Let Q be a connected solvable Lie group of polynomial growth. Let also E1, …, Ep be left invariant vector fields on G that satisfy Hοrmander’s condition and denote by L = -(E1 2 +… + Ep 2) the associated sub-Laplacian and ...

We prove a Harnack inequality on connected Lie groups of polynomial volume growth. We use this inequality to study the large time behavior of the heat kernels associated to centered sub-Laplacians. Thus, we obtain Gaussian ...

We prove Gaussian estimates for heat kernels on semisimple Lie groups by using the method of Block wave representation. We also give a large time asymptotic expansion for heat kernels on compact extensions of abelian Lie groups.

We prove Lpestimates for oscillating spectral multipliers on Lie groups of polynomial volume growth and Riemannian manifolds of nonnegative curvature. We apply these results to obtain Lpestimates for the Riesz means of the ...

We prove a Sobolev inequality for functions not necessarily with compact support, on a connected Lie group G of polynomial volume growth. To prove this inequality we have to characterise the harmonic functions of polynomial ...

The classical Mikhlin-Hörmander multiplier theorem is generalised to the context of discrete groups of polynomial volume growth.

We prove a parabolic Harnack inequality for a centered sub-Laplacian L on a connected Lie group G of polynomial volume growth by using ideas from Homogenisation theory and by adapting the method of Krylov and Safonov. We ...