The boundedness of the bilinear oscillatory integral along a parabola

Pub Date : 2023-02-01 DOI:10.1017/S0013091523000032

Guoliang Li, Junfeng Li

引用次数: 0

Abstract

Abstract In this paper, the $L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})$ boundedness of the bilinear oscillatory integral along parabola \begin{equation*} T_\beta(f, g)(x)=p.v.\int_{{\mathbb R}} f(x-t)g(x-t^{2})\,{\rm e}^{i |t|^{\beta}}\,\frac{{\rm d}t}{t} \end{equation*}is set up, where β > 1 or β < 0, $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$ and $\frac{1}{2}\lt r\lt\infty$, p > 1 and q > 1. The result for the case β < 0 extends the $L^\infty\times L^2\to L^2$ boundedness obtained by Fan and Li (D. Fan and X. Li, A bilinear oscillatory integral along parabolas, Positivity 13(2) (2009), 339–366) by confirming an open question raised in it.

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双线性振荡积分沿抛物线的有界性

摘要本文建立了沿抛物线\begin{equation*}T_\beta(f, g)(x)=p.v.\int_{{\mathbb R}} f(x-t)g(x-t^{2})\,{\rm e}^{i |t|^{\beta}}\,\frac{{\rm d}t}{t}\end{equation*}的双线性振荡积分的$L^p(\mathbb{R})\times L^q(\mathbb{R})\rightarrow L^r(\mathbb{R})$有界性，其中β > 1或β < 0, $\frac{1}{p}+\frac{1}{q}=\frac{1}{r}$和$\frac{1}{2}\lt r\lt\infty$, p > 1和q > 1。β < 0情况下的结果扩展了Fan和Li (D. Fan和X. Li, A沿抛物线的双线性振荡积分，正13(2)(2009)，339-366)得到的$L^\infty\times L^2\to L^2$有界性，证实了其中提出的一个开放问题。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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