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

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 \((1,1)\).