  Let $H=-\Delta+V$, where $V$ is a real valued potential on $\R^2$ satisfying
$|V(x)|\les \la x\ra^{-3-}$. We prove that if zero is a regular point of the
spectrum of $H=-\Delta+V$, then $$ \|w^{-1}
e^{itH}P_{ac}f\|_{L^\infty(\R^2)}\les \f1{|t|\log^2(|t|)} \|w f\|_{L^1(\R^2)},
|t| >2, $$ with $w(x)=\log^2(2+|x|)$. This decay rate was obtained by Murata in
the setting of weighted $L^2$ spaces with polynomially growing weights.
