Reverse Carleson measures for Hardy spaces in the unit ball

TL;DR

This paper characterizes reverse Carleson measures for Hardy spaces in the unit ball, providing conditions under which the Hardy space norm is controlled by a measure, and explores related measures for de Branges-Rovnyak spaces.

Contribution

It offers a complete characterization of reverse Carleson measures for Hardy spaces in the unit ball and investigates their properties in de Branges-Rovnyak spaces associated with non-inner functions.

Findings

Characterization of reverse Carleson measures for Hardy spaces in the unit ball.

Conditions under which Hardy space norms are dominated by measures.

Properties of measures for de Branges-Rovnyak spaces associated with non-inner functions.

Abstract

Let $H^p=H^p(B_d)$ denote the Hardy space in the open unit ball $B_d$ of $\mathbb{C}^d$, $d\ge 1$. We characterize the reverse Carleson measures for $H^p$, $1<p<\infty$, that is, we describe all finite positive Borel measures $\mu$, defined on the closed ball $\overline{B}_d$, such that \[ \|f \|_{H^p} \le c \|f\|_{L^p(\overline{B}_d,\mu)} \] for all $f\in H^p(B_d) \cap C(\overline{B}_d)$ and a universal constant $c>0$. Given a non-inner holomorphic function $b: B_d \to B_1$, we obtain properties of the reverse Carleson measures for the de Branges-Rovnyak space $\mathcal{H}(b)$.