In this talk, we shall discuss some dispersive estimates for
wave equations on black hole backgrounds.  In particular, we look at
the localized energy estimates and Strichartz estimates.  An initial
localized energy estimate is proved using a differential multiplier in
rather standard ways.  Then, using a rough WKB approximation, this
localized energy estimate is improved near the region with trapping,
namely the photon sphere, and the loss there can be taken to be merely
logarithmic.  Finally by combining the localized energy estimates and
local in time Strichartz estimates on a compact region and by using
the parametrix of Metcalfe-Tataru outside of this compact region,
Strichartz estimates are shown.  This is a joint work with Marzuola,
Tataru, and Tohaneanu.