Bilinear oscillatory integrals and boundedness for new bilinear multipliers

TL;DR

This paper studies bilinear oscillatory integrals with oscillating symbols, establishing Lebesgue space bounds that extend linear oscillatory integral theory, using time-frequency analysis and phase estimates.

Contribution

It introduces new bilinear multipliers with non-smooth symbols and provides decay estimates for their boundedness as oscillation increases.

Findings

Lebesgue space inequalities with decay for bilinear oscillatory integrals

Extension of linear oscillatory integral theory to bilinear setting

Boundedness results for new bilinear multipliers with non-smooth symbols

Abstract

We consider bilinear oscillatory integrals, i.e. pseudo-product operators whose symbol involves an oscillating factor. Lebesgue space inequalities are established, which give decay as the oscillation becomes stronger ; this extends the well-known linear theory of oscillatory integral in some directions. The proof relies on a combination of time-frequency analysis of Coifman-Meyer type with stationary and non-stationary phase estimates. As a consequence of this analysis, we obtain Lebesgue estimates for new bilinear multipliers defined by non-smooth symbols.