in this paper we study properties of symbols such that these belong to class of symbols sitting insidesm ρ,φ that we shall introduce as the following. so for because hypoelliptic pseudodifferential operatorsplays a key role in quantum mechanics we will investigate some properties of m−hypoelliptic pseudodifferential operators for which define base on this class of symbols. also we consider maxi...

We consider the fundamental solution of a simple hypoelliptic stochastic partial differential equation in which the first-order term is modulated by white noise. We derive some short-time asymptotic formulæ. We discover that the form of the dominant short-time asymptotics depends nontrivially upon the interplay between the geometry of the noisy first-order term and the geometry defined by the h...

The Atiyah-Singer index theorem gives a topological formula for the index of an elliptic differential operator. The topological index depends on a cohomology class that is constructed from the principal symbol of the operator. On contact manifolds, the important Fredholm operators are not elliptic, but hypoelliptic. Their symbolic calculus is noncommutative, and is closely related to analysis o...

For the hypoelliptic differential operators P = ∂ x + ( x∂y − x∂t 2 introduced by T. Hoshiro, generalizing a class of M. Christ, in the cases of k and l left open in the analysis, the operators P also fail to be analytic hypoelliptic (except for (k, l) = (0, 1)), in accordance with Treves’ conjecture. The proof is constructive, suitable for generalization, and relies on evaluating a family of e...

In this paper we construct a noncommutative residue for the Heisenberg calculus, that is, for the hypoelliptic calculus on Heisenberg man-ifolds, including on CR and contact manifolds. This noncommutative residue as the residual induced on operators of integer orders by the analytic extension of the usual trace to operators of non-integer orders and it agrees with the integral of the density de...

We consider a second order operator with analytic coefficients whose principal symbol vanishes exactly to order two on a symplectic real analytic manifold. We assume that the first (non degenerate) eigenvalue vanishes on a symplectic submanifold of the characteristic manifold. In the C∞ framework this situation would mean a loss of 3/2 derivatives (see [5]). We prove that this operator is analy...

In this paper we study a Landau-type operator with an external force. It is a linear model of the Landau equation near Maxwellian distributions. Making use of multiplier method, we get the global hypoelliptic estimate under suitable assumptions on the external potential.

We present a uniform L-stability estimate for mild solutions to the Fokker-Planck-Boltzmann equation. For stability estimate, we derive a Gronwall type estimate using dispersion estimates for mild solutions due to the hypoelliptic structure of the Vlasov-Fokker-Planck operator.

We show that for a hypoelliptic Dirichlet form operatorA on a stratified complex Lie group, if the logarithmic Sobolev inequality holds, then a holomorphic projection of A is strongly hypercontractive in the sense of Janson. This extends previous results of Gross to a setting in which the operator A is not holomorphic.

We define and study pseudo-differential operators on a class of fractals that include the post-critically finite self-similar sets and Sierpinski carpets. Using the sub-Gaussian estimates of the heat operator we prove that our operators have kernels that decay and, in the constant coefficient case, are smooth off the diagonal. Our analysis can be extended to products of fractals. While our resu...