  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.
