Tangential convergence of Poisson integrals is proved for certain spaces of regular functions which contain the spaces of Bessel potentials of
L
p
{L^p}
functions,
1
>
p
>
∞
1 > p > \infty
, and of functions in the local Hardy space
h
1
{h^1}
, and the corresponding tangential maximal functions are shown to be of strong
p
p
type,
p
⩾
1
p \geqslant 1
.