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.