The Beurling operator for the hyperbolic plane (Q2908727)

Let \(\mathbb C_+\) denote the upper half plane. The author introduces the Beurling-type operators NEWLINENEWLINENEWLINE\[NEWLINEB^{\downarrow}[f](z):=p.v. \int_{\mathbb C_+} \big[(z-\overline{w})^{-2} - (z-w)^{-2}\big]f(w) dA(w),\quad z\in\mathbb C_+,NEWLINE\]NEWLINENEWLINENEWLINENEWLINE\[NEWLINE B^{\uparrow}[f](z):=p.v. \int_{\mathbb C_+} \big[(\overline{z}-w)^{-2} - (z-w)^{-2}\big]f(w) dA(w), \quad z\in \mathbb C_+.NEWLINE\]NEWLINE NEWLINENEWLINEFor \(1<p<\infty\) and real \(q\), the author introduces the space \(L^p_q(\mathbb C_+)\) endowed with the norm \(\int_{\mathbb C_+}|f(z)|^p (Im z)^q dA(z)\). It is proved that the operator \(B^{\uparrow}\) is bounded and transforms \(L^p_p(\mathbb C_+)\) into itself. Cauchy-type operators are introduced and discussed. Estimates of the norms are obtained.