We study the densities of the semigroup generated by the operator 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 . We give explicit spectral decomposition of images of in representations.
The notion of the extremal length and the module of families of curves has been studied extensively and has given rise to a lot of applications to complex analysis and the potential theory. In particular, the coincidence of the p-module and the p-capacity plays an mportant role. We consider this problem on the Carnot group. The Carnot group G is a simply connected nilpotent Lie group equipped vith an appropriate family of dilations. Let omega be a bounded domain on G and Ko, K1 be disjoint non-empty...
We give new and simple sufficient conditions for Gaussian upper bounds for a convolution semigroup on a unimodular locally compact group. These conditions involve certain semigroup estimates in L²(G). We describe an application for estimates of heat kernels of complex subelliptic operators on unimodular Lie groups.
Let G be a homogeneous Lie group. We prove that for every closed, homogeneous subset Γ of G* which is invariant under the coadjoint action, there exists a regular kernel P such that P goes to 0 in any representation from Γ and P satisfies the Rockland condition outside Γ. We prove a subelliptic estimate as an application.
Let G be a homogeneous Lie group with a left Haar measure dg and L the action of G as left translations on . Further, let H = dL(C) denote a homogeneous operator associated with L. If H is positive and hypoelliptic on we prove that it is closed on each of the -spaces, p ∈ 〈 1,∞〉, and that it generates a semigroup S with a smooth kernel K which, with its derivatives, satisfies Gaussian bounds. The semigroup is holomorphic in the open right half-plane on all the -spaces, p ∈ [1,∞]. Further extensions...