\(L^p\) boundedness of Carleson \& Hilbert transforms along plane curves with certain curvature constraints (Q2077370)

In the main result of this paper, which is Theorem 1.1, the authors show that if \(p\in(1,\infty)\), \(u:\mathbb{R}\rightarrow\mathbb{R}\) is a measurable function and \(\gamma\) is a plane curve with certain constraints then the Carleson transform \[ \mathcal{C}_{u,\gamma}f(x)=p.v.\int_{-\infty}^{\infty}e^{iu(x)\gamma(t)}f(x-t)\frac{dt}{t}\qquad x\in\mathbb{R} \] and the Hilbert transform \[ H_{u,\gamma}f(x_{1},x_{2})=p.v.\int_{-\infty}^{\infty}f(x_{1}-t,x_{2}-u(x_{1})\gamma(t))\frac{dt}{t}\qquad(x_{1},x_{2})\in\mathbb{R}^{2} \] are bounded on \(L^{p}\).