Integrability of the Poisson integral Is your $f=P_y*g$ for some $g$? Maybe the result is not true for every $p$. Is the result true for $p=2$, i.e., given $g\in H^1(\mathbb{R})$, is $P_y*g\in L^2(\mathbb{R}\times\mathbb{R}^+)$?

A bound for a product in BMO Thank you for your answer. However, your answer only says that my approach is wrong. It doesn't say anything on the bound, right?