论文标题
通过生成内核来建造庞加莱型系列
Construction of Poincaré-type series by generating kernels
论文作者
论文摘要
令$γ\ subset \ textrm {psl} _2（{\ mathbb r}）$是第一种具有有限多余区域的基本域的fuchsian组，让$ \widetildeγ$为$ \ dextrm {sl} _2（sl} _2（Math} $）。考虑双曲线上半平面上的两次连续可区分的，可与方形的函数，在$ \widetildeγ$的动作下，相对于重量$ k \ in {\ mathbb r} $的乘数$ k \的乘数系统，它以适当的方式转换。这种功能的空间承认双曲线拉普拉斯$Δ_k$ a $ k $的动作。遵循Jorgenson的方法，Von Pippich和Smajlović（其中$ k = 0 $），我们使用与$Δ_K$相关的光谱膨胀来构建波浪分布，然后确定其代表自动态内核的测试功能的条件，并进一步提高poincaré-poincaré-type系列。该方法的一个优点是，所得序列可以自然地持续到整个复杂平面。此外，我们在$Δ_K$的离散频谱中得出了特征函数的sup-norm界限。
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 $Δ_k$.
