Endpoint regularity of general Fourier integral operators

TL;DR

This paper proves endpoint regularity results for a class of Fourier integral operators, showing their boundedness from local Hardy spaces to Lebesgue spaces under specific amplitude and phase conditions.

Contribution

It establishes new boundedness results for Fourier integral operators on Hardy and Lebesgue spaces, improving recent work by Staubach and collaborators.

Findings

Boundedness from $h^1$ to $L^1$ for certain Fourier integral operators.

Extension of results to $L^p$ spaces for $1<p<2$.

Rigorous improvement over previous theorems by Staubach et al.

Abstract

Let $n\geq 1,0<\rho<1, \max\{\rho,1-\rho\}\leq \delta\leq 1$ and $$m_1=\rho-n+(n-1)\min\{\frac 12,\rho\}+\frac {1-\delta}{2}.$$ If the amplitude $a$ belongs to the H\"{o}rmander class $S^{m_1}_{\rho,\delta}$ and $\phi\in \Phi^{2}$ satisfies the strong non-degeneracy condition, then we prove that the following Fourier integral operator $T_{\phi,a}$ defined by \begin{align*} T_{\phi,a}f(x)=\int_{\mathbb{R}^{n}}e^{i\phi(x,\xi)}a(x,\xi)\widehat{f}(\xi)d\xi, \end{align*} is bounded from the local Hardy space $h^1(\mathbb{R}^n)$ to $L^1(\mathbb{R}^n)$. As a corollary, we can also obtain the corresponding $L^p(\mathbb{R}^n)$-boundedness when $1<p<2$. These theorems are rigorous improvements on the recent works of Staubach and his collaborators. When $0\leq \rho\leq 1,\delta\leq \max\{\rho,1-\rho\}$, by using some similar techniques in this note, we can get the corresponding theorems which coincide with the known results.