Reverse Carleson measures for spaces of analytic functions

Abstract

Let $X$ be a quasi-Banach space of analytic functions in the unit disc and let $q>0$. A finite positive Borel measure $\hat{I}¼$ 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 $$\Vert f{\Vert }_X\le C\Vert f{\Vert }_{L^q(\overline{\mathbb{D}},d\hat{I}¼)}$$ 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 $0<p\le \infty$, 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 $H{F}_0^{q,r}$ and the holomorphic Besov spaces $H{B}_0^{q,r}$. Related results are obtained for the Hardy spaces and certain holomorphic Triebel--Lizorkin spaces in the unit ball of ${\mathbb{C}}^d$.

25 pages, no figures. Comments are welcome !