In this paper, the L 2 × L ∞ → L 2 and L 2 × L 2 → L 1 boundedness of bilinear Fourier multiplier operators is discussed under weak smoothness conditions on multipliers. As an application, we prove the L 2 × BM O → L 2 and L 2 × L 2 → H 1 boundedness of bilinear operators with multipliers of limited smoothness satisfying vanishing conditions.

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Abstract. In this paper, we prove certain L 2 -estimate for multilinear Fourier multiplier operators with multipliers of limited smoothness. As a result, we extend the result of Calderón and Torchinsky in the linear theory to the multilinear case. The sharpness of our results and some related estimates in Hardy spaces are also discussed.

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Abstract. We study the action on modulation spaces of Fourier multipliers with symbols e iµ(ξ) , for real-valued functions µ having unbounded second derivatives. In a simplified form our result reads as follows: if µ satisfies the usual symbol estimates of order α ≥ 2, or if µ is a positively homogeneous function of degree α, then the corresponding Fourier multiplier is bounded as an operator between the weighted modulation spaces M p,q s and M p,q , for all 1 ≤ p, q ≤ ∞ and s ≥ (α − 2)n|1/p − 1/2|. Here s represents the loss of derivatives. The above threshold is shown to be sharp for any homogeneous function µ whose Hessian matrix is non-degenerate at some point.