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 \( D_α(\mathbb{B}_n) \). Specifically, we show that a multiplier \( f \) analytic on a neighborhood of the closed unit ball, whose zero set on the unit sphere is a compact, smooth, complex tangential submanifold of real dimension \( m \leq n - 1 \), is cyclic in \( D_α(\mathbb{B}_n) \) if and only if \( α\leq \frac{2n - m}{2} \), where \( m \) is the real dimension of the zero set of \( f \) on the boundary. Our approach combines classical results on peak sets in \( A^\infty(\mathbb{B}_n) \) due to Chaumat and Chollet with a Corona-type theorem with two generators for the multiplier algebra.