Boundedness Properties of Pseudo-Differential and Calderón-Zygmund Operators on Modulation Spaces

In this article, we study the boundedness of pseudo-differential operators with symbols in Sρ,δm on the modulation spaces Mp,q. We discuss the order m for the boundedness Op(Sρ,δm)⊂ℒ(Mp,q) to be true. We also prove the existence of a Calderón-Zygmund operator which is not bounded on the modulation space Mp,q with q≠2. This unboundedness is still true even if we assume a generalized T(1) condition. These results are induced by the unboundedness of pseudo-differential operators on Mp,q whose symbols are of the class S1,δ0 with 0<δ<1.