By using the Hba's expression of the inverse Abel transform for the Riemannian symmetric space SU* (6)/SP(3) , we obtain the analytic expression of the heat kernal e(t Delta) for this space, and then deduce the we...By using the Hba's expression of the inverse Abel transform for the Riemannian symmetric space SU* (6)/SP(3) , we obtain the analytic expression of the heat kernal e(t Delta) for this space, and then deduce the weak (1-1) boundedness of the maximal operator associated to the heat kernel, we obtain also the asymptotic behavious of the Riesz potential (Delta)(-1/2) near infinite and near the origin. Finally we study the integrability of the Riesz transform Brad (Delta)(-1/2).展开更多
The authors obtain an explicit expression of the heat kernel for the Cayley Heisenberg group of order n by using the stochastic integral method of Gaveau. Apart from the standard Heisenberg group and the quaternionic ...The authors obtain an explicit expression of the heat kernel for the Cayley Heisenberg group of order n by using the stochastic integral method of Gaveau. Apart from the standard Heisenberg group and the quaternionic Heisenberg group, this is the only nilpotent Lie group on which an explicit formula for the heat kernel has been obtained.展开更多
We give heat kernel estimates on Julia sets J(f;) for quadratic polynomials f c(z) = z;+ c for c in the main cardioid or the ±1/k-bulbs where k ≥ 2. First we use external ray parametrization to construct a r...We give heat kernel estimates on Julia sets J(f;) for quadratic polynomials f c(z) = z;+ c for c in the main cardioid or the ±1/k-bulbs where k ≥ 2. First we use external ray parametrization to construct a regular, strongly local and conservative Dirichlet form on Julia set. Then we show that this Dirichlet form is a resistance form and the corresponding resistance metric induces the same topology as Euclidean metric. Finally, we give heat kernel estimates under the resistance metric.展开更多
This article obtains an explicit expression of the heat kernels on H-type groups and then follow the estimate of heat kernels to deduce the Hardy's uncertainty principle on the nilpotent Lie groups.
We apply the Davies method to give a quick proof for the upper estimate of the heat kernel for the non-local Dirichlet form on the ultra-metric space. The key observation is that the heat kernel of the truncated Diric...We apply the Davies method to give a quick proof for the upper estimate of the heat kernel for the non-local Dirichlet form on the ultra-metric space. The key observation is that the heat kernel of the truncated Dirichlet form vanishes when two spatial points are separated by any ball of a radius larger than the truncated range. This new phenomenon arises from the ultra-metric property of the space.展开更多
Let be a Schr?dinger operator on . We show that gradient estimates for the heat kernel of with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay...Let be a Schr?dinger operator on . We show that gradient estimates for the heat kernel of with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists in the Schwartz class such that the long time gradient heat kernel estimate fails.展开更多
Many websites use verification codes to prevent users from using the machine automatically to register,login,malicious vote or irrigate but it brought great burden to the enterprises involved in internet marketing as ...Many websites use verification codes to prevent users from using the machine automatically to register,login,malicious vote or irrigate but it brought great burden to the enterprises involved in internet marketing as entering the verification code manually.Improving the verification code security system needs the identification method as the corresponding testing system.We propose an anisotropic heat kernel equation group which can generate a heat source scale space during the kernel evolution based on infinite heat source axiom,design a multi-step anisotropic verification code identification algorithm which includes core procedure of building anisotropic heat kernel,settingwave energy information parameters,combing outverification codccharacters and corresponding peripheral procedure of gray scaling,binarizing,denoising,normalizing,segmenting and identifying,give out the detail criterion and parameter set.Actual test show the anisotropic heat kernel identification algorithm can be used on many kinds of verification code including text characters,mathematical,Chinese,voice,3D,programming,video,advertising,it has a higher rate of 25%and 50%than neural network and context matching algorithm separately for Yahoo site,49%and 60%for Captcha site,20%and 52%for Baidu site,60%and 65%for 3DTakers site,40%,and 51%.for MDP site.展开更多
It is well known that for a Brownian motion, if we change the medium to be inhomogeneous by a measure μ, then the new motion(the time-changed process) will diffuse according to a different metric D(·, ·).In...It is well known that for a Brownian motion, if we change the medium to be inhomogeneous by a measure μ, then the new motion(the time-changed process) will diffuse according to a different metric D(·, ·).In 2009, Kigami initiated a general scheme to construct such metrics through some self-similar weight functions g on the symbolic space. In order to provide concrete models to Kigami’s theoretical construction, in this paper,we give a thorough study of his metric on two classes of fractals of primary importance: the nested fractals and the generalized Sierpinski carpets;we further assume that the weight functions g := ga are generated by“symmetric” weights a. Let M be the domain of a such that Dgadefines a metric, and let S be the boundary of M. One of our main results is that the metrics from ga satisfy the metric chain condition if and only if a ∈ S.To determine M and S, we provide a recursive weight transfer construction on the nested fractals, and a basic symmetric argument on the Sierpinski carpet. As an application, we use the metric chain condition to obtain the lower estimate of the sub-Gaussian heat kernel. This together with the upper estimate obtained by Kigami allows us to have a concrete class of metrics for the time change, and the two-sided sub-Gaussian heat kernel estimate on the fundamental fractals.展开更多
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-si...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 Hlder 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.展开更多
In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.
