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Abstract

We study Fourier integral operators of Hörmander's type acting on the spaces $\mathcal F L^p(\mathbb R^d)_{\rm comp} $, $1\leq p\leq\infty$, 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 ${\mit\Phi}(x,\eta)$ 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(\mathbb R^d)_{\rm comp} $ if the mapping $x\mapsto\nabla_x{\mit\Phi}(x,\eta)$ is constant on the fibres, of codimension $r$, of an affine fibration.