Multilinear Fourier multipliers related to time–frequency localization

We consider multilinear multipliers associated in a natural way with localization operators. Boundedness and compactness results are obtained. In particular, we get a geometric condition on a subset A ⊂ R 2 d which guarantees that, for a fixed synthesis window ψ ∈ L 2 ( R d ) , the family of localization operators L φ , ψ A obtained when the analysis window φ varies on the unit ball of L 2 ( R d ) is a relatively compact set of compact operators.