Pseudo-differential operators on Sobolev and Lipschitz spaces

In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo-differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the authors establish the boundedness of pseudo-differential operators with symbols in S m .,. , on Sobolev spaces, where m . ., . . 1 and . . 1. As its applications, the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced. Moreover, the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.