Global low regularity solutions to the Benjamin equation in weighted spaces

Abstract

We show that the Benjamin equation is globally well-posed for real-valued data in the weighted space $H^s\cap H_r^{s-2r}≔\left\{u|{\Vert u\Vert }_{H^s\left(R_x\right)}+{\Vert \hat{u}\Vert }_{H^r(R_{\xi }^{+},{(1+\left|\xi \right|)}^{2(s-2r)}d\xi )}<\infty \right\},$where $0\le r$ and $-\frac{3}{4}+r<s$. The proof is based on direct extensions of standard linear and bilinear estimates originated in Kenig et al. (1993), Kenig et al. (1996), Linares (1999), Kozono et al. (2001), Colliander et al. (2003), Li and Wu (2010) to the weighted settings.