Reverse Carleson measures for spaces of analytic functions

TL;DR

This paper characterizes $q$-reverse Carleson measures for various spaces of analytic functions, providing a comprehensive understanding of when these measures control the norm of functions in these spaces.

Contribution

It offers a complete characterization of $q$-reverse Carleson measures for Hardy, BMOA, Bloch, Triebel--Lizorkin, and Besov spaces, extending known results to new function spaces.

Findings

Characterization for Hardy spaces $H^p$ for all $0<p extless=ty$.

Characterization for $ ext{BMOA}$ and Bloch spaces.

Results for holomorphic Triebel--Lizorkin and Besov spaces.

Abstract

Let $X$ be a quasi-Banach space of analytic functions in the unit disc and let $q>0$. A finite positive Borel measure $\mu$ in the closed unit disc $\overline{\mathbb{D}}$ is called a $q$-reverse Carleson measure for $X$ if and only if there exists a constant $C>0$ such that $$\|f\|_{X}\leq C \|f\|_{L^q(\overline{\mathbb D},d\mu)} $$ for all $f\in X\cap C(\overline{\mathbb D})$. We fully characterize the $q$-reverse Carleson measures with all $q>0$ for Hardy spaces $H^p(\mathbb D)$ with all $0<p\leq \infty$, for the space $\mathrm{BMOA}(\mathbb D)$ and for the Bloch space. In addition, we describe $q$-reverse Carleson measures for the holomorphic Triebel--Lizorkin spaces $HF_0^{q,r}$ and the holomorphic Besov spaces $HB_0^{q,r}$. Related results are obtained for the Hardy spaces and certain holomorphic Triebel--Lizorkin spaces in the unit ball of $\mathbb{C}^d$.