Highly oscillatory unimodular Fourier multipliers on modulation spaces

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We study the continuity on the modulation spaces Mᵖ,q of Fourier multipliers with symbols of the type eⁱμ⁽ξ⁾, for some real-valued function μ(ξ). A number of results are known, assuming that the derivatives of order ≥ 2 of the phase μ(ξ) are bounded or, more generally, that its second derivatives belong to the Sjöstrand class M∞,¹. Here we extend those results, by assuming that the second derivatives lie in the bigger Wiener amalgam space W(F L¹,L∞); in particular they could have stronger oscillations at infinity such as |ξ|². Actually our main result deals with the more general case of possibly unbounded second derivatives. In that case we have boundedness on weighted modulation spaces with a sharp loss of derivatives.