Boundedness properties of pseudo-differential operators and Calderón-Zygmund operators on modulation spaces

Details Boundedness properties of pseudo-differential operators and Calderón-Zygmund operators on modulation spaces Boundedness properties of pseudo-differential operators and Calderón-Zygmund operators on modulation spaces

2007-01-29

arxiv.org/abs/math/0701832v1

Mathematics

Functional Analysis

Scientific paper

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 $0<\delta<1$.

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