The boundedness of multi-linear and multi-parameter pseudo-differential operators

Liang Huang; Jiao Chen

doi:10.3934/cpaa.2020291

Article Contents

2021, Volume 20, Issue 2: 801-815. Doi: 10.3934/cpaa.2020291

This issue Previous Article The anisotropic fractional isoperimetric problem with respect to unconditional unit balls Next Article Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity

The boundedness of multi-linear and multi-parameter pseudo-differential operators

Liang Huang^1, and
Jiao Chen^2, ,

1.
School of Science, Xi'an University of Posts and Telecommunications, Xi'an, Shanxi 710121, China
2.
School of Mathematical Sciences, Chongqing Normal University, Chongqing 400000, China

^* Corresponding author
^* Corresponding author

Received: May 31, 2020

Revised: September 30, 2020

Early access: December 2020

Published: February 2021

The authors were supported partly by NNSF of China (Grant No.11801049), the Open Project of Key Laboratory (No.CSSXKFKTZ202004), School of Mathematical Sciences, Chongqing Normal University, the Natural Science Foundation of Chongqing (cstc2019jcyjmsxmX0374, cstc2019jcyj-msxmX0295), Technology Project of Chongqing Education Committee (Grant No. KJQN201800514)

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Abstract

In this paper, we establish the boundedness on $ L^r(\mathbb{R}^{n_1}\times\mathbb{R}^{n_2}) $ of bilinear and bi-parameter pseudo-differential operators whose symbols $ \sigma(x,\xi,\eta)\in S^{(0,0)}_{(1,1),(\delta_1,\delta_2)} $ for $ x,\xi,\eta\in\mathbb{R}^{n_1}\times\mathbb{R}^{n_2} $ and $ 0\leq\delta_1,\delta_2<1 $, which extends the result of Dai and Lu in [8].

Keywords:

Maximal function,
$ L^p $ estimates,
bi-parameter pseudo-differential operators.

Mathematics Subject Classification: Primary:42B15, 42B25.

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References

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