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

Abstract

We study the oscillatory hyper-Hilbert transform H n , α , β f ( x ) = ∫ 0 1 f ( x − Γ ( t ) ) e i t − β t − 1 − α d t \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 Γ ( t ) = ( t p 1 , t p 2 , ⋯ , t p n ) \Gamma (t)=(t^{p_1},t^{p_2},\cdots ,t^{p_n}) , where p 1 , p 2 , ⋯ , p n , α , β p_1,p_2,\cdots ,p_n,\alpha ,\beta are some real positive numbers. We prove that if β > ( n + 1 ) α \beta >(n+1)\alpha , then H n , α , β H_{n,\alpha ,\beta } is bounded on L p L^p whenever p ∈ ( 2 β 2 β − ( n + 1 ) α , 2 β ( n + 1 ) α ) p \in (\frac {2\beta }{2\beta -(n+1)\alpha },\frac {2\beta }{(n+1)\alpha }) . Furthermore, we also prove that H n , α , β H_{n,\alpha ,\beta } is bounded on L 2 L^2 when β = ( n + 1 ) α \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 L^p boundedness result for some strongly parabolic singular integrals with rough kernels.