  In this paper, we define the Green function for the Dirac operator under two
local boundary conditions: the condition associated with a chirality operator
(also called the chiral bag boundary condition) and the $\MIT$ bag boundary
condition. Then we give some applications of these constructions for each Green
function. From the existence of the chiral Green function, we derive an
inequality on a spin conformal invariant which, in particular, solve the Yamabe
problem on manifolds with boundary in some cases. Finally, using the $\MIT$
Green function, we give a simple proof of a positive mass theorem previously
proved by Escobar.
