# Reproducing kernel Hilbert spaces on manifolds: Sobolev and Diffusion spaces

## TL;DR

This paper investigates the structure of reproducing kernel Hilbert spaces on Riemannian manifolds, characterizing Sobolev spaces as RKHS and introducing diffusion spaces with detailed examples.

## Contribution

It provides conditions under which Sobolev spaces are RKHS and introduces a new class of smoother RKHS called diffusion spaces.

## Key findings

- Sobolev spaces can be characterized as RKHS under specific conditions
- Diffusion spaces are a new class of smoother RKHS
- Detailed examples illustrate the theoretical results

## Abstract

We study reproducing kernel Hilbert spaces (RKHS) on a Riemannian manifold. In particular, we discuss under which condition Sobolev spaces are RKHS and characterize their reproducing kernels. Further, we introduce and discuss a class of smoother RKHS that we call diffusion spaces. We illustrate the general results with a number of detailed examples.