Madani Moussai

On the Fourier multipliers of the space $L^{p}$

Abstract:
We prove that the function $m(\xi)=\psi(\xi)e^{i\phi(\xi)}$ is not the Fourier multiplier of the space $L^{p}$, where the real phase $\phi $ has the property $\phi^{\prime\prime}\geq c>0$, the amplitude $\psi$ vanishes near the origin, $\psi(\xi)=O(\vert \xi \vert ^{-2})$ as $\xi\rightarrow \infty $, and $\psi ^{\prime }\in L^{1}.$

Keywords:
Fourier multiplier, oscillatory integral.

MSC 2000: 42A15, 35S30