Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation

In the first part of this paper, we get new Li–Yau type gradient estimates for positive solutions of heat equation on Riemannian manifolds with Ricci(M)?−k, k∈R. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type Li–Yau–Hamilton differential Harnack inequality for heat kernels on manifolds with Ricci(M)?−k, which generalizes a result of L. Ni (2004, 2006) [20] and [21]. As applications, we obtain new Harnack inequalities and heat kernel estimates on general manifolds. We also obtain various entropy monotonicity formulas for all compact Riemannian manifolds.     

A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces

On doubling metric measure spaces endowed with a strongly local regular Dirichlet form, we show some characterisations of pointwise upper bounds of the heat kernel in terms of global scale-invariant inequalities that correspond respectively to the Nash inequality and to a Gagliardo–Nirenberg type inequality when the volume growth is polynomial. This yields a new proof and a generalisation of the well-known equivalence between classical heat kernel upper bounds and relative Faber–Krahn inequalities or localised Sobolev or Nash inequalities. We are able to treat more general pointwise estimates, where the heat kernel rate of decay is not necessarily governed by the volume growth. A crucial role is played by the finite propagation speed property for the associated wave equation, and our main result holds for an abstract semigroup of operators satisfying the Davies–Gaffney estimates.     

Nonlinear potential theory for Sobolev spaces on Carnot groups

Considering Bessel kernels on a Carnot group, we establish the main facts of nonlinear potential theory: a Wolff-type inequality, capacity estimates, and a strong capacity inequality. Deriving corollaries, we give an inequality of Sobolev-Adams type and relations between the capacity and Hausdorff measure, as well as lower bounds on the Teichmüller capacity. These yield the continuity of monotone functions of a Sobolev class and some estimates applicable to studying the fine properties of functions.     

Lower inequalities of heat semigroups by using parabolic maximum principle

Using parabolic maximum principle, we apply the analytic method to obtain lower comparison inequalities for non-negative weak supersolutions of the heat equation associated with a regular strongly ρ-local Dirichle form on the abstract metric measure space. As an application, we obtain lower estimates for heat kernels on some Riemannian manifolds.     

Equivalence conditions for on-diagonal upper bounds of heat kernels on self-similar spaces

We obtain the equivalence conditions for an on-diagonal upper bound of heat kernels on self-similar measure energy spaces. In particular, this upper bound of the heat kernel is equivalent to the discreteness of the spectrum of the generator of the Dirichlet form, and to the global Poincaré inequality. The key ingredient of the proof is to obtain the Nash inequality from the global Poincaré inequality. We give two examples of families of spaces where the global Poincaré inequality is easily derived. They are the post-critically finite (p.c.f.) self-similar sets with harmonic structure and the products of self-similar measure energy spaces.     

Dimension-Independent Estimates for Heat Operators and Harmonic Functions

We establish dimension-independent estimates related to heat operators e ^tL on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply these results to two examples of interest: when L is the Laplace–Beltrami operator on a Riemannian manifold with Ricci curvature bounded from below, and when L is an invariant subelliptic operator of Hörmander type on a Lie group. In the former example, we also obtain pointwise bounds on harmonic and subharmonic functions, while in the latter example, we obtain pointwise bounds on harmonic functions when a generalized curvature-dimension inequality is satisfied.     

紧Riemann流形的高阶特征值估计

对Ricci曲率具负下界的紧Riemann流形,本文获得了热方程正解优化的梯度估计及Harnack不等式,证明了高阶特征值下界定量估计的猜想．     

Long-time asymptotics of heat kernels for one-dimensional elliptic operators with periodic coefficients

We give long-time asymptotics of heat kernels generated by one-dimensionalsecond-order elliptic operators with periodic coefficients.As a by-product Gaussian bounds are also derived.     

Localization and Schrödinger Perturbations of Kernels

We study iterations of integral kernels satisfying a transience-type condition and we prove exponential estimates analogous to Gronwall’s inequality. As a consequence we obtain estimates of Schrödinger perturbations of integral kernels, including Markovian semigroups.     

Sous-laplaciens et densités centrées sur les groupes de Lie à croissance polynomiale du volume

We prove a Harnack inequality on connected Lie groups of polynomial volume growth. We use this inequality to study the large time behavior of the heat kernels associated to centered sub-Laplacians. Thus, we obtain Gaussian estimates and estimates of the type Berry—Esseen. We also obtain similar results for the convolution powers of centered densities.     

