Boundedness properties of pseudo-differential operators and Calder\'on-Zygmund operators on modulation spaces

Abstract

In this paper, we study the boundedness of pseudo-differential operators with symbols in $S_{\rho,\delta}^m$ on the modulation spaces $M^{p,q}$. We discuss the order $m$ for the boundedness $\mathrm{Op}(S_{\rho,\delta}^m) \subset \calL(M^{p,q}(\R^n))$ to be true. We also prove the existence of a Calder\'on-Zygmund operator which is not bounded on the modulation space $M^{p,q}$ with $q \neq 2$. 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 $M^{p,q}$ whose symbols are of the class $S_{1,\delta}^0$ with $0<\delta<1$.