  In this paper, the sharp maximal theorem is generalized to mixed-norm ball
Banach function spaces, which is defined as Definition 2.7. As an application,
we give a characterization of BMO via the boundedness of commutators of
fractional integral operators on mixed-norm Lebesgue spaces. Moreover, the
characterization of homogeneous Lipschitz space is also given by the
boundedness of commutators of fractional integral operators on mixed-norm
Lebesgue spaces. Finally, two applications of Corollary 6.4 are given.
