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

Maximal functions of multipliers on compact manifolds without boundary

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.
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,, \emph{Invent. Math.}, 159 (2005), 187.  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,, \emph{Phase Space Analysis of Partial Differential Equations.} Vol. I, (2004), 21.   Google Scholar

[3]

N. Burq, P. Gérard and N. Tzvetkov, Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds,, \emph{Duke Math. J.}, 138 (2007), 445.  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,, \emph{Comment. Math. Helv.}, 60 (1985), 217.  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,, \emph{Trans. Amer. Math. Soc.}, 328 (1991), 73.  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,, \emph{Math. Z.}, 249 (2005), 223.   Google Scholar

[9]

C. Fefferman and E. M. Stein, Some maximal inequalities,, \emph{Amer. J. Math.}, 93 (1971), 107.   Google Scholar

[10]

G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups,, Mathematical Notes, 28 (1982).   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces,, \emph{Acta Math.}, 104 (1960), 93.   Google Scholar

[16]

G. Mauceri and S. Meda, Vector-valued multipliers on stratified groups,, \emph{Rev. Mat. Iberoamericana}, 6 (1990), 141.  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,, \emph{Duke Math. J.}, 59 (1989), 709.  doi: 10.1215/S0012-7094-89-05932-2.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

M. Taylor, Pseudo-differential Operators,, Princeton Univ. Press, (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,, \emph{Invent. Math.}, 159 (2005), 187.  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,, \emph{Phase Space Analysis of Partial Differential Equations.} Vol. I, (2004), 21.   Google Scholar

[3]

N. Burq, P. Gérard and N. Tzvetkov, Restrictions of the Laplace-Beltrami eigenfunctions to submanifolds,, \emph{Duke Math. J.}, 138 (2007), 445.  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,, \emph{Comment. Math. Helv.}, 60 (1985), 217.  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,, \emph{Trans. Amer. Math. Soc.}, 328 (1991), 73.  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,, \emph{Math. Z.}, 249 (2005), 223.   Google Scholar

[9]

C. Fefferman and E. M. Stein, Some maximal inequalities,, \emph{Amer. J. Math.}, 93 (1971), 107.   Google Scholar

[10]

G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups,, Mathematical Notes, 28 (1982).   Google Scholar

[11]

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

[12]

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

[13]

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

[14]

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

[15]

L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces,, \emph{Acta Math.}, 104 (1960), 93.   Google Scholar

[16]

G. Mauceri and S. Meda, Vector-valued multipliers on stratified groups,, \emph{Rev. Mat. Iberoamericana}, 6 (1990), 141.  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,, \emph{Duke Math. J.}, 59 (1989), 709.  doi: 10.1215/S0012-7094-89-05932-2.  Google Scholar

[18]

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

[19]

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

[20]

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

[21]

M. Taylor, Pseudo-differential Operators,, Princeton Univ. Press, (1981).   Google Scholar

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