Plancherel-type estimates and sharp spectral multipliers

Journal of Functional Analysis

Volumn 196, Issue 2, 2002, Pages 443-485

  Duong, Xuan Thinh ^a   Ouhabaz, El Maati ^b   Sikora, Adam ^c  

^a MACQUARIE UNIVERSITY   (Australia)

^b UNIVERSITÉ DE BORDEAUX   (France)

^c NEW MEXICO STATE UNIVERSITY   (United States)

Author keywords

Spectral multipliers

Indexed keywords

EID: 0037147379     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0022-1236(02)00009-5     Document Type: Article

References (62)

1. S. Agmon, Y. Kannai, On the asymptotic behavior of spectral functions and resolvant kernels of elliptic operators, Israel J. Math. 5 (1967) 1-30.
2. G. Alexopoulos, Spectral multipliers on Lie groups of polynomial growth, Proc. Amer. Math. Soc. 120 (3) (1994) 973-979.
3. p bounds for spectral multipliers from Gaussian estimates of the heat kernel, preprint, 2000.
4. V.G. Avakumović, Über die Eigenfunktionen auf geschlossenen Riemannschen Mannigfaltigkeiten, Math. Z. 65 (1956) 327-344.
5. M.T. Barlow, E.A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields 79 (4) (1988) 543-623.
6. p-boundedness of Riesz means, J. Evol. Equs. 2 (2002) 299-317.
7. J. Cheeger, M. Gromov, M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1) (1982) 15-53.
8. 1 convergence of eigenfunction expansions for pseudodifferential operators, Invent. Math. 94 (2) (1988) 421-453.
9. M. Christ, Weak type (1, 1) bounds for rough operators, Ann. of Math. (2) 128 (1) (1988) 19-42.
10. p bounds for spectral multipliers on nilpotent groups, Trans. Amer. Math. Soc. 328 (1) (1991) 73-81.
11. T. Coulhon, Xuan Thinh Duong, Riesz transforms for 1≤p≤2, Trans. Amer. Math. Soc. 351 (3) (1999) 1151-1169.
12. ∞ functional calculus, J. Austral. Math. Soc. Ser. A 60 (1) (1996) 51-89.
13. M. Cowling, A. Sikora, A spectral multiplier theorem on su(2), Math. Z. 238 (1) (2001) 1-36.
14. E.B. Davies, Heat Kernels and Spectral Theory, Cambridge University Press, Cambridge, 1989.
15. E.B. Davies, Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1) (1995) 141-169.
16. p multipliers on stratified groups, Ann. Mat. Pura Appl. (4) 148 (1987) 353-366.
17. J.J. Duistermaat, VW. Guillemin, The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1) (1975) 39-79.
18. Xuan T. Duong, D.W. Robinson, Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal. 142 (1) (1996) 89-128.
19. 1 norms of the complex heat kernels to a Hörmander multiplier theorem for sub-Laplacians on nilpotent Lie groups, Pacific J. Math. 173 (2) (1996) 413-424.
20. Xuan Thinh Duong, A. McIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana 15 (2) (1999) 233--265.
21. Xuan Thinh Duong, E.M. Ouhabaz, Heat kernel bounds and spectral multipliers on spaces of polynomial growth and irregular domains, preprint (1999).
22. C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 124 (1970) 9-36.
23. G.B. Folland, E.M. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press, Princeton, NJ, 1982.
24. J.E. Galé, T. Pytlik, Functional calculus for infinitesimal generators of holomorpbic semigroups, J. Funct. Anal. 150 (2) (1997) 307-355.
25. W. Hebisch, Almost everywhere summability of eigenfunction expansions associated to elliptic operators, Studia Math. 96 (3) (1990) 263-275.
26. W. Hebisch, A multiplier theorem for Schrödinger operators, Colloq. Math. 60/61 (2) (1990) 659-664.
27. W. Hebisch, Multiplier theorem on generalized Heisenberg groups, Colloq. Math. 65 (2) (1993) 231-239.
28. W. Hebisch, Functional calculus for slowly decaying kernels, preprint, 1995.
29. p spaces, Acta Math. 104 (1960) 93- 140.
30. L. Hörmander, The spectral function of an elliptic operator, Acta Math. 121 (1968) 193-218.
31. L. Hörmander, On the Riesz means of spectral functions and eigenfunction expansions for elliptic differential operators, in: Some Recent Advances in the Basic Sciences, Vol. 2 (Proc. Annual Sci. Conf., Belfer Grad. School Sci., Yeshiva Univ., New York, 1965-1966), Beffier Graduate School of Science, Yeshiva University, New York, 1969, pp. 155 202.
