American Institute of Mathematical Sciences

Advanced
September  2015, 14(5): 1885-1902. doi: 10.3934/cpaa.2015.14.1885

Maximal functions of multipliers on compact manifolds without boundary

Woocheol Choi 1,

1. 

Korea Institute for Advanced Study School of Mathematics, Hoegiro 85, Dongdaemun-gu, Seoul 130-722, South Korea

Received  October 2014 Revised  March 2015 Published  June 2015

Let $P$ be a self-adjoint positive elliptic (-pseudo) differential operator on a smooth compact manifold $M$ without boundary. In this paper, we obtain a refined $L^p$ bound of the maximal function of the multiplier operators associated to $P$ satisfying the Hörmander-Mikhlin condition.
Keywords: the Hörmander-Mikhlin condition, compact manifold., Maximal multipliers.
Mathematics Subject Classification: Primary: 58J40, 42B1.
Citation: Woocheol Choi. Maximal functions of multipliers on compact manifolds without boundary. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1885-1902. doi: 10.3934/cpaa.2015.14.1885
References:
[1]

N. Burq, P. Gérard and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math., 159 (2005), 187-223. doi: 10.1007/s00222-004-0388-x.  Google Scholar

[2]

N. Burq, P. Gérard and N. Tzvetkov, The Cauchy problem for the nonlinear Schrödinger equation on compact manifolds, Phase Space Analysis of Partial Differential Equations. Vol. I, 21-52, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004.  Google Scholar

[3]

N. Burq, P. Gérard and N. Tzvetkov, Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J., 138 (2007), 445-486. doi: 10.1215/S0012-7094-07-13834-1.  Google Scholar

[4]

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

[5]

W. Choi, Maximal functions for multipliers on stratified groups,, \emph{Math. Nachr.}, ().  doi: 10.1002/mana.201300305.  Google Scholar

[6]

M. Christ, $L^p$ bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc., 328 (1991), 73-81. doi: 10.2307/2001877.  Google Scholar

[7]

M. Christ, Lectures on Singular Integral Operators, CBMS Regional Conference Series in Mathematics, 77, 1990.  Google Scholar

[8]

M. Christ, L. Grafakos, P. Honzik and A. Seeger, Maximal functions associated with Fourier multipliers of Mikhlin-Hörmander type, Math. Z., 249 (2005), 223-240. Google Scholar

[9]

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

[10]

G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes, 28 Princeton University Press.; University of Tokyo Press, 1982.  Google Scholar

[11]

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

[12]

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

[13]

P. Honzik, Maximal Marcinkiewicz multipliers, Ark. Mat., 52 (2014), 135-147. doi: 10.1007/s11512-013-0189-9.  Google Scholar

[14]

A. Hassell and M. Tacy, Semiclassical $L^p$ estimates of quasimodes on curved hypersurfaces, J. Geom. Anal., 22 (2012), 74-89. doi: 10.1007/s12220-010-9191-7.  Google Scholar

[15]

L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math., 104 (1960), 93-139.  Google Scholar

[16]

G. Mauceri and S. Meda, Vector-valued multipliers on stratified groups, Rev. Mat. Iberoamericana, 6 (1990), 141-154. doi: 10.4171/RMI/100.  Google Scholar

[17]

A. Seeger and C. D. Sogge, On the boundedness of functions of (pseudo-) differential operators on compact manifolds, Duke Math. J., 59 (1989), 709-736. doi: 10.1215/S0012-7094-89-05932-2.  Google Scholar

[18]

C. D. Sogge, On the convergence of Riesz means on compact manifolds, Ann. of Math., 126 (1987), 439-447. doi: 10.2307/1971356.  Google Scholar

[19]

C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511530029.  Google Scholar

[20]

M. Tacy, Semiclassical $L^p$ estimates of quasimodes on submanifolds, Comm. Partial Differential Equations, 35 (2010), 1538-1562. doi: 10.1080/03605301003611006.  Google Scholar

[21]

M. Taylor, Pseudo-differential Operators, Princeton Univ. Press, Princeton N.J., 1981.  Google Scholar

show all references

References:
[1]

