The $\mathcal{Q}_p$ Carleson Measure Problem

TL;DR

This paper characterizes when the space alQ_p, a Möbius invariant function space, is boundedly or compactly embedded into a quadratic tent space, and applies this to analyze related operators.

Contribution

It provides criteria for embedding alQ_p into tent spaces and uses these to study boundedness and compactness of Volterra and multiplication operators.

Findings

Established embedding conditions for alQ_p into tent spaces.

Derived criteria for boundedness of Volterra and multiplication operators.

Connected embedding properties to operator compactness.

Abstract

Let $\mu$ be a nonnegative Borel measure on the open unit disk $\mathbb{D}\subset\mathbb{C}$. This note shows how to decide that the M\"obius invariant space $\mathcal{Q}_p$, covering $\mathcal{BMOA}$ and $\mathcal{B}$, is boundedly (resp., compactly) embedded in the quadratic tent-type space $T^\infty_p(\mu)$. Interestingly, the embedding result can be used to determine the boundedness (resp., the compactness) of the Volterra-type and multiplication operators on $\mathcal{Q}_p$.