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

Studia Mathematica (2004)

• Volume: 160, Issue: 3, page 249-265
• ISSN: 0039-3223

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## Abstract

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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 $\sigma \left(A\right)={\int }_{B}{\chi }_{A}\left(x,\phi \left(x\right)\right)dx$ 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 ${L}^{p}\left(ℝ³\right)$ to ${L}^{q}\left(\Sigma ,d\sigma \right)$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.

## How to cite

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E. Ferreyra, T. Godoy, and M. Urciuolo. "Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³." Studia Mathematica 160.3 (2004): 249-265. <http://eudml.org/doc/284895>.

@article{E2004,
abstract = {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 $σ(A) = ∫_\{B\} χ_\{A\}(x,φ(x))dx$ 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 $L^\{p\}(ℝ³)$ to $L^\{q\}(Σ,dσ)$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.},
author = {E. Ferreyra, T. Godoy, M. Urciuolo},
journal = {Studia Mathematica},
keywords = {restriction theorems; Fourier transform; Littlewood-Paley decomposition; Littlewood-Paley inequality},
language = {eng},
number = {3},
pages = {249-265},
title = {Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³},
url = {http://eudml.org/doc/284895},
volume = {160},
year = {2004},
}

TY - JOUR
AU - E. Ferreyra
AU - T. Godoy
AU - M. Urciuolo
TI - Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ℝ³
JO - Studia Mathematica
PY - 2004
VL - 160
IS - 3
SP - 249
EP - 265
AB - 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 $σ(A) = ∫_{B} χ_{A}(x,φ(x))dx$ 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 $L^{p}(ℝ³)$ to $L^{q}(Σ,dσ)$ for certain p,q. For m ≥ 6 the results are sharp except for some border points.
LA - eng
KW - restriction theorems; Fourier transform; Littlewood-Paley decomposition; Littlewood-Paley inequality
UR - http://eudml.org/doc/284895
ER -

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