The Bloch space and the dual space of a Luecking-type subspace of A¹

Let $X$ be the dual space of a Luecking-type subspace of the Bergman space $A^1$. It is known that the Bloch space $\mathcal{B}$ is a subset of $X$. In 1990, Ghatage and Sun asked whether $\mathcal{B}$ is dense in $X$. They also asked whether the little version of $X$ is a subset of $\mathcal{B}$. In this note, based on results and methods of Girela, Pel\'aez, P\'erez-Gonz\'alez and R\"atty\"a in 2008, we answer the two questions in the negative.