The asymptotic expansion of the heat kernel Θ(t)=sum from ∞ to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞ are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R^n(n=2 or 3) is studied for sho...The asymptotic expansion of the heat kernel Θ(t)=sum from ∞ to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞ are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R^n(n=2 or 3) is studied for short-time t for a generalbounded domain Ω with a smooth boundary (?)Ω.In this paper,we consider the case of a finite number of theDirichlet conditions φ=0 on Γ_i (i=1,...,J) and the Neumann conditions (?)=0 on Γ_i (i=J+1,...,k) andthe Robin conditions ((?)+γ_i)φ=0 on Γ_i (i=k+1,...,m) where γ_i are piecewise smooth positive impedancefunctions,such that (?)Ω consists of a finite number of piecewise smooth components Γ_i(i=1,...,m) where(?)Ω=(?)Γ_i.We construct the required asymptotics in the form of a power series over t.The senior coefficients inthis series are specified as functionals of the geometric shape of the domain Ω.This result is applied to calculatethe one-particle partition function of a“special ideal gas”,i.e.,the set of non-interacting particles set up in abox with Dirichlet,Neumann and Robin boundary conditions for the appropriate wave function.Calculationof the thermodynamic quantities for the ideal gas such as the internal energy,pressure and specific heat revealsthat these quantities alone are incapable of distinguishing between two different shapes of the domain.Thisconclusion seems to be intuitively clear because it is based on a limited information given by a one-particlepartition function;nevertheless,its formal theoretical motivation is of some interest.展开更多
The asymptotic expansions of the trace of the heat kernel Θ(t) = Σ_(v =1)~∞exp(-tλ_v) for small positive t, where {λ_v} are the eigenvalues of the negative Laplacian-△_n = -Σ_(i = 1)~n(partial deriv/(partial de...The asymptotic expansions of the trace of the heat kernel Θ(t) = Σ_(v =1)~∞exp(-tλ_v) for small positive t, where {λ_v} are the eigenvalues of the negative Laplacian-△_n = -Σ_(i = 1)~n(partial deriv/(partial deriv)x^i)~2 in R^n (n = 2 or 3), are studied for ageneral annular bounded domain Ω with a smooth inner boundary (partial deriv)Ω_1 and a smoothouter boundary (partial deriv)Ω_2, where a finite number of piecewise smooth Robin boundaryconditions (partial deriv/(partial deriv)n_j + γ_j)φ = 0 on the components Γ_j(j = 1, …, k) of(partial deriv)Ω_1 and on the components Γ_j(j = k + 1, …, m) of (partial deriv)Ω_2 areconsidered such that (partial deriv)Ω_1 = ∪_(j = 1)~kΓ_j and (partial deriv)Ω_2 = ∪_(j = k +1)~mΓ_j and where the coefficients γ_j(j = 1, …, m) are piecewise smooth positive functions. Someapplications of Θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given.Further results are also obtained.展开更多
The heat kernel in the setting of classical Fourier-Bessel expansions is defined by an os- cillatory series which cannot be computed explicitly. We prove qualitatively sharp estimates of this kernel. Our method relies...The heat kernel in the setting of classical Fourier-Bessel expansions is defined by an os- cillatory series which cannot be computed explicitly. We prove qualitatively sharp estimates of this kernel. Our method relies on establishing a connection with a situation of expansions based on Jacobi polynomials and then transferring known sharp bounds for the related Jacobi heat kernel.展开更多
In this paper,we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation S^(b):=△^(α/2)+b·▽,where △^(α/2) is the...In this paper,we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation S^(b):=△^(α/2)+b·▽,where △^(α/2) is the truncated fractional Laplacian,α∈(1,2) and b ∈ K_(d)^(α-1).In the second part,for a more general finite range jump process,we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance |x-y|in short time.展开更多
Using Duhamel’s formula,we prove sharp two-sided estimates for the spectral fractional Laplacian’s heat kernel with time-dependent gradient perturbation in bounded C^1,1 domains.In addition,we obtain a gradient esti...Using Duhamel’s formula,we prove sharp two-sided estimates for the spectral fractional Laplacian’s heat kernel with time-dependent gradient perturbation in bounded C^1,1 domains.In addition,we obtain a gradient estimate as well as the Holder continuity of the heat kernel’s gradient.展开更多
We present some improvements of the Li-Yau heat kernel estimate on a Riemannian manifold with Ricci curvature bounded below.As a consequence we prove a gradient estimate to the heat kernel with an optimal leading term.
