In the first part of this paper we study the local and global solvability and the hypoellipticity of a family of left-invariant sublaplacians ${\mathcal{L}}_{\alpha }$ on the spheres $S^{2n+1}\simeq U\left(n+1\right)/U\left(n\right)$. In the second part, we introduce a larger family of left-invariant sublaplacians ${\mathcal{L}}_{\alpha ,\beta }$ on $S^3\simeq SU\left(2\right)$ and study the corresponding properties by means of a Lie group contraction to the Heisenberg group.

We solve the Kato square root problem for bounded measurable perturbations of subelliptic operators on connected Lie groups. The subelliptic operators are divergence form operators with complex bounded coefficients, which may have lower order terms. In this general setting we deduce inhomogeneous estimates. In case the group is nilpotent and the subelliptic operator is pure second order, we prove stronger homogeneous estimates. Furthermore, we prove Lipschitz stability of the estimates under small...

We obtain (weighted) Poincaré type inequalities for vector fields satisfying the Hörmander condition for p < 1 under some assumptions on the subelliptic gradient of the function. Such inequalities hold on Boman domains associated with the underlying Carnot- Carathéodory metric. In particular, they remain true for solutions to certain classes of subelliptic equations. Our results complement the earlier results in these directions for p ≥ 1.

In this paper we furnish mean value characterizations for subharmonic functions related to linear second order partial differential operators with nonnegative characteristic form, possessing a well-behaved fundamental solution $\Gamma$. These characterizations are based on suitable average operators on the level sets of $\Gamma$. Asymptotic characterizations are also considered, extending classical results of Blaschke, Privaloff, Radó, Beckenbach, Reade and Saks. We analyze as well the notion of subharmonic function...