$L^p$ bounds for oscillatory hyper-Hilbert transform along curves

We study the oscillatory hyper-Hilbert transform

$H_{n,\alpha,\beta}f(x)=\int^1_0 f(x-\Gamma(t))e^{it^{-\beta}}t^{-1-\alpha}dt$

(1)

along the curve $\Gamma(t)=(t^{p_1},t^{p_2},\cdots,t^{p_n})$, where $p_1,p_2,\cdots,p_n,\alpha,\beta$ are some real positive numbers. We prove that if $\beta>(n+1)\alpha$, then $H_{n,\alpha,\beta}$ is bounded on $L^p$ whenever $p \in(\frac{2\beta}{2\beta-(n+1)\alpha},\frac{2\beta}{(n+1)\alpha})$. Furthermore, we also prove that $H_{n,\alpha,\beta}$ is bounded on $L^2$ when $\beta=(n+1)\alpha$. Our work improves and extends some known results by Chandarana in 1996 and in a preprint. As an application, we obtain an $L^p$ boundedness result for some strongly parabolic singular integrals with rough kernels.