Fourier multipliers of classical modulation spaces

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Local well‐posedness of nonlinear dispersive equations on modulation spaces (2007, background): “As proved in [3] and [7], a function σ(ξ) is a symbol of a bounded Fourier multiplier on Mp,q for 1 ≤ p, q ≤ ∞ if σ ∈ W (FL1, l∞) (see the proofs of the following two lemmas for a definition of this space).”
Modulation Spaces: Looking Back and Ahead (2006, background): “Results concerning duality and pointwise multipliers were even given in a more general form by describing pointwise multipliers of decomposition spaces [73, 63], see also [11, 87].”
Changes of variables in modulation and Wiener amalgam spaces (2008, background): “(See also [15] for other interesting and related results on Fourier multipliers.”