Abstract:

This paper investigates the boundedness of a Toeplitz operator $Q_\mu$ acting on the Hardy space $H^1(\mathbb{B}_{n})$. Let $\mu$ be a positive Borel measure on $\mathbb{B}_{n}$ and $0. The main results are following

1. If $\mu$ is a $(1,1)$-logarithmic Carleson measure, then $Q_\mu: H^1(\mathbb{B}_{n})\to H^1(\mathbb{B}_{n})$ is bounded;

2. If $Q_\mu: H^1(\mathbb{B}_{n})\to H^1(\mathbb{B}_{n})$ is bounded, then $\mu$ is a Carleson measure;

3. $Q_\mu: H^p(\mathbb{B}_{n})\to H^q(\mathbb{B}_{n})$ is bounded if and only if $\mu$ is a $(1+\frac1p-\frac1q)$-Carleson measure.

Key words: Hardy spaces, Toeplitz-type operators, Carleson measures