On cyclicity in weighted Dirichlet spaces (Q1970277)

Let \(D_\alpha(0\leq\alpha\leq 1)\) be the weighted Dirichlet space on the open unit disk \(\Delta\). If \(\alpha=1\), this is the classical Dirichlet space \(D_1\) and if \(\alpha=0\), this is the Hardy space \(H^2\). \textit{L. Brown} and \textit{A. L. Shields} [Trans. Am. Math. Soc. 285, 269-304 (1984; Zbl 0537.30040)] as whether \(f,g\in D_\alpha\) \((0<\alpha\leq 1)g\) is cyclic and \(|f(z)|\geq |g(z)|\) for all \(x\in\Delta\) implies that \(f\) is cyclic. In this present paper, the authors answer positively for the question when \(0<\alpha<1\) and \(g\in D_1\cap H^\infty\).