On the L2-boundedness of pseudo-differential operators and their commutators with symbols in α-modulation spaces

The results of Sjöstrand [J. Sjöstrand, An algebra of pseudodifferential operators, Math. Res. Lett. 1 (1994) 185–192] and Sugimoto [M. Sugimoto, Lp-boundedness of pseudo-differential operators satisfying Besov estimates, I, J. Math. Soc. Japan 40 (1988) 105–122] on a mapping property of pseudo-differential operators are two different kinds of extensions of the pioneering work by Calderón and Vaillancourt [A.P. Calderón, R. Vaillancourt, On the boundedness of pseudo-differential operators, J. Math. Soc. Japan 23 (1971) 374–378]. The objective of this paper is to show that these two results, which appeared to be independent ones, can be proved based on the same principle. For the purpose, we use the α-modulation spaces, a parameterized family of function spaces, which include Besov spaces and modulation spaces as special cases. As an application, we also discuss the L2-boundedness of the commutator [σ(X,D),a], where a(x) is a Lipschitz function and σ belongs to an α-modulation space.