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