Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-19T03:50:30.036Z Has data issue: false hasContentIssue false
  • English
  • Français

Convolution kernels of (n + 1)-fold Marcinkiewicz multipliers on the Heisenberg group

Published online by Cambridge University Press:  17 April 2009

A. J. Fraser
A. J. Fraser
Affiliation:
University of New South Wales, Sydney NSW 2052, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove a characterisation, in terms of regularity and cancellation conditions, of the convolution kernels of Marcinkiewicz multiplier operators m (𝔏1,…,𝔏n, iT) on the Heisenberg group ℍn, where 𝔏1,…,𝔏n are the n partial sub-Laplacians. The necessity of these regularity and cancellation conditions was established by Fraser (2001); here, we prove their sufficiency.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 64 , Issue 3 , December 2001 , pp. 353 - 376
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Chang, S.Y. and Fefferman, R., ‘Some recent developments in Fourier analysis and H p theory on product domains’, Bull. Amer. Math. Soc. 12 (1985), 1–43.CrossRefGoogle Scholar
[2]Erdélyi, A., Magnus, W., Oberhettinger, F. and Tricomi, F.G., Higher transcendental functions, II (McGraw-Hill, New York, Toronto, London, 1953).Google Scholar
[3]Fraser, A.J., ‘An (n+1)–fold Marcinkiewicz multiplier theorem on the Heisenberg group’, Bull. Austral. Math. Soc. 63 (2001), 35–38.CrossRefGoogle Scholar
[4]Geller, D., ‘Fourier analysis on the Heisenberg group’, Proc. Nat. Acad. Sci. U. S. A. 74 (1977), 1328–1331.CrossRefGoogle ScholarPubMed
[5]Journé, J.L., ‘Calderón-Zygmund operators on product spaces’, Rev. Mat. Iberoamericana 1 (1985), 55–92.CrossRefGoogle Scholar
[6]Müller, D., Ricci, F. and Stein, E.M., ‘Marcinkiewicz multipliers and two-parameter structures on Heisenberg groups, I’, Invent. Math. 119 (1995), 199–233.CrossRefGoogle Scholar
[7]Stein, E.M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series 30 (Princeton Univ. Press, Princeton, N.J., 1970).Google Scholar