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Time-Frequency Analysis and PDE's Tabacco, Anita Fourier multipliers modulation spaces short-time Fourier transform We study the action on modulation spaces of Fourier multipliers with symbols $e^{imu(xi)}$, for real-valued functions $mu$ having unbounded second derivatives. We show that if $mu$ satisfies the usual symbol estimates of order $alphageq2$, or if $mu$ is a positively homogeneous function of degree $alpha$, the corresponding Fourier multiplier is bounded as an operator between the weighted modulation spaces $mathcal{M}^{p,q}_delta$ and $mathcal{M}^{p,q}$, for every $1leq p,qleqinfty$ and $deltageq d(alpha-2)|frac{1}{p}-frac{1}{2}|$. Here $delta$ represents the loss of derivatives. The above threshold is shown to be sharp for {it all} homogeneous functions $mu$ whose Hessian matrix is non-degenerate at some point. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Anita Tabacco 2009 Is Part Of Dagstuhl Seminar Proceedings, Volume 8492, Structured Decompositions and Efficient Algorithms (2009) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/DagSemProc.08492.10 urn:nbn:de:0030-drops-18792 https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08492.10 eng https://creativecommons.org/licenses/by/4.0/legalcode