Handling the inverse Fourier transform term supported on the boundary ∂Ω

Determine a rigorous procedure to treat the inverse Fourier transform F^{-1}[f] that yields a Dirac delta distribution supported on the boundary ∂Ω when transforming back from the Fourier domain, and clarify its implications for deriving and analyzing the equivalent integral equation Z = T(Z), including consequences for local existence and uniqueness.

Background

When inverting certain Fourier-domain boundary terms, the authors obtain an expression for F^{-1}[f] involving the Dirac delta in space-time, which, after restricting to the boundary, effectively leads to a distribution supported on ∂Ω. This complicates completing the inverse transform and undermines the straightforward derivation of local existence and uniqueness for the integral equation representation Z = T(Z).

They explicitly state that they do not know how to proceed at this step, highlighting a gap in the transform/inversion scheme in the presence of a boundary-supported delta distribution.