Restriction theorems for the Fourier transform to homogeneous polynomial surfaces in ${\Bbb R}^{3}$
Tom 160 / 2004
Studia Mathematica 160 (2004), 249-265
MSC: Primary 42B10, 26D10.
DOI: 10.4064/sm160-3-4
Streszczenie
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.
