Nonlocal scalar fields in static spacetimes via heat kernels

We solve the nonlocal equation -e- 2□□φ=J for (i) static scalar fields in static spacetimes and (ii) time-dependent scalar fields in ultrastatic spacetimes. Corresponding equations are rewritten as nonlocal Poisson/inhomogeneous Helmholtz equations in compact and noncompact weighted/Riemannian manifolds using static/frequency-domain Green's functions, which can be computed from the heat kernels in the respective manifolds. With the help of the heat kernel estimates, we derive the static Green's function estimates and use them to discuss the regularity. We also present several examples of exact and estimated static/frequency-domain Green's functions.