American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Ground state and nodal solutions for fractional Schrödinger-Maxwell-Kirchhoff systems with pure critical growth nonlinearity
  • CPAA Home
  • This Issue
  • Next Article
    The anisotropic fractional isoperimetric problem with respect to unconditional unit balls
February  2021, 20(2): 801-815. doi: 10.3934/cpaa.2020291

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

Received  June 2020 Revised  October 2020 Published  February 2021 Early access  December 2020

Fund Project: 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)

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.
Citation: Liang Huang, Jiao Chen. The boundedness of multi-linear and multi-parameter pseudo-differential operators. Communications on Pure & Applied Analysis, 2021, 20 (2) : 801-815. doi: 10.3934/cpaa.2020291
References:
[1]

Á. Bényi, D. Maldonado, V. Naibo and R. H. Torres, On the Hörmander classes of bilinear pseudodifferential operators, Integral Equ. Oper. Theory, 67 (2010), 341-264.  doi: 10.1007/s00020-010-1782-y.  Google Scholar

[2]

Á. Bényi and R. H. Torres, Symbolic calculus and the transposes of bilinear pseudodifferential operators, Commun. Partial Differ. Equ., 28 (2003), 1161-1181.  doi: 10.1081/PDE-120021190.  Google Scholar

[3]

F. Bernicot, Local estimates and global continuities in Lebesgue spaces for bilinear operators, Anal. PDE, 1 (2008), 1-27.  doi: 10.2140/apde.2008.1.1.  Google Scholar

[4]

J. Chen and G. Lu, Hörmander type theorems for multi-linear and multi-parameter Fourier multiplier operators with limited smoothness, Nonlinear Anal., 101 (2014), 98-112.  doi: 10.1016/j.na.2014.01.005.  Google Scholar

[5]

J. Chen and G. Lu, Hömander type theorem on Bi-parameter Hardy spaces for Fourier multipliers with optimal smoothness, Rev. Mat. Iberoam., 34 (2018), 1541-1561.  doi: 10.4171/rmi/1035.  Google Scholar

[6]

M. Christ and J. L. Journé, Polynomial growth estimates for multilinear singular integral operators, Acta Math., 159 (1987), 51-80.  doi: 10.1007/BF02392554.  Google Scholar

[7]

R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331.  doi: 10.2307/1998628.  Google Scholar

[8]

W. Dai and G. Lu, $L^p$ estimates for multi-linear and multi-parameter pseudo-differential operators, Bull. Soc. Math. France., 143 (2013), 567-597.  doi: 10.24033/bsmf.2698.  Google Scholar

[9]

W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear. Anal., 184 (2019), 352-380.  doi: 10.1016/j.na.2019.02.014.  Google Scholar

[10]

C. Fefferman, $L^p$ bounds for pseudo-differential operators, Israel J. Math., 14 (1973), 413-417.  doi: 10.1007/BF02764718.  Google Scholar

[11]

C. Fefferman and E. M. Stein, Some maximal inequalities, Am. J. Math., 93 (1971), 107-115.  doi: 10.2307/2373450.  Google Scholar

[12]

L. Grafakos and R. H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math., 165 (2002), 124-164.  doi: 10.1006/aima.2001.2028.  Google Scholar

[13]

Y. Han, G. Lu and E. Sawyer, Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group, Anal. PDE, 7 (2014), 1465-1534.  doi: 10.2140/apde.2014.7.1465.  Google Scholar

[14]

L. Hörmander, On the $L^2$ continuity of pseudo-differential operators, Commun. Pure Appl. Math., 24 (1971), 529-535.  doi: 10.1002/cpa.3160240406.  Google Scholar

[15]

C. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett., 6 (1999), 1-15.  doi: 10.4310/MRL.1999.v6.n1.a1.  Google Scholar

[16]

K. Koezuka and N. Tomita, Bilinear pseudo-differential operators with symbols in $BS^{m}_{1,1}$ on Triebel-Lizorkin spaces, J. Fourier Anal. Appl., 24 (2018), 309-319.  doi: 10.1007/s00041-016-9518-2.  Google Scholar

[17]

G. Lu and L. Zhang, $L^p$ estimates for a trilinear pseudo-differential operator with flag symbols, Indiana Univ. Math. J., 66 (2017), 877-900.  doi: 10.1512/iumj.2017.66.6069.  Google Scholar

[18]

A. Miyachi and N. Tomita, Estimates for trilinear flag paraproducts on $L^{\infty}$ and Hardy spaces, Math. Z., 282 (2016), 577-613.  doi: 10.1007/s00209-015-1554-0.  Google Scholar

[19]

