Volume 7, Issue 2
The Fourier Transform and Its Weyl Symbol on Two-step Nilpotent Lie Groups

Jiang Yaping

J. Part. Diff. Eq.,7(1994),pp.183-192

Published online: 1994-07

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  • Abstract
In this paper, we give all equivalence classes of irreducible unitary representations for the group H_n ⊗ R^m thereby formulate the Fourier transform on H_n ⊗ R^m (n ≥ 0, m ≥ 0}, which naturally unifies the Fourier transform between the Euclidean group and the Heisenberg group, more generally, between Abelian groups and two-step nilpotent Lie groups. Moreover, by the Plancberel formula for H_n ⊗ R^m we produce the Weyl symbol association with functions of the harmonic oscillator so that to derive the heat kernel on H_n ⊗ R^m.
  • Keywords

Nilpotent representation group-Fourier transform Weyl symbol heat kernel hypoellipticity

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@Article{JPDE-7-183, author = {Jiang Yaping}, title = {The Fourier Transform and Its Weyl Symbol on Two-step Nilpotent Lie Groups}, journal = {Journal of Partial Differential Equations}, year = {1994}, volume = {7}, number = {2}, pages = {183--192}, abstract = { In this paper, we give all equivalence classes of irreducible unitary representations for the group H_n ⊗ R^m thereby formulate the Fourier transform on H_n ⊗ R^m (n ≥ 0, m ≥ 0}, which naturally unifies the Fourier transform between the Euclidean group and the Heisenberg group, more generally, between Abelian groups and two-step nilpotent Lie groups. Moreover, by the Plancberel formula for H_n ⊗ R^m we produce the Weyl symbol association with functions of the harmonic oscillator so that to derive the heat kernel on H_n ⊗ R^m.}, issn = {2079-732X}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jpde/5681.html} }
TY - JOUR T1 - The Fourier Transform and Its Weyl Symbol on Two-step Nilpotent Lie Groups AU - Jiang Yaping JO - Journal of Partial Differential Equations VL - 2 SP - 183 EP - 192 PY - 1994 DA - 1994/07 SN - 7 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jpde/5681.html KW - Nilpotent KW - representation KW - group-Fourier transform KW - Weyl symbol KW - heat kernel KW - hypoellipticity AB - In this paper, we give all equivalence classes of irreducible unitary representations for the group H_n ⊗ R^m thereby formulate the Fourier transform on H_n ⊗ R^m (n ≥ 0, m ≥ 0}, which naturally unifies the Fourier transform between the Euclidean group and the Heisenberg group, more generally, between Abelian groups and two-step nilpotent Lie groups. Moreover, by the Plancberel formula for H_n ⊗ R^m we produce the Weyl symbol association with functions of the harmonic oscillator so that to derive the heat kernel on H_n ⊗ R^m.
Jiang Yaping. (1970). The Fourier Transform and Its Weyl Symbol on Two-step Nilpotent Lie Groups. Journal of Partial Differential Equations. 7 (2). 183-192. doi:
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