用生成核构造Poincaré型级数
Construction of Poincaré-type series by generating kernels
摘要：
设$Γ\subset \textrm{PSL}_2({\mathbb R})$是第一类富克斯群，其基本区具有有限的双曲面积，并设$\widetildeΓ$是它在$\textrm{SL}_2({\mathbb R})$中的覆盖.考虑双曲上半平面上的二次连续可微、平方可积函数空间，在$\widetildeΓ$的作用下，它们以适当的方式关于权乘子系统$k\in{\mathbb R}$变换。此类函数的空间允许权重为$k$的双曲拉普拉斯算子$Δ_k$的作用。遵循Jorgenson，von Pippich和Smajlović（其中$k=0$）的方法，我们使用与$Δ_k$相关的谱展开来构造波分布，然后确定其测试函数上表示自守核的条件，并进一步产生Poincaré型级数。这种方法的一个优点是所得级数可以自然地亚纯延拓到整个复平面。此外，我们推导出超范数界的特征函数的离散谱的$Δ_k$。
英文摘要：
Let $Γ\subset \textrm{PSL}_2({\mathbb R})$ be a Fuchsian group of the first kind having a fundamental domain with a finite hyperbolic area, and let $\widetildeΓ$ be its cover in $\textrm{SL}_2({\mathbb R})$. Consider the space of twice continuously differentiable, square-integrable functions on the hyperbolic upper half-plane, which transform in a suitable way with respect to a multiplier system of weight $k\in{\mathbb R}$ under the action of $\widetildeΓ$. The space of such functions admits the action of the hyperbolic Laplacian $Δ_k$ of weight $k$. Following an approach of Jorgenson, von Pippich and Smajlović (where $k=0$), we use the spectral expansion associated to $Δ_k$ to construct a wave distribution and then identify the conditions on its test functions under which it represents automorphic kernels and further gives rise to Poincaré-type series. An advantage of this method is that the resulting series may be naturally meromorphically continued to the whole complex plane. Additionally, we derive sup-norm bounds for the eigenfunctions in the discrete spectrum of $Δ_k$.
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