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Fourier inversion of distributions supported by a hypersurface. (English) Zbl 1215.42017
The authors consider a compact oriented (N-1)-dimensional analytic submanifold $\Sigma$ of $\mathbb{R}^N$, with $N\geq 3$, and define the natural measure $\mu _\Sigma$ on $\Sigma$, which can be seen as a distribution on $\mathbb{R}^N$ of order 0 and compact support included in $\Sigma$. In the main result, they give a sufficient condition in order that the Fourier integral of the distribution $P(D)\psi \mu _\Sigma$ at a point outside $\Sigma$ is $(C,\lambda)$-summable to zero. Here $P(D)$ is a partial differential operator with constant coefficients of order $m$ and $\psi \in C^\infty (\mathbb{R}^N,\mathbb{R})$. As an example, they consider an ellipsoid $\Sigma$ in $\mathbb{R}^3$ with axes of different lengths.

42B10Fourier type transforms, several variables
46F12Integral transforms in distribution spaces
Full Text: DOI
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