Determine the full parameter range for the exact weighted Bergman norm formula

Determine the full range of parameters alpha>-1, bgeq0, and 1<p<infty satisfying alpha+2<p(b+1) for which the generalized Hilbert operator mathcal{H}_b on the weighted Bergman space A^p_alpha satisfies the exact norm identity ||\mathcal{H}_b||_{A^p_\alpha\to A^p_\alpha}=Bleft(\frac{\alpha+2}{p},b+1-\frac{\alpha+2}{p}\right).

Background

The paper studies the generalized Hilbert operator mathcal{H}b on weighted Bergman spaces A^p\alpha, where bgeq0 and alpha>-1. Under the boundedness condition alpha+2<p(b+1), the authors establish several upper and lower norm estimates. In particular, they prove the exact norm formula ||\mathcal{H}_b||=B((\alpha+2)/p,b+1-(\alpha+2)/p) for all alpha>-1 and pgeq2(\alpha+2), as well as additional parameter ranges when p<2(\alpha+2).

The unresolved issue is to determine whether the displayed Beta-function value remains the exact operator norm throughout the entire admissible parameter region, including parameter ranges not covered by the paper’s sufficient conditions. The question is explicitly posed after the paper’s main results and is not resolved by the subsequent theorems.