N. Burq, P. Gérard and N. Tzvetkov, Bilinear eigenfunction estimates and the nonlinear Schrödinger equation on surfaces, Invent. Math., 159 (2005), 187-223. doi: 10.1007/s00222-004-0388-x.  Google Scholar

[2]

N. Burq, P. Gérard and N. Tzvetkov, The Cauchy problem for the nonlinear Schrödinger equation on compact manifolds, Phase Space Analysis of Partial Differential Equations. Vol. I, 21-52, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Norm. Sup., Pisa, 2004.  Google Scholar

[3]

N. Burq, P. Gérard and N. Tzvetkov, Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds, Duke Math. J., 138 (2007), 445-486. doi: 10.1215/S0012-7094-07-13834-1.  Google Scholar

[4]

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

[5]

W. Choi, Maximal functions for multipliers on stratified groups,, \emph{Math. Nachr.}, ().  doi: 10.1002/mana.201300305.  Google Scholar

[6]

M. Christ, $L^p$ bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc., 328 (1991), 73-81. doi: 10.2307/2001877.  Google Scholar

[7]

M. Christ, Lectures on Singular Integral Operators, CBMS Regional Conference Series in Mathematics, 77, 1990.  Google Scholar

[8]

M. Christ, L. Grafakos, P. Honzik and A. Seeger, Maximal functions associated with Fourier multipliers of Mikhlin-Hörmander type, Math. Z., 249 (2005), 223-240. Google Scholar

[9]

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

[10]

G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Mathematical Notes, 28 Princeton University Press.; University of Tokyo Press, 1982.  Google Scholar

[11]

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

[12]

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

[13]

P. Honzik, Maximal Marcinkiewicz multipliers, Ark. Mat., 52 (2014), 135-147. doi: 10.1007/s11512-013-0189-9.  Google Scholar

[14]

A. Hassell and M. Tacy, Semiclassical $L^p$ estimates of quasimodes on curved hypersurfaces, J. Geom. Anal., 22 (2012), 74-89. doi: 10.1007/s12220-010-9191-7.  Google Scholar

[15]

L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math., 104 (1960), 93-139.  Google Scholar

[16]

G. Mauceri and S. Meda, Vector-valued multipliers on stratified groups, Rev. Mat. Iberoamericana, 6 (1990), 141-154. doi: 10.4171/RMI/100.  Google Scholar

[17]

A. Seeger and C. D. Sogge, On the boundedness of functions of (pseudo-) differential operators on compact manifolds, Duke Math. J., 59 (1989), 709-736. doi: 10.1215/S0012-7094-89-05932-2.  Google Scholar

[18]

C. D. Sogge, On the convergence of Riesz means on compact manifolds, Ann. of Math., 126 (1987), 439-447. doi: 10.2307/1971356.  Google Scholar

[19]

C. D. Sogge, Fourier Integrals in Classical Analysis, Cambridge Tracts in Mathematics, 105. Cambridge University Press, Cambridge, 1993. doi: 10.1017/CBO9780511530029.  Google Scholar

[20]

M. Tacy, Semiclassical $L^p$ estimates of quasimodes on submanifolds, Comm. Partial Differential Equations, 35 (2010), 1538-1562. doi: 10.1080/03605301003611006.  Google Scholar

[21]

M. Taylor, Pseudo-differential Operators, Princeton Univ. Press, Princeton N.J., 1981.  Google Scholar

[1]

María José Beltrán, José Bonet, Carmen Fernández. Classical operators on the Hörmander algebras. Discrete & Continuous Dynamical Systems, 2015, 35 (2) : 637-652. doi: 10.3934/dcds.2015.35.637
[2]

Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations & Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391
[3]

Valerii Los, Vladimir A. Mikhailets, Aleksandr A. Murach. An isomorphism theorem for parabolic problems in Hörmander spaces and its applications. Communications on Pure & Applied Analysis, 2017, 16 (1) : 69-98. doi: 10.3934/cpaa.2017003
[4]

