论文标题
关于Lieb-Solovej的猜想
About a conjecture of Lieb-Solovej
论文作者
论文摘要
最近，E。H。Lieb和J. P. Solovej提出了一个关于嵌入到上半平面两个伯格曼空间之间的常数的猜想。与Axpine $ ax+b $组的WERHL型熵不等式有关的问题。更准确地说，对于上半平面上的任何全体形态函数$ f $ $π^+$，$$ \ int_ {π^+} | \ frac {π^{1-s}} {（2S-1）2^{2S-2}}} \ left（\ int_ {π^+} | f（x+iy）|^2 dxdy \ right）^s $ $ \ frac {π^{1-s}}} {（2S-1）2^{2S-2}} $是尖锐的。我们以不同的方式证明，每当$ s $是一个整数时，我们都会证明它在$ s \ rightarrow \ infty $时所持有。我们还证明，当仅限于伯格曼内核的权力时，猜想就会存在。下一步，我们研究了$ s $接近$ 1的案例，此后，我们将猜想转移到了单位光盘中，我们表明，当猜想仅限于分析单元时。最后，我们概述了我们为证明猜想而获得的界限。
Very recently, E. H. Lieb and J. P. Solovej stated a conjecture about the constant of embedding between two Bergman spaces of the upper-half plane. A question in relation with a Werhl-type entropy inequality for the affine $AX+B$ group. More precisely, that for any holomorphic function $F$ on the upper-half plane $Π^+$, $$\int_{Π^+}|F(x+iy)|^{2s}y^{2s-2}dxdy\le \frac{π^{1-s}}{(2s-1)2^{2s-2}}\left(\int_{Π^+}|F(x+iy)|^2 dxdy\right)^s $$ for $s\ge 1$, and the constant $\frac{π^{1-s}}{(2s-1)2^{2s-2}}$ 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.
