  We show that linear scalar waves are bounded and continuous up to the Cauchy
horizon of Reissner-Nordstr\"om-de Sitter and Kerr-de Sitter spacetimes, and in
fact decay exponentially fast to a constant along the Cauchy horizon. We obtain
our results by modifying the spacetime beyond the Cauchy horizon in a suitable
manner, which puts the wave equation into a framework in which a number of
standard as well as more recent microlocal regularity and scattering theory
results apply. In particular, the conormal regularity of waves at the Cauchy
horizon - which yields the boundedness statement - is a consequence of radial
point estimates, which are microlocal manifestations of the blue-shift and
red-shift effects.
