Fractional Sobolev spaces on Riemannian manifolds (Q6634487)

Let \(M\) be a Riemannian manifold and \( s \in (0, 2) \). The paper studies three equivalent definitions for the fractional Hilbert space \( H^{s/2}(M) \) in terms of the heat kernel, a spectral decomposition, or an extension problem in a higher dimension. The precise singular behavior of the kernel associated with the fractional Hilbert norm is obtained. The authors also provide a monotonicity formula for stationary points of some fractional functionals (which includes the case of nonlocal \( s \)-minimal surfaces) and some estimates of independent interest for the extension problem.