A Fatou theorem for \(\alpha\)-harmonic functions. (Q1414109)

Let \(H\) be the upper half space of \(\mathbb R^d\), \(\alpha \in (0,2)\), \(\beta\in (0,1)\) and \(p,p_0\in (0,\infty]\). Let \(f\) be the restriction to \(H^c\) of an \(L^p\)-Hölder continuous function of order \(\beta\) and let \(u_f\) be the \(\alpha\)-harmonic function in \(H\), solution of the Dirichlet problem associated to \(f\). If \(f\) is locally bounded, \(p_0>1/(1-\alpha/2)\) and \(\beta p>1\) the authors prove the existence almost everywhere of the nontangential limit of \(u_f\) on \(\partial H\). Moreover, the conditions imposed by the authors in order to prove this Fatou theorem for \(\alpha\)-harmonic functions, are sharp.