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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 and 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.

[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.

[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.

[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.

[5]

W. Choi, Maximal functions for multipliers on stratified groups, Math. Nachr., 10.1002/mana.201300305 doi: 10.1002/mana.201300305.

[6]

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

[7]

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

[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.

[9]

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

[10]

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

[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.

[12]

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

[13]

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

[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.

[15]

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

[16]

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

[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.

[18]

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

[19]

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

[20]

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

[21]

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

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.

[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.

[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.

[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.

[5]

W. Choi, Maximal functions for multipliers on stratified groups, Math. Nachr., 10.1002/mana.201300305 doi: 10.1002/mana.201300305.

[6]

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

[7]

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

[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.

[9]

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

[10]

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

[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.

[12]

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

[13]

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

[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.

[15]

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

[16]

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

[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.

[18]

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

[19]

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

[20]

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

[21]

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

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