resolve an open question raised in Xiao&#39;s 2001 book, as well as the diagonal case of Zhao&#39;s 2009 question, a longstanding problem in the theory of $\mathcal Q_p$ spaces. As an application of our main theorems, we also obtain boundedness and compactness criteria for the $\mathcal Q_p$-Carleson measure embedding $\textrm{id}:\mathcal Q_p\to L^2(WdA)$ for weights $W$ in the Littlewood--Paley class.