On the weighted estimate of the solution associated with the Schrödinger equation

Let $u(x,t)$ be the solution of the Schrödinger equation with initial data $f$ in the Sobolev space ${H^{ - 1 + a/2}}({\mathbb{R}^n})$ with $a > 1$. This paper shows that the weighted inequality $\int_{{\mathbb{R}^n}} {\int_\mathbb{R} {{{\left\vert {u(x,t)} \right\vert}^2}d... ...a}}dx \leq C{{\left\Vert f \right\Vert}_{{H^{ - 1 + a/2}}({\mathbb{R}^n})}}} }$ is false. Another improved weighted inequality is proved for the general case.