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

[21]

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

[21]

C. MuscaluJ. PipherT. 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
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