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

TL;DR

This paper investigates the continuity of strongly singular integral operators at extreme $p$ values, extending known boundedness results to weighted spaces and providing alternative proofs for weighted $L^p$ estimates.

Contribution

It generalizes Miyachi's boundedness results to Muckenhoupt weights and offers an alternative proof of Chanillo's weighted $L^p$ estimates using extrapolation.

Findings

Weighted $L^ abla$-$BMO$ boundedness for singular integral operators.

Extension of Miyachi's results to Muckenhoupt weights.

Alternative proof of Chanillo's weighted $L^p$ estimates.

Abstract

In this work, we establish continuity properties of strongly singular integral operators for extreme values of $p$. Particularly, weighted $L^\infty$-$BMO$ 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.