A note for Carleson measure on bounded $\mathbb{C}$-convex domains

Let $0<q<p<\infty$, $\Omega$ be a bounded $\bbC$-convex domains in $\bbC^n$. We establish several equivalent characterizations for the boundedness of Carleson embedding $J_\mu:A_\alpha^p\hookrightarrow L^q(\mu)$ on $\Omega$ with sharp $\cB$ condition. Furthermore, we prove that the boundedness of $J_\mu$ is equivalent to its compactness.