Poisson integrals and singular weights (Q920244)

For \(\alpha\geq 0\), let \(C_{\alpha}\) denote the class of functions f such that \[ \int_{{\mathbb{R}}}| f(t)|^ 2(1+t^{-2})^{\alpha /2} dt<\infty. \] Clearly, \(C_{\alpha}\subseteq L^ 2({\mathbb{R}})\), so that each \(f\in C_{\alpha}\) has a Poisson integral F on \({\mathbb{R}}\times]0,\infty [\). The author shows that the harmonic functions on \({\mathbb{R}}\times]0,\infty [\), which arise in this way, are characterized by the condition \[ \sup_{y>0}\int_{{\mathbb{R}}}| F(x,y)|^ 2 W_{\alpha}(x,y) dx\quad < \infty, \] where \(W_{\alpha}\) is the reciprocal of the Poisson integral of \((1+t^{-2})^{-\alpha /2}\). For \(\alpha <3\), a more explicit characterization is also presented. The proofs are given in full detail, and are very clearly argued.