Necessary and sufficient conditions for hypoelliptic of certain left invariant operators on nilpotent Lie groups II.
New smooth disintegration of for exponential solvable groups. (Nouvelle désintégration lisse de pour les groupes résolubles exponentiels.)
Noncolliding Brownian motions and Harish-Chandra formula.
Nonlinear potential theory for Sobolev spaces on Carnot groups.
Nonlocal Poincaré inequalities on Lie groups with polynomial volume growth and Riemannian manifolds
Let G be a real connected Lie group with polynomial volume growth endowed with its Haar measuredx. Given a C² positive bounded integrable function M on G, we give a sufficient condition for an L² Poincaré inequality with respect to the measure M(x)dx to hold on G. We then establish a nonlocal Poincaré inequality on G with respect to M(x)dx. We also give analogous Poincaré inequalities on Riemannian manifolds and deal with the case of Hardy inequalities.
Note on analytic regularity of heat kernels on nilpotent Lie groups
Let G be the simplest nilpotent Lie group of step 3. We prove that the densities of the semigroup generated by the sublaplacian on G are not real-analytic.
Note on coarea formulae in the Heisenberg group.
We show a first nontrivial example of coarea formula for vector-valued Lipschitz maps defined on the three dimensional Heisenberg group. In this coarea formula, integration on level sets is performed with respect to the 2-dimensional spherical Hausdorff measure, built by the Carnot-Carathéodory distance. The standard jacobian is replaced by the so called horizontal jacobian, corresponding to the jacobian of the Pansu differential of the Lipschitz map. Joining previous results, we achieve all possible...
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 a semigroup of measures with irregular densities
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.
On coincidence of p-module of a family of curves and p-capacity on the Carnot group.
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...
On commutators on nilpotent Lie group
On convexity, the Weyl group and the Iwasawa decomposition
On Gaussian kernel estimates on groups
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.
On group algebras of nilpotent Lie groups
On Haar measure of .
On Haar Measure on Sl(n,r)
On heat kernels on Lie groups.
On Lp Spectral Multipliers for a Solvable Lie Group.
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.
On positive Rockland operators
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...