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October, 2022 Global $L^{2}$-boundedness of a New Class of Rough Fourier Integral Operators
Jiawei Dai, Qiang Huang
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Taiwanese J. Math. 26(5): 1029-1043 (October, 2022). DOI: 10.11650/tjm/220403

Abstract

In this paper, we investigate the $L^{2}$ boundedness of Fourier integral operator $T_{\phi,a}$ with rough symbol $a \in L^{\infty} S^{m}_{\rho}$ and rough phase $\phi \in L^{\infty} \Phi^{2}$ which satisfies $\big| \{ x: |\nabla_{\xi} \phi(x,\xi) - y| \leq r \} \big| \leq C(r^{n-1}+r^{n})$ for any $\xi,y \in \mathbb{R}^{n}$ and $r \gt 0$. We obtain that $T_{\phi,a}$ is bounded on $L^2$ if $m \lt \rho(n-1)/2 - n/2$ when $0 \leq \rho \leq 1/2$ or $m \lt -(n+1)/4$ when $1/2 \leq \rho \leq 1$. When $\rho = 0$ or $n = 1$, the condition of $m$ is sharp. Moreover, the maximal wave operator is a special class of $T_{\phi,a}$ which is studied in this paper. Thus, our main theorem substantially extends and improves some known results about the maximal wave operator.

Funding Statement

This work was supported by Scientific Research Fund of Zhejiang Provincial Education Department (No. Y201738640) and the National Natural Science Foundation of China (No. 11801518).

Citation

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Jiawei Dai. Qiang Huang. "Global $L^{2}$-boundedness of a New Class of Rough Fourier Integral Operators." Taiwanese J. Math. 26 (5) 1029 - 1043, October, 2022. https://doi.org/10.11650/tjm/220403

Information

Received: 9 December 2021; Revised: 3 March 2022; Accepted: 10 April 2022; Published: October, 2022
First available in Project Euclid: 21 April 2022

Digital Object Identifier: 10.11650/tjm/220403

Subjects:
Primary: 35S30 , 42B37

Keywords: $L^{2}$ boundedness , maximal wave operator , rough Fourier integral operator

Rights: Copyright © 2022 The Mathematical Society of the Republic of China

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Vol.26 • No. 5 • October, 2022
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