A Restriction Theorem for a k-Surface in ℝn

2005 ◽  
Vol 48 (2) ◽  
pp. 260-266 ◽  
Author(s):  
Daniel M. Oberlin
Keyword(s):  
Restriction Theorem ◽  
Fourier Restriction ◽  
Restriction Estimate

AbstractWe establish a sharp Fourier restriction estimate for a measure on a k-surface in ℝn, where n = k(k + 3)/2.

Reduction to Restriction Estimates near the Principal Root Jet

2016 ◽  
Author(s):  
Isroil A. Ikromov ◽  
Detlef Müller
Keyword(s):  
Newton Polyhedron ◽  
Small Letter ◽  
Fourier Restriction ◽  
Restriction Estimate

This chapter shows that one may reduce the desired Fourier restriction estimate to a piece Ssubscript Greek small letter psi of the surface S lying above a small, “horn-shaped” neighborhood Dsubscript Greek small letter psi of the principal root jet ψ‎, on which ∣x₂ − ψ‎(x₁)∣ ≤ ε‎xᵐ₁. Here, ε‎ > 0 can be chosen as small as one wishes. The proof then provides the opportunity to introduce some of the basic tools which will be applied frequently, such as dyadic domain decompositions, rescaling arguments based on the dilations associated to a given edge of the Newton polyhedron, in combination with Greenleaf's restriction and Littlewood–Paley theory, hence summing the estimates that have been obtained for the dyadic pieces.

scholarly journals Convolution Powers of Salem Measures With Applications

2017 ◽  
Vol 69 (02) ◽  
pp. 284-320 ◽  
Author(s):  
Xianghong Chen ◽  
Andreas Seeger
Keyword(s):  
Fourier Multipliers ◽  
Restriction Theorem ◽  
Fourier Restriction ◽  
Convolution Powers ◽  
Regularity Results

AbstractWe study the regularity of convolution powers for measures supported on Salemsets, and prove related results on Fourier restriction and Fourier multipliers. In particular we show that for α of the form d/n, n = 2, 3, … there exist α-Salem measures for which the L2Fourier restriction theorem holds in the range. The results rely on ideas of Körner. We extend some of his constructions to obtain upper regular α-Salem measures, with sharp regularity results forn-fold convolutions for all n ∈ ℕ.

How to Go beyond the Case hlin(φ‎) ≥ 5

2016 ◽  
Author(s):  
Isroil A. Ikromov ◽  
Detlef Müller
Keyword(s):  
Lower Bounds ◽  
The Other ◽  
Complex Interpolation ◽  
Restriction Theorem ◽  
Fourier Restriction

This chapter mostly considers the domains of type Dsubscript (l), which are in some sense “closest” to the principal root jet, since it turns out that the other domains Dsubscript (l) with l ≥ 2 are easier to handle. In a first step, by means of some lower bounds on the r-height, this chapter establishes favorable restriction estimates in most situations, with the exception of certain cases where m = 2 and B = 3 or B = 4. In some cases the chapter applies interpolation arguments in order to capture the endpoint estimates for p = psubscript c. Sometimes this can be achieved by means of a variant of the Fourier restriction theorem. However, in most of these cases the chapter applies complex interpolation in a similar way as has been done in Chapter 5.

scholarly journals Slicing surfaces and the Fourier restriction conjecture

2009 ◽  
Vol 52 (2) ◽  
pp. 515-527 ◽  
Author(s):  
Fabio Nicola
Keyword(s):  
Fourier Transform ◽  
Finite Type ◽  
Fourier Restriction ◽  
The Fourier Transform ◽  
Restriction Estimate

AbstractWe deal with the restriction phenomenon for the Fourier transform. We prove that each of the restriction conjectures for the sphere, the paraboloid and the elliptic hyperboloid in ℝn implies that for the cone in ℝn+1. We also prove a new restriction estimate for any surface in ℝ3 locally isometric to the plane and of finite type.

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