Estimates for strongly singular operators along curves

TL;DR

This paper investigates the boundedness of a class of strongly singular operators along curves in the plane, establishing conditions on functions defining the operator for boundedness on L^2 and some L^p spaces.

Contribution

It provides new sufficient regularity and growth conditions on the phase and amplitude functions ensuring the operator's boundedness on L^2 and extends results to certain L^p spaces.

Findings

Operator bounded on L^2 under specified conditions

Conditions on  and  ensure boundedness of the multiplier

Extension of boundedness results to some L^p spaces

Abstract

For a proper function $f$ on the plane, we study the operator \[ Tf(x,y) = \lim_{\varepsilon\to 0} \int_\varepsilon^1 f(x-t,y-t^k) \frac{e^{2\pi i \gamma(t)}}{\psi(t)} dt, \] where $k\ge1$ 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 $p's$.