Restriction of the Fourier transform to some oscillating curves

Abstract

Let $\varphi$ be a smooth function on a compact interval $I$. Let $$\gamma (t)=\left(t,t^2,\cdots ,t^{n-1},\varphi (t)\right).$$ In this paper, we show that $${\left({\int }_I|\hat{f}(\gamma (t)){|}^q|{\varphi }^{(n)}(t){|}^{\frac{2}{n(n+1)}}dt\right)}^{1/q}\le C\Vert f{\Vert }_{L^p({\mathbb{R}}^n)}$$ holds in the range $$1\le p<\frac{n^2+n+2}{n^2+n},1\le q<\frac{2}{n^2+n}{p}^{\prime }.$$ This generalizes an affine restriction theorem of Sjölin (1974) for $n=2$. Our proof relies on ideas of Sjölin (1974) and Drury (1985), and more recently Bak-Oberlin-Seeger (2008) and Stovall (2016), as well as a variation bound for smooth functions.

17 pages