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

TL;DR

This paper characterizes when Carleson embeddings are bounded and compact on bounded -convex domains in ^n, providing sharp conditions and equivalences between boundedness and compactness.

Contribution

It establishes sharp -conditions for Carleson embedding boundedness and proves their equivalence to compactness on bounded -convex domains.

Findings

Sharp -conditions for boundedness of Carleson embeddings.

Equivalence between boundedness and compactness of these embeddings.

Characterizations applicable to -convex domains in ^n.

Abstract

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.