Wydawnictwa / Czasopisma IMPAN / Studia Mathematica / Wszystkie zeszyty

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

Tom 198 / 2010

Studia Mathematica 198 (2010), 207-219 MSC: Primary 35S30, Secondary 42B35. DOI: 10.4064/sm198-3-1

Streszczenie

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.