Reproducing kernel Hilbert spaces on manifolds: Sobolev and Diffusion spaces

Among different notions of function spaces, reproducing kernel Hilbert spaces (RKHS) play a central role in a number of diverse contexts, including stochastic analysis [13]- where they are also known as Cameron-Martin spaces [12], harmonic analysis [10], [19], physics [3], numerical analysis [46]- where they are also known as native spaces, statistics [11], and machine learning [18, 41], to name a few. RKHS are Hilbert spaces of functions with continuous evaluation functionals, a property that naturally yields a number of implications and characterizations, where positive kernels and corresponding integral operators are key objects. Among other references [5] is a classic. Examples of RKHS and kernels abound and include functions defined in Euclidean spaces [11] but also for functions on less structured space, for example discrete space [39]. In many modern applications it is relevant to consider functions depending on a large, if not huge, number of variables potentially related to each others.