Insensitizing controls for a forward stochastic heat equation

The existence of insensitizing controls for a forward stochastic heat equation is considered. To develop the duality, we obtain observability estimates for linear forward and backward coupled stochastic heat equations with general coefficients, by means of some global Carleman estimates. Furthermore, the constant in the observability inequality is estimated by an explicit function of the norm of the involved coefficients in the equation. As far as we know, our paper is the first one to address the problem of insensitizing controls for stochastic partial differential equations.     

Some new properties of biharmonic heat kernels

Contrary to the second-order case, biharmonic heat kernels are sign-changing. A deep knowledge of their behaviour may however allow us to prove positivity results for solutions of the Cauchy problem. We establish further properties of these kernels, we prove some Lorch–Szegö-type monotonicity results and we give some hints on how to obtain similar results for higher order polyharmonic parabolic problems.     

Growth of heat trace and heat content asymptotic coefficients

We show in the smooth category that the heat trace asymptotics and the heat content asymptotics can be made to grow arbitrarily rapidly. In the real analytic context, however, this is not true and we establish universal bounds on their growth.     

Revisiting the heat kernel on isotropic and nonisotropic Heisenberg groups*

The aim of this paper is threefold. First, we obtain the precise bounds for the heat kernel on isotropic Heisenberg groups by using well-known results in the three-dimensional case. Second, we study the asymptotic estimates at infinity for the heat kernel on nonisotropic Heisenberg groups. As a consequence, we give uniform upper and lower estimates of the heat kernel, and complete its short-time behavior obtained by Beals–Gaveau–Greiner. Third, we prove that the uniform asymptotic behaviour at infinity (so the small-time asymptotic behaviour) of the heat kernel for Grushin operators, obtained by the first author, are still valid in two and three dimensions.     

Volume and distance comparison theorems for sub-Riemannian manifolds

In this paper we study global distance estimates and uniform local volume estimates in a large class of sub-Riemannian manifolds. Our main device is the generalized curvature dimension inequality introduced by the first and the third author in [3] and its use to obtain sharp inequalities for solutions of the sub-Riemannian heat equation. As a consequence, we obtain a Gromov type precompactness theorem for the class of sub-Riemannian manifolds whose generalized Ricci curvature is bounded from below in the sense of [3].     

Yamabe流下黎曼流形上热方程的梯度估计

研究了在Yamabe流下演化的一个完备非紧黎曼流形,对流形上热方程的正解给出了两种局部的梯度估计.作为应用,可以得到这个热方程的Harnack不等式.     

Off-Diagonal Heat Kernel Lower Bounds Without Poincare

On a manifold with polynomial volume growth satisfying Gaussianupper bounds of the heat kernel, a simple characterization ofthe matching lower bounds is given in terms of a certain Sobolevinequality. The method also works in the case of so-called sub-Gaussianor sub-diffusive heat kernels estimates, which are typical offractals. Together with previously known results, this yieldsa new characterization of the full upper and lower Gaussianor sub-Gaussian heat kernel estimates.     

Ultracontractivity for Brownian motion with Markov switching

Consider the transition density functions for Brownian motion with two-state Markov switching. The characteristic functions for transition density functions are presented. Then, we show that the semigroup-associated Brownian motion with Markov switching is ultracontractive. And an explicit time-dependent upper bound for heat kernels are presented. Moreover, we prove that the Dirichlet form associated Brownian motion with Markov switching satisfies the Nash inequality.     

Intrinsic ultracontractivity on Riemannian manifolds with infinite volume measures

By establishing the intrinsic super-Poincar'e inequality,some explicit conditions are presented for diffusion semigroups on a non-compact complete Riemannian manifold to be intrinsically ultracontractive.These conditions,as well as the resulting uniform upper bounds on the intrinsic heat kernels,are sharp for some concrete examples.     

Dimension dependent hypercontractivity for Gaussian kernels

We derive sharp, local and dimension dependent hypercontractive bounds on the Markov kernel of a large class of diffusion semigroups. Unlike the dimension free ones, they capture refined properties of Markov kernels, such as trace estimates. They imply classical bounds on the Ornstein?CUhlenbeck semigroup and a dimensional and refined (transportation) Talagrand inequality when applied to the Hamilton?CJacobi equation. Hypercontractive bounds on the Ornstein?CUhlenbeck semigroup driven by a non-diffusive Lévy semigroup are also investigated. Curvature-dimension criteria are the main tool in the analysis.