Multiplier characterization for the Drury-Arveson space

Let $H^2_n$ be the Drury-Arveson space on the unit ball {\bf B} in ${\bold C}^n$, and suppose that $n \geq 2$. Let $k_z$, $z \in $ {\bf B}, be the normalized reproducing kernel for $H^2_n$. In this talk we will discuss the following rather basic question in the theory of the Drury-Arveson space: For $f \in H^2_n$, does the condition $\sup _{|z|\u3c1}\|fk_z\| \u3c \infty $ imply that $f$ is a multiplier of $H^2_n$? We show that the answer is negative and the analogue of the familiar norm inequality $\|H_\varphi \| \leq C\|\varphi \|_{\text{BMO}}$ for Hankel operators fails in the Drury-Arveson space. This is joint work with Jingbo Xia