Reverse Carleson measures on the closed polydisk

Abstract

Let ${\mathbb {D}}$ denote the open unit disk and ${\mathbb {T}}=\partial {\mathbb {D}}$. We characterize the reverse Carleson measures for the Hardy space $H^p({\mathbb {D}}^n)$, $1<p<\infty$, $n\ge 1$, that is, we describe those finite positive Borel measures $\mu$ defined on the closed, in the sense of Shilov, polydisk $\overline{{\mathbb {D}}}^n:= {\mathbb {D}}^n \cup {\mathbb {T}}^n$, for which

$$\begin{aligned} \Vert f \Vert _{H^p} \le c \Vert f\Vert _{L^p(\overline{{\mathbb {D}}}^n,\mu )} \end{aligned}$$

for all $f\in H^p({\mathbb {D}}^n) \cap C(\overline{{\mathbb {D}}}^n)$ and a constant $c>0$. Also, we prove that the reverse Carleson inequality for a pluriharmonic measure $\mu$ defined on the torus ${\mathbb {T}}^n$ is equivalent to the uniform reverse inequalities for almost all slice-measures $\mu _\zeta$, $\zeta \in {\mathbb {T}}^n$.