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 Schrodinger equation $ i u_t + u_{x x} + ia(|u|^2u)_x = 0, \quad (t,x) \in \mathbf{R}\times \mathbf{R},$ $ u(0,x) = u_0 (x), \quad x\in \mathbf{R},\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$(DNLS) where $a \in \mathbf{R}$. We prove that if $ ||u_0||_{ H^{1,\gamma}} + ||u_0||_{ H^{1+\gamma,0}}$ is sufficiently small with $\gamma > 1/2$, then the solution of (DNLS) satisfies the time decay estimate $ ||u(t)||_{L^\infty} + ||u_x(t)||_{L^\infty}\le C(1+|t|)^{-1/2}, $ where $H^{m,s}= \{f\in \mathcal{S}'; ||f||_{m,s}= ||(1+|x|^2)^{s/2}(1-\partial_x^2)^{m/2}f||_{L^2} < \infty\}$, $m,s\in \mathbf{R}$. In the previous paper [4,Theorem 1.1] we showed the same result under the condition that $\gamma \ge 2$. Furthermore we show the asymptotic behavior in time of solutions involving the previous result [4,Theorem 1.2].