  Cauchy problem for 3D incompressible Hall-magnetohydrodynamics (Hall-MHD)
system with fractional Laplacians is studied. First, global well-posedness of
small-energy solutions with general initial data in $H^s$, $s>\frac{5}{2}$, is
proved. Second, a special class of large-energy initial data is constructed,
with which the Cauchy problem is globally well-posed. The proofs rely upon a
new global bound of energy estimates involving Littlewood-Paley decomposition
and Sobolev inequalities, which enables one to overcome the $\frac{1}{2}$-order
derivative loss of the magnetic field.
