Hilbert transforms along double variable fractional monomials

In this paper, we obtain the \begin{document}$L^2(\mathbb{R}^2)$\end{document} boundedness and single annulus \begin{document}$L^p(\mathbb{R}^2)$\end{document} estimate for the Hilbert transform \begin{document}$H_{α,β}$\end{document} along double variable fractional monomial \begin{document}$u_1(x_1)[t]^α+u_2(x_1)[t]^β$\end{document} \begin{document}$H_{α,β}f(x_1,x_2): = \mathit{\rm{p.\,v.}}∈\int_{ - \infty }^\infty {} f(x_1-t,x_2-u_1(x_1)[t]^α-u_2(x_1)[t]^β)\,\frac{\textrm{d}t}{t}$ \end{document} with the bounds are independent of the measurable function \begin{document}$u_1$\end{document} and \begin{document}$u_2$\end{document} . At the same time, we also obtain the \begin{document}$L^p(\mathbb{R})$\end{document} boundedness of the corresponding Carleson operator \begin{document}$\mathcal{C}_{α,β}f(x):=\mathop {\sup }\limits_{{N_1},{N_2} \in \mathbb{R}} |{\rm{p.\,v.}}\int_{ - \infty }^\infty {} e^{iN_1[t]^α+iN_2[t]^β}f(x-t)\,\frac{\textrm{d}t}{t}|,$ \end{document} where \begin{document}$[t]^α$\end{document} stands for either \begin{document}$|t|^α$\end{document} or \begin{document}$\textrm{sgn}(t)|t|^α$\end{document} , \begin{document}$[t]^β$\end{document} stands for either \begin{document}$|t|^β$\end{document} or \begin{document}$\textrm{sgn}(t)|t|^β$\end{document} and \begin{document}$α,β,p∈ (1,∞)$\end{document} .