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

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 $(1,1)$.