  This paper is dedicated to study weighted $L^p$ inequalities for
pseudo-differential operators with amplitudes and their commutators by using
the new class of weights $A_p^\vc$ and the new BMO function space BMO$_\vc$
which are larger than the Muckenhoupt class of weights $A_p$ and classical BMO
space BMO, respectively. The obtained results therefore improve substantially
some well-known results.
