Nash and log-Sobolev inequalities for hypoelliptic operators

By estimating the intrinsic distance and using known heat kernel upper bounds, the global Nash inequality with exact dimension is established for a class of square fields with algebraic growth induced by vector fields satisfying the Hörmander condition. As an application, a sufficient condition is presented for the log-Sobolev inequality to hold. Typical examples for Gruschin type operators and generalized Kohn-Lapacians on Heisenberg groups are provided.     

Anisotropic hypoelliptic estimates for Landau-type operators

We establish global hypoelliptic estimates for linear Landau-type operators. Linear Landau-type equations are a class of inhomogeneous kinetic equations with anisotropic diffusion whose study is motivated by the linearization of the Landau equation near the Maxwellian distribution. By introducing a microlocal method by multiplier which can be adapted to various linear inhomogeneous kinetic equations, we establish for linear Landau-type operators optimal global hypoelliptic estimates with loss of 4/3 derivatives in a Sobolev scale which is exactly related to the anisotropy of the diffusion.     

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.

     

Harnack inequalities and sub-Gaussian estimates for random walks

We show that the -parabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to the sub-Gaussian estimate for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order . The latter condition can be replaced by a certain estimate of the resistance of annuli. Received: 15 November 2001 / Revised version: 21 February 2002 / Published online: 6 August 2002     

On a class of maximally hypoelliptic operators

The maximal hypoellipticity of the operator X ₁ ^2m , ... X _k ^2m is proved, where X_j are first-order pseudodifferential operators and m is a natural number. It is assumed that the set of these operators and their first-order commutators is an elliptic system. The results obtained generalize the well-known results of Hormander, Oleinik, and Radkevich, Kohn, Helffer, and Nourrigat, and Rothshield and Stein. Bibliography: 9 titles. Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 17, pp. 3–26, 1994.     

Harnack inequalities for non-local operators of variable order

We consider harmonic functions with respect to the operator

Under suitable conditions on we establish a Harnack inequality for functions that are nonnegative and harmonic in a domain. The operator is allowed to be anisotropic and of variable order.

     

Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds

A gradient-entropy inequality is established for elliptic diffusion semigroups on arbitrary complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived.     

Pointwise local estimates and Gaussian upper bounds for a class of uniformly subelliptic ultraparabolic operators

We consider a class of second order ultraparabolic differential equations in the form

     

Harnack inequalities and difference estimates for random walks with infinite range

Difference estimates and Harnack inequalities for mean zero, finite variance random walks with infinite range are considered. An example is given to show that such estimates and inequalities do not hold for all mean zero, finite variance random walks. Conditions are then given under which such results can be proved.Research supported by the National Science Foundation.     

Global positivity estimates and Harnack inequalities for the fast diffusion equation

We investigate local and global properties of positive solutions to the fast diffusion equation u_t=Δu^m in the range (d−2)₊/d<m<1, corresponding to general nonnegative initial data. For the Cauchy problem posed in the whole Euclidean space we prove sharp local positivity estimates (weak Harnack inequalities) and elliptic Harnack inequalities; we use them to derive sharp global positivity estimates and a global Harnack principle. For the mixed initial and boundary value problem posed in a bounded domain of with homogeneous Dirichlet condition, we prove weak and elliptic Harnack inequalities. Our work shows that these fast diffusion flows have regularity properties comparable and in some senses better than the linear heat flow.     

Gradient estimates and Harnack inequalities for diffusion equation on Riemannian manifolds

We derive the gradient estimates and Harnack inequalities for positive solutions of the diffusion equation u _t = Δu ^m on Riemannian manifolds. Then, we prove a Liouville type theorem.     

齐次群上带漂移项亚椭圆算子的整体Sobolev-Morrey 估计

设G是一个齐次群,X0,X1,X2,...,Xp0为G上满足Hormander秩条件的实左不变向量场且X1,X2,...,Xp0是1次齐次的,X0是2次齐次的.在本文中,我们研究如下带有漂移项的算子：L=∑p0i,j=1aijXiXj＋a0X0,其中（aij）是一个常数矩阵且满足椭圆条件,a0∈R／{0}.对算子L,通过建立齐型空间上的奇异积分Morrey有界性和关于此向量场的插值不等式,我们在群G上获得了整体Sobolev-Morrey估计.     

Gradient estimates and Harnack inequalities for a Yamabe-type parabolic equation under the Yamabe flow

In this paper, we derive a series of gradient estimates and Harnack inequalities for positive solutions of a Yamabe-type parabolic partial differential equation (△-?t)u=pu+qu~(a+1) under the Yamabe flow. Here p,q∈C~(2,1)(M~n×[0,T]) and a is a positive constant.     

Gradient estimates and differential Harnack inequalities for a nonlinear parabolic equation on Riemannian manifolds

Let (M ⁿ , g) be an n-dimensional complete Riemannian manifold. We consider gradient estimates for the positive solutions to the following nonlinear parabolic equation: $$u_t=\Delta u+au\log u+bu$$ on M ⁿ  × [0,T], where a, b are two real constants. We derive local gradient estimates of the Li-Yau type for positive solutions of the above equations on Riemannian manifolds with Ricci curvature bounded from below. As applications, several parabolic Harnack inequalities are obtained. In particular, our results extend the ones of Davies in Heat Kernels and Spectral Theory, Cambridge Tracts in Mathematics, vol 92, Cambridge University Press, Cambridge,1989, and Li and Xu in Adv Math 226:4456–4491 (2011).