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

Fabio Nicola¹

¹ Dipartimento di Matematica Politecnico di Torino Corso Duca degli Abruzzi 24 10129 Torino, Italy

Studia Mathematica, Tome 198 (2010) no. 3, pp. 207-219

Résumé

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.

Détail

DOI : 10.4064/sm198-3-1

Keywords: study fourier integral operators rmanders type acting spaces mathcal mathbb comp leq leq infty compactly supported distributions whose fourier transform sharp loss derivatives operator bounded these spaces related rank hessian phase mit phi eta respect space variables indeed operators order r bounded mathcal mathbb comp mapping mapsto nabla mit phi eta constant fibres codimension affine fibration