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Estimations des fonctions du Laplacien sur les variétés cuspidales
Author:
Hong-Quan Li
Journal:
Trans. Amer. Math. Soc. 357 (2005), 337-354
MSC (2000):
Primary 42B15, 42B20
Posted:
August 24, 2004
MathSciNet review:
2098098
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
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Abstract: Le but de cet article est d'étudier la continuité des fonctions du Laplacien sur les variétés cuspidales.
- 1.
G. Alexopoulos, Spectral multipliers on Lie groups of polynomial growth, Proc. Amer. Math. Soc. 120 (1994) 973-979. MR 1172944 (95j:22016)
- 2.
G. Alexopoulos,
bounds for spectral multipliers from Gaussian estimates of the heat kernel, Prépublication 1999.
- 3.
G. Alexopoulos, Spectral multipliers on discrete groups, Bull. London Math. Soc. 33 (2001) 417-424. MR 1832553 (2002d:22010)
- 4.
G. Alexopoulos, Spectral multipliers for Markov chains. A paraître au J. Math. Soc. Japan.
- 5.
J.-P. Anker, N. Lohoué, Multiplicateurs sur certains espaces symétriques, Amer. J. Math. 108 (1986) 1303-1354. MR 0868894 (88c:43008)
- 6.
J.-P. Anker,
Fourier multipliers on Riemannian symmetric spaces of the non-compact type, Ann. of Math. 132 (1990) 597-628. MR 1078270 (92e:43006)
- 7.
J.-P. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Mathematical Journal 65 (1992) 257-297. MR 1150587 (93b:43007)
- 8.
J.-P. Anker, E. Damek, C. Yacoub, Spherical Analysis on Harmonic
Groups, Ann. Sc. Norm. Super. Pisa. Cl. Sci. IV. Sen. 23 (1996) 643-679. MR 1469569 (99a:22014)
- 9.
F. Astengo, Multipliers for a distinguished Laplacean on solvable extensions of H-type groups, Monatsh. Math. 120 (1995) 179-188.MR 1363136 (97c:22012)
- 10.
P. Bérard, On the Wave Equation on Riemannian Manifold without Conjugate Points, Math. Z., 155 (1977) 249-276. MR 0455055 (56:13295)
- 11.
J. Cheeger, M. Gromov, M. E. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982) 15-53.MR 0658471 (84b:58109)
- 12.
J. Cheeger, M. E. Taylor, On the diffraction of wave by conical singularities. I, Comm. Pure Appl. Math. XXV (1982) 275-331.MR 0649347 (84h:35091a)
- 13.
J. Clerc, E. M. Stein,
-multipliers for noncompact symmertric spaces, Proc. Nat. Acad. Sci. U.S.A. 71 (1974) 3911-3912. MR 0367561 (51:3803)
- 14.
M. Christ,
bound for spectral multiplier on nilpotent groups, Trans. Amer. Math. Soc. 328 (1991) 73-81. MR 1104196 (92k:42017)
- 15.
M. Christ, D. Müller, On
spectral multipliers for a solvable Lie group, Geom. Funct. Anal. 6 (1996) 860-876. MR 1415763 (98b:22017)
- 16.
R. Coifman, G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Springer L.N.
242, 1971. MR 0499948 (58:17690)
- 17.
M. Cowling, S. Giulini, A. Hulanicki, G. Mauceri, Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth, Studia Math. 111 (1994) 103-121.MR 1301761 (95j:43002)
- 18.
M. Cowling, A. Sikora, A spectral multiplier theorem for a sublaplacian on SU(2), Math. Z. 238 (2001) 1-36. MR 1860734 (2002h:42016)
- 19.
L. De Michèle, G. Mauceri,
-multipliers on the Heisenberg group, Michigan Math. J., 26 (1979) 361-373. MR 0544603 (80j:43003b)
- 20.
L. De Michèle, G. Mauceri,
multipliers on stratified groups, Ann. Mat. Pura Appl. 148 (1987) 353-366. MR 0932770 (89e:43009)
- 21.