In this note,we use Schr?dinger representations and the Fourier transform on two step nilpotent Lie groups to compute the explicit formula of the sub-Laplacian operator and its symbol,which is associated with the resc...In this note,we use Schr?dinger representations and the Fourier transform on two step nilpotent Lie groups to compute the explicit formula of the sub-Laplacian operator and its symbol,which is associated with the rescaled harmonic oscillator.Then we can give an explicit formula for the heat kernel of the rescaled harmonic oscillator for the singularity at the origin.Our results are useful for the general two step nilpotent Lie groups,including the Heisenberg group and H-type group.展开更多
We establish explicit and sharp on-diagonal heat kernel estimates for Schrödinger semigroups with unbounded potentials corresponding to a large class of symmetric jump processes.The approach is based on recent de...We establish explicit and sharp on-diagonal heat kernel estimates for Schrödinger semigroups with unbounded potentials corresponding to a large class of symmetric jump processes.The approach is based on recent developments on the two-sided(Dirichlet)heat kernel estimates and intrinsic contractivity properties for symmetric jump processes.As a consequence,we present a more direct argument to yield asymptotic behaviors for eigenvalues of associated nonlocal operators.展开更多
In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associatedwith Jacobi transform, and get some analogues of Hardy's Theorem for Jacobi transform byusing the sharp estimate of the ...In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associatedwith Jacobi transform, and get some analogues of Hardy's Theorem for Jacobi transform byusing the sharp estimate of the heat kernel.展开更多
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...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-E′mery curvature Γ2 on the radial functions and some associated semigroup technics.展开更多
文摘By using the Hba's expression of the inverse Abel transform for the Riemannian symmetric space SU* (6)/SP(3) , we obtain the analytic expression of the heat kernal e(t Delta) for this space, and then deduce the weak (1-1) boundedness of the maximal operator associated to the heat kernel, we obtain also the asymptotic behavious of the Riesz potential (Delta)(-1/2) near infinite and near the origin. Finally we study the integrability of the Riesz transform Brad (Delta)(-1/2).
文摘The authors obtain an explicit expression of the heat kernel for the Cayley Heisenberg group of order n by using the stochastic integral method of Gaveau. Apart from the standard Heisenberg group and the quaternionic Heisenberg group, this is the only nilpotent Lie group on which an explicit formula for the heat kernel has been obtained.
文摘We give heat kernel estimates on Julia sets J(f;) for quadratic polynomials f c(z) = z;+ c for c in the main cardioid or the ±1/k-bulbs where k ≥ 2. First we use external ray parametrization to construct a regular, strongly local and conservative Dirichlet form on Julia set. Then we show that this Dirichlet form is a resistance form and the corresponding resistance metric induces the same topology as Euclidean metric. Finally, we give heat kernel estimates under the resistance metric.
基金supported by National Science Foundation of China (10571044)
文摘This article obtains an explicit expression of the heat kernels on H-type groups and then follow the estimate of heat kernels to deduce the Hardy's uncertainty principle on the nilpotent Lie groups.
基金supported by National Natural Science Foundation of China(11871296).
文摘We apply the Davies method to give a quick proof for the upper estimate of the heat kernel for the non-local Dirichlet form on the ultra-metric space. The key observation is that the heat kernel of the truncated Dirichlet form vanishes when two spatial points are separated by any ball of a radius larger than the truncated range. This new phenomenon arises from the ultra-metric property of the space.
文摘Let be a Schr?dinger operator on . We show that gradient estimates for the heat kernel of with upper Gaussian bounds imply polynomial decay for the kernels of certain smooth dyadic spectral operators. The latter decay property has been known to play an important role in the Littlewood-Paley theory for and Sobolev spaces. We are able to establish the result by modifying Hebisch and the author’s recent proofs. We give a counterexample in one dimension to show that there exists in the Schwartz class such that the long time gradient heat kernel estimate fails.
