Green's functions associated with the edth operators (Q791883)

The operator \({\tilde \partial}\) acting on \(f(\zeta,{\bar \zeta})\) of spin weight s being defined by \({\tilde \partial}f=(1+\zeta {\bar \zeta})^{1-s}((1+\zeta {\bar \zeta})^ sf)_{\zeta}\) where \(\zeta\) and \({\bar \zeta}\) are the complex stereographic coordinates on the sphere \(S^ 2 (\zeta =e^{i\phi} \cot frac{1}{2}\theta)\), the author gives the Green's function for the following operators, acting on spin weight-0 quantities: i) \({\tilde \partial}^ 2U=V\) arising in the study of asymptotic flat spacetimes admitting a subgroup of BMS group. The case when \(U=(1+\zeta {\bar \zeta})^{-2}\) is examined at the end of the paper, ii) \({\tilde \partial}_{\zeta}X(\zeta,{\bar \zeta})=Y(\zeta,{\bar \zeta})\) involved in Yang-Mills equations when self-dual solutions are considered in the asymptotic domain, iii) the Laplace operator.