  We are concerned with the study of the Cauchy problem to the 3D compressible
Hall-magnetohydrodynamic system. We first establish the unique global
solvability of strong solutions to the system when the initial data are close
to a stable equilibrium state in critical Besov spaces. Furthermore, under a
suitable additional condition involving only the low frequencies of the data
and in $L^{2}$-critical regularity framework, we exhibit the optimal time decay
rates for the constructed global solutions. The proof relies on an application
of Fourier analysis to a mixed parabolic-hyperbolic system, and on a refined
time-weighted energy functional.
