  We solve the non-local equation ${-e^{-\ell^2\square}\square{\phi}=J}$ for i)
static scalar fields in static spacetimes and ii) time-dependent scalar fields
in ultrastatic spacetimes. Corresponding equations are rewritten as non-local
Poisson/inhomogeneous Helmholtz equations in compact and non-compact
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.

General Relativity and Quantum Cosmology;High Energy Physics - Theory