On some bilinear Fourier multipliers with oscillating factors, II

TL;DR

This paper investigates bilinear Fourier multipliers with oscillating exponential factors involving fractional powers, establishing criteria for their boundedness on various Lebesgue space products, extending understanding of such operators in harmonic analysis.

Contribution

It provides new boundedness criteria for bilinear Fourier multipliers with oscillating factors, particularly for fractional powers, in different Lebesgue space settings.

Findings

Derived conditions on $m$ for boundedness on $L^{ ext{infinity}} imes L^{ ext{infinity}}$

Established criteria for $L^{1} imes L^{ ext{infinity}}$ and $L^{ ext{infinity}} imes L^{1}$ boundedness

Extended analysis to multipliers with oscillating exponential factors involving fractional powers

Abstract

For $s > 0$, $s \neq 1$, bilinear Fourier multipliers of the form $e^{i (|\xi|^s + |\eta|^s+ |\xi + \eta|^s)} \sigma (\xi, \eta)$ are considered, where $\sigma(\xi, \eta)$ belongs to the H\"ormander class $S^{m}_{1, 0}(\mathbb{R}^{2n})$. A criterion for $m$ to ensure the $L^{\infty}\times L^{\infty} \to L^\infty$, $L^{1} \times L^{\infty} \to L^{1}$, and $L^{\infty} \times L^{1} \to L^{1}$ boundedness of the corresponding bilinear operators is given.