标题： 傅里叶变换到某些振荡曲线的限制 显示英文标题

标题： Restriction of the Fourier transform to some oscillating curves

摘要： 设$\phi$是在紧区间$I$上的光滑函数。 设 $$\gamma(t)=\left (t,t^2,\cdots,t^{n-1},\phi(t)\right).$$在本文中，我们证明在范围$$1\le p<\frac{n^2+n+2}{n^2+n},\quad 1\le q<\frac{2}{n^2+n}p'.$$内$$\left(\int_I \big|\hat f(\gamma(t))\big|^q \big|\phi^{(n)}(t)\big|^{\frac{2}{n(n+1)}} dt\right)^{1/q}\le C\|f\|_{L^p(\mathbb R^n)}$$成立。这推广了 Sjölin（1974）针对$n=2$的仿射限制定理。 我们的证明依赖于 Sjölin（1974）和 Drury（1985）的思想，以及最近 Bak-Oberlin-Seeger（2008）和 Stovall（2016）的工作，以及对光滑函数的变分界。 显示英文摘要

摘要： 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.