Restriction of the Fourier transform to some oscillating curves

Let $ϕ$ be a smooth function on a compact interval $I$. Let $γ(t)=\left (t,t^2,\cdots,t^{n-1},ϕ(t)\right).$ In this paper, we show that $\left(\int_I \big|\hat f(γ(t))\big|^q \big|ϕ^{(n)}(t)\big|^{\frac{2}{n(n+1)}} dt\right)^{1/q}\le C\|f\|_{L^p(\mathbb R^n)}$ holds in the range $1\le p<\frac{n^2+n+2}{n^2+n},\quad 1\le q<\frac{2}{n^2+n}p'.$ 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.

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