X.-T. Duong, E. M. Ouhabaz, Heat kernel bounds and spectral multipliers on spaces of polynomial volume growth and irregular domains. Prépublication 1999.
- 22.
X.-T. Duong, E. M. Ouhabaz, A. Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal. 196 (2002) 443-485.MR 1943098 (2003k:43012)
- 23.
J. García-Cuerva, G. Mauceri, P. Sjögren, J.-L. Torrea, Spectral multipliers for the Ornstein-Uhlenbeck semigroup, J. Anal. Math. 78 (1999) 281-305. MR 1714425 (2000m:42015)
- 24.
J. García-Cuerva, G. Mauceri, S. Meda, P. Sjögren, J.-L. Torrea, Functional calculus for the Ornstein-Uhlenbeck operator, J. Funct. Anal. 183 (2001) 413-450. MR 1844213 (2002d:47023)
- 25.
I. M. Gelfand, G. Shilov, Generalized Functions I, Academic Press, New York and London, 1964. MR 0166596 (29:3869)
- 26.
S. Giulini, G. Mauceri, S. Meda,
multipliers on noncompact symmetric spaces, J. Reine Angew. Math. 482 (1997) 151-175.MR 1427660 (98g:43006)
- 27.
M. Gnewuch, Spectral Multipliers for sub-Laplacians on amenable Lie groups with exponential volume growth, Math. Z. 246 (2004) 69-83.MR 2031445
- 28.
W. Hebisch, The subalgebra of
generated by the laplacean, Proc. Amer. Math. Soc. 117 (1993) 547-549. MR 1111218 (93d:22009)
- 29.
W. Hebisch, Multiplier theorem on generalized Heisenberg groups, Coll. Math. 65 (1993) 231-239. MR 1240169 (94m:43013)
- 30.
W. Hebisch, J. Zienkiewicz, Multiplier theorem on generalized Heisenberg groups II, Coll. Math. 69 (1995) 29-36. MR 1341678 (96h:43007)
- 31.
W. Hebisch, Boundedness of
spectral multipliers for an exponential solvable Lie group, Coll. Math. 73 (1997) 155-164. MR 1436956 (98e:22005)
- 32.
W. Hebisch, Spectral multipliers on exponential growth solvable Lie groups, Math. Z. 229 (1998) 435-441. MR 1658573 (2000a:43004)
- 33.
W. Hebisch, Spectral multipliers on metabelian groups, Revista Math. Iberoamericana 16 (2000) 597-604. MR 1813330 (2001m:22016)
- 34.
W. Hebisch, T. Steger, Multipliers and singular integrals on exponential growth groups, Math. Z. 245 (2003) 37-61. MR 2023952
- 35.
L. Hörmander, Estimates for translation invariant operators in
spaces, Acta Math. 104 (1960) 93-139. MR 0121655 (22:12389)
- 36.
L. Hörmander, The Analysis of Linear Partial Differential Operator III : Pseudo - Differential Operators, Springer-Verlag, Berlin Heidelberg, 1985. MR 0781536 (87d:35002a)
- 37.
A. Hulanicki, Subalgebra of
associated with Laplacian on a Lie group, Coll. Math. 31 (1974) 259-287. MR 0372536 (51:8743)
- 38.
A. Ionescu, Singular integrals on symmetric spaces of real rank one, Duke Math. J. 114 (2002) 101-122. MR 1915037 (2003c:43008)
- 39.
A. Ionescu, Fourier integral operators on noncompact symmetric spaces of real rank one, J. Funct. Anal. 174 (2000) 274-300. MR 1767376 (2001h:43009)
- 40.
A. Ionescu, Singular integrals on symmetric spaces II, Trans. Amer. Math. Soc. 355 (2003) 3359-3378. MR 1974692 (2004b:43009)
- 41.
C. E. Kenig, R. J. Stanton, P. A. Tomas, Divergence of eigenfunction expansions, J. Funct. Anal. 46 (1982) 28-44. MR 0654463 (83k:22025)
- 42.
