Cyclicity of Multipliers on the Unit Ball of ${\mathbb{C}}^n$: A Corona-Based Approach

We study the cyclicity of multipliers in Dirichlet-type spaces $D_{\alpha }({\mathbb{B}}_n)$. Specifically, we show that a multiplier $f$ analytic on a neighborhood of ${\bar{\mathbb{B}}}_n$, whose zero set on the unit sphere is a compact, smooth, complex tangential submanifold of real dimension $m\le n-1$, is cyclic in $D_{\alpha }({\mathbb{B}}_n)$ if and only if $\alpha \le \frac{2n-m}{2}$. Our approach combines classical results on peak sets in $A^{\mathrm{\infty }}({\mathbb{B}}_n)$ due to Chaumat and Chollet with a Corona-type theorem with two generators for the multiplier algebra.