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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.

MSC:

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