Weak-type (1,1) estimates for strongly singular operators

TL;DR

This paper proves that a class of strongly singular oscillatory convolution operators are of weak type (1,1) under certain regularity and growth conditions on their defining functions.

Contribution

It establishes weak type (1,1) bounds for a new class of strongly singular oscillatory operators with general phase functions.

Findings

Operators are of weak type (1,1) under specified conditions.

Results extend classical singular integral theory to oscillatory, strongly singular kernels.

Provides a framework for analyzing similar operators with oscillatory behavior.

Abstract

Let $\psi$ be a positive function defined near the origin such that $\lim_{t\to 0^{+}}\psi(t)=0$. We consider the operator \begin{equation*} T_\theta f(x) = \lim_{\varepsilon\to 0^+} \int_\varepsilon^1 e^{i\gamma(t)}f(x-t) \frac{dt}{t^{\theta}\psi(t)^{1-\theta}}, \end{equation*} where $\gamma$ is a real function with $\lim_{t\to 0^+}|\gamma(t)| = \infty$ and $0 \le \theta \le 1$. Assuming certain regularity and growth conditions on $\psi$ and $\gamma$, we show that $T_1$ is of weak type $(1,1)$.