On a conjecture about contractive inequalities for weighted Bergman spaces

TL;DR

This paper investigates the inclusion norms between weighted Bergman spaces, addressing a conjecture about their contractivity, and offers partial results and improvements on recent findings.

Contribution

It provides partial answers and improvements regarding the conjectured contractive inequalities between specific weighted Bergman spaces.

Findings

Partial results supporting the conjecture for certain cases

Improved bounds on the inclusion norms

Insights into the structure of weighted Bergman spaces

Abstract

We are interested in the norm of the inclusion between the standard weighted Bergman spaces $A^2_\alpha$ and $A^p_{\frac{p}{2} (\alpha + 2)-2}$, $p > 2$, which is conjectured to be contractive by O.F. Brevig, J. Ortega-Cerd\`a, K. Seip and J. Zhao (in the limit case $\alpha = -1$) and E. Lieb and J.P. Solovej (for $\alpha > -1$). We provide some partial answers and improvements of some recent results.