Heat kernel estimates for pseudodifferential operators,fractional Laplacians and Dirichlet-to-Neumann operators

The purpose of this article is to establish upper and lower estimates for the integral kernel of the semigroup exp(?t P) associated with a classical, strongly elliptic pseudodifferential operator P of positive order on a closed manifold. The Poissonian bounds generalize those obtained for perturbations of fractional powers of the Laplacian. In the selfadjoint case, extensions to ${t \in{\mathbb C}_+}$ are studied. In particular, our results apply to the Dirichlet-to-Neumann semigroup.     

Harnack inequality and heat kernel estimates on manifolds with curvature unbounded below

Using the coupling by parallel translation, along with Girsanov's theorem, a new version of a dimension-free Harnack inequality is established for diffusion semigroups on Riemannian manifolds with Ricci curvature bounded below by , where c>0 is a constant and ρo is the Riemannian distance function to a fixed point o on the manifold. As an application, in the symmetric case, a Li-Yau type heat kernel bound is presented for such semigroups.     

Strichartz type estimates for fractional heat equations

We obtain Strichartz estimates for the fractional heat equations by using both the abstract Strichartz estimates of Keel-Tao and the Hardy-Littlewood-Sobolev inequality. We also prove an endpoint homogeneous Strichartz estimate via replacing by BMOx(Rn) and a parabolic homogeneous Strichartz estimate. Meanwhile, we generalize the Strichartz estimates by replacing the Lebesgue spaces with either Besov spaces or Sobolev spaces. Moreover, we establish the Strichartz estimates for the fractional heat equations with a time dependent potential of an appropriate integrability. As an application, we prove the global existence and uniqueness of regular solutions in spatial variables for the generalized Navier-Stokes system with Lr(Rn) data.     

Global hypoelliptic estimates for fractional order kinetic equation

In this paper we study a class of fractional order kinetic equation, which is a linear model of spatially inhomogeneous Boltzmann equation without angular cutoff. Using the multiplier method introduced by F. Hérau and K. Pravda‐Starov (J. Math. Pures et Appl., 2011), we establish the optimal global hypoelliptic estimates with weights for the linear model operator.     

Two-sided heat kernel estimates for censored stable-like processes

In this paper, we study the precise behavior of the transition density functions of censored (resurrected) α-stable-like processes in C ^1,1 open sets in ${\mathbb R^d}$ , where d ≥ 1 and ${\alpha\in (1, 2)}$ . We first show that the semigroup of the censored α-stable-like process in any bounded Lipschitz open set is intrinsically ultracontractive. We then establish sharp two-sided estimates for the transition density functions of a large class of censored α-stable-like processes in C ^1,1 open sets. We further obtain sharp two-sided estimates for the Green functions of these censored α-stable-like processes in bounded C ^1,1 open sets.     

Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians

Abstract connections between integral kernels of positivity preserving semigroups and suitable L^p contractivity properties are established. Then these questions are studied for the semigroups generated by $\text{?Δ + V}\text{and}\text{H}{}_{\mathrm{\Omega }}$, the Dirichlet Laplacian for an open, connected region Ω. As an application under a suitable hypothesis, Sobolev estimates are proved valid up to ?Ω, of the form $\text{¦η(x)¦? c?}{}_0\text{(x) ∥H}{}_{\mathrm{\Omega }}{}^{\mathrm{k}}\text{η∥}{}_2$, where ?₀ is the unique positive L² eigenfunction of $\text{H}{}_{\mathrm{\Omega }}$.     

Lipschitz estimates for generalized commutators of fractional integrals with rough kernel

Let (0 < α < n) be the generalized commutator generated by fractional integral with rough kernel and the m–th order remainder of the Taylor formula of a function A. In this paper, the (L^p, L^r) (r > 1) boundedness, the weak (L¹, L^{n/(n–α–β)}) boundedness and the (L^p, ?^{β, ∞}_p) boundedness of are discussed, when D^γA belongs to the Lipschitz function spaces.     

