Let \(u(x,y)\) be the Poisson integral in \(\mathbb{R}_ +^{n+1}= \bigl\{ (x,y):x \in \mathbb{R}^ n,\;y>0\bigr\}\) of a function \(f\in L^ p(\mathbb{R}^ n)\). The authors investigate questions of boundary convergence in non- nontangential and nonadmissible approach regions. For proofs of these results they use interesting methods connected with a Carleson measure.