  We derive the exact law for three-dimensional (3D) homogeneous compressible
isothermal Hall magnetohydrodynamics (CHMHD) turbulence, without the assumption
of isotropy. The Hall current is shown to introduce new flux and sources terms
that act at the small scales (comparable or smaller than the ion skin depth) to
significantly impact the turbulence dynamics. The new law provides an accurate
means to estimate for the first time the energy cascade rate over a broad range
of scales covering the MHD inertial range and the sub-ion dispersive range in
3D numerical simulations and {\it in situ} spacecraft observations of
compressible turbulence. This work is particularly relevant to astrophysical
flows in which small scale density fluctuations cannot be ignored such as the
solar wind, planetary magnetospheres and the interstellar medium (ISM).
