I have been reading the paper "Carleson Measures in Hardy and Weighted Bergman Spaces of Polydiscs" by F. Jafari and there are a few things that going on that I am not entirely convinced of.
The paper deals with continuous and compact embeddings $H^p(\mathbb{D}^2) \subseteq L^p(\mathbb{D}^2,\mu)$. Where $H^p(\mathbb{D}^2) = H^p$ is the Hardy space consisting of holomorphic functions.
The author claims that the characterization of $\mu$ such that the embedding is continuous is an immediate conseuqence of Chang's proof using standard arguments. Refering to the characterization of $\mu$ such that $h^p(\mathbb{D}^2) \subseteq L^2(\mu)$ where $h^p$ is the Hardy space consisting of harmonic functions. I do not see that.
Secondly, (which in fact is closely related) in the proof of Theorem 2.3 the author says that if there exists $\varepsilon>0$ and a sequence of open sets $V_j \subseteq \mathbb{T}^2$ of vanishing area such that $$ \mu(S(V_j))>\varepsilon |V_j|, \forall j $$ where $S(V_j)$ is the Carleson area and $|\cdot |$ is the Lebesgue measure, then there exist rectangles $R_j\subseteq \mathbb{T}^2$ that satisfy the same conditions. I am also skeptical about this claim.
Any thoughts ? Am I completely wrong here and these things are obvious ?
