On a class of partially hypoelliptic operators in weighted Sobolev spaces

The set of smooth solutions of a class of regular, partially hypoelliptic with respect to the plane x ₁ = 0 equations in a rather wide strip is extracted.     

On a class of almost hypoelliptic operators in generalized Sobolev spaces

The paper considers differential operators in the generalized Sobolev spaces. Differring necessary and sufficient conditions on the polyhedron ? are found, under which the function h _? ^s satisfies the Beurling condition for s great enough. They coincide (become necessary and sufficient) for polyhedrons in ? ₊ ² . The regularity of a class of differential equations is investigated.     

On a class of almost hypoelliptic polynomials

The paper gives criteria for almost hypoelliptic polynomials P of two and three variables, providing that P(ξ) → ∞ as |ξ| → +∞.     

Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators

We prove a global Harnack inequality for a class of degenerate evolution operators by repeatedly using an invariant local Harnack inequality. As a consequence we obtain an accurate Gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators.

     

T-product of hypoelliptic operators

Asymptotics as t0 for the solution of the Cauchy problem for hypoelliptic equations are obtained. Using these results, the existence of the T-product for hypoelliptic operators is proved as is a theorem on the removal of autonomous brackets in the T-product.Translated from Itogi i Nauki i Tekhniki, Sovremennye Problemy Matematiki, Vol. 8, pp. 137–197, 1977.     

On smooth solutions of a class of almost hypoelliptic equations

For a class of non-degenerate (regular) almost hypoelliptic equations, which are partially hypoelliptic with respect to certain variables, infinitely differentiable in a specific strip solutions are extracted.     

Spectral projections of hypoelliptic operators

Translated from Matematicheskie Zametki, Vol. 49, No. 5, pp. 16–25, May, 1991.     

Dirichlet problem for some hypoelliptic operators

In this paper, the Dirichlet problem for hypoelliptic operators verifying Hörmander condition and the maximum principle is considered.     

On the essential spectrum of hypoelliptic pseudo‐differential operators

We describe the essential spectrum of a hypoelliptic pseudo‐differential operator which is the sum of a constantcoefficients operator and an operator with coefficients vanishing at infinity. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Complex powers of hypoelliptic pseudodifferential operators

Complex powers of a class of hypoelliptic pseudodifferential operators in Rn, as well as their heat kernels are studied. An application to the Schatten-von Neumann property of pseudodifferential operators is given.     

On the Harnack inequality for a class of hypoelliptic evolution equations

We give a direct proof of the Harnack inequality for a class of degenerate evolution operators which contains the linearized prototypes of the Kolmogorov and Fokker-Planck operators. We also improve the known results in that we find explicitly the optimal constant of the inequality.

     

Regularity near the initial state in the obstacle problem for a class of hypoelliptic ultraparabolic operators

This paper is devoted to a proof of regularity, near the initial state, for solutions to the Cauchy-Dirichlet and obstacle problem for a class of second order differential operators of Kolmogorov type. The approach used here is general enough to allow us to consider smooth obstacles as well as non-smooth obstacles.