C. Muscalu, Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoam, 23 (2007), 705-742.  doi: 10.4171/RMI/510.  Google Scholar

[20]

C. Muscalu, J. Pipher, T. Tao and C. Thiele, Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296.  doi: 10.1007/BF02392566.  Google Scholar

[21]

C. Muscalu, J. Pipher, T. Tao and C. Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam, 22 (2006), 963-976.  doi: 10.4171/RMI/480.  Google Scholar

[22] C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis II, Cambridge Univ. Press, 2013.   Google Scholar

show all references

References:
[1]

Á. Bényi, D. Maldonado, V. Naibo and R. H. Torres, On the Hörmander classes of bilinear pseudodifferential operators, Integral Equ. Oper. Theory, 67 (2010), 341-264.  doi: 10.1007/s00020-010-1782-y.  Google Scholar

[2]

Á. Bényi and R. H. Torres, Symbolic calculus and the transposes of bilinear pseudodifferential operators, Commun. Partial Differ. Equ., 28 (2003), 1161-1181.  doi: 10.1081/PDE-120021190.  Google Scholar

[3]

F. Bernicot, Local estimates and global continuities in Lebesgue spaces for bilinear operators, Anal. PDE, 1 (2008), 1-27.  doi: 10.2140/apde.2008.1.1.  Google Scholar

[4]

J. Chen and G. Lu, Hörmander type theorems for multi-linear and multi-parameter Fourier multiplier operators with limited smoothness, Nonlinear Anal., 101 (2014), 98-112.  doi: 10.1016/j.na.2014.01.005.  Google Scholar

[5]

J. Chen and G. Lu, Hömander type theorem on Bi-parameter Hardy spaces for Fourier multipliers with optimal smoothness, Rev. Mat. Iberoam., 34 (2018), 1541-1561.  doi: 10.4171/rmi/1035.  Google Scholar

[6]

M. Christ and J. L. Journé, Polynomial growth estimates for multilinear singular integral operators, Acta Math., 159 (1987), 51-80.  doi: 10.1007/BF02392554.  Google Scholar

[7]

R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331.  doi: 10.2307/1998628.  Google Scholar

[8]

W. Dai and G. Lu, $L^p$ estimates for multi-linear and multi-parameter pseudo-differential operators, Bull. Soc. Math. France., 143 (2013), 567-597.  doi: 10.24033/bsmf.2698.  Google Scholar

[9]

W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear. Anal., 184 (2019), 352-380.  doi: 10.1016/j.na.2019.02.014.  Google Scholar

[10]

C. Fefferman, $L^p$ bounds for pseudo-differential operators, Israel J. Math., 14 (1973), 413-417.  doi: 10.1007/BF02764718.  Google Scholar

[11]

C. Fefferman and E. M. Stein, Some maximal inequalities, Am. J. Math., 93 (1971), 107-115.  doi: 10.2307/2373450.  Google Scholar

[12]

L. Grafakos and R. H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math., 165 (2002), 124-164.  doi: 10.1006/aima.2001.2028.  Google Scholar

[13]

Y. Han, G. Lu and E. Sawyer, Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group, Anal. PDE, 7 (2014), 1465-1534.  doi: 10.2140/apde.2014.7.1465.  Google Scholar

[14]

L. Hörmander, On the $L^2$ continuity of pseudo-differential operators, Commun. Pure Appl. Math., 24 (1971), 529-535.  doi: 10.1002/cpa.3160240406.  Google Scholar

[15]

C. Kenig and E. M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett., 6 (1999), 1-15.  doi: 10.4310/MRL.1999.v6.n1.a1.  Google Scholar

[16]

K. Koezuka and N. Tomita, Bilinear pseudo-differential operators with symbols in $BS^{m}_{1,1}$ on Triebel-Lizorkin spaces, J. Fourier Anal. Appl., 24 (2018), 309-319.  doi: 10.1007/s00041-016-9518-2.  Google Scholar

[17]

G. Lu and L. Zhang, $L^p$ estimates for a trilinear pseudo-differential operator with flag symbols, Indiana Univ. Math. J., 66 (2017), 877-900.  doi: 10.1512/iumj.2017.66.6069.  Google Scholar

[18]

A. Miyachi and N. Tomita, Estimates for trilinear flag paraproducts on $L^{\infty}$ and Hardy spaces, Math. Z., 282 (2016), 577-613.  doi: 10.1007/s00209-015-1554-0.  Google Scholar

[19]

C. Muscalu, Paraproducts with flag singularities. I. A case study, Rev. Mat. Iberoam, 23 (2007), 705-742.  doi: 10.4171/RMI/510.  Google Scholar

[20]

