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TBDmathscidoc:1701.333085

Arkiv for Matematik, 44, (2), 241-259, 2005.2
The α-modulation spaces\$M\$^{\$s\$,α}_{\$p\$,\$q\$}(\$R\$^{\$d\$}), α∈[0,1], form a family of spaces that contain the Besov and modulation spaces as special cases. In this paper we prove that a pseudodifferential operator σ(\$x\$,\$D\$) with symbol in the Hörmander class\$S\$^{\$b\$}_{ρ,0}extends to a bounded operator σ(\$x\$,\$D\$):\$M\$^{\$s\$,α}_{\$p\$,\$q\$}(\$R\$^{\$d\$})→\$M\$^{\$s\$-\$b\$,α}_{\$p\$,\$q\$}(\$R\$^{\$d\$}) provided 0≤α≤ρ≤1, and 1<\$p\$,\$q\$<∞. The result extends the well-known result that pseudodifferential operators with symbol in the class\$S\$^{\$b\$}_{1,0}maps the Besov space\$B\$^{\$s\$}_{\$p\$,\$q\$}(\$R\$^{\$d\$}) into\$B\$^{\$s\$-\$b\$}_{\$p\$,\$q\$}(\$R\$^{\$d\$}).
```@inproceedings{lasse2005boundedness,
title={Boundedness for pseudodifferential operators on multivariate α-modulation spaces},
author={Lasse Borup, and Morten Nielsen},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203617516723894},
booktitle={Arkiv for Matematik},
volume={44},
number={2},
pages={241-259},
year={2005},
}
```
Lasse Borup, and Morten Nielsen. Boundedness for pseudodifferential operators on multivariate α-modulation spaces. 2005. Vol. 44. In Arkiv for Matematik. pp.241-259. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203617516723894.