On the non-tangential convergence of Poisson and modified Poisson semigroups at the smoothness points of $$L_{p}$$-functions

The high-dimensional version of Fatou’s classical theorem asserts that the Poisson semigroup of a function $$f\in L_{p}(\mathbb {R}^{n}), \ 1\le p \le \infty $$, converges to f non-tangentially at Lebesque points. In this paper we investigate the rate of non-tangential convergence of Poisson and metaharmonic semigroups at $$\mu $$-smoothness points of f.