32. L. Hörmander, Differential operators with constant coefficients, in: The Analysis of Linear Partial Differential Operators. II, Springer-Verlag, Berlin, 1983.
33. A. Hulanicki, E.M. Stein, Marcinkiewicz multiplier theorem for stratified groups, manuscript.
34. V.Ja. Ivriǐ, The second term of the spectral asymptotics for a Laplace-Beltrami operator on manifolds with boundary, Funktsional. Anal. Prilozhen. 14 (2) (1980) 25-34.
35. J. Kigami, Analysis on Fractals, Cambridge University Press, Cambridge, 2001.
36. C. Markett, Mean Cesàro summability of Laguerre expansions and norm estimates with shifted parameter, Anal. Math. 8 (1) (1982) 19-37.
37. G. Mauceri, S. Meda, Vector-valued multipliers on stratified groups, Rev. Mat. Iberoamericana 6 (3-4) (1990) 141-154.
38. R. Melrose, The trace of the wave group, in: Microlocal Analysis (Boulder, Colo., 1983), Amer. Math. Soc., Providence, RI, 1984, pp. 127-167, and Contemp. Math. 27.
39. S.G. Mikhlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, Oxford, 1965 (translated from the Russian by W. J. A. Whyte. Translation edited by I. N. Sneddon).
40. B. Muckenhoupt, Mean convergence of Hermite and Laguerre series. I, II. Trans. Amer. Math. Soc. 147 (1970) 419-431, 433-460.
41. D. Müller, E.M. Stein, On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl. (9) 73 (4) (1994) 413-440.
42. E.M. Ouhabaz, Gaussian estimates and holomorphy of semigroups, Proc. Amer. Math. Soc. 123 (5) (1995) 1465-1474.
43. M. Reed, B. Simon, Scattering theory, in: Methods of Modern Mathematical Physics. Vol. III, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1979.
44. M. Riesz, Sur la sommation des séries de fourier, A.S 1 (1923) 104-113.
45. D.W. Robinson, Elliptic Operators and Lie Groups, The Clarendon Press, Oxford University Press, New York, 1991 (Oxford Science Publications).
46. A. Seeger, C.D. Sogge, On the boundedness of functions of (pseudo-) differential operators on compact manifolds, Duke Math. J. 59 (3) (1989) 709-736.
47. A. Sikora, Sharp pointwise estimates on heat kernels, Quart. J. Math. Oxford Ser. (2) 47 (187) (1996) 371-382.
48. ∞ norms of spectral multipliers of "quasi -homogeneous" operators on homogeneous groups, Trans. Amer. Math. Soc. 351 (9) (1999) 3743-3755.
49. A. Sikora, Riesz transform and the method of wave equation, preprint, 2001.
50. A. Sikora, J. Wright, Imaginary powers of Laplace operators, Proc. Amer. Math. Soc. 129 (6) (2001) 1745-1754.
51. B. Simon, Maximal and minimal Schrödinger forms, J. Operator Theory 1 (1) (1979) 37-47.
52. P. Sjögren, An estimate for a first-order Riesz operator on the affine group, Trans. Amer. Math. Soc. 351 (8) (1999) 3301-3314.
53. C.D. Sogge, On the convergence of Riesz means on compact manifolds, Ann. of Math. (2) 126 (2) (1987) 439-447.
54. C.D. Sogge, Fourier Integrals in Classical Analysis, Cambridge University Press, Cambridge, 1993.
55. E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton Mathematical Series, No. 30. Princeton University Press, Princeton, NJ, 1970.
56. E.M. Stein, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, Princeton, NJ, 1993 (with the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III).
57. K. Stempak, J. Zienkiewicz, Twisted convolution and Riesz means, J. Anal. Math. 76 (1998) 93-107.
58. R.S. Strichartz, Function spaces on fractals, preprint, 2001.
59. T. Tao, Weak-type endpoint bounds for Riesz means, Proc. Amer. Math. Soc. 124 (9) (1996) 2797-2805.
60. S. Thangavelu, Summability of Hermite expansions. I, II. Trans. Amer. Math. Soc. 314 (1) (1989) 119-142, 143-170.
61. S Thangavelu, Lectures on Hermite and Laguerre Expansions, Princeton University Press, Princeton, NJ, 1993 (with a preface by Robert S. Strichartz).
62. N. Th. Varopoulos, L. Saloff-Coste, T. Coulbon, Analysis and Geometry on Groups, Cambridge University Press, Cambridge, 1992.