Hilbert transforms along double variable fractional monomials

In this paper, we obtain the $L^2(\mathbb{R}^2)$ boundedness and single annulus $L^p(\mathbb{R}^2)$ estimate for the Hilbert transform $H_{α,β}$ along double variable fractional monomial $u_1(x_1)[t]^α+u_2(x_1)[t]^β$

$H_{α,β}f(x_1,x_2): = \mathit{\rm{p.\,v.}}∈\int_{ - \infty }^\infty {} f(x_1-t,x_2-u_1(x_1)[t]^α-u_2(x_1)[t]^β)\,\frac{\textrm{d}t}{t}$

with the bounds are 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

$\mathcal{C}_{α,β}f(x):=\mathop {\sup }\limits_{{N_1},{N_2} \in \mathbb{R}} |{\rm{p.\,v.}}\int_{ - \infty }^\infty {} e^{iN_1[t]^α+iN_2[t]^β}f(x-t)\,\frac{\textrm{d}t}{t}|,$

where $[t]^α$ stands for either $|t|^α$ or $\textrm{sgn}(t)|t|^α$, $[t]^β$ stands for either $|t|^β$ or $\textrm{sgn}(t)|t|^β$ and $α,β,p∈ (1,∞)$.