The uniformly boundedness of the riesz transforms on the cayley heisenberg groups

In this article,the authors estimate some functions by using the explicit expression of the heat kernels for the Cayley Heisenberg groups,and then prove the uniform boundedness of the Riesz transforms on these nilpotent Lie groups.     

Gradient Estimates for the Heat Kernels in Higher Dimensional Heisenberg Groups

The author obtains sharp gradient estimates for the heat kernels in two kinds of higher dimensional Heisenberg groups -- the non-isotropic Heisenberg group and the Heisenberg type group Hn,m. The method used here relies on the positive property of the Bakry-Emery curvature F2 on the radial functions and some associated semigroup technics.     

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 Dirichlet form on the abstract metric measure space. As an application, we obtain lower estimates for heat kernels on some Riemannian manifolds.     

Some Estimates for Convolution Operators with Kernels of Type （l, r） on Homogeneous Groups

Some estimates for the convolution operators with kernels of type （l, r） on Lebesgue spaces with power weights and Herz-type spaces in the setting of homogeneous groups are established.     

THE ACTION OF THE LINEAR CHARACTER GROUP ON THE SET OF NON-LINEAR IRREDUCIBLE CHARACTERS

Let G be a nonabelian finite group. Then Irr(G/G′) is an abelian group under the multiplication of characters and acts on the set of non-linear irreducible characters of G via the multiplication of characters. The purpose of this paper is to establish some facts about the action of linear character group on non-linear irreducible characters and determine the structures of groups G for which either all the orbit kernels are trivial or the number of orbits is at most two. Using the established results on this action, it is very easy to classify groups G having at most three nomlinear irreducible characters.     

Morrey Estimates for Nondivergence Parabolic Operators with Discontinuous Coefficients on Homogeneous Groups

In this paper, by establishing the boundedness of singular integral operators with variable kernels and their commutators with BMO functions on Morrey spaces of homogeneous groups, we prove a local a prior estimate in Sobolev-Morrey space for solutions to the nondivergence parabolic equation with discontinuous coefficients.     

RECURSIVE KERNELS

This paper is an extension of earlier papers [8, 9] on the ＂native＂ Hilbert spaces of functions on some domain Ωbelong toR^d Rd in which conditionally positive definite kernels are reproducing kernels. Here, the focus is on subspaces of native spaces which are induced via subsets of Ω, and we shall derive a recursive subspace structure of these, leading to recur- sively defined reproducing kernels. As an application, we get a recursive Neville-Aitken- type interpolation process and a recursively defined orthogonal basis for interpolation by translates of kernels.     

Improved Gagliardo-Nirenberg inequalities on Heisenberg type groups

Motivated by the idea of M. Ledoux who brings out the connection between Sobolev embeddings and heat kernel bounds, we prove an analogous result for Kohn’s sub-Laplacian on the Heisenberg type groups. The main result includes features of an inequality of either Sobolev or Galiardo-Nirenberg type.     

Weakly strongly singular integral operators on anisotropic Hardy spaces and their dual operators

Let A be an expansive dilation. We define weakly strongly singular integral kernels and study the action of the operators induced by these kernels on anisotropic Hardy spaces associated with A.     

Time-decay Estimates for Keller–Segel System

For the Cauchy problem to Keller–Segel system, we show well-posedness and time-decay estimates in the critical scaling-invariant Besov spaces by using Littlewood–Paley analysis together with the decay estimates of heat kernels.     

Sur le comportement asymptotique des puissances de convolution et du noyau de la chaleur sur les groupes de Lie semisimples

We give an asymptotic expansion for the central value of convolution powers and heat kernels on semisimple Lie groups.     

Gaussian-type upper bound for the evolution kernels on nilpotent meta-abelian groups

We give a Gaussian-type upper bound for the transition kernels of the time-inhomogeneous diffusion processes on a nilpotent meta-abelian Lie group N generated by the family of time dependent second order left-invariant differential operators. These evolution kernels are related to the heat kernel for the left-invariant second order differential operators on higher rank NA groups.     

紧致对称空间的热核、特征值和特征函数

本文利用群表示论研究李群以及紧致对称空间的热核，特征值与特征函数．特别讨论了复格拉斯曼流形以及流形Sp（n）／U（n）上特征值及特征函数。     

An MRA approach to surface completion and image inpainting

The objective of this paper is to introduce a multi-resolution approximation (MRA) approach to the study of continuous function extensions with emphasis on surface completion and image inpainting. Along the line of the notion of diffusion maps introduced by Coifman and Lafon with some “heat kernels” as integral kernels of these operators in formulating the diffusion maps, we apply the directional derivatives of the heat kernels with respect to the inner normal vectors (on the boundary of the hole to be filled in) as integral kernels of the “propagation” operators. The extension operators defined by propagations followed by the corresponding sequent diffusion processes provide the MRA continuous function extensions to be discussed in this paper. As a case study, Green's functions of some “anisotropic” differential operators are used as heat kernels, and the corresponding extension operators provide a vehicle to transport the surface or image data, along with some mixed derivatives, from the exterior of the hole to recover the missing data in the hole in an MRA fashion, with the propagated mixed derivative data to provide the surface or image “details” in the hole. An error formula in terms of the heat kernels is formulated, and this formula is applied to give the exact order of approximation for the isotropic setting.     

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.     

Comparison inequalities for heat semigroups and heat kernels on metric measure spaces

We prove a certain inequality for a subsolution of the heat equation associated with a regular Dirichlet form. As a consequence of this inequality, we obtain various interesting comparison inequalities for heat semigroups and heat kernels, which can be used for obtaining pointwise estimates of heat kernels. As an example of application, we present a new method of deducing sub-Gaussian upper bounds of the heat kernel from on-diagonal bounds and tail estimates.     

高维Heisenberg群上热算子的分析

用留数定理及轨道积分方法,讨论了2n+1(n≥1)维Heisenberg群上热核及Green核的渐近性,并给出了更简明的渐近公式,彻底解决了Hueber,H等人遗留的问题.     

Fundamental solutions for hermite and subelliptic operators

In this article, we introduce a geometric method based on multipliers to compute heat kernels for operators with potentials. Using the heat kernel, we compute the fundamental solution for the Hermite operator with singularity at an arbitrary point on Euclidean space and on Heisenberg groups. As a consequence, we obtain the fundamental solutions for the sub-laplacian □ _J in a family of quadratic submanifolds. The research is partially supported by a William Fulbright Reserch Grant and a Competitive Research Grant at Georgetown University.     

An identification problem for a linear integrodifferential equation occurring in heat flow

A linear integrodifferential equation describing the heat flow in a material with memory is considered. This equation contains a pair of time-dependent convolution kernels that are unknown. Such kernels are recovered by using an overdetermined set of data. A local existence theorem for small times and the continuous dependence of the solutions on the data are shown.