H.-Q. Li, Certaines fonctions maximales sur les variétés cuspidales. Prépublication de l'IHES, décembre 2002.
- 43.
H.-Q. Li, Estimations
des fonctions du Laplacien sur les variétés cuspidales I. Prépublication de l'IHES, décembre 2002.
- 44.
H.-Q. Li, Analyse sur les variétés cuspidales, Math. Ann. 326 (2003) 625-647. MR 2003446
- 45.
H.-Q. Li, N. Lohoué, Estimations
des solutions de l'équation des ondes sur certaines variétés coniques, Trans. Amer. Math. Soc. 355 (2003) 689-711. MR 1932721 (2003i:58052)
- 46.
N. Lohoué, A. Nilsson, Spectral multipliers on quotients by amenable subgroups, Manuscripta Math. 104 (2001) 295-299. MR 1828875 (2002e:43002)
- 47.
J. Ludwig, D. Müller, Sub-Laplacians of holomorphic
-type on rank one -groups and related solvable groups, J. Funct. Anal. 170 (2000) 366-427. MR 1740657 (2001g:43010)
- 48.
W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for Special Functions of Mathematical Physics, Springer-Verlag, Berlin Heidelberg, 1966. MR 0232968 (38:1291)
- 49.
G. Mauceri, S. Meda, Vector-valued multipliers on stratified groups, Revista Math. Iberoamericana 6 (1990) 141-154. MR 1125759 (92j:42014)
- 50.
S. Mikhlin, Multidimensional singular integral equation, Pergamon Press, New York, 1965. MR 0185399 (32:2866)
- 51.
D. Müller, E. M. Stein, On spectral multipliers for Heisenberg and related groups, J. Math. Pures et Appl. 73 (1994) 413-440. MR 1290494 (96m:43004)
- 52.
W. Müller, Spectral theory for Riemannian manifolds with cusps and a related trace formula, Math. Nachr. 111 (1983) 197-288. MR 0725778 (85i:58121)
- 53.
S. Mustapha, Multiplicateurs spectraux sur certains groupes non-unimodulaires, Harmonic Analysis and Number Theory, CMS Conf. Proceedings, Vol. 21, 1997, 11-30.MR 1472776 (98k:22032)
- 54.
S. Mustapha, Multiplicateurs de Mikhlin pour une classe particulière de groupes non-unimodulaires, Ann. Inst. Fourier 48 (1998) 957-966. MR 1656003 (99j:22013)
- 55.
A. Seeger, C.D. Sogge, On the boundedness of function of (pseudo-)Differential operators on compact manifolds, Duke Math. J. 59 (1989) 709-736.MR 1046745 (91d:58244)
- 56.
A. Seeger, Endpoint estimates for multiplier transformations on compact manifolds, Indiana Univ. Math. J. 40 (1991) 471-533. MR 1119186 (92f:58166)
- 57.
A. Sikora, Multiplicateurs associés aux sous-Laplaciens sur les groupes homogènes, C. R. Acad. Sci. Paris 315 (1992) 417-419. MR 1179049 (93e:43010)
- 58.
A. Sikora, Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth (a remark on the paper of M. Cowling et al.), Studia Math. 152 (2002) 125-130. MR 1916545 (2003g:22006)
- 59.
R. Stanton, P. Tomas, Expansions for spherical functions on noncompact symmetric spaces, Acta Math. 140 (1978) 251-276. MR 0511124 (58 #23365)
- 60.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970. MR 0290095 (44:7280)
- 61.
E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, 1971. MR 0304972 (46:4102)
- 62.
E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatary Integrals, Princeton Univ. Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)
- 63.
M. E. Taylor,
-Estimates on functions of the Laplace operator, Duke Math. J. 58 (1989) 773-793. MR 1016445 (91d:58253)
- 1.
- G. Alexopoulos, Spectral multipliers on Lie groups of polynomial growth, Proc. Amer. Math. Soc. 120 (1994) 973-979. MR 1172944 (95j:22016)
- 2.
