Continuity properties of strongly singular integral operators for extreme values of $p$

In this work, we establish continuity properties of strongly singular integral operators for extreme values of $p$. Particularly, weighted $L^{\mathrm{\infty }}$-$BMO$ boundedness is obtained, generalizing Miyachi’s result to the context of Muckenhoupt weights. As an application, we get an alternative proof of Chanillo’s weighted $L^p$ estimates via extrapolation techniques.