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May  2017, 16(3): 883-898. doi: 10.3934/cpaa.2017042

$L^p$ boundedness for maximal functions associated with multi-linear pseudo-differential operators

JIAO CHEN 1, , WEI DAI 2, and  GUOZHEN LU 3,,

1. 

School of Mathematical Sciences, Beijing Normal University, Beijing, 100875, China
2. 

School of Mathematics and Systems Science, Beihang University (BUAA), Beijing, 100191, China
3. 

Department of Mathematics, University of Connecticut, Storrs, CT 06269, USA

* Corresponding author: Guozhen Lu

Received  July 2016 Revised  December 2016 Published  February 2017

Fund Project: The first two authors were partly supported by the NNSF of China (No. 11371056), the second author was also supported by the NNSF of China (No. 11501021), the third author was partly supported by a US NSF grant and a Simons Fellowship from the Simons Foundation.

In this paper, we establish the $L^p$ estimates for the maximal functions associated with the multilinear pseudo-differential operators. Our main result is Theorem 1.2. There are several major different ingredients and extra difficulties in our proof from those in Grafakos, Honzík and Seeger [15] and Honzík [22] for maximal functions generated by multipliers. First, in order to eliminate the variable $x$ in the symbols, we adapt a non-smooth modification of the smooth localization method developed by Muscalu in [26,30]. Then, by applying the inhomogeneous Littlewood-Paley dyadic decomposition and a discretization procedure, we can reduce the proof of Theorem 1.2 into proving the localized estimates for localized maximal functions generated by discrete paraproducts. The non-smooth cut-off functions in the localization procedure will be essential in establishing localized estimates. Finally, by proving a key localized square function estimate (Lemma 4.3) and applying the good-$\lambda$ inequality, we can derive the desired localized estimates.

Keywords: Multi-linear pseudo-differential operators, $L^p$ estimates, maximal functions, paraproducts.
Mathematics Subject Classification: Primary: 35S05; Secondary: 42B15, 42B20.
Citation: 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
References:
[1]

Á. Bényi, D. Maldonado, V. Naibo and H. Torres, On the Hörmander classes of bilinear pseudodifferential operators, Integr. Equ. oper. Theory, 67 (2010), 341-364.  doi: 10.1007/s00020-010-1782-y.  Google Scholar

[2]

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

[3]

D. L. Burkholder, Distribution function inequalities for martingales, Ann. Prob., 1 (1973), 19-42.   Google Scholar

[4]

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

[5]

M. Christ, L. Grafakos, P. Honzík and A. Seeger, Maximal functions associated with multipliers of Mikhlin-Hörmander type, Math. Zeit., 249 (2005), 223-240.  doi: 10.1007/s00209-004-0698-0.  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]

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

[8]

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

[9]

R. Coifman and Y. Meyer, Au delá des opérateurs pseudo-différentiels, Astérisque, 57 (1978).   Google Scholar

[10]

S. Y. A. Chang, M. Wilson and T. Wolff, Some weighted norm inequalities concerning the Schrödinger operator, Comment. Math. Helv., 60 (1985), 217-246.  doi: 10.1007/BF02567411.  Google Scholar

[11]

J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math., vol. 29, Amer. Math. Soc., Providence, RI, (2001).   Google Scholar

[12]

W. Dai and G. Lu, Lp estimates for multi-linear and multi-parameter pseudo-differential operators, Bull. Soc. Math. France, 143 (2015), 567-597.   Google Scholar

[13]

H. Dappa and W. Trebels, On maximal functions generated by Fourier multipliers, Ark. Mat., 23 (1985), 241-259.  doi: 10.1007/BF02384428.  Google Scholar

[14]

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

[15]

L. Grafakos, P. Honzík and A. Seeger, On maximal functions for Mikhlin-Hörmander multipliers, Adv. Math., 204 (2006), 363-378.  doi: 10.1016/j.aim.2005.05.010.  Google Scholar

