Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations

Abstract

We carry on the study of Fourier integral operators of H{ö}rmander's type acting on the spaces $(\mathcal{F}{L}^p{)}_{comp}$, $1\le p\le \infty$, of compactly supported distributions whose Fourier transform is in $L^p$. We show that the sharp loss of derivatives for such an operator to be bounded on these spaces is related to the rank $r$ of the Hessian of the phase $\Phi (x,\eta )$ with respect to the space variables $x$. Indeed, we show that operators of order $m=-r|1/2-1/p|$ are bounded on $(\mathcal{F}{L}^p{)}_{comp}$, if the mapping $x⟼{\nabla }_x\Phi (x,\eta )$ is constant on the fibers, of codimension $r$, of an affine fibration.

13 pages

References in corpus (2)

• Changes of variables in modulation and Wiener amalgam spaces
• On the global boundedness of Fourier integral operators