Poisson maximal function is dominated by the Hardy-Littlewood maximal function (${\sup }_{z\in {\Gamma }_A(\xi )}|P[g](z)|\le C_{A}Mg(\xi )$)

For integrable boundary data $g$ on the unit circle, this follows from the Poisson kernel bound $C_A\delta /({\delta }^2+d^2)$ in a fixed Stolz region. The arc of radius $\delta$ and successive dyadic annuli contribute at most $C_A{2}^{-k}Mg(\xi )$, which sum to the displayed estimate. This links harmonic extension to the Hardy-Littlewood maximal function and its strong $L^p$ bounds for $p>1$.