Spherical Fourier transform and uncertainty principle on H-type groups

The present work is devoted to the study of the spherical Fourier transform on H-type groups G. We first determine the bounded spherical functions and introduce the corresponding spherical Fourier transform. Then, by exploiting the inversion formula of this transform, we derive explicit formulas for the heat kernel associated with the sub-Laplacian of G. Finally, we formulate and prove some uncertainty principles associated with the spherical Fourier transform on G, including analogues of the Heisenberg–Pauli–Weyl and Donoho–Stark inequalities.