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

Magali Folch-Gabayet, Ricardo A. Sáenz

Published 2018-02-13Version 1

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_\beta f(x) = \lim_{\varepsilon\to 0^+} \int_\varepsilon^1 e^{i\gamma(t)}f(x-t) \frac{dt}{t^{1+i\beta}\psi(t)^{i\beta}}, \end{equation*} where $\gamma$ is a real function. Assuming certain regularity conditions on $\psi$ and $\gamma$, we show that $T_{\beta}$ is of weak type $(1,1)$.