\(L^p\) mapping properties of the Bergman projection on the Hartogs triangle (Q2790271)

Let \(\mathbb H\) be the Hartogs triangle consisting of all points \((z_1, z_2)\in\mathbb C^2\) such that \(|z_2|<|z_1|<1\). For \(1<p<\infty\) and a positive continuous function \(\omega\) on \(\mathbb H\), let \(A^p(\mathbb H,\omega)\) be the \(\omega\)-weighted \(L^p\)-Bergman space consisting of all holomorphic functions in \(L^p(\mathbb H, \omega dV)\) where \(dV\) is the Lebesgue measure on \(\mathbb H\). Let \(B_{\mathbb H}: L^2(\mathbb H)\to A^2(\mathbb H)\) be the Bergman projection. Here, \(L^p(\mathbb H)=L^p(\mathbb H, dV)\) and \(A^p(\mathbb H)=A^p(\mathbb H, 1)\). Let \(\delta_1\) be the function on \(\mathbb C^2\) defined by \(\delta_1(z_1, z_2)=|z_1|\), which is comparable to the distance to the singular boundary point \(0\).NEWLINENEWLINEThe authors first prove that \(B_{\mathbb H}:L^p(\mathbb H)\to A^p(\mathbb H, \delta_1^{p-2})\) is bounded and surjective for every \(p\geq 2\). As applications they recover the following folk results: (a) If \(\frac{4}{3}<p<4\), then \(B_{\mathbb H}\) is bounded (and surjective) from \(L^p(\mathbb H)\) to \(A^p(\mathbb H)\); (b) If \(p\geq 4\), then \(B_{\mathbb H}\) does not map \(L^p(\mathbb H)\) into \(A^p(\mathbb H)\). More interestingly, for the remaining parameter range \(1<p<\frac{4}{3}\), they obtain quite a pathological behaviour of \(B_{\mathbb H}\) showing that there is absolutely no way to control \(B_{\mathbb H}\) on \(L^p(\mathbb H)\) by means of a weight depending on \(\delta_1\). More precisely, they show: If \(1<p<\frac{4}{3}\) and \(\lambda>0\) is a continuous function on \((0,1]\), then \(B_{\mathbb H}\) does not map \(L^p(\mathbb H)\) into \(A^p(\mathbb H,\lambda\circ\delta_1)\).