Estimates for the Fourier transform of the indicator function for nonconvex domains
type: Item
instance of: scholarly article
title: Estimates for the Fourier transform of the indicator function for nonconvex domains
zbMATH Open document ID: 1056.42503
DOI: 10.1007/BF01077049
publication date: 1995-01-01T00:00:00Z
review text: 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 …
reviewed by: Elijah Liflyand
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 42B25
Mathematics Subject Classification ID: 58C25
zbMATH DE Number: 898971
author: Isroil A. Ikromov
MaRDI profile type: Publication
cites work: Fourier transforms related to convex sets
cites work: On the Fourier Transform of the Indicator Function of a Planar Set
cites work: Estimates for the Fourier transform of the characteristic function of a convex set
cites work: Nombre de points entiers dans une famille homothétique de domaines de ${R}$
cites work: Complete biorthogonal systems in Banach spaces
cites work: Singularities of differentiable maps, Volume 1. Classification of critical points, caustics and wave fronts. Transl. from the Russian by Ian Porteous, edited by V. I. Arnol'd
cites work: Q5566770
cites work: Oscillatory integrals, lagrange immersions and unfolding of singularities
cites work: Stability of \(C^\infty\) mappings. II: Infinitesimal stability implies stability
cites work: Q3314329
cites work: Q3994690
Recommended article: Q4323971
Recommended article: Spherical convergence of the Fourier integral of the indicator function of an $ N$-dimensional domain
Recommended article: An endpoint estimate for some maximal operators
Recommended article: Q4428829
Recommended article: Continuity of pseedo‐differential operators of type 1,1
Recommended article: The support theorem for the single radius spherical mean transform
Recommended article: Q4867468
Recommended article: Weighted Fourier Transform Inequalities for Radially Decreasing Functions
Recommended article: On radial and conical Fourier multipliers
Recommended article: Convex hypersurfaces and Fourier transforms
published in: Functional Analysis and Its Applications
description: scientific article; zbMATH DE number 898971