- G. Alexopoulos,
bounds for spectral multipliers from Gaussian estimates of the heat kernel, Prépublication 1999.
- 3.
- G. Alexopoulos, Spectral multipliers on discrete groups, Bull. London Math. Soc. 33 (2001) 417-424. MR 1832553 (2002d:22010)
- 4.
- G. Alexopoulos, Spectral multipliers for Markov chains. A paraître au J. Math. Soc. Japan.
- 5.
- J.-P. Anker, N. Lohoué, Multiplicateurs sur certains espaces symétriques, Amer. J. Math. 108 (1986) 1303-1354. MR 0868894 (88c:43008)
- 6.
- J.-P. Anker,
Fourier multipliers on Riemannian symmetric spaces of the non-compact type, Ann. of Math. 132 (1990) 597-628. MR 1078270 (92e:43006)
- 7.
- J.-P. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Mathematical Journal 65 (1992) 257-297. MR 1150587 (93b:43007)
- 8.
- J.-P. Anker, E. Damek, C. Yacoub, Spherical Analysis on Harmonic
Groups, Ann. Sc. Norm. Super. Pisa. Cl. Sci. IV. Sen. 23 (1996) 643-679. MR 1469569 (99a:22014)
- 9.
- F. Astengo, Multipliers for a distinguished Laplacean on solvable extensions of H-type groups, Monatsh. Math. 120 (1995) 179-188.MR 1363136 (97c:22012)
- 10.
- P. Bérard, On the Wave Equation on Riemannian Manifold without Conjugate Points, Math. Z., 155 (1977) 249-276. MR 0455055 (56:13295)
- 11.
- J. Cheeger, M. Gromov, M. E. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982) 15-53.MR 0658471 (84b:58109)
- 12.
- J. Cheeger, M. E. Taylor, On the diffraction of wave by conical singularities. I, Comm. Pure Appl. Math. XXV (1982) 275-331.MR 0649347 (84h:35091a)
- 13.
- J. Clerc, E. M. Stein,
-multipliers for noncompact symmertric spaces, Proc. Nat. Acad. Sci. U.S.A. 71 (1974) 3911-3912. MR 0367561 (51:3803)
- 14.
- M. Christ,
bound for spectral multiplier on nilpotent groups, Trans. Amer. Math. Soc. 328 (1991) 73-81. MR 1104196 (92k:42017)
- 15.
- M. Christ, D. Müller, On
spectral multipliers for a solvable Lie group, Geom. Funct. Anal. 6 (1996) 860-876. MR 1415763 (98b:22017)
- 16.
- R. Coifman, G. Weiss, Analyse harmonique non-commutative sur certains espaces homogènes, Springer L.N.
242, 1971. MR 0499948 (58:17690)
- 17.
- M. Cowling, S. Giulini, A. Hulanicki, G. Mauceri, Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth, Studia Math. 111 (1994) 103-121.MR 1301761 (95j:43002)
- 18.
- M. Cowling, A. Sikora, A spectral multiplier theorem for a sublaplacian on SU(2), Math. Z. 238 (2001) 1-36. MR 1860734 (2002h:42016)
- 19.
- L. De Michèle, G. Mauceri,
-multipliers on the Heisenberg group, Michigan Math. J., 26 (1979) 361-373. MR 0544603 (80j:43003b)
- 20.
- L. De Michèle, G. Mauceri,
multipliers on stratified groups, Ann. Mat. Pura Appl. 148 (1987) 353-366. MR 0932770 (89e:43009)
- 21.
- X.-T. Duong, E. M. Ouhabaz, Heat kernel bounds and spectral multipliers on spaces of polynomial volume growth and irregular domains. Prépublication 1999.
- 22.
- X.-T. Duong, E. M. Ouhabaz, A. Sikora, Plancherel-type estimates and sharp spectral multipliers, J. Funct. Anal. 196 (2002) 443-485.MR 1943098 (2003k:43012)
- 23.
- J. García-Cuerva, G. Mauceri, P. Sjögren, J.-L. Torrea, Spectral multipliers for the Ornstein-Uhlenbeck semigroup, J. Anal. Math. 78 (1999) 281-305. MR 1714425 (2000m:42015)
- 24.
