Restriction of the Fourier Transform to Some Oscillating Curves

Abstract

Let $$\phi $$ be a smooth function on a compact interval I. Let $$\begin{aligned} \gamma (t)=\left( t,t^2,\ldots ,t^{n-1},\phi (t)\right) . \end{aligned}$$ In this paper, we show that $$\begin{aligned} \left( \int _I \big |\hat{f}(\gamma (t))\big |^q \big |\phi ^{(n)}(t)\big |^{\frac{2}{n(n+1)}} \mathrm{{d}}t\right) ^{1/q}\le C\Vert f\Vert _{L^p(\mathbb R^n)} \end{aligned}$$ holds in the range $$\begin{aligned} 1\le p<\frac{n^2+n+2}{n^2+n},\quad 1\le q<\frac{2}{n^2+n}p'. \end{aligned}$$ This generalizes an affine restriction theorem of Sjolin (Stud Math 51:169–182, 1974) for $$n=2$$ . Our proof relies on ideas of Sjolin (Stud Math 51:169–182, 1974) and Drury (Ann Inst Fourier (Grenoble) 35(1):117–123, 1985), and more recently Bak, Oberlin and Seeger (J Aust Math Soc 85(1):1–28, 2008) and Stovall (Am J Math 138(2):449–471, 2016), as well as a variation bound for smooth functions.