  In this paper, we first give some new characterizations of Muckenhoupt type
weights through establishing the boundedness of maximal operators on the
weighted Lorentz and Morrey spaces. Secondly, we establish the boundedness of
sublinear operators including many interesting in harmonic analysis and its
commutators on the weighted Morrey spaces. Finally, as an application, the
boundedness of strongly singular integral operators and commutators with
symbols in BMO space are also given.
