The Fourier Transform and Its Weyl Symbol on Two-step Nilpotent Lie Groups (Journal of Partial Differential Equations, 1994)
In this paper, we give all equivalence classes of irreducible unitary representations for the group H_n &#8855 R^m thereby formulate the Fourier transform on H_n &#8855 R^m (n &#8805 0, m &#8805 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 &#8855 R^m we produce the Weyl symbol association with functions of the harmonic oscillator so that to derive the heat kernel on H_n &#8855 R^m.