Wave operators to a quadratic nonlinear Klein-Gordon equation in two space dimensions revisited

We continue to study the existence of the wave operators for the nonlinear Klein-Gordon equation with quadratic nonlinearity in two space dimensions (partial derivative(2)(t) - Delta + m())(2)u = lambda u(2), (t, x) is an element of R x R-2. We prove that if u(1)(+) is an element of H-3/2+3 gamma,H-1 (R-2), u(2)(+) is an element of H-1/2+3 gamma,H-1 (R-2), where gamma is an element of (0, 1/4) and the norm parallel to u(1)(+)parallel to(H13/2+gamma) + parallel to u(2)(+)parallel to(H11/2+gamma) <= rho, then there exist rho > 0 and T > 1 such that the nonlinear Klein-Gordon equation has a unique global solution u is an element of C([T, infinity); H-1/2 (R-2)) satisfying the asymptotics parallel to u(t) - u(0)(t)parallel to(H1/2) <= Ct-1/2-gamma for all t > T, where u (0) denotes the solution of the free Klein-Gordon equation.