A sharp norm estimate for weighted Bergman projections on the minimal ball
The authors prove the following theorem: For an arbitrary \(s>-1\), consider the weighted Bergman projection with the weight \[ \bigg (1-|z|^2-\bigg|\sum_{j=1}^{n}z_{j}^2\bigg|\bigg)^s \] on the so-called minimal ball \[ \bigg\{ z\in \mathbb C^{n}: |z|^2+\bigg|\sum_{j=1}^{n}z_{j}^2\bigg|<1\bigg\}. \] This projection sends a certain \(L^2\) space into the space of \(L^2\) of holomorphic functions, but also can be seen as a linear operator on the space \[ L^{p}\Bigg(\cdot, \bigg(1-|z|^2-\bigg|\sum_{j=1}^{n}z_{j}^2\bigg|\bigg)^s\bigg(\bigg|\sum_{j=1}^{n}z_{j}^2\bigg|\bigg)^{\frac{p-2}{2}}\Bigg). \] Then, for any \(0<p<\infty\), the norm of this projection is greater than a constant times \(\text{csc\,}(\frac{\pi}{p})\), and it is less than another constant times \(\text{csc\,}(\frac{\pi}{p})\). The authors also consider the hypersurface in the Euclidean unit ball in \(\mathbb C^{n+1}\) defined by \[ \bigg\{ z\in \mathbb C^{n+1}: |z|<1, \sum_{j=1}^{n}z_{j}^2=0\bigg\}. \] Also on this hypersurface, for any \(1<p<\infty\), an analogous estimate for the weighted Bergman projection is shown.
scientific article; zbMATH DE number 5935419