Monatshefte für Mathematik

, Volume 188, Issue 4, pp 667–687 | Cite as

Pseudodifferential operators with symbols in the Hörmander class \(S^0_{\alpha ,\alpha }\) on \(\alpha \)-modulation spaces

  • Tomoya KatoEmail author
  • Naohito Tomita


In this paper, we study the boundedness of pseudodifferential operators with symbols in the Hörmander class \(S^0_{\rho ,\rho }\) on \(\alpha \)-modulation spaces \(M_{p,q}^{s,\alpha }\), and consider the relation between \(\alpha \) and \(\rho \). In particular, we show that pseudodifferential operators with symbols in \(S^0_{\alpha ,\alpha }\) are bounded on all \(\alpha \)-modulation spaces \(M^{s,\alpha }_{p,q}\), for arbitrary \(s\in \mathbb {R}\) and for the whole range of exponents \(0 < p,q \le \infty \).


Pseudodifferential operators Hörmander class \(\alpha \)-Modulation spaces 

Mathematics Subject Classification

35S05 42B35 



The authors sincerely express deep gratitude to the anonymous referees for their careful reading and giving fruitful suggestions and comments. The first author is supported by Grant-in-Aid for JSPS Research Fellow (No. 17J00359). The second author is partially supported by Grant-in-aid for Scientific Research from JSPS (No. 16K05201).


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© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonakaJapan

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