Restriction of Fourier transforms to some complex curves

The purpose of this paper is to prove a Fourier restriction estimate for certain 2-dimensional surfaces in \(\bbR^{2d}\), \(d\ge 3\). These surfaces are defined by a complex curve \(\gamma(z)\) of simple type, which is given by a mapping of the form % \[ z\mapsto \gamma (z) = \big(z, \, z^2,..., \,...

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Bibliographic Details
Published inarXiv.org
Main Authors Bak, Jong-Guk, Ham, Seheon
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 29.03.2013
Subjects
Analytic functions
Fourier transforms
Mapping
Polynomials
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Summary:The purpose of this paper is to prove a Fourier restriction estimate for certain 2-dimensional surfaces in \(\bbR^{2d}\), \(d\ge 3\). These surfaces are defined by a complex curve \(\gamma(z)\) of simple type, which is given by a mapping of the form % \[ z\mapsto \gamma (z) = \big(z, \, z^2,..., \, z^{d-1}, \, \phi(z) \big) \] % where \(\phi(z)\) is an analytic function on a domain \(\Omega \subset \bbC\). This is regarded as a real mapping \(z=(x,y) \mapsto \gamma(x,y)\) from \(\Omega \subset \bbR^2\) to \(\bbR^{2d}\). Our results cover the case \(\phi(z) = z^N\) for any nonnegative integer \(N\), in all dimensions \(d\ge 3\). Furthermore, when \(d=3\), we have a uniform estimate, where \(\phi(z)\) may be taken to be an arbitrary polynomial of degree at most \(N\). These results are analogues of the uniform restricted strong type estimate in \cite{BOS3}, valid for polynomial curves of simple type and some other classes of curves in \(\bbR^d\), \(d\ge 3\).
ISSN:2331-8422