Weighted norm inequalities for oscillatory integrals with finite type phases on the line

TL;DR

This paper establishes optimal two-weighted L^2 norm inequalities for oscillatory integral operators with finite type phases on the real line, advancing understanding of their boundedness properties.

Contribution

It introduces new two-weight inequalities involving geometric maximal functions, extending previous work to cover finite type phase oscillatory integrals.

Findings

Inequalities are best-possible and imply full L^p-L^q mapping properties.

Conditions involve geometrically-defined maximal functions.

Results build on and extend prior foundational work.

Abstract

We obtain two-weighted $L^2$ norm inequalities for oscillatory integral operators of convolution type on the line whose phases are of finite type. The conditions imposed on the weights involve geometrically-defined maximal functions, and the inequalities are best-possible in the sense that they imply the full $L^p(\mathbb{R})\rightarrow L^q(\mathbb{R})$ mapping properties of the oscillatory integrals. Our results build on work of Carbery, Soria, Vargas and the first author.