Endpoint regularity of general Fourier integral operators

Abstract

Let $n\ge 1,0<\ddot{I}<1,\max \{\ddot{I},1-\ddot{I}\}\le \hat{I}´\le 1$ and $$m_1=\ddot{I}-n+(n-1)\min \{\frac{1}{2},\ddot{I}\}+\frac{1-\hat{I}´}{2}.$$ If the amplitude $a$ belongs to the Hörmander class $S_{\ddot{I},\hat{I}´}^{m_1}$ and $\ddot{I}•\in \hat{I}{¦}^2$ satisfies the strong non-degeneracy condition, then we prove that the following Fourier integral operator $T_{\ddot{I}•,a}$ defined by \begin{align*} T_{ϕ,a}f(x)=\int_{\mathbb{R}^{n}}e^{iϕ(x,ξ)}a(x,ξ)\widehat{f}(ξ)dξ, \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 $0\le \ddot{I}\le 1,\hat{I}´\le \max \{\ddot{I},1-\ddot{I}\}$, by using some similar techniques in this note, we can get the corresponding theorems which coincide with the known results.

22 pages. arXiv admin note: substantial text overlap with arXiv:2406.03076