An $ L^{q}\rightarrow  L^{r} $ estimate for rough Fourier integral operators and its applications

Jinhui Li; Guangqing Wang

doi:10.3934/dcds.2022102

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2022, Volume 42, Issue 11: 5387-5397. Doi: 10.3934/dcds.2022102

This issue Previous Article Monotonicity of entropy for unimodal real quadratic rational maps Next Article Spectral theory of spin substitutions

An $ L^{q}\rightarrow L^{r} $ estimate for rough Fourier integral operators and its applications

Jinhui Li and
Guangqing Wang^,

School of Mathematics and Statistics, Fuyang Normal University, 236041 Fuyang, China

^*Corresponding author: Guangqing Wang
^*Corresponding author: Guangqing Wang

Received: December 2021

Revised: June 2022

Early access: August 12, 2022

Published online: August 12, 2022

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Abstract

Let $ T_{a, \varphi} $ be a Fourier integral operator defined by the oscillatory integral

$ \begin{eqnarray*} T_{a, \varphi}u(x) & = &\frac{1}{(2\pi)^n}\int_{\mathbb{R}^n} e^{ i \varphi(x, \xi)}a(x, \xi) \hat{u}(\xi)d\xi, \end{eqnarray*} $

where $ a\in L^{p} S^{m}_{\varrho} $ ($ 1\leq p\leq\infty, $ $ 0\leq\varrho\leq1 $) and $ \varphi\in L^{\infty}\Phi^{2} $, satisfying the rough non-degenerate condition. It is showed that if $ 0<r\leq \infty, $ $ 1\leq q\leq\infty $ satisfying the relation $ \frac{1}{r} = \frac{1}{q}+\frac{1}{p} $, then $ T_{a, \varphi} $ is a bounded operator from $ L^{q}(\mathbb{R}^n) $ to $ L^{r}(\mathbb{R}^n) $ provided

$ m<-\varrho\frac{(n-1)}{2}\big(\frac{1}{s}+\frac{1}{\min(p, s')}\big)+\frac{n(\varrho-1)}{s}, $

where $ s = \min(2, p, q) $ and $ \frac{1}{s}+\frac{1}{s'} = 1. $

Keywords:

Rough Fourier integral operators,
bilinear Fourier integral operators.

Mathematics Subject Classification: Primary: 42B20; Secondary: 42B37.

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