A remark on Carleson measures of domains in ℂⁿ

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 trianglesHk≔{(z1,z2)∈C2:|z1|k>|z2|>1},k∈Z+.\begin{equation*} \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}^{+}. \end{equation*}