Exponential divergence estimates and heat kernel tail

We obtain a lower bound for the density of a real random variable on the Wiener space under an exponential moment condition of the divergence. We apply this result to the solution of a non-linear SDE. To cite this article: E. Nualart, C. R. Acad. Sci. Paris, Ser. I 338 (2004).     

Spectral asymptotics for Laplacians on self-similar sets

Given a self-similar Dirichlet form on a self-similar set, we first give an estimate on the asymptotic order of the associated eigenvalue counting function in terms of a ‘geometric counting function’ defined through a family of coverings of the self-similar set naturally associated with the Dirichlet space. Secondly, under (sub-)Gaussian heat kernel upper bound, we prove a detailed short time asymptotic behavior of the partition function, which is the Laplace-Stieltjes transform of the eigenvalue counting function associated with the Dirichlet form. This result can be applicable to a class of infinitely ramified self-similar sets including generalized Sierpinski carpets, and is an extension of the result given recently by B.M. Hambly for the Brownian motion on generalized Sierpinski carpets. Moreover, we also provide a sharp remainder estimate for the short time asymptotic behavior of the partition function.     

Green function estimates for relativistic stable processes in half-space-like open sets

In this paper, we establish sharp two-sided estimates for the Green functions of relativistic stable processes (i.e. Green functions for non-local operators m−(m^2/α−Δ)^α/2) in half-space-like C^1,1 open sets. The estimates are uniform in m∈(0,M] for each fixed M∈(0,∞). When m↓0, our estimates reduce to the sharp Green function estimates for −(−Δ)^α/2 in such kind of open sets that were obtained recently in Chen and Tokle [12]. As a tool for proving our Green function estimates, we show that a boundary Harnack principle for X^m, which is uniform for all m∈(0,∞), holds for a large class of non-smooth open sets.     

Gradient estimates for the subelliptic heat kernel on H-type groups

We prove the following gradient inequality for the subelliptic heat kernel on nilpotent Lie groups G of H-type:

|∇Ptf|?KPt(|∇f|),     

Symmetric jump processes and their heat kernel estimates

We survey the recent development of the DeGiorgi-Nash-Moser-Aronson type theory for a class of symmetric jump processes (or equivalently, a class of symmetric integro-differential operators). We focus on the sharp two-sided estimates for the transition density functions (or heat kernels) of the processes, a priori Hölder estimate and parabolic Harnack inequalities for their parabolic functions. In contrast to the second order elliptic differential operator case, the methods to establish these properties for symmetric integro-differential operators are mainly probabilistic.     

Evolving sets, mixing and heat kernel bounds

We show that a new probabilistic technique, recently introduced by the first author, yields the sharpest bounds obtained to date on mixing times of Markov chains in terms of isoperimetric properties of the state space (also known as conductance bounds or Cheeger inequalities). We prove that the bounds for mixing time in total variation obtained by Lovász and Kannan, can be refined to apply to the maximum relative deviation |pⁿ(x,y)/π(y)−1| of the distribution at time n from the stationary distribution π. We then extend our results to Markov chains on infinite state spaces and to continuous-time chains. Our approach yields a direct link between isoperimetric inequalities and heat kernel bounds; previously, this link rested on analytic estimates known as Nash inequalities.Research supported in part by NSF Grants #DMS-0104073 and #DMS-0244479.     

Uniqueness sets for unbounded spectra

For every set $S?\mathbb{R}$ of finite measure, we construct a system of exponentials ${\{e^{i\lambda t}\}}_{\lambda \in \Lambda }$ which is complete in $L^2(S)$ and such that the set of frequencies Λ has the critical density $D(\Lambda )=\mathrm{mes}(S)/2\pi$.     

Global heat kernel bounds via desingularizing weights

We obtain global heat kernel bounds for semigroups which need not be ultracontractive by transferring them to appropriately chosen weighted spaces where they become ultracontractive. Our construction depends upon two assumptions: the classical Sobolev imbedding and a “desingularizing” (L¹,L¹) bound on the weighted semigroup.