基金The national natural science foundation(61273290,61373147)Xiamen Scientific Plan Project(2014S0048,3502Z20123037)+1 种基金Fujian Scientific Plan Project(2013HZ0004-1)FuJian provincial education office A-class project(-JA13238)
文摘Many websites use verification codes to prevent users from using the machine automatically to register,login,malicious vote or irrigate but it brought great burden to the enterprises involved in internet marketing as entering the verification code manually.Improving the verification code security system needs the identification method as the corresponding testing system.We propose an anisotropic heat kernel equation group which can generate a heat source scale space during the kernel evolution based on infinite heat source axiom,design a multi-step anisotropic verification code identification algorithm which includes core procedure of building anisotropic heat kernel,settingwave energy information parameters,combing outverification codccharacters and corresponding peripheral procedure of gray scaling,binarizing,denoising,normalizing,segmenting and identifying,give out the detail criterion and parameter set.Actual test show the anisotropic heat kernel identification algorithm can be used on many kinds of verification code including text characters,mathematical,Chinese,voice,3D,programming,video,advertising,it has a higher rate of 25%and 50%than neural network and context matching algorithm separately for Yahoo site,49%and 60%for Captcha site,20%and 52%for Baidu site,60%and 65%for 3DTakers site,40%,and 51%.for MDP site.
基金supported by National Natural Science Foundation of China(Grant Nos.12101303 and 12171354)supported by National Natural Science Foundation of China(Grant No.12071213)+4 种基金supported by National Natural Science Foundation of China(Grant No.11771391)supported by the Hong Kong Research Grant Councilthe Natural Science Foundation of Jiangsu Province in China(Grant No.BK20211142)Zhejiang Provincial National Science Foundation of China(Grant No.LY22A010023)the Fundamental Research Funds for the Central Universities of China(Grant No.2021FZZX001-01)。
文摘It is well known that for a Brownian motion, if we change the medium to be inhomogeneous by a measure μ, then the new motion(the time-changed process) will diffuse according to a different metric D(·, ·).In 2009, Kigami initiated a general scheme to construct such metrics through some self-similar weight functions g on the symbolic space. In order to provide concrete models to Kigami’s theoretical construction, in this paper,we give a thorough study of his metric on two classes of fractals of primary importance: the nested fractals and the generalized Sierpinski carpets;we further assume that the weight functions g := ga are generated by“symmetric” weights a. Let M be the domain of a such that Dgadefines a metric, and let S be the boundary of M. One of our main results is that the metrics from ga satisfy the metric chain condition if and only if a ∈ S.To determine M and S, we provide a recursive weight transfer construction on the nested fractals, and a basic symmetric argument on the Sierpinski carpet. As an application, we use the metric chain condition to obtain the lower estimate of the sub-Gaussian heat kernel. This together with the upper estimate obtained by Kigami allows us to have a concrete class of metrics for the time change, and the two-sided sub-Gaussian heat kernel estimate on the fundamental fractals.
基金supported by National Science Foundation of USA(Grant No.DMS-0600206)
文摘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 Hlder 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.
基金supported by NSF (Grant No. DMS-0600206)supported by the Korea Science Engineering Foundation (KOSEF) Grant funded by the Korea government (MEST) (No. R01-2008-000-20010-0)supported by the Grant-in-Aid for Scientific Research (B) 18340027
文摘In this paper, we discuss necessary and sufficient conditions on jumping kernels for a class of jump-type Markov processes on metric measure spaces to have scale-invariant finite range parabolic Harnack inequality.
文摘The asymptotic expansion of the heat kernel Θ(t)=sum from ∞ to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞ are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R^n(n=2 or 3) is studied for short-time t for a generalbounded domain Ω with a smooth boundary (?)Ω.In this paper,we consider the case of a finite number of theDirichlet conditions φ=0 on Γ_i (i=1,...,J) and the Neumann conditions (?)=0 on Γ_i (i=J+1,...,k) andthe Robin conditions ((?)+γ_i)φ=0 on Γ_i (i=k+1,...,m) where γ_i are piecewise smooth positive impedancefunctions,such that (?)Ω consists of a finite number of piecewise smooth components Γ_i(i=1,...,m) where(?)Ω=(?)Γ_i.We construct the required asymptotics in the form of a power series over t.The senior coefficients inthis series are specified as functionals of the geometric shape of the domain Ω.This result is applied to calculatethe one-particle partition function of a“special ideal gas”,i.e.,the set of non-interacting particles set up in abox with Dirichlet,Neumann and Robin boundary conditions for the appropriate wave function.Calculationof the thermodynamic quantities for the ideal gas such as the internal energy,pressure and specific heat revealsthat these quantities alone are incapable of distinguishing between two different shapes of the domain.Thisconclusion seems to be intuitively clear because it is based on a limited information given by a one-particlepartition function;nevertheless,its formal theoretical motivation is of some interest.
