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Operateurs de convolution definis à partir d’une forme quadratique. (English) Zbl 0536.47037

Journ. Équ. Dériv. Partielles, Saint-Jean-De-Monts 1982, Exp. No. 16, 8 p. (1982).
The starting point for the paper is a preceding result of the author [C. R. Acad. Sci., Paris, Sér. I 293, 121-124 (1981; Zbl 0475.35070)] concerning the inverse scattering problem for the nonlinear Klein-Gordon equation \[ u_{tt}-\Delta_ xu+m^ 2u=q(x)| u|^ 2u,\quad x\in {\mathbb{R}}^ 3. \] The proof of the author was based on an auxiliary proposition for families of convolution operators; such a proposition, which has independent interest, is stated and proved here under a more general formulation. The result concerns an arbitrary non-degenerate quadratic form Q in \({\mathbb{R}}^ n\), and the measure \(d\sigma_ r\), \(r\in {\mathbb{R}}\), supported by the manifold \(\{Q(x)=r\}\) and defined by \[ d\sigma_ r=dx_ 1...dx_{n-1}/| \partial Q/\partial x_ n|. \] The convergence for convolution operators \(d\sigma_ r*\to d\sigma_ t*\) as \(r\to t\) is studied in the frame of the \(L^ p\) spaces (the case considered in connection with the Klein-Gordon equation was \(Q(x)=x^ 2_ 1+...+x^ 2_{n-1}-x^ 2_ n)\).
Reviewer: L.Rodino

MSC:

47Gxx Integral, integro-differential, and pseudodifferential operators
44A35 Convolution as an integral transform
35P25 Scattering theory for PDEs

Citations:

Zbl 0475.35070
Full Text: Numdam EuDML