Estimates for the Fourier transform of the indicator function for nonconvex domains
This paper continues a series of works on the behavior of the Fourier transform of the indicator of a compact domain \(D\subset\mathbf R^{n+1}\). More precisely, for the integral \(\widehat u_D(r\xi)=\int_Du(x)e^{ir(x,\xi)}\,dx\), \(u\in C^\infty(\mathbf R^{n+1})\), with \(r\geq 0\), \(\xi\in S^n\), the unit sphere in \(\mathbf R^{n+1}\), and \((x,\xi)\) the scalar product of the vectors \(x\) and \(\xi\), the maximal function \(M(\xi)=\sup_{r>0}r^{(n+2)/2}|\widehat u_D(r\xi)|\) is studied. For \(u(x)\equiv 1\) the function \(\widehat u_D(r\xi)\) coincides with the Fourier transform of the indicator of \(D\). Earlier works concentrated mainly on the case of convex domains \(D\). It is worth adding the following items to the list of references: \textit{I. M. Gel'fand, M. I. Graev} and \textit{N. Ya. Vilenkin} [``Generalized functions'', Vol. 5, English translation (1966; Zbl 0144.17202)]; \textit{A. N. Podkorytov} [Vestn. Leningr. Univ., Math. 24, No. 2, 57--65 (1991); translation from Vestn. Leningr. Univ., Ser. I 1991, No. 2, 50--57, 125 (1991; Zbl 0741.42012)]; \textit{A. G. Ramm} and \textit{A. I. Zaslavsky} [C. R. Acad. Sci., Paris, Sér. I 316, No. 6, 541--545 (1993; Zbl 0781.42016)]. In the two papers where the non-convex case was considered, assumptions were rather restrictive. In the paper under review, for \(n\leq 6\), less restrictive assumptions are given in order that \(M\in L_{4-0}(S^n):=\bigcap_{0<\varepsilon <4}L_{p-\varepsilon}(S^n)\). The sharpness of this result is also shown. The proof is based on a general theory of differentiable maps [see \textit{V. I. Arnol'd, S. M. Gusejn-Zade} and \textit{A. N. Varchenko}, ``Singularities of differentiable maps'', Vol. I, translated from the Russian (1985; Zbl 0554.58001)]. The restriction \(n\leq 6\) is implied by the fact that some properties of the space of all the embeddings of a manifold in \(\mathbf R^{n+1}\) are known only for \(n\leq 6\). Any progress in this will yield an extension of the results of the paper under review. Note also that in the two-dimensional case the results of this work as well as of some previous ones are more developed.
scientific article; zbMATH DE number 898971
scientific article