$L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators

JIAO CHEN; WEI DAI; GUOZHEN LU

doi:10.3934/cpaa.2017042

Article Contents

2017, Volume 16, Issue 3: 883-898. Doi: 10.3934/cpaa.2017042

This issue Previous Article On weighted mixed-norm Sobolev estimates for some basic parabolic equations Next Article Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients

$L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators

1.
School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China
2.
School of Mathematics and Systems Science, Beihang University (BUAA), Beijing, 100191, China
3.
Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA

^* Corresponding author: Guozhen Lu
^* Corresponding author: Guozhen Lu

Received: July 2016

Revised: December 2016

Published: January 2018

The first two authors were partly supported by the NNSF of China (No. 11371056), the second author was also supported by the NNSF of China (No. 11501021), the third author was partly supported by a US NSF grant and a Simons Fellowship from the Simons Foundation.

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Abstract

In this paper, we establish the $L^p$ estimates for the maximal functions associated with the multilinear pseudo-differential operators. Our main result is Theorem 1.2. There are several major different ingredients and extra difficulties in our proof from those in Grafakos, Honzík and Seeger ^[15] and Honzík ^[22] for maximal functions generated by multipliers. First, in order to eliminate the variable $x$ in the symbols, we adapt a non-smooth modification of the smooth localization method developed by Muscalu in ^[26,30]. Then, by applying the inhomogeneous Littlewood-Paley dyadic decomposition and a discretization procedure, we can reduce the proof of Theorem 1.2 into proving the localized estimates for localized maximal functions generated by discrete paraproducts. The non-smooth cut-off functions in the localization procedure will be essential in establishing localized estimates. Finally, by proving a key localized square function estimate (Lemma 4.3) and applying the good-$\lambda$ inequality, we can derive the desired localized estimates.

Keywords:

Multi-linear pseudo-differential operators,
$L^p$ estimates,
maximal functions,
paraproducts.

Mathematics Subject Classification: Primary: 35S05; Secondary: 42B15, 42B20.

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References

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