Aylin Aydoğdu, Sean T. McQuade, Nastassia Pouradier Duteil. Opinion Dynamics on a General Compact Riemannian Manifold. Networks & Heterogeneous Media, 2017, 12 (3) : 489-523. doi: 10.3934/nhm.2017021
[5]

Andrea Bonfiglioli, Ermanno Lanconelli. Lie groups related to Hörmander operators and Kolmogorov-Fokker-Planck equations. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1587-1614. doi: 10.3934/cpaa.2012.11.1587
[6]

César Augusto Bortot, Wellington José Corrêa, Ryuichi Fukuoka, Thales Maier Souza. Exponential stability for the locally damped defocusing Schrödinger equation on compact manifold. Communications on Pure & Applied Analysis, 2020, 19 (3) : 1367-1386. doi: 10.3934/cpaa.2020067
[7]

Saikat Mazumdar. Struwe's decomposition for a polyharmonic operator on a compact Riemannian manifold with or without boundary. Communications on Pure & Applied Analysis, 2017, 16 (1) : 311-330. doi: 10.3934/cpaa.2017015
[8]

Matti Lassas, Teemu Saksala, Hanming Zhou. Reconstruction of a compact manifold from the scattering data of internal sources. Inverse Problems & Imaging, 2018, 12 (4) : 993-1031. doi: 10.3934/ipi.2018042
[9]

Shengbing Deng, Zied Khemiri, Fethi Mahmoudi. On spike solutions for a singularly perturbed problem in a compact riemannian manifold. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2063-2084. doi: 10.3934/cpaa.2018098
[10]

Erwann Delay, Pieralberto Sicbaldi. Extremal domains for the first eigenvalue in a general compact Riemannian manifold. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5799-5825. doi: 10.3934/dcds.2015.35.5799
[11]

Martin Pinsonnault. Maximal compact tori in the Hamiltonian group of 4-dimensional symplectic manifolds. Journal of Modern Dynamics, 2008, 2 (3) : 431-455. doi: 10.3934/jmd.2008.2.431
[12]

Russell Ricks. The unique measure of maximal entropy for a compact rank one locally CAT(0) space. Discrete & Continuous Dynamical Systems, 2021, 41 (2) : 507-523. doi: 10.3934/dcds.2020266
[13]

Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4353-4390. doi: 10.3934/dcds.2018190
[14]

Angelo Favini, Rabah Labbas, Stéphane Maingot, Hiroki Tanabe, Atsushi Yagi. Necessary and sufficient conditions for maximal regularity in the study of elliptic differential equations in Hölder spaces. Discrete & Continuous Dynamical Systems, 2008, 22 (4) : 973-987. doi: 10.3934/dcds.2008.22.973
[15]

Haiyang He. Asymptotic behavior of the ground state Solutions for Hénon equation with Robin boundary condition. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2393-2408. doi: 10.3934/cpaa.2013.12.2393
[16]

Karla L. Cortez, Javier F. Rosenblueth. Normality and uniqueness of Lagrange multipliers. Discrete & Continuous Dynamical Systems, 2018, 38 (6) : 3169-3188. doi: 10.3934/dcds.2018138
[17]

E. Camouzis, H. Kollias, I. Leventides. Stable manifold market sequences. Journal of Dynamics & Games, 2018, 5 (2) : 165-185. doi: 10.3934/jdg.2018010
[18]

Camillo De Lellis, Emanuele Spadaro. Center manifold: A case study. Discrete & Continuous Dynamical Systems, 2011, 31 (4) : 1249-1272. doi: 10.3934/dcds.2011.31.1249
[19]

Zhiguo Feng, Ka-Fai Cedric Yiu. Manifold relaxations for integer programming. Journal of Industrial & Management Optimization, 2014, 10 (2) : 557-566. doi: 10.3934/jimo.2014.10.557
[20]

Noriaki Kawaguchi. Maximal chain continuous factor. Discrete & Continuous Dynamical Systems, 2021, 41 (12) : 5915-5942. doi: 10.3934/dcds.2021101

2020 Impact Factor: 1.916

Tools

Metrics

  • PDF downloads (51)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences