## 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.

### MSC:

 42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type 46F12 Integral transforms in distribution spaces

### Keywords:

Fourier transform; distribution; hypersurface; Cesàro means
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### References:

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