Asymptotic behavior in time of solutions to the derivative nonlinear schrödinger equation revisited

We continue to study the asymptotic behavior in time of solutions to the derivative nonlinear Schrödinger equation {iu_t + u_xx + ia(|u|²u)_x = 0, (t, x) ∈ R × R, (DNLS) {u(0, x) = u₀(x), x ∈ R, where a ∈ R. We prove that if ||u₀||_H1,y + ||u₀||_H1+y,0 is sufficiently small with γ > 1/2, then the solution of (DNLS) satisfies the time decay estimate ||u(t)||_L∞ + ||u_x(t)||_L∞ ≤ C(1 + |t|)^-1/2, where H^m,s = {f ∈ S′; ||f||_m,s = ||(1 + |x|²)^s/2(1 - ∂²_x)^m/2 f||_L2 < ∞}, m, s ∈ R. In the previous paper [4,Theorem 1.1] we showed the same result under the condition that γ ≥ 2. Furthermore we show the asymptotic behavior in time of solutions involving the previous result [4,Theorem 1.2].