Estimates for strongly singular operators along curves

For a proper function $f$ on the plane, we study the operator $$Tf(x,y)=\underset{\epsilon \to 0}{\lim }{\int }_{\epsilon }^{1}f(x-t,y-t^k)\frac{e^{2\pi i\gamma (t)}}{\psi (t)}dt,$$ where $k\ge 1$ and $\psi$ and $\gamma$ are functions defined near the origin such that $\psi (t)\to 0$ and $\mathrm{\mid }\gamma (t)\mathrm{\mid }\to \mathrm{\infty }$ as $t\to 0$. We give sufficient regularity and growth conditions on $\psi$ and $\gamma$ for its multiplier to be a bounded function, and thus for the operator to be bounded on $L^2({\mathbb{R}}^2)$. We consider an extension to $L^p({\mathbb{R}}^2)$, for certain $p^{\mathrm{\prime }}s$.