In this paper, we derive the early-time asymptotics for fixed-frequency solutions phi(l) to the wave equation square(g)phi(l) = 0 on a fixed Schwarzschild background (M > 0) arising from the no incoming radiation condition on I- and polynomially decaying data, r phi(l) similar to t(-1) as t -> -infinity, on either a timelike boundary of constant area radius r > 2M (I) or an ingoing null hypersurface (II). In case (I), we show that the asymptotic expansion of partial derivative(v)(r phi(l)) along outgoing null hypersurfaces near spacelike infinity i(0) contains logarithmic terms at order r(-3-l) log r. In contrast, in case (II), we obtain that the asymptotic expansion of partial derivative(v)(r phi(l)) near spacelike infinity i(0) contains logarithmic terms already at order r(-3) log r (unless l = 1). These results suggest an alternative approach to the study of late-time asymptotics near future timelike infinity i(+) that does not assume conformally smooth or compactly supported Cauchy data: In case (I), our results indicate a logarithmically modified Price's law for each l-mode. On the other hand, the data of case (II) lead to much stronger deviations from Price's law. In particular, we conjecture that compactly supported scattering data on H- and I- lead to solutions that exhibit the same late-time asymptotics on I+ for each l: r phi(l)vertical bar (I+) similar to u(-2) as u -> infinity.