文摘The asymptotic expansions of the trace of the heat kernel Θ(t) = Σ_(v =1)~∞exp(-tλ_v) for small positive t, where {λ_v} are the eigenvalues of the negative Laplacian-△_n = -Σ_(i = 1)~n(partial deriv/(partial deriv)x^i)~2 in R^n (n = 2 or 3), are studied for ageneral annular bounded domain Ω with a smooth inner boundary (partial deriv)Ω_1 and a smoothouter boundary (partial deriv)Ω_2, where a finite number of piecewise smooth Robin boundaryconditions (partial deriv/(partial deriv)n_j + γ_j)φ = 0 on the components Γ_j(j = 1, …, k) of(partial deriv)Ω_1 and on the components Γ_j(j = k + 1, …, m) of (partial deriv)Ω_2 areconsidered such that (partial deriv)Ω_1 = ∪_(j = 1)~kΓ_j and (partial deriv)Ω_2 = ∪_(j = k +1)~mΓ_j and where the coefficients γ_j(j = 1, …, m) are piecewise smooth positive functions. Someapplications of Θ(t) for an ideal gas enclosed in the general annular bounded domain Ω are given.Further results are also obtained.
基金supported by MNiSW(Grant No.N201 417839)supported by(Grant No.MTM2012-36732-C03-02)from Spanish Government
文摘The heat kernel in the setting of classical Fourier-Bessel expansions is defined by an os- cillatory series which cannot be computed explicitly. We prove qualitatively sharp estimates of this kernel. Our method relies on establishing a connection with a situation of expansions based on Jacobi polynomials and then transferring known sharp bounds for the related Jacobi heat kernel.
基金Partially supported by NSFC(Grant Nos.11731009 and 11401025)。
文摘In this paper,we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation S^(b):=△^(α/2)+b·▽,where △^(α/2) is the truncated fractional Laplacian,α∈(1,2) and b ∈ K_(d)^(α-1).In the second part,for a more general finite range jump process,we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance |x-y|in short time.
基金supported by the Simons Foundation(Grant No.#429343)supported by the Alexander-von-Humboldt Foundation+3 种基金National Natural Science Foundation of China(Grant No.11701233)National Science Foundation of Jiangsu(Grant No.BK20170226)supported by National Natural Science Foundation of China(Grant No.11771187)The Priority Academic Program Development of Jiangsu Higher Education Institutions。
文摘Using Duhamel’s formula,we prove sharp two-sided estimates for the spectral fractional Laplacian’s heat kernel with time-dependent gradient perturbation in bounded C^1,1 domains.In addition,we obtain a gradient estimate as well as the Holder continuity of the heat kernel’s gradient.
基金Supported by National Natural Science Foundation of China(Grant Nos.11926339,11821101)。
文摘We present some improvements of the Li-Yau heat kernel estimate on a Riemannian manifold with Ricci curvature bounded below.As a consequence we prove a gradient estimate to the heat kernel with an optimal leading term.
基金Supported by the National Natural Science Foundation of China (Grant Nos.12101546,11771385)。
文摘In this note,we use Schr?dinger representations and the Fourier transform on two step nilpotent Lie groups to compute the explicit formula of the sub-Laplacian operator and its symbol,which is associated with the rescaled harmonic oscillator.Then we can give an explicit formula for the heat kernel of the rescaled harmonic oscillator for the singularity at the origin.Our results are useful for the general two step nilpotent Lie groups,including the Heisenberg group and H-type group.
基金supported by the National Natural Science Foundation of China(Nos.11522106 and 11831014)the Fok Ying Tung Education Foundation(No.151002)+1 种基金the Program for Probability and Statistics:Theory and Application(No.IRTL1704)the Program for Innovative Research Team in Science and Technology in Fujian Province University(IRTSTFJ).
文摘We establish explicit and sharp on-diagonal heat kernel estimates for Schrödinger semigroups with unbounded potentials corresponding to a large class of symmetric jump processes.The approach is based on recent developments on the two-sided(Dirichlet)heat kernel estimates and intrinsic contractivity properties for symmetric jump processes.As a consequence,we present a more direct argument to yield asymptotic behaviors for eigenvalues of associated nonlocal operators.
基金Project supported by the National Natural Science Foundation of China(No.10001002).
文摘In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associatedwith Jacobi transform, and get some analogues of Hardy's Theorem for Jacobi transform byusing the sharp estimate of the heat kernel.
基金Project supported by China Scholarship Council (No. 2007U13020)
文摘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-E′mery curvature Γ2 on the radial functions and some associated semigroup technics.