Some one-sided estimates for oscillatory singular integrals

TL;DR

This paper establishes one-sided estimates for oscillatory singular integrals, proving boundedness on weighted Hardy spaces and demonstrating limitations of the theory for certain spaces and weights.

Contribution

It introduces new one-sided estimates for oscillatory singular integrals and explores their boundedness on weighted Hardy spaces, highlighting the theory's limitations.

Findings

Boundedness of oscillatory singular integrals on $H^{1}_{+}(w)$ is proved.

The theory cannot be extended to $H^{q}_{+}(w)$ for $0<q<1$ with $w otin A_{p}^{+}$.

A criterion for weighted $L^{p}$-boundedness of these integrals is provided.

Abstract

The purpose of this paper is to establish some one-sided estimates for oscillatory singular integrals. The boundedness of certain oscillatory singular integral on weighted Hardy spaces $H^{1}_{+}(w)$ is proved. It is here also show that the $H^{1}_{+}(w)$ theory of oscillatory singular integrals above cannot be extended to the case of $H^{q}_{+}(w)$ when $0<q<1$ and $w\in A_{p}^{+}$, a wider weight class than the classical Muckenhoupt class. Furthermore, a criterion on the weighted $L^{p}$-boundednesss of the oscillatory singular integral is given.