Weyl transform and generalized spectrogram associated with quaternion Heisenberg group

In this paper, we define the Weyl transform and the generalized spectrogram associated with the quaternion Heisenberg group. We show that the Weyl transform with symbols in Lq(Q×IQ\{0},Sq,dν⊗dσ) acting on Lp(Q,dν) with 1⩽p⩽∞ is bounded if and only if q⩽2 and q⩽p⩽q′. We also prove Liebʼs uncertainty principle and the Lp boundedness for the generalized spectrogram.