- J. García-Cuerva, G. Mauceri, S. Meda, P. Sjögren, J.-L. Torrea, Functional calculus for the Ornstein-Uhlenbeck operator, J. Funct. Anal. 183 (2001) 413-450. MR 1844213 (2002d:47023)
- 25.
- I. M. Gelfand, G. Shilov, Generalized Functions I, Academic Press, New York and London, 1964. MR 0166596 (29:3869)
- 26.
- S. Giulini, G. Mauceri, S. Meda,
multipliers on noncompact symmetric spaces, J. Reine Angew. Math. 482 (1997) 151-175.MR 1427660 (98g:43006)
- 27.
- M. Gnewuch, Spectral Multipliers for sub-Laplacians on amenable Lie groups with exponential volume growth, Math. Z. 246 (2004) 69-83.MR 2031445
- 28.
- W. Hebisch, The subalgebra of
generated by the laplacean, Proc. Amer. Math. Soc. 117 (1993) 547-549. MR 1111218 (93d:22009)
- 29.
- W. Hebisch, Multiplier theorem on generalized Heisenberg groups, Coll. Math. 65 (1993) 231-239. MR 1240169 (94m:43013)
- 30.
- W. Hebisch, J. Zienkiewicz, Multiplier theorem on generalized Heisenberg groups II, Coll. Math. 69 (1995) 29-36. MR 1341678 (96h:43007)
- 31.
- W. Hebisch, Boundedness of
spectral multipliers for an exponential solvable Lie group, Coll. Math. 73 (1997) 155-164. MR 1436956 (98e:22005)
- 32.
- W. Hebisch, Spectral multipliers on exponential growth solvable Lie groups, Math. Z. 229 (1998) 435-441. MR 1658573 (2000a:43004)
- 33.
- W. Hebisch, Spectral multipliers on metabelian groups, Revista Math. Iberoamericana 16 (2000) 597-604. MR 1813330 (2001m:22016)
- 34.
- W. Hebisch, T. Steger, Multipliers and singular integrals on exponential growth groups, Math. Z. 245 (2003) 37-61. MR 2023952
- 35.
- L. Hörmander, Estimates for translation invariant operators in
spaces, Acta Math. 104 (1960) 93-139. MR 0121655 (22:12389)
- 36.
- L. Hörmander, The Analysis of Linear Partial Differential Operator III : Pseudo - Differential Operators, Springer-Verlag, Berlin Heidelberg, 1985. MR 0781536 (87d:35002a)
- 37.
- A. Hulanicki, Subalgebra of
associated with Laplacian on a Lie group, Coll. Math. 31 (1974) 259-287. MR 0372536 (51:8743)
- 38.
- A. Ionescu, Singular integrals on symmetric spaces of real rank one, Duke Math. J. 114 (2002) 101-122. MR 1915037 (2003c:43008)
- 39.
- A. Ionescu, Fourier integral operators on noncompact symmetric spaces of real rank one, J. Funct. Anal. 174 (2000) 274-300. MR 1767376 (2001h:43009)
- 40.
- A. Ionescu, Singular integrals on symmetric spaces II, Trans. Amer. Math. Soc. 355 (2003) 3359-3378. MR 1974692 (2004b:43009)
- 41.
- C. E. Kenig, R. J. Stanton, P. A. Tomas, Divergence of eigenfunction expansions, J. Funct. Anal. 46 (1982) 28-44. MR 0654463 (83k:22025)
- 42.
- H.-Q. Li, Certaines fonctions maximales sur les variétés cuspidales. Prépublication de l'IHES, décembre 2002.
- 43.
- H.-Q. Li, Estimations
des fonctions du Laplacien sur les variétés cuspidales I. Prépublication de l'IHES, décembre 2002.
- 44.
- H.-Q. Li, Analyse sur les variétés cuspidales, Math. Ann. 326 (2003) 625-647. MR 2003446
- 45.
