Restriction of Fourier transforms to some complex curves

2011/11/28 by Jong-Guk Bak, Bak, Jong-Guk, Seheon Ham +1
Mathematics · #42B10 #42B99 #Advanced Harmonic Analysis Research #Analytic and geometric function theory #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Partial Differential Equations #math.CA #msc:42B10 #msc:42B99

paper · pdf · doi:10.48550/arxiv.1111.6409

openalex publication_date 2011/11/28 · arxiv created 2013/03/29 · arxiv updated 2013/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The purpose of this paper is to prove a Fourier restriction estimate for certain 2-dimensional surfaces in \bbR^2d, d≥ 3. These surfaces are defined by a complex curve γ(z) of simple type, which is given by a mapping of the form % z↦ γ(z) = (z, z²,..., z^d-1, ϕ(z) ) % where ϕ(z) is an analytic function on a domain Ω⊂ \bbC. This is regarded as a real mapping z=(x,y) ↦ γ(x,y) from Ω⊂ \bbR² to \bbR^2d. Our results cover the case ϕ(z) = z^N for any nonnegative integer N, in all dimensions d≥ 3. Furthermore, when d=3, we have a uniform estimate, where ϕ(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 \citeBOS3, valid for polynomial curves of simple type and some other classes of curves in \bbR^d, d≥ 3.

Related

• A geometric lemma for complex polynomial curves with applications in Fourier restriction theory
• A Uniform Estimate for Fourier Restriction to Simple Curves
• Uniform Convolution and Fourier Restriction estimates for complex polynomial curves in ℂ³
• Fourier restriction for smooth hyperbolic 2-surfaces
• Fourier restriction estimates for surfaces of co-dimension two in ℝ⁵