On Fourier multipliers with rapidly oscillating symbols

Abstract

We provide asymptotically sharp bounds for the $L_p$ norms of the Fourier multipliers with the symbols $e^{i\lambda \phi (\xi /|\xi |)}$, where $\lambda \in \mathbb{R}$ is a large parameter.

10 pages; second version contains the proofs of lower bounds for arbitrary d, some typos are also corrected