-
Previous Article
Weighted lorentz estimates for nondivergence linear elliptic equations with partially BMO coefficients
- CPAA Home
- This Issue
-
Next Article
On weighted mixed-norm Sobolev estimates for some basic parabolic equations
$L^p$ 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 |
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 [
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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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] | |
[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. Muscalu, J. Pipher, T. Tao and C. Thiele,
Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296.
doi: 10.1007/BF02392566. |
[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. |
[30] |
C. Muscalu and W. Schlag, Classical and multilinear Harmonic Analysis, Ⅱ, Cambridge Studies in Advanced Mathematics, vol. 138, Cambridge University Press, Cambridge, 2013.
![]() ![]() |
[31] |
C. Muscalu, T. Tao and C. Thiele,
Multilinear operators given by singular multipliers, J. Amer. Math. Soc., 15 (2002), 469-496.
|
[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. |
[33] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Priceton Univ. Press, Princeton, NJ, 1970.
![]() ![]() |
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. |
[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. 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. |
[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. 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. |
[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. 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. |
[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] | |
[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. Muscalu, J. Pipher, T. Tao and C. Thiele,
Bi-parameter paraproducts, Acta Math., 193 (2004), 269-296.
doi: 10.1007/BF02392566. |
[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. |
[30] |
C. Muscalu and W. Schlag, Classical and multilinear Harmonic Analysis, Ⅱ, Cambridge Studies in Advanced Mathematics, vol. 138, Cambridge University Press, Cambridge, 2013.
![]() ![]() |
[31] |
C. Muscalu, T. Tao and C. Thiele,
Multilinear operators given by singular multipliers, J. Amer. Math. Soc., 15 (2002), 469-496.
|
[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. |
[33] |
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Priceton Univ. Press, Princeton, NJ, 1970.
![]() ![]() |
[1] |
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 |
[2] |
Ildoo Kim. An $L_p$-Lipschitz theory for parabolic equations with time measurable pseudo-differential operators. Communications on Pure and Applied Analysis, 2018, 17 (6) : 2751-2771. doi: 10.3934/cpaa.2018130 |
[3] |
Lanzhe Liu. Mean oscillation and boundedness of Toeplitz Type operators associated to pseudo-differential operators. Communications on Pure and Applied Analysis, 2015, 14 (2) : 627-636. doi: 10.3934/cpaa.2015.14.627 |
[4] |
Thomas Kappeler, Riccardo Montalto. Normal form coordinates for the Benjamin-Ono equation having expansions in terms of pseudo-differential operators. Discrete and Continuous Dynamical Systems, 2022 doi: 10.3934/dcds.2022048 |
[5] |
N. V. Krylov. Some $L_{p}$-estimates for elliptic and parabolic operators with measurable coefficients. Discrete and Continuous Dynamical Systems - B, 2012, 17 (6) : 2073-2090. doi: 10.3934/dcdsb.2012.17.2073 |
[6] |
Simona Fornaro, Giorgio Metafune, Diego Pallara, Roland Schnaubelt. Multi-dimensional degenerate operators in $L^p$-spaces. Communications on Pure and Applied Analysis, 2022, 21 (6) : 2115-2145. doi: 10.3934/cpaa.2022052 |
[7] |
Dinh Nguyen Duy Hai. Hölder-Logarithmic type approximation for nonlinear backward parabolic equations connected with a pseudo-differential operator. Communications on Pure and Applied Analysis, 2022, 21 (5) : 1715-1734. doi: 10.3934/cpaa.2022043 |
[8] |
İsmail Aslan, Türkan Yeliz Gökçer. Approximation by pseudo-linear discrete operators. Mathematical Foundations of Computing, 2021 doi: 10.3934/mfc.2021037 |
[9] |
Giuseppe Da Prato, Alessandra Lunardi. Maximal dissipativity of a class of elliptic degenerate operators in weighted $L^2$ spaces. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 751-760. doi: 10.3934/dcdsb.2006.6.751 |
[10] |
Samer Dweik. $ L^{p, q} $ estimates on the transport density. Communications on Pure and Applied Analysis, 2019, 18 (6) : 3001-3009. doi: 10.3934/cpaa.2019134 |
[11] |
Marta García-Huidobro, Raul Manásevich, J. R. Ward. Vector p-Laplacian like operators, pseudo-eigenvalues, and bifurcation. Discrete and Continuous Dynamical Systems, 2007, 19 (2) : 299-321. doi: 10.3934/dcds.2007.19.299 |
[12] |
Wolfgang Arendt, Patrick J. Rabier. Linear evolution operators on spaces of periodic functions. Communications on Pure and Applied Analysis, 2009, 8 (1) : 5-36. doi: 10.3934/cpaa.2009.8.5 |
[13] |
Fabrice Planchon, John G. Stalker, A. Shadi Tahvildar-Zadeh. $L^p$ Estimates for the wave equation with the inverse-square potential. Discrete and Continuous Dynamical Systems, 2003, 9 (2) : 427-442. doi: 10.3934/dcds.2003.9.427 |
[14] |
Seung-Yeal Ha, Mitsuru Yamazaki. $L^p$-stability estimates for the spatially inhomogeneous discrete velocity Boltzmann model. Discrete and Continuous Dynamical Systems - B, 2009, 11 (2) : 353-364. doi: 10.3934/dcdsb.2009.11.353 |
[15] |
Antonio Vitolo. $H^{1,p}$-eigenvalues and $L^\infty$-estimates in quasicylindrical domains. Communications on Pure and Applied Analysis, 2011, 10 (5) : 1315-1329. doi: 10.3934/cpaa.2011.10.1315 |
[16] |
Peter Weidemaier. Maximal regularity for parabolic equations with inhomogeneous boundary conditions in Sobolev spaces with mixed $L_p$-norm. Electronic Research Announcements, 2002, 8: 47-51. |
[17] |
Dalila Azzam-Laouir, Warda Belhoula, Charles Castaing, M. D. P. Monteiro Marques. Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators. Evolution Equations and Control Theory, 2020, 9 (1) : 219-254. doi: 10.3934/eect.2020004 |
[18] |
Karen Yagdjian, Anahit Galstian. Fundamental solutions for wave equation in Robertson-Walker model of universe and $L^p-L^q$ -decay estimates. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 483-502. doi: 10.3934/dcdss.2009.2.483 |
[19] |
Masahiro Ikeda, Takahisa Inui, Mamoru Okamoto, Yuta Wakasugi. $ L^p $-$ L^q $ estimates for the damped wave equation and the critical exponent for the nonlinear problem with slowly decaying data. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1967-2008. doi: 10.3934/cpaa.2019090 |
[20] |
Tadeusz Iwaniec, Gaven Martin, Carlo Sbordone. $L^p$-integrability & weak type $L^{2}$-estimates for the gradient of harmonic mappings of $\mathbb D$. Discrete and Continuous Dynamical Systems - B, 2009, 11 (1) : 145-152. doi: 10.3934/dcdsb.2009.11.145 |
2020 Impact Factor: 1.916
Tools
Metrics
Other articles
by authors
[Back to Top]