IUMJ

Title: On the problem of characterizing multipliers for the Drury-Arveson space

Authors: Jingbo Xia and Quanlei Fang

Issue: Volume 64 (2015), Issue 3, 663-696

Abstract:

Let $H^2_n$ be the Drury-Arveson space on the unit ball $\mathbb{B}$ in $\mathbb{C}^n$, and suppose that $n\geq2$. Let $k_z$, $z\in\mathbb{B}$ be the normalized reproducing kernel for $H^2_n$. In this paper, we consider the following rather basic question in the theory of the Drury-Arveson space: for $f\in H^2_n$, does the condition $\sup_{|z|<1}\|fk_z\|<\infty$ imply that $f$ is a multiplier of $H^2_n$? We show that the answer is negative. We further show that the analogue of the familiar norm inequality $\|H_{\phi}\|\leq C\|\phi\|_{\mbox{\scriptsize BMO}}$ for Hankel operators fails in the Drury-Arveson space.