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

Abstract:

We study the oscillatory hyper-Hilbert transform \begin{equation} H_{n,\alpha ,\beta }f(x)=\int ^1_0 f(x-\Gamma (t))e^{it^{-\beta }}t^{-1-\alpha }dt \end{equation} 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.