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

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Nicola F.

Studia Mathematica

2010

tom 198

nr 3

207-219

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.

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1 Studia Mathematica

1 Nicola F.

1 2010

Wyniki wyszukiwania

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