Oracle inequalities and upper bounds for kernel density estimators on manifolds and more general metric spaces

We prove oracle inequalities and upper bounds for kernel density estimators on a very broad class of metric spaces. Precisely we consider the setting of a doubling measure metric space in the presence of a non-negative self-adjoint operator whose heat kernel enjoys Gaussian regularity. Many classical settings like Euclidean spaces, spheres, balls, cubes as well as general Riemannian manifolds, are contained in our framework. Moreover the rate of convergence we achieve is the optimal one in these special cases. Finally we provide the general methodology of constructing the proper kernels when the manifold under study is given and we give precise examples for the case of the sphere.