Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³

E. Ferreyra; T. Godoy; M. Urciuolo

E. Ferreyra ; T. Godoy ; M. Urciuolo

Studia Mathematica, Tome 162 (2004), p. 249-265 / Harvested from The Polish Digital Mathematics Library

Résumé

Let φ:ℝ² → ℝ be a homogeneous polynomial function of degree m ≥ 2, let Σ = (x,φ(x)): |x| ≤ 1 and let σ be the Borel measure on Σ defined by $σ (A) = \int_{B} χ_{A} (x, φ (x)) d x$ where B is the unit open ball in ℝ² and dx denotes the Lebesgue measure on ℝ². We show that the composition of the Fourier transform in ℝ³ followed by restriction to Σ defines a bounded operator from $L^{p} (ℝ ³)$ to $L^{q} (Σ, d σ)$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.

Publié le : 2004-01-01

Zbl 1045.42002

EUDML-ID : urn:eudml:doc:284895

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     title = {Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in $\mathbb{R}$$^3$},
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     pages = {249-265},
     zbl = {1045.42002},
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E. Ferreyra; T. Godoy; M. Urciuolo. Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³. Studia Mathematica, Tome 162 (2004) pp. 249-265. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm160-3-4/