  [Abridged] We present the results of a Hubble Space Telescope snapshot
program to survey the inner region of the globular cluster NGC 6441 for its
variable stars. A total of 57 variable stars was found including 38 RR Lyrae
stars, 6 Population II Cepheids, and 12 long period variables. Of the RR Lyrae
stars observed in this survey, 26 are pulsating in the fundamental mode with a
mean period of 0.753d and 12 are first-overtone mode pulsators with a mean
period of 0.365d. These values match up very well with those found in
ground-based surveys. Combining all the available data for NGC 6441, we find
mean periods of 0.759d and 0.375d for the RRab and RRc stars, respectively. We
also find that the RR Lyrae in this survey are located in the same regions of a
period-amplitude diagram as those found in ground-based surveys. Although NGC
6441 is a metal-rich globular cluster, its RR Lyrae more closely resemble those
in Oosterhoff type II globular clusters. However, even compared to typical
Oosterhoff type II systems, the mean period of its RRab stars is unusually
long. We also derived I-band period-luminosity relations for the RR Lyrae
stars. Of the six Population II Cepheids, five are of W Virginis type and one
is a BL Herculis variable stars. This makes NGC 6441, along with NGC 6388, the
most metal-rich globular cluster known to contain these types of variable
stars. Another variable, V118, may also be a Population II Cepheid given its
long period and its separation in magnitude from the RR Lyrae stars. We argue
that there does not appear to be a change in the period-luminosity relation
slope between the BL Herculis and W Virginis stars, but that a change of slope
does occur when the RV Tauri stars are added to the period-luminosity relation.
