Boundedness of Fourier Integral Operators with complex phases on Fourier Lebesgue spaces

TL;DR

This work proves sharp boundedness results for Fourier integral operators with complex phases on Fourier Lebesgue spaces under a spatial smooth factorization condition of rank $\varkappa$. By conjugating to a real-like phase $\Phi_{\tau}$ and working in the Hörmander class $S^m_{1/2,1/2}$, the authors implement a dyadic Littlewood–Paley and angular decomposition to exploit the geometry of the complex canonical relation. The main theorem states that $T: \mathcal{F}L^p \to \mathcal{F}L^p$ is bounded for $1\le p\le \infty$ when $m\le -\varkappa|\frac{1}{p}-\frac{1}{2}|$, with endpoint analysis and sharpness established; the result extends the real-phase theory (Nicola 2010) to complex phases and accommodates degenerate spatial directions. The findings have implications for wave propagation and parametrix constructions in settings where real-phase FIOs are insufficient, highlighting the role of complex phases in achieving precise regularity on Fourier Lebesgue scales.

Abstract

In this paper, we develop boundedness estimates for Fourier integral operators on Fourier Lebesgue spaces when the associated canonical relation is parametrised by a complex phase function. Our result constitutes the complex analogue of those obtained for real canonical relations by Rodino, Nicola, and Cordero. We prove that, under the spatial factorization condition of rank $\varkappa$, the corresponding Fourier integral operator is bounded on the Fourier Lebesgue space $\mathcal{F}L^p,$ provided that the order $m$ of the operator satisfies that $ m \leq -\varkappa\left|\frac{1}{p}-\frac{1}{2}\right|, 1 \leq p \leq \infty. $ This condition on the order $m$ is sharp.

Theorems & Definitions (23)

• Theorem 1.1
• Remark 1.2
• Remark 1.4
• Remark 1.5
• Remark 1.6
• Definition 2.1: Spatial smooth factorization condition (SSFC)
• Remark 2.2
• Remark 2.3
• Lemma 3.1
• proof