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

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

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.

Citation: 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
References:
[1]

Á. BényiD. MaldonadoV. 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.

[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.

[3]

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

[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.

[5]

M. ChristL. GrafakosP. 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.

[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]

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.

[8]

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

[9]

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

[10]

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

[11]

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

[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. 

[13]

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

[14]

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

[15]

L. GrafakosP. 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.

[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.

[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.

[18]

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

[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. 

[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.

[21]

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

[22]

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

[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.

[24]

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

[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.

[26]

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

[27]

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

[28]

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

[29]

C. MuscaluJ. PipherT. Tao and C. Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam., 23 (2007), 705-742.  doi: 10.4171/RMI/480.

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

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

[32]

C. MuscaluT. 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.

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

show all references

References:
[1]

Á. BényiD. MaldonadoV. 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.

[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.

[3]

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

[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.

[5]

M. ChristL. GrafakosP. 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.

[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]

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.

[8]

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

[9]

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

[10]

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

[11]

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

[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. 

[13]

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

[14]

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

[15]

L. GrafakosP. 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.

[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.

[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.

[18]

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

[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. 

[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.

[21]

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

[22]

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

[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.

[24]

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

[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.

[26]

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

[27]

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

[28]

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

[29]

C. MuscaluJ. PipherT. Tao and C. Thiele, Multi-parameter paraproducts, Rev. Mat. Iberoam., 23 (2007), 705-742.  doi: 10.4171/RMI/480.

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

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

[32]

C. MuscaluT. 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.

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