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  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]

Jong Yoon Hyun, Yoonjin Lee, Yansheng Wu. Connection of $ p $-ary $ t $-weight linear codes to Ramanujan Cayley graphs with $ t+1 $ eigenvalues. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020133
[2]

Lei Liu, Li Wu. Multiplicity of closed characteristics on $ P $-symmetric compact convex hypersurfaces in $ \mathbb{R}^{2n} $. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020378
[3]

Chaoqian Li, Yajun Liu, Yaotang Li. Note on $ Z $-eigenvalue inclusion theorems for tensors. Journal of Industrial & Management Optimization, 2021, 17 (2) : 687-693. doi: 10.3934/jimo.2019129
[4]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113
[5]

Emma D'Aniello, Saber Elaydi. The structure of $ \omega $-limit sets of asymptotically non-autonomous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 903-915. doi: 10.3934/dcdsb.2019195
[6]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1631-1648. doi: 10.3934/dcdss.2020447
[7]

Raj Kumar, Maheshanand Bhaintwal. Duadic codes over $ \mathbb{Z}_4+u\mathbb{Z}_4 $. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020135
[8]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[9]

Ondrej Budáč, Michael Herrmann, Barbara Niethammer, Andrej Spielmann. On a model for mass aggregation with maximal size. Kinetic & Related Models, 2011, 4 (2) : 427-439. doi: 10.3934/krm.2011.4.427
[10]

Braxton Osting, Jérôme Darbon, Stanley Osher. Statistical ranking using the $l^{1}$-norm on graphs. Inverse Problems & Imaging, 2013, 7 (3) : 907-926. doi: 10.3934/ipi.2013.7.907
[11]

Horst R. Thieme. Remarks on resolvent positive operators and their perturbation. Discrete & Continuous Dynamical Systems - A, 1998, 4 (1) : 73-90. doi: 10.3934/dcds.1998.4.73
[12]

Jiaquan Liu, Xiangqing Liu, Zhi-Qiang Wang. Sign-changing solutions for a parameter-dependent quasilinear equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1779-1799. doi: 10.3934/dcdss.2020454
[13]

Sara Munday. On the derivative of the $\alpha$-Farey-Minkowski function. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 709-732. doi: 10.3934/dcds.2014.34.709
[14]

Hakan Özadam, Ferruh Özbudak. A note on negacyclic and cyclic codes of length $p^s$ over a finite field of characteristic $p$. Advances in Mathematics of Communications, 2009, 3 (3) : 265-271. doi: 10.3934/amc.2009.3.265
[15]

Wei Liu, Pavel Krejčí, Guoju Ye. Continuity properties of Prandtl-Ishlinskii operators in the space of regulated functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (10) : 3783-3795. doi: 10.3934/dcdsb.2017190
[16]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009
[17]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25
[18]

Ralf Hielscher, Michael Quellmalz. Reconstructing a function on the sphere from its means along vertical slices. Inverse Problems & Imaging, 2016, 10 (3) : 711-739. doi: 10.3934/ipi.2016018
[19]

Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181
[20]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (46)
  • HTML views (68)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences