On a semigroup of measures with irregular densities
We study the densities of the semigroup generated by the operator $-X^2+|Y|$ on the 3-dimensional Heisenberg group. We show that the 7th derivatives of the densities have a jump discontinuity. Outside the plane x=0 the densities are $C^∞$. We give explicit spectral decomposition of images of $-X^2+|Y|$ in representations.