Reverse Carleson measures for spaces of analytic functions

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 $\bar{\mathbb{D}}$ is called a $q$-reverse Carleson measure for $X$ if and only if there exists a constant $C>0$ such that $$\mathrm{\parallel }f{\mathrm{\parallel }}_X\le C\mathrm{\parallel }f{\mathrm{\parallel }}_{L^q(\bar{\mathbb{D}},d\mu )}$$ for all $f\in X\cap C(\bar{\mathbb{D}})$. We fully characterize the $q$-reverse Carleson measures with all $q>0$ for Hardy spaces $H^p(\mathbb{D})$ with all $0<p\le \mathrm{\infty }$, for the space $\mathrm{B}\mathrm{M}\mathrm{O}\mathrm{A}(\mathbb{D})$ 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 ${\mathbb{C}}^d$.