C. Muscalu, J. Pipher, T. Tao and C. Thiele, Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296.  doi: 10.1007/BF02392566.  Google Scholar

[21]

C. Muscalu, J. Pipher, T. Tao and C. Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam, 22 (2006), 963-976.  doi: 10.4171/RMI/480.  Google Scholar

[22] C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis II, Cambridge Univ. Press, 2013.   Google Scholar
[1]

Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130
[2]

JIAO CHEN, WEI DAI, GUOZHEN LU. $L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators. Communications on Pure & Applied Analysis, 2017, 16 (3) : 883-898. doi: 10.3934/cpaa.2017042
[3]

Li Wang, Qiang Xu, Shulin Zhou. $ L^p $ Neumann problems in homogenization of general elliptic operators. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 5019-5045. doi: 10.3934/dcds.2020210
[4]

Woocheol Choi, Yong-Cheol Kim. $L^p$ mapping properties for nonlocal Schrödinger operators with certain potentials. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5811-5834. doi: 10.3934/dcds.2018253
[5]

Jinrui Huang, Wenjun Wang, Huanyao Wen. On $ L^p $ estimates for a simplified Ericksen-Leslie system. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1485-1507. doi: 10.3934/cpaa.2020075
[6]

Xinghong Pan, Jiang Xu. Global existence and optimal decay estimates of the compressible viscoelastic flows in $ L^p $ critical spaces. Discrete & Continuous Dynamical Systems, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085
[7]

Umberto De Maio, Peter I. Kogut, Gabriella Zecca. On optimal $ L^1 $-control in coefficients for quasi-linear Dirichlet boundary value problems with $ BMO $-anisotropic $ p $-Laplacian. Mathematical Control & Related Fields, 2020, 10 (4) : 827-854. doi: 10.3934/mcrf.2020021
[8]

Guangfeng Dong, Changjian Liu, Jiazhong Yang. On the maximal saddle order of $ p:-q $ resonant saddle. Discrete & Continuous Dynamical Systems, 2019, 39 (10) : 5729-5742. doi: 10.3934/dcds.2019251
[9]

Fang Liu. The eigenvalue problem for a class of degenerate operators related to the normalized $ p $-Laplacian. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021155
[10]

Yamin Wang. On nonexistence of extremals for the Trudinger-Moser functionals involving $ L^p $ norms. Communications on Pure & Applied Analysis, 2020, 19 (9) : 4257-4268. doi: 10.3934/cpaa.2020191
[11]

Junjie Zhang, Shenzhou Zheng, Haiyan Yu. $ L^{p(\cdot)} $-regularity of Hessian for nondivergence parabolic and elliptic equations with measurable coefficients. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2777-2796. doi: 10.3934/cpaa.2020121
[12]

Qunyi Bie, Haibo Cui, Qiru Wang, Zheng-An Yao. Incompressible limit for the compressible flow of liquid crystals in $ L^p$ type critical Besov spaces. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124
[13]

Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations: $ L^p $-spaces. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021028
[14]

Jiahao Qiu, Jianjie Zhao. Maximal factors of order $ d $ of dynamical cubespaces. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 601-620. doi: 10.3934/dcds.2020278
[15]

Tuan Anh Dao, Michael Reissig. $ L^1 $ estimates for oscillating integrals and their applications to semi-linear models with $ \sigma $-evolution like structural damping. Discrete & Continuous Dynamical Systems, 2019, 39 (9) : 5431-5463. doi: 10.3934/dcds.2019222
[16]

Niklas Sapountzoglou, Aleksandra Zimmermann. Well-posedness of renormalized solutions for a stochastic $ p $-Laplace equation with $ L^1 $-initial data. Discrete & Continuous Dynamical Systems, 2021, 41 (5) : 2341-2376. doi: 10.3934/dcds.2020367
[17]

Yu-Zhao Wang. $ \mathcal{W}$-Entropy formulae and differential Harnack estimates for porous medium equations on Riemannian manifolds. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2441-2454. doi: 10.3934/cpaa.2018116
[18]

Genni Fragnelli, Jerome A. Goldstein, Rosa Maria Mininni, Silvia Romanelli. Operators of order 2$ n $ with interior degeneracy. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3417-3426. doi: 10.3934/dcdss.2020128
[19]

Jiangang Qi, Bing Xie. Extremum estimates of the $ L^1 $-norm of weights for eigenvalue problems of vibrating string equations based on critical equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3505-3516. doi: 10.3934/dcdsb.2020243
[20]

Gyula Csató. On the isoperimetric problem with perimeter density $r^p$. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2729-2749. doi: 10.3934/cpaa.2018129

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (98)
  • HTML views (94)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences