• 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

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].

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

Á. BényiD. MaldonadoV. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

Y. HanG. 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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

C. MuscaluJ. PipherT. Tao and C. Thiele, Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296.  doi: 10.1007/BF02392566.

[21]

C. MuscaluJ. PipherT. Tao and C. Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam, 22 (2006), 963-976.  doi: 10.4171/RMI/480.

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

show all references

References:
[1]

Á. BényiD. MaldonadoV. 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9]

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

[10]

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

[11]

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

[12]

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

[13]

Y. HanG. 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.

[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.

[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.

[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.

[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.

[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.

[19]

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

[20]

C. MuscaluJ. PipherT. Tao and C. Thiele, Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296.  doi: 10.1007/BF02392566.

[21]

C. MuscaluJ. PipherT. Tao and C. Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam, 22 (2006), 963-976.  doi: 10.4171/RMI/480.

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

Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems, 2019, 39 (4) : 2021-2057. doi: 10.3934/dcds.2019085

[7]

Luisa Malaguti, Stefania Perrotta, Valentina Taddei. $ L^p $-exact controllability of partial differential equations with nonlocal terms. Evolution Equations and Control Theory, 2021  doi: 10.3934/eect.2021053

[8]

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 and Related Fields, 2020, 10 (4) : 827-854. doi: 10.3934/mcrf.2020021

[9]

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

[10]

Xuerui Gao, Yanqin Bai, Shu-Cherng Fang, Jian Luo, Qian Li. A new hybrid $ l_p $-$ l_2 $ model for sparse solutions with applications to image processing. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021211

[11]

Yusuke Ishigaki. On $ L^1 $ estimates of solutions of compressible viscoelastic system. Discrete and Continuous Dynamical Systems, 2022, 42 (4) : 1835-1853. doi: 10.3934/dcds.2021174

[12]

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

[13]

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

[14]

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 and Continuous Dynamical Systems, 2018, 38 (6) : 2879-2910. doi: 10.3934/dcds.2018124

[15]

Khalid Latrach, Hssaine Oummi, Ahmed Zeghal. Existence results for nonlinear mono-energetic singular transport equations: $ L^p $-spaces. Discrete and Continuous Dynamical Systems - S, 2022, 15 (1) : 179-195. doi: 10.3934/dcdss.2021028

[16]

Boya Li, Hongjie Ju, Yannan Liu. A flow method for a generalization of $ L_{p} $ Christofell-Minkowski problem. Communications on Pure and Applied Analysis, 2022, 21 (3) : 785-796. doi: 10.3934/cpaa.2021198

[17]

Anis Dhifaoui. $ L^p $-strong solution for the stationary exterior Stokes equations with Navier boundary condition. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1403-1420. doi: 10.3934/dcdss.2022086

[18]

Fang Liu. The eigenvalue problem for a class of degenerate operators related to the normalized $ p $-Laplacian. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2701-2720. doi: 10.3934/dcdsb.2021155

[19]

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

[20]

Antonio G. García. Sampling in $ \Lambda $-shift-invariant subspaces of Hilbert-Schmidt operators on $ L^2(\mathbb{R}^d) $. Mathematical Foundations of Computing, 2021, 4 (4) : 281-297. doi: 10.3934/mfc.2021019

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (232)
  • HTML views (99)
  • Cited by (0)

Other articles
by authors

[Back to Top]