= Non-tangential maximal function
{title2=$F^*_a(x)=\sup_{y>0,\ |x-x'|\leq ay}|F(x',y)|$}
This supremum function takes a supremum over a cone approaching the boundary point $x$; the positive number $a$ is its aperture. For an analytic upper-half-plane $H^1$ function, it belongs to $L^1$. To prove this without first assuming boundary absolute continuity, shift the function upwards, apply subharmonic Poisson comparison to its square root, and use the strong $L^2$ bound for the <Hardy-Littlewood maximal function>. <Fatou lemma> removes the shift.
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