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

Mitsuru Sugimoto·Naohito Tomita
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Abstract

In this paper, we study the boundedness of pseudo-differential operators with symbols in $S_{ρ,δ}^m$ on the modulation spaces $M^{p,q}$. We discuss the order $m$ for the boundedness $\mathrm{Op}(S_{ρ,δ}^m) \subset \calL(M^{p,q}(\R^n))$ to be true. We also prove the existence of a Calderón-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 $S_{1,δ}^0$ with $0<δ<1$.

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Mitsuru Sugimoto, Naohito Tomita. 2007-01-29. Boundedness properties of pseudo-differential operators and Calderón-Zygmund operators on modulation spaces. https://arxiv.org/abs/math/0701832

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