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

Abstract

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