[16]

L. Grafakos and N. J. Kalton, The Marcinkiewicz multiplier condition for bilinear operators, Studia Math., 146 (2001), 115-156.  doi: 10.4064/sm146-2-2.  Google Scholar

[17]

L. Grafakos and T. Tao, Multilinear interpolation between adjoint operators, J. Funct. Anal., 199 (2003), 379-385.  doi: 10.1016/S0022-1236(02)00098-8.  Google Scholar

[18]

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

[19]

Q. Hong and G. Lu, Symbolic calculus and boundedness of multi-parameter and multi-linear pseudo-differential operators, Adv. Nonlinear Stud., 14 (2014), 1055-1082.   Google Scholar

[20]

Q. Hong and L. Zhang, Lp estimates for bi-parameter and bilinear Fourier integral operators, Acta Mathematica Sinica -English Series, (2016).  doi: 10.1007/s10114-016-6269-6.  Google Scholar

[21]

L. Hörmander, Estimates for translation invariant operators in Lp spaces, Acta Math., 104 (1960), 93-140.  doi: 10.1007/BF02547187.  Google Scholar

[22]

P. Honzík, Maximal functions of multilinear multipliers, Math. Res. Lett., 16 (2009), 995-1006.  doi: 10.4310/MRL.2009.v16.n6.a7.  Google Scholar

[23]

C. E. 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

[24]

G. Lu and L. Zhang, Lp estimates for a trilinear pseudo-differential operator with flag symbols, Indiana University Mathematics Journal, to appear. Google Scholar

[25]

G. Lu and P. Zhang, Multilinear Calderón-Zygmund operators with kernels of Dinios type and applications, Nonlinear Anal., 107 (2014), 92-117.  doi: 10.1016/j.na.2014.05.005.  Google Scholar

[26]

C. Muscalu, Unpublished notes Ⅰ, IAS Princeton, 2003. Google Scholar

[27]

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

[28]

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

[29]

C. Muscalu, J. Pipher, T. Tao and C. Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam., 23 (2007), 705-742.  doi: 10.4171/RMI/480.  Google Scholar

[30] C. Muscalu and W. Schlag, Classical and multilinear Harmonic Analysis, Ⅱ, Cambridge Studies in Advanced Mathematics, vol. 138, Cambridge University Press, Cambridge, 2013.   Google Scholar
[31]

C. Muscalu, T. Tao and C. Thiele, Multilinear operators given by singular multipliers, J. Amer. Math. Soc., 15 (2002), 469-496.   Google Scholar

[32]

C. Muscalu, T. Tao and C. Thiele, Lp estimates for the biest Ⅱ, The Fourier case, Math. Ann., 329 (2004), 427-461.  doi: 10.1007/s00208-003-0508-8.  Google Scholar

[33] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Priceton Univ. Press, Princeton, NJ, 1970.   Google Scholar

show all references

References:
[1]

Á. Bényi, D. Maldonado, V. Naibo and H. Torres, On the Hörmander classes of bilinear pseudodifferential operators, Integr. Equ. oper. Theory, 67 (2010), 341-364.  doi: 10.1007/s00020-010-1782-y.  Google Scholar

[2]

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

[3]

D. L. Burkholder, Distribution function inequalities for martingales, Ann. Prob., 1 (1973), 19-42.   Google Scholar

[4]

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

[5]

M. Christ, L. Grafakos, P. Honzík and A. Seeger, Maximal functions associated with multipliers of Mikhlin-Hörmander type, Math. Zeit., 249 (2005), 223-240.  doi: 10.1007/s00209-004-0698-0.  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]

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

[8]

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

[9]

R. Coifman and Y. Meyer, Au delá des opérateurs pseudo-différentiels, Astérisque, 57 (1978).   Google Scholar

[10]

