AI 中文总结
在纯指数体积增长的非正曲率调和流形\(\mathbb{X}\)上，研究特征函数和上调和函数边界行为。通过势理论，对\(L^2\)谱外特征函数获加权非切向极限等结果，对正上调和函数研究其非切向和切向边界行为，多数结果新颖。
AI 中文摘要
在维度\(n\geq3\)的纯指数体积增长的非正曲率调和流形\(\mathbb{X}\)上，研究特征函数和上调和函数边界行为的某些定量方面。首先关注\(\Delta\)的\(L^2\)谱外的复值特征函数，得到加权非切向极限几乎处处存在、径向极限边界例外集的精确豪斯多夫维数和豪斯多夫测度估计。接着研究正上调和函数的非切向和切向边界行为。即使在非紧型秩一黎曼对称空间和达梅克 - 里奇空间的齐次情形下，多数结果也是新的。论证基于适应\(\mathbb{X}\)内在格罗莫夫双曲几何的势理论。
英文摘要
On $\mathbb{X}$, a non-positively curved harmonic manifold of purely exponential volume growth, of dimension $n \ge 3$, we study certain quantitative aspects of the boundary behaviour of eigenfunctions and superharmonic functions. We first focus on complex-valued eigenfunctions lying outside the $L^2$-spectrum of $Δ$ and obtain the almost everywhere existence of weighted non-tangential limits, sharp Hausdorff dimension and Hausdorff measure estimates of the boundary exceptional sets for radial limits. Then in the second part, we shift our attention to non-tangential and tangential boundary behaviour of positive superharmonic functions. Most of our results are new even for the homogeneous setting of rank one Riemannian symmetric spaces of non-compact type and Damek-Ricci spaces. Our arguments are based on potential theory adapted to the intrinsic Gromov hyperbolic geometry of $\mathbb{X}$.
