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Marcinkiewicz multipliers on the Heisenberg group

Published online by Cambridge University Press:  17 April 2009

Alessandro Veneruso
Alessandro Veneruso
Affiliation:
Dipartimento di Matematica, Università di Genova, Via Dodecaneso 35, 16146 Genova, Italy, e-mail: [email protected]
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Abstract

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Let Hn be the Heisenberg group of dimension 2n + 1. Let ℒ1,…,ℒn be the partial sub-Laplacians on Hn and T the central element of the Lie algebra of Hn. We prove that the operator m (ℒ1,…,ℒn,−iT) is bounded on Lp (Hn), 1 < p < +∞, if the function m satisfies a Marcinkiewicz-type condition in Rn+1.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 61 , Issue 1 , February 2000 , pp. 53 - 68
Copyright
Copyright © Australian Mathematical Society 2000

References

REFERENCES

[1]Benson, C., Jenkins, J. and Ratcliff, G., ‘The spherical transform of a Schwartz function on the Heisenberg group’, J. Funct. Anal. 151 (1998), 379–423.CrossRefGoogle Scholar
[2]Benson, C., Jenkins, J., Ratcliff, G. and Worku, T., ‘Spectra for Gelfand pairs associated with the Heisenberg group’, Colloq. Math. 71 (1996), 305–328.CrossRefGoogle Scholar
[3]Christ, M., ‘Lp bounds for spectral multipliers on nilpotent groups’, Trans. Amer. Math. Soc. 328 (1991), 73–81.Google Scholar
[4]Coifman, R.R. and Weiss, G., Transference methods in Analysis,CBMS Regional Conference Series in Mathematics 31 (Amer. Math. Soc., Providence, R.I., 1977).CrossRefGoogle Scholar
[5]De Michele, L. and Mauceri, G., ‘Lp multipliers on the Heisenberg group’, Michigan Math. J. 26 (1979), 361–371.CrossRefGoogle Scholar
[6]De Michele, L. and Mauceri, G., ‘Hp multipliers on stratified groups’, Ann. Mat. Pura Appl. 148 (1987), 353–366.CrossRefGoogle Scholar
[7]Folland, G.B. and Stein, E.M., Hardy spaces on homogeneous groups, Mathematical Notes 28 (Princeton University Press, Princeton, 1982).Google Scholar
[8]Fraser, A.J., Marcinkiewicz multipliers on the Heisenberg group, Ph.D. Thesis (Princeton University, 1997).Google Scholar
[9]Hebisch, W., ‘Multiplier theorem on generalized Heisenberg groups’, Colloq. Math. 65 (1993), 231–239.CrossRefGoogle Scholar
[10]Hulanicki, A. and Jenkins, J., ‘Almost everywhere summability on nilmanifolds’, Trans. Amer. Math. Soc. 278 (1983), 703–715.CrossRefGoogle Scholar
[11]Hulanicki, A. and Ricci, F., ‘A Tauberian theorem and tangential convergence for bounded harmonic functions on balls in Cn’, Invent. Math. 62 (1980), 325–331.CrossRefGoogle Scholar
[12]Korányi, A., Vági, S. and Welland, G.V., ‘Remarks on the Cauchy integral and the conjugate function in generalized half-planes’, J. Math. Mech. 19 (1970), 1069–1081.Google Scholar
[13]Lin, C.-C., ‘Lp multipliers and their H 1 − L 1 estimates on the Heisenberg group’, Rev. Mat. Iberoamericana 11 (1995), 269–308.CrossRefGoogle Scholar
[14]Mauceri, G., ‘Zonal multipliers on the Heisenberg group’, Pacific J. Math. 95 (1981), 143–159.CrossRefGoogle Scholar
[15]Mauceri, G., ‘Maximal operators and Riesz means on stratified groups’, Symposia Math. 29 (1987), 47–62.Google Scholar
[16]Mauceri, G. and Meda, S., ‘Vector-valued multipliers on stratified groups’, Rev. Mat. Iberoamericana 6 (1990), 141–154.CrossRefGoogle Scholar
[17]Müller, D., Ricci, F. and Stein, E.M., ‘Marcinkiewicz multipliers and multi-parameter structure on Heisenberg(-type) groups, I’, Invent. Math. 119 (1995), 199–233.CrossRefGoogle Scholar
[18]Müller, D., Ricci, F. and Stein, E.M., ‘Marcinkiewicz multipliers and multi-parameter structure on Heisenberg(-type) groups, II’, Math. Z. 221 (1996), 267–291.Google Scholar
[19]Müller, D. and Stein, E.M., ‘On spectral multipliers for Heisenberg and related groups’, J. Math. Pures Appl. 73 (1994), 413–440.Google Scholar
[20]Stein, E.M., Singular integrals and differentiablity properties of functions (Princeton University Press, Princeton, 1970).Google Scholar
[21]Zygmund, A., Trigonometric series I (Cambridge University Press, Cambridge, 1959).Google Scholar