S. Y. A. Chang, M. Wilson and T. Wolff, Some weighted norm inequalities concerning the Schrödinger operator, Comment. Math. Helv., 60 (1985), 217-246.  doi: 10.1007/BF02567411.  Google Scholar

[11]

J. Duoandikoetxea, Fourier Analysis, Grad. Stud. Math., vol. 29, Amer. Math. Soc., Providence, RI, (2001).   Google Scholar

[12]

W. Dai and G. Lu, Lp estimates for multi-linear and multi-parameter pseudo-differential operators, Bull. Soc. Math. France, 143 (2015), 567-597.   Google Scholar

[13]

H. Dappa and W. Trebels, On maximal functions generated by Fourier multipliers, Ark. Mat., 23 (1985), 241-259.  doi: 10.1007/BF02384428.  Google Scholar

[14]

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

[15]

L. Grafakos, P. Honzík and A. Seeger, On maximal functions for Mikhlin-Hörmander multipliers, Adv. Math., 204 (2006), 363-378.  doi: 10.1016/j.aim.2005.05.010.  Google Scholar

[16]

L. Grafakos and N. J. Kalton, The Marcinkiewicz multiplier condition for bilinear operators, Studia Math., 146 (2001), 115-156.  doi: 10.4064/sm146-2-2.  Google Scholar

[17]

L. Grafakos and T. Tao, Multilinear interpolation between adjoint operators, J. Funct. Anal., 199 (2003), 379-385.  doi: 10.1016/S0022-1236(02)00098-8.  Google Scholar

[18]

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

[19]

Q. Hong and G. Lu, Symbolic calculus and boundedness of multi-parameter and multi-linear pseudo-differential operators, Adv. Nonlinear Stud., 14 (2014), 1055-1082.   Google Scholar

[20]

Q. Hong and L. Zhang, Lp estimates for bi-parameter and bilinear Fourier integral operators, Acta Mathematica Sinica -English Series, (2016).  doi: 10.1007/s10114-016-6269-6.  Google Scholar

[21]

L. Hörmander, Estimates for translation invariant operators in Lp spaces, Acta Math., 104 (1960), 93-140.  doi: 10.1007/BF02547187.  Google Scholar

[22]

P. Honzík, Maximal functions of multilinear multipliers, Math. Res. Lett., 16 (2009), 995-1006.  doi: 10.4310/MRL.2009.v16.n6.a7.  Google Scholar

[23]

C. E. 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

[24]

G. Lu and L. Zhang, Lp estimates for a trilinear pseudo-differential operator with flag symbols, Indiana University Mathematics Journal, to appear. Google Scholar

[25]

G. Lu and P. Zhang, Multilinear Calderón-Zygmund operators with kernels of Dinios type and applications, Nonlinear Anal., 107 (2014), 92-117.  doi: 10.1016/j.na.2014.05.005.  Google Scholar

[26]

C. Muscalu, Unpublished notes Ⅰ, IAS Princeton, 2003. Google Scholar

[27]

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

[28]

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

[29]

C. Muscalu, J. Pipher, T. Tao and C. Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam., 23 (2007), 705-742.  doi: 10.4171/RMI/480.  Google Scholar

[30] C. Muscalu and W. Schlag, Classical and multilinear Harmonic Analysis, Ⅱ, Cambridge Studies in Advanced Mathematics, vol. 138, Cambridge University Press, Cambridge, 2013.   Google Scholar
[31]

C. Muscalu, T. Tao and C. Thiele, Multilinear operators given by singular multipliers, J. Amer. Math. Soc., 15 (2002), 469-496.   Google Scholar

[32]

C. Muscalu, T. Tao and C. Thiele, Lp estimates for the biest Ⅱ, The Fourier case, Math. Ann., 329 (2004), 427-461.  doi: 10.1007/s00208-003-0508-8.  Google Scholar

[33] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Priceton Univ. Press, Princeton, NJ, 1970.   Google Scholar
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