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Zbl 1215.42017
González Vieli, F.J.; Seifert, E.
Fourier inversion of distributions supported by a hypersurface.
(English)
[J] J. Fourier Anal. Appl. 16, No. 1, 34-51 (2010). ISSN 1069-5869; ISSN 1531-5851/e

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.
[Lourdes Rodr\'iguez-Mesa (La Laguna)]
MSC 2000:
*42B10 Fourier type transforms, several variables
46F12 Integral transforms in distribution spaces

Keywords: Fourier transform; distribution; hypersurface; Cesàro means

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