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 $\mathbb{ C}^d$.
Categories: math.CV math.FA