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

Fabio Nicola. "Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations." Studia Mathematica 198.3 (2010): 207-219. <http://eudml.org/doc/285591>.

@article{FabioNicola2010,
abstract = {We study Fourier integral operators of Hörmander’s type acting on the spaces $ℱL^\{p\}(ℝ^\{d\})_\{comp\}$, 1 ≤ p ≤ ∞, 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 Φ(x,η) with respect to the space variables x. Indeed, we show that operators of order m = -r|1/2-1/p| are bounded on $ℱ L^\{p\}(ℝ^\{d\})_\{comp\}$ if the mapping $x ↦ ∇_\{x\}Φ(x,η)$ is constant on the fibres, of codimension r, of an affine fibration.},
author = {Fabio Nicola},
journal = {Studia Mathematica},
language = {eng},
number = {3},
pages = {207-219},
title = {Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations},
url = {http://eudml.org/doc/285591},
volume = {198},
year = {2010},
}

TY - JOUR
AU - Fabio Nicola
TI - Boundedness of Fourier integral operators on Fourier Lebesgue spaces and affine fibrations
JO - Studia Mathematica
PY - 2010
VL - 198
IS - 3
SP - 207
EP - 219
AB - We study Fourier integral operators of Hörmander’s type acting on the spaces $ℱL^{p}(ℝ^{d})_{comp}$, 1 ≤ p ≤ ∞, 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 Φ(x,η) with respect to the space variables x. Indeed, we show that operators of order m = -r|1/2-1/p| are bounded on $ℱ L^{p}(ℝ^{d})_{comp}$ if the mapping $x ↦ ∇_{x}Φ(x,η)$ is constant on the fibres, of codimension r, of an affine fibration.
LA - eng
UR - http://eudml.org/doc/285591
ER -