# The Bloch space and the dual space of a Luecking-type subspace of $A^1$

## TL;DR

This paper investigates the relationship between the Bloch space and the dual space of a Luecking-type subspace of the Bergman space, providing negative answers to longstanding questions about their density and subset relations.

## Contribution

It demonstrates that the Bloch space is neither dense in nor a subset of the dual space of the Luecking-type subspace, resolving two open questions from 1990.

## Key findings

- The Bloch space is not dense in the dual space.
- The little Bloch space is not a subset of the dual space.
- Answers to Ghatage and Sun's questions are negative.

## Abstract

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.