= H1 non-tangential maximal inequality on the disk
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{title2=$\|N_Af\|_{L^1}\le C_A\|f\|_{H^1}$}
For an analytic <H1 space on the unit disk> function, its fixed-aperture <non-tangential maximal function> obeys the displayed strong $L^1$ bound. First replace $f$ by $f_R(z)=f(Rz)$, $R<1$. The fact that <modulus powers of holomorphic functions are subharmonic> gives $|f_R|^{1/2}\le P[|f(R\xi)|^{1/2}]$. Since a <Poisson maximal function is dominated by the Hardy-Littlewood maximal function>, the strong $L^2$ <Hardy-Littlewood maximal inequality> yields $\int N_Af_R\le C_A\int|f(R\xi)|\le C_A\|f\|_1$. Pointwise interior convergence and the <Fatou lemma> remove $R$. Analyticity permits this endpoint estimate; general $L^1$ boundary data have only a weak $L^1$ maximal bound.
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