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1994 | 111 | 2 | 103-121
Tytuł artykułu

Spectral multipliers for a distinguished Laplacian on certain groups of exponential growth

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
We prove that on Iwasawa AN groups coming from arbitrary semisimple Lie groups there is a Laplacian with a nonholomorphic functional calculus, not only for $L^1(AN),$ but also for $L^p(AN)$, where 1 < p < ∞. This yields a spectral multiplier theorem analogous to the ones known for sublaplacians on stratified groups.
Słowa kluczowe
Czasopismo
Rocznik
Tom
111
Numer
2
Strony
103-121
Opis fizyczny
Daty
wydano
1994
otrzymano
1993-11-02
poprawiono
1994-04-20
Twórcy
  • School of Mathematics, University of New South Wales, Kensington, New South Wales 2033, Australia
  • Dipartimento di Matematica, Università di Genova, Via L. B. Alberti 4, 16132 Genova, Italia
  • Mathematical Institute, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
  • Dipartimento di Matematica, Università di Genova, Via L. B. Alberti 4, 16132 Genova, Italia
Bibliografia
  • [1] G. Alexopoulos, Spectral multipliers on Lie groups of polynomial growth, preprint.
  • [2] J. Ph. Anker, $L^p$ Fourier multipliers on Riemannian symmetric spaces of the non-compact type, Ann. of Math. 132 (1990), 597-628.
  • [3] J. Ph. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J. 65 (1992), 257-297.
  • [4] J. Ph. Anker, A short proof of a classical covering lemma, Monatsh. Math. 107 (1989), 5-7.
  • [5] J. Ph. Anker et N. Lohoué, Multiplicateurs sur certains espaces symétriques, Amer. J. Math. 108 (1986), 1303-1354.
  • [6] A. Bonami et J.-L. Clerc, Sommes de Cesàro et multiplicateurs de développements en harmoniques sphériques, Trans. Amer. Math. Soc. 183 (1973), 223-263.
  • [7] P. Bougerol, Exemples de théorèmes locaux sur les groupes résolubles, Ann. Inst. H. Poincaré 19 (1983), 369-391.
  • [8] J. Cheeger, M. Gromov and M. E. Taylor, Finite propagation speed, kernel estimates for functions of the Laplacian, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), 15-53.
  • [9] M. Christ, $L^p$ bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc. 328 (1991), 73-81.
  • [10] J.-L. Clerc and E. M. Stein, $L^p$ multipliers for noncompact symmetric spaces, Proc. Nat. Acad. Sci. U.S.A. 71 (1974), 3911-3912.
  • [11] R. R. Coifman and G. Weiss, Analyse Harmonique Non-commutative sur Certains Espaces Homogènes, Lecture Notes in Math. 242, Springer, Berlin, 1971.
  • [12] M. G. Cowling, S. Giulini, G. I. Gaudry and G. Mauceri, Weak type (1,1) estimates for heat kernel maximal functions on Lie groups, Trans. Amer. Math. Soc. 323 (1991), 637-649.
  • [13] L. De Michele and G. Mauceri, $L^p$ multipliers on the Heisenberg group, Michigan J. Math. 26 (1979), 361-371.
  • [14] L. De Michele and G. Mauceri, $H^p$ multipliers on stratified groups, Ann. Mat. Pura Appl. 148 (1987), 353-366.
  • [15] G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Math. Notes 28, Princeton University Press, Princeton, N.J., 1982.
  • [16] G. I. Gaudry, T. Qian and P. Sjögren, Singular integrals associated to the Laplacian on the affine group ax+b, preprint.
  • [17] S. Giulini and G. Mauceri, Analysis of a distinguished Laplacean on solvable Lie groups, preprint.
  • [18] W. Hebisch, The subalgebra of $L^1(AN)$ generated by the Laplacean, Proc. Amer. Mat. Soc., to appear.
  • [19] S. Helgason, Groups and Geometric Analysis, Academic Press, New York, 1984.
  • [20] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Grundlehren Math. Wiss. 115, Springer, Berlin, 1963.
  • [21] L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta Math. 104 (1960), 93-140.
  • [22] A. Hulanicki, Subalgebra of $L^1(G)$ associated with laplacian on a Lie group, Colloq. Math. 31 (1974), 259-287.
  • [23] A. Hulanicki and E. M. Stein, Marcinkiewicz multiplier theorem for stratified groups, manuscript.
  • [24] G. Mauceri and S. Meda, Vector valued multipliers on stratified groups, Rev. Mat. Iberoamericana 6 (1990), 141-154.
  • [25] S. Mikhlin, Multidimensional Singular Integral Equations, Pergamon Press, 1965.
  • [26] D. Müller and E. M. Stein, announcement at a conference, August 1992.
  • [27] J.-P. Pier, Amenable Locally Compact Groups, Wiley, New York, 1984.
  • [28] R. J. Stanton and P. A. Tomas, Expansions for spherical functions on noncompact symmetric spaces, Acta Math. 140 (1978), 251-276.
  • [29] M. E. Taylor, $L^p$ estimates on functions of the Laplace operator, Duke Math. J. 58 (1989), 773-793.
  • [30] N. Th. Varopoulos, Analysis on Lie groups, J. Funct. Anal. 76 (1988), 346-410.
  • [31] L. Vretare, On $L^p$ Fourier multipliers on certain symmetric spaces, Math. Scand. 37 (1975), 111-121.
  • [32] N. J. Weiss, $L^p$ estimates for bi-invariant operators on compact Lie groups, Amer. J. Math. 94 (1972), 103-118.
Typ dokumentu
Bibliografia
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bwmeta1.element.bwnjournal-article-smv111i2p103bwm
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