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{If} one replaces the Poisson kernel of the unit disc by its square root, then normalised Poisson integrals of $L^{p}$ boundary functions converge along approach regions wider than the ordinary nontangential cones, as proved by Ronning ($1\leq p<\infty$) and Sjogren ($p=1$ and $p=\infty$). In this paper we present new and simplified proofs of these results. We also generalise the $L^{\infty}$ result to higher dimensions.

Approach regions for $l^{p}$ potentials with respect to the square root of the Poisson kernel