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's 2001 book, as well as the diagonal case of Zhao'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 $\text{id}:{\mathcal{Q}}_p\to L^2(WdA)$ for weights $W$ in the Littlewood--Paley class.

25 pages, 1 figure. Include a new section on an application of the main results. Comments are welcome!