A remark on Carleson measures of domains in $\mathbb{C}^{n}$

TL;DR

This paper characterizes Carleson measures on certain bounded pseudoconvex domains and provides an example of a vanishing Carleson measure with a non-vanishing Berezin transform on the boundary of Hartogs triangles.

Contribution

It offers new characterizations of Carleson measures on pseudoconvex domains and presents a novel example involving Hartogs triangles.

Findings

Characterizations of Carleson measures on pseudoconvex domains

Existence of a vanishing Carleson measure with non-vanishing Berezin transform

Analysis of Hartogs triangles in complex analysis

Abstract

We provide characterizations of Carleson measures on a certain class of bounded pseudoconvex domains. An example of a vanishing Carleson measure whose Berezin transform does not vanish on the boundary is given in the class of the Hartogs triangles $$\mathbb{H}_{k}:=\left\{ \left(z_{1},z_{2}\right)\in\mathbb{C}^{2}:\left|z_{1}\right|^{k}<\left|z_{2}\right|<1\right\},\;k\in\mathbb{Z}^{+}.$$