Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ${\Bbb R}^{3}$

E. Ferreyra¹ ; T. Godoy¹ ; M. Urciuolo¹

¹FaMAF, Universidad Nacional de Córdoba and CIEM-CONICET Ciudad Universitaria 5000 Córdoba, Argentina

Studia Mathematica, Tome 160 (2004) no. 3, pp. 249-265

Résumé

Let $\varphi:\mathbb{R}^{2}\rightarrow \mathbb{R}$ be a homogeneous polynomial function of degree $m\geq 2,$ let ${{\mit\Sigma}} =\{( x,\varphi (x)):|x| \leq 1\} $ and let $ \sigma $ be the Borel measure on ${{\mit\Sigma}} $ defined by $\sigma( A) =\int_{B}\chi _{A}(x,\varphi (x)) \,dx$ where $B$ is the unit open ball in $\mathbb{R}^{2}$ and $dx$ denotes the Lebesgue measure on $\mathbb{R}^{2}.$ We show that the composition of the Fourier transform in $\mathbb{R}^{3}$ followed by restriction to ${{\mit\Sigma}} $ defines a bounded operator from $L^{p}( \mathbb{R}^{3}) $ to $L^{q}({{\mit\Sigma}},d\sigma) $ for certain $p,q.$ For $m\geq 6$ the results are sharp except for some border points.

Détail

DOI : 10.4064/sm160-3-4

Keywords: varphi mathbb rightarrow mathbb homogeneous polynomial function degree geq mit sigma varphi leq sigma borel measure mit sigma defined sigma int chi varphi where unit ball mathbb denotes lebesgue measure mathbb composition fourier transform mathbb followed restriction mit sigma defines bounded operator mathbb mit sigma sigma certain geq results sharp except border points