An operator van der Corput estimate arising from oscillatory Riemann-Hilbert problems

TL;DR

This paper develops a uniform operator van der Corput estimate for oscillatory integrals, applicable to phases with low regularity, singularities, and degenerate stationary points, with implications for harmonic analysis and PDEs.

Contribution

It introduces a general, real-variable approach to estimate oscillatory operators with complex phase behaviors, extending classical results to broader contexts.

Findings

Achieves strongly uniform decay estimates for oscillatory operators

Handles phases with low regularity and singularities

Accommodates degenerate stationary points of various orders

Abstract

We study an operator analogue of the classical problem of finding the rate of decay of an oscillatory integral on the real line. This particular problem arose in the analysis of oscillatory Riemann-Hilbert problems associated with partial differential equations in the Ablowitz-Kaup-Newell-Segur hierarchy, but is interesting in its own right as a question in harmonic analysis and oscillatory integrals. As was the case in earlier work of the first author, the approach is general and purely real-variable. The resulting estimates we achieve are strongly uniform as a function of the phase and can simultaneously accommodate phases with low regularity (as low as $C^{1,\alpha}$), local singularities, and essentially arbitrary sets of stationary points that degenerate to finite or infinite order.