  This paper deals with the applications of weighted Besov spaces to elliptic
equations on asymptotically flat Riemannian manifolds, and in particular to the
solutions of Einstein's constraints equations. We establish existence theorems
for the Hamiltonian an momentum constraints with constant mean curvature and
with a background metric which satisfies very low regularity assumptions. These
results extend the regularity results of Holst, Nagy and Tsogtgerel about the
constraint equations on compact manifolds in the Besov space $B_{p,p}^s$, to
asymptotically flat manifolds. We also consider the Brill--Cantor criterion in
the weighted Besov spaces. Our results improve the regularity assumptions on
asymptotically flat manifolds Choquet--Bruhat, Isenberg and Pollack, and
Maxwell, as well as they enable us to construct the initial data for the
Einstein--Euler system.
