 Non-tangential maximal function

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.