On heat kernels on Lie groups.
Estimates on the semigroups generated by left invariant operators on Lie groups.
Boundedness of spectral multipliers for an exponential solvable Lie group
Multiplier theorem on generalized Heisenberg groups
Almost everywhere summability of eigenfunction expansions associated to elliptic operators
On operators satisfying the Rockland condition
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.
Sharp pointwise estimate for the kernels of the semigroup generated by sums of even powers of vector fields on homogeneous groups
A multiplier theorem for Schrödinger operators
Spectral multipliers on metabelian groups.
Let G be a Lie group, Xj right invariant vector fields on G, which generate (as a Lie algebra) the Lie algebra of G, L = -Σ Xj 2. (...) In this paper we consider L1(G) boundedness of F(L) for (some) metabelian G and a distinguished L on G. Of the main interest is that the group is of exponential growth, and possibly higher rank. Previously positive results about higher rank groups...
Pointwise estimates for densities of stable semigroups of measures
Let be a symmetric α-stable semigroup of probability measures on a homogeneous group N, where 0 < α < 2. Assume that are absolutely continuous with respect to Haar measure and denote by the corresponding densities. We show that the estimate , x≠0, holds true with some integrable function Ω on the unit sphere Σ if and only if the density of the Lévy measure of the semigroup belongs locally to the Zygmund class LlogL(N╲e). The problem turns out to be related to the properties of the maximal...
Multiplier theorem on generalized Heisenberg groups II
We prove that on a product of generalized Heisenberg groups, a Hörmander type multiplier theorem for Rockland operators is true with the critical index n/2 + ϵ, ϵ>0, where n is the euclidean (topological) dimension of the group.
A smooth subadditive homogeneous norm on a homogeneous group
Note on semigroups generated by positive Rockland operators on graded homogeneous groups
Let L be a positive Rockland operator of homogeneous degree d on a graded homogeneous group G and let be the convolution kernels of the semigroup generated by L. We prove that if τ(x) is a Riemannian distance of x from the unit element, then there are constants c>0 and C such that . Moreover, if G is not stratified, more precise estimates of at infinity are given.
On the relation between elliptic and parabolic Harnack inequalities
We show that, if a certain Sobolev inequality holds, then a scale-invariant elliptic Harnack inequality suffices to imply its a priori stronger parabolic counterpart. Neither the relative Sobolev inequality nor the elliptic Harnack inequality alone suffices to imply the parabolic Harnack inequality in question; both are necessary conditions. As an application, we show the equivalence between parabolic Harnack inequality for on , (i.e., for ) and elliptic Harnack inequality for on .
Page 1