About a conjecture of Lieb-Solovej

David Békollè, Jocelyn Gonessa, Benoît F. Sehba

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.