The weak (1,1) boundedness of Fourier integral operators with complex phases

TL;DR

This work establishes weak $(1,1)$ boundedness for Fourier integral operators of order $-(n-1)/2$ whose canonical relations are locally parametrised by complex phases of positive type. The authors blend Tao’s degenerate/non-degenerate factorisation with Melin–Sjostrand’s global complex-phase parametrisation and a local graph condition to overcome obstacles that do not arise in the real-phase setting. The main result extends endpoint control to complex phases, showing that under $ ext{Im}( ext{Phi}(x, heta))>0$ for $| heta| eq 0$ and a local graph assumption, such operators map $L^1$ to $L^{1, abla}$ in a weak sense. These findings have implications for Cauchy problems with complex characteristics and broaden the $L^p$ theory of FIOs beyond real phases. The approach offers a robust framework for endpoint estimates in the complex-phase regime, with potential applications to hyperbolic PDEs and global harmonic analysis on manifolds.

Abstract

Let $T$ be a Fourier integral operator of order $-(n-1)/2$ associated with a canonical relation locally parametrised by a real-phase function. A fundamental result due to Seeger, Sogge, and Stein proved in the 90's, gives the boundedness of $T$ from the Hardy space $H^1$ into $L^1.$ Additionally, it was shown by T. Tao the weak (1,1) type of $T$. In this work, we establish the weak (1,1) boundedness of a Fourier integral operator $T$ of order $-(n-1)/2$ when it has associated a canonical relation parametrised by a complex phase function. This result in the complex-valued setting, cannot be derived from its counterpart in the real-valued case.

Figures (2)

• Figure 1: Here we illustrate the partition of the $\omega$ variable smoothly into about $2^{(n-1)k/2}$ disks $D$ is radius $2^{-k/2}$ when $|\omega|\lesssim 1.$ First, in the system of coordinates $(\overline{\xi},\xi_n),$ the projective coordinates $(\lambda,\omega)$ determine each point $(\overline{\xi},\xi_n)\in \mathscr{C}.$ The unit ball $B_{n-1}=\{|\omega|\leq 1\}$ on $\mathbb{R}^{n-1}$ is saturated to the 'segment' $\overline{AB}.$ Each partition of the ball $B_{n-1}$ in a family of disks $\{D\}_{D\in \mathcal{I}}$ is saturated to a partition of the segment $\overline{AB}.$ In particular if the radii of the disks are proportional to $r=2^{-k/2},$ one can estimate $|\mathcal{I}|\sim 2^{(n-1)k/2}.$
• Figure 2: Note that the dyadic partition of the variable $\xi_n=\lambda$ together with the family of 'ellipsoids' $\{D\}_{D\in \mathcal{I}}$ decomposing the variable $\omega,$ provide a partition of the cone bundle $\mathscr{C}.$

Theorems & Definitions (43)

• Definition 1.1
• Theorem 1.2: Tao Tao, 2004
• Theorem 1.3: Seeger, Sogge and Stein SSS, 1991
• Remark 1.5
• Theorem 1.6
• Theorem 1.7
• Remark 1.8
• Remark 1.9
• Remark 1.10
• Definition 2.1: Real-valued phase functions