- H.-Q. Li, N. Lohoué, Estimations
des solutions de l'équation des ondes sur certaines variétés coniques, Trans. Amer. Math. Soc. 355 (2003) 689-711. MR 1932721 (2003i:58052)
- 46.
- N. Lohoué, A. Nilsson, Spectral multipliers on quotients by amenable subgroups, Manuscripta Math. 104 (2001) 295-299. MR 1828875 (2002e:43002)
- 47.
- J. Ludwig, D. Müller, Sub-Laplacians of holomorphic
-type on rank one -groups and related solvable groups, J. Funct. Anal. 170 (2000) 366-427. MR 1740657 (2001g:43010)
- 48.
- W. Magnus, F. Oberhettinger, R. P. Soni, Formulas and Theorems for Special Functions of Mathematical Physics, Springer-Verlag, Berlin Heidelberg, 1966. MR 0232968 (38:1291)
- 49.
- G. Mauceri, S. Meda, Vector-valued multipliers on stratified groups, Revista Math. Iberoamericana 6 (1990) 141-154. MR 1125759 (92j:42014)
- 50.
- S. Mikhlin, Multidimensional singular integral equation, Pergamon Press, New York, 1965. MR 0185399 (32:2866)
- 51.
- D. Müller, E. M. Stein, On spectral multipliers for Heisenberg and related groups, J. Math. Pures et Appl. 73 (1994) 413-440. MR 1290494 (96m:43004)
- 52.
- W. Müller, Spectral theory for Riemannian manifolds with cusps and a related trace formula, Math. Nachr. 111 (1983) 197-288. MR 0725778 (85i:58121)
- 53.
- S. Mustapha, Multiplicateurs spectraux sur certains groupes non-unimodulaires, Harmonic Analysis and Number Theory, CMS Conf. Proceedings, Vol. 21, 1997, 11-30.MR 1472776 (98k:22032)
- 54.
- S. Mustapha, Multiplicateurs de Mikhlin pour une classe particulière de groupes non-unimodulaires, Ann. Inst. Fourier 48 (1998) 957-966. MR 1656003 (99j:22013)
- 55.
- A. Seeger, C.D. Sogge, On the boundedness of function of (pseudo-)Differential operators on compact manifolds, Duke Math. J. 59 (1989) 709-736.MR 1046745 (91d:58244)
- 56.
- A. Seeger, Endpoint estimates for multiplier transformations on compact manifolds, Indiana Univ. Math. J. 40 (1991) 471-533. MR 1119186 (92f:58166)
- 57.
- A. Sikora, Multiplicateurs associés aux sous-Laplaciens sur les groupes homogènes, C. R. Acad. Sci. Paris 315 (1992) 417-419. MR 1179049 (93e:43010)
- 58.
- A. Sikora, Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth (a remark on the paper of M. Cowling et al.), Studia Math. 152 (2002) 125-130. MR 1916545 (2003g:22006)
- 59.
- R. Stanton, P. Tomas, Expansions for spherical functions on noncompact symmetric spaces, Acta Math. 140 (1978) 251-276. MR 0511124 (58 #23365)
- 60.
- E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1970. MR 0290095 (44:7280)
- 61.
- E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, 1971. MR 0304972 (46:4102)
- 62.
- E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatary Integrals, Princeton Univ. Press, Princeton, NJ, 1993. MR 1232192 (95c:42002)
- 63.
- M. E. Taylor,
-Estimates on functions of the Laplace operator, Duke Math. J. 58 (1989) 773-793. MR 1016445 (91d:58253)
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Additional Information
Hong-Quan Li
Affiliation:
Institut für Angewandte Mathematik, Universität Bonn, Wegelerstr. 6, D-53115 Bonn, Germany
Email:
li-hq@wiener.iam.uni-bonn.de
DOI:
http://dx.doi.org/10.1090/S0002-9947-04-03695-5
PII:
S 0002-9947(04)03695-5
Received by editor(s):
August 9, 2003
Posted:
August 24, 2004
Article copyright:
© Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
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