Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian
Asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian
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DOI：
10.3934/dcds.2022085
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发表时间：
2021-07
影响因子：
1.1
通讯作者：
Martin Gebhard Mayer;C. B. Ndiaye
中科院分区：
文献类型：
--
作者：
Martin Gebhard Mayer;C. B. Ndiaye
研究了渐近双曲流形的共形无穷远的分数次共形拉普拉斯函数的Poisson核和格林函数的渐近性。我们得到了共形拉普拉斯函数的泊松核函数和格林函数在奇点附近的精确展开式。我们的格林函数展开式回答了在Poincare-Einstein流形的局部平坦共形无限的情况下Kim-Musso-wei[21]猜想的第一部分，并与Poisson核渐近一起被用来证明分数次Yamabe问题在这种情况下的可解性。在[29]中，我们在渐近双曲空间的共形无穷远的一般情形下的格林函数的渐近性也被用来证明分式Yamabe问题的可解性，其中的分式Yamabe问题对应于前人留下的一个整体情形。
We study the asymptotics of the Poisson kernel and Green's functions of the fractional conformal Laplacian for conformal infinities of asymptotically hyperbolic manifolds. We derive sharp expansions of the Poisson kernel and Green's functions of the conformal Laplacian near their singularities. Our expansions of the Green's functions answer the first part of the conjecture of Kim-Musso-Wei[21] in the case of locally flat conformal infinities of Poincare-Einstein manifolds and together with the Poisson kernel asymptotic is used also in our paper [25] to show solvability of the fractional Yamabe problem in that case. Our asymptotics of the Green's functions on the general case of conformal infinities of asymptotically hyperbolic space is used also in [29] to show solvability of the fractional Yamabe problem for conformal infinities of dimension \begin{document}$ 3 $\end{document} and fractional parameter in \begin{document}$ (\frac{1}{2}, 1) $\end{document} corresponding to a global case left by previous works.