About a conjecture of Lieb-Solovej

TL;DR

This paper investigates a conjecture related to embedding constants between Bergman spaces on the upper-half plane, proving it for integer and large s, and exploring special cases and bounds.

Contribution

The authors provide a new proof of the conjecture for integer s and as s approaches infinity, and verify it for powers of the Bergman kernel and monomials.

Findings

Conjecture holds for integer s and as s approaches infinity.

Verified the conjecture for powers of the Bergman kernel.

Confirmed the conjecture for monomials in the unit disk.

Abstract

Very recently, E. H. Lieb and J. P. Solovej stated a conjecture about the constant of embedding between two Bergman spaces of the upper-half plane. A question in relation with a Werhl-type entropy inequality for the affine $AX+B$ group. More precisely, that for any holomorphic function $F$ on the upper-half plane $\Pi^+$, $$\int_{\Pi^+}|F(x+iy)|^{2s}y^{2s-2}dxdy\le \frac{\pi^{1-s}}{(2s-1)2^{2s-2}}\left(\int_{\Pi^+}|F(x+iy)|^2 dxdy\right)^s $$ for $s\ge 1$, and the constant $\frac{\pi^{1-s}}{(2s-1)2^{2s-2}}$ is sharp. We prove differently that the above holds whenever $s$ is an integer and we prove that it holds when $s\rightarrow\infty$. We also prove that when restricted to powers of the Bergman kernel, the conjecture holds. We next study the case where $s$ is close to $1.$ Hereafter, we transfer the conjecture to the unit disc where we show that the conjecture holds when restricted to analytic monomials. Finally, we overview the bounds we obtain in our attempts to prove the conjecture.