Restriction of the Fourier Transform to Some Oscillating Curves

Abstract Let %$\phi %$ be a smooth function on a compact interval I. Let γ(t)=t,t2,…,tn-1,ϕ(t).%$\begin{aligned} \gamma (t)=\left( t,t^2,\ldots ,t^{n-1},\phi (t)\right) . \end{aligned}%$In this paper, we show that ∫I|f^(γ(t))|q|ϕ(n)(t)|2n(n+1)dt1/q≤C‖f‖Lp(Rn)%$\begin{aligned} \left( \int _I \big |\h...
Ausführliche Beschreibung

Abstract Let %$\phi %$ be a smooth function on a compact interval I. Let γ(t)=t,t2,…,tn-1,ϕ(t).%$\begin{aligned} \gamma (t)=\left( t,t^2,\ldots ,t^{n-1},\phi (t)\right) . \end{aligned}%$In this paper, we show that ∫I|f^(γ(t))|q|ϕ(n)(t)|2n(n+1)dt1/q≤C‖f‖Lp(Rn)%$\begin{aligned} \left( \int _I \big |\hat{f}(\gamma (t))\big |^q \big |\phi ^{(n)}(t)\big |^{\frac{2}{n(n+1)}} \mathrm{{d}}t\right) ^{1/q}\le C\Vert f\Vert _{L^p(\mathbb R^n)} \end{aligned}%$holds in the range 1≤p<n2+n+2n2+n,1≤q<2n2+np′.%$\begin{aligned} 1\le p<\frac{n^2+n+2}{n^2+n},\quad 1\le q<\frac{2}{n^2+n}p'. \end{aligned}%$This generalizes an affine restriction theorem of Sjölin (Stud Math 51:169–182, 1974) for %$n=2%$. Our proof relies on ideas of Sjölin (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. Ausführliche Beschreibung