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_θf(x) = \lim_{\varepsilon\to 0^+} \int_\varepsilon^1 e^{iγ(t)}f(x-t) \frac{dt}{t^θψ(t)^{1-θ}}, \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 $(1,1)$.

10 pages