  We study the linear wave equation $\Box_{g}u=0$ in Bondi-Sachs coordinates,
for an asymptotically flat Lorentz metric $g$. We consider the null-timelike
boundary problem, where an initial value is given on the null surface $\tau=0$
and a boundary value on the timelike surface $r=r_{0}$. We obtain spacetime
$H^{p}$-estimates of $ru$ for $r>r_0$ and derive an asymptotic exapnsion of
$ru$ in terms of ${1}/{r}$ as $r\to\infty$.
