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

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 Möbius invariant space ${\mathcal{Q}}_p$, covering $\mathcal{BM}\mathcal{O}\mathcal{A}$ and $\mathcal{B}$, is boundedly (resp., compactly) embedded in the quadratic tent-type space $T_p^{\infty }(\mu )$. Interestingly, the embedding result can be used to determine the boundedness (resp., the compactness) of the Volterra-type and multiplication operators on ${\mathcal{Q}}_p$.

12 pages