= Poisson maximal function is dominated by the Hardy-Littlewood maximal function
{c}
{title2=$\sup_{z\in\Gamma_A(\xi)}|P[g](z)|\le C_A Mg(\xi)$}
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 $C_A2^{-k}Mg(\xi)$, which sum to the displayed estimate. This links harmonic extension to the <Hardy-Littlewood maximal function> and its strong $L^p$ bounds for $p>1$.
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