Asymptotics for null-timelike boundary problems for general linear wave equations

We study the linear wave equation□gu=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 surfaceτ=0 and a boundary value on the timelike surface r=r0.We obtain spacetime Hp-estimates of ru for r>r0 and derive an asymptotic expansion of ru in terms of 1/r as r→∞.