Restriction of the Fourier transform to some oscillating curves

TL;DR

This paper extends restriction theorems for Fourier transforms to oscillating curves in higher dimensions, establishing new bounds that generalize earlier results for the case n=2.

Contribution

It generalizes an affine restriction theorem to higher dimensions for a class of oscillating curves, providing new bounds and proof techniques.

Findings

Established Fourier restriction bounds for oscillating curves in higher dimensions.

Generalized Sjölin's affine restriction theorem from 2D to n-dimensional case.

Utilized advanced harmonic analysis techniques and variation bounds in proofs.

Abstract

Let $\phi$ be a smooth function on a compact interval $I$. Let $$\gamma(t)=\left (t,t^2,\cdots,t^{n-1},\phi(t)\right).$$ In this paper, we show that $$\left(\int_I \big|\hat f(\gamma(t))\big|^q \big|\phi^{(n)}(t)\big|^{\frac{2}{n(n+1)}} dt\right)^{1/q}\le C\|f\|_{L^p(\mathbb R^n)}$$ 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 Sj\"olin (1974) for $n=2$. Our proof relies on ideas of Sj\"olin (1974) and Drury (1985), and more recently Bak-Oberlin-Seeger (2008) and Stovall (2016), as well as a variation bound for smooth functions.