We are studying the problem of estimating density in a wide range of metric spaces, including the Euclidean space, the sphere, the ball, and various Riemannian manifolds. Our framework involves a metric space with a doubling measure and a self-adjoint operator, whose heat kernel exhibits Gaussian behaviour. We begin by reviewing the construction of kernel density estimators and the related background information. As a novel result, we present a pointwise kernel density estimation for probability density functions that belong to general H\"{o}lder spaces. The study is accompanied by an application in Seismology. Precisely, we analyze a globally-indexed dataset of earthquake occurrence and compare the out-of-sample performance of several approximated kernel density estimators indexed on the sphere.

翻译：我们研究在广泛度量空间（包括欧氏空间、球面、球体及各种黎曼流形）中进行密度估计的问题。该框架包含一个具有倍测度的度量空间以及一个自伴算子，其热核呈现高斯行为。首先回顾核密度估计器的构造及相关背景知识。作为一项新成果，我们提出了一种针对属于一般Hölder空间的概率密度函数的逐点核密度估计方法。该研究伴随地震学应用实例：具体而言，我们分析了全球索引的地震发生数据集，并比较了球面上索引的几种近似核密度估计器的样本外表现性能。