Time-Frequency Analysis and PDE's

We study the action on modulation spaces of Fourier multipliers with symbols e, for real-valued functions μ having unbounded second derivatives. We show that if μ satisfies the usual symbol estimates of order α ≥ 2, or if μ is a positively homogeneous function of degree α, the corresponding Fourier multiplier is bounded as an operator between the weighted modulation spaces M δ and M, for every 1 ≤ p, q ≤ ∞ and δ ≥ d(α − 2)| 1 p − 1 2 |. Here δ represents the loss of derivatives. The above threshold is shown to be sharp for all homogeneous functions μ whose Hessian matrix is non-degenerate at some point.