Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-02T21:08:58.916Z Has data issue: false hasContentIssue false
  • English
  • Français

A note on the heat kernel on the Heisenberg group

Published online by Cambridge University Press:  17 April 2009

Adam Sikora  and
Jacek Zienkiewicz
Adam Sikora
Affiliation:
Department of Mathematical Sciences, New Mexico State University, PO Box 30001, Las Cruces NM 88003, Untied States of America e-mail: [email protected]
Jacek Zienkiewicz
Affiliation:
Instytut Matematyczuy, Uniwersytet Wroclawski, 50–384 Wroclaw, pl. Grunwaldzki 2/4, Poland e-mail: [email protected]
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 describe the analytic continuation of the heat kernel on the Heisenberg group ℍn(ℝ). As a consequence, we show that the convolution kernel corresponding to the Schrödinger operater eisL is a smooth function on ℍn(ℝ) \ Ss, where Ss = {(0, 0, ±sk) ∈ ℍn(ℝ) : k = n, n + 2, n + 4,…}. At every point of Ss the convolution kernel of eisL has a singularity of Calderón–Zygmund type.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 1 , February 2002 , pp. 115 - 120
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Christ, M., ‘Analytic hypoellipticity, representation of nilpotent groups, and a nonlinear eigenvalue problem’, Duke Math. J. 72 (1993), 595–639.CrossRefGoogle Scholar
[2]Christ, M., ‘Nonexistence of invariant analytic hypoelliptic differential operators on nilpotent groups of step greater than two’, in Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton Math. Series 42 (Princeton Univ. Press, Princeton, NJ, 1995), pp. 127–145.CrossRefGoogle Scholar
[3]Folland, G.B. and Stein, E.M., ‘Estimates for the complex and analysis on the Heisenberg group’, Comm. Pure Appl. Math. 27 (1974), 429–522.CrossRefGoogle Scholar
[4]Sikora, A., ‘On the L 2 → L ∞ norms of spectral multipliers of “quasi-homogeneous” operators on homogeneous groups’, Trans. Amer. Math. Soc. 351 (1999), 3743–3755.CrossRefGoogle Scholar
[5]Strichartz, R.S., ‘Lp harmonic analysis and Radon transforms on the Heisenberg group’, J. Funct. Anal. 96 (1991), 350–406.CrossRefGoogle Scholar
[6]Taylor, M.E., Noncommutative harmonic analysis, Mathematical Surveys and Monographs 22 (American Mathematical Society, Providence, R.I., 1986).CrossRefGoogle Scholar