On The Boundedness Of Pseudodifferential Operators And Their Commutators

In this thesis,we study four problems for pseudo-differential operators T with smooth symbols in S?,?m(Rn),i.e.,the boundedness on Hardy type spaces for the commutators of pseudo-differential operators,the boundedness on weighted Hardy spaces for the pseudo-differential operators and their commutators,the two-weighted boundedness for commutators and the L?p bounds for pseudo-differential operators and their commutators with Ap(?)weights.In Chapter 2,we present some lemmas and well-known results used later,for example,the estimate for the kernel associated with the pseudo-differential operators,the Lp and weighted Lp bounds for the pseudo-differential operators and so on.In Chapter 3,we mainly obtain the bounds from H1 to weak L1 and from Hbp to Lp for commutators generated by the pseudo-differential operators and BMO functions.In Chapter 4,we discuss some bound problem about the pseudo-differential operators and their commutators on weighted Hardy spaces H?1,where ?? Ap and obtain the sufficient conditions which lead to the boundedness from H?1 to L?1 and H?1 into itself.We also get an estimate from H?1 to L?1 for the commutators generated by the pseudo-differential operators and the BMO functions,where BMO(?)BMO.In Chapter 5,we establish a two-weight inequality for commutators generated by pseudo-differential operators T with weighted BMO(Rn)functions b.In Chapter 6,we establish weighted Lp inequalities for pseudo-differential operators T and their commutators[b,T],where b ? BMO(Rn)and the weight ?belongs to the new class Ap(?).It is well known that the Muckenhoupt class Ap falls within the class Ap(?)and we extend the related results.