The boundedness of multi-linear and multi-parameter pseudo-differential operators

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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

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 and 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.
[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. 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.
[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. 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.
[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. Muscalu, J. Pipher, T. Tao and C. Thiele, Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296. doi: 10.1007/BF02392566.
[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.
[22]	C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis II, Cambridge Univ. Press, 2013.

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.
[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. 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.
[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. 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.
[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. Muscalu, J. Pipher, T. Tao and C. Thiele, Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296. doi: 10.1007/BF02392566.
[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.
[22]	C. Muscalu and W. Schlag, Classical and Multilinear Harmonic Analysis II, Cambridge Univ. Press, 2013.

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