Estimates for strongly singular operators along curves

Abstract

For a proper function $f$ on the plane, we study the operator \[ Tf(x,y) = \lim_{\varepsilon\to 0} \int_\varepsilon^1 f(x-t,y-t^k) \frac{e^{2πi γ(t)}}{ψ(t)} dt, \] where $k\ge 1$ and $\psi$ and $\gamma$ are functions defined near the origin such that $\psi (t)\to 0$ and $|\gamma (t)|\to \infty$ 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 $p^{\prime }s$.