Heat Kernel and Hardy’s Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower bounds for the heat kernel associated with Jacobi transform, and get some analogues of Hardy's Theorem for Jacobi transform by using the sharp estimate of the heat kernel.     

Heat Kernel and Hardy’s Theorem for Jacobi Transform

In this paper, the authors obtain sharp upper and lower hounds for the heat kernel associated with Jacobi transform, and get some analogues of Hardy‘s Theorem for Jacobi transform by using the sharp estimate of the heat kernel.     

Sharp Heat Kernel Estimates in the Fourier–Bessel Setting for a Continuous Range of the Type Parameter

The heat kernel in the setting of classical Fourier–Bessel expansions is defined by an oscillatory 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.Keywords Fourier–Bessel expansions,heat kernel,Bessel process,transition density     

本刊英文版Vol.37(2021),No.2论文摘要

Heat Kernel Estimates for Non-symmetric Finite Range Jump Processes Jie Ming WANG Abstract 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.     

Heat Kernel Estimates for Non-symmetric Finite Range Jump Processes

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 ■,where ■ 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.     

Sharp heat kernel bounds and entropy in metric measure spaces

We establish the sharp upper and lower bounds of Gaussian type for the heat kernel in the metric measure space satisfying the RCD(0, N)(equivalently, RCD~*(0, N), condition with N∈N\ {1} and having the maximum volume growth, and then show its application on the large-time asymptotics of the heat kernel, sharp bounds on the(minimal) Green function, and above all, the large-time asymptotics of the Perelman entropy and the Nash entropy, where for the former the monotonicity of the Perelman entropy is proved. The results generalize the corresponding ones in the Riemannian manifolds, and some of them appear more explicit and sharper than the ones in metric measure spaces obtained recently by Jiang et al.(2016).     

THE HEAT KERNEL ON THE CAYLEY HEISENBERG GROUP

1IntroductionThe aim of the present paper is to construct an explicit expression of an heat kernel forthe Cayley Heisenberg group of order n.Hulanicki[1]and Gaveau[2]constructed the explicit expression of the heat kernel for theHeisenberg group by using p…     

HEAT KERNEL ESTIMATES ON JULIA SETS

We give heat kernel estimates on Julia sets J(f_c) for quadratic polynomials f c(z) = z~2+ 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.     

A LAW OF THE ITERATED LOGARITHM FOR NONPARAMETRIC CONDITIONAL DENSITY ESTIMATORS

We establish a law of the itearted logarithm for a triangular array of indepentent random variables, and apply it to obtain sharp pointwise rates of strong consistency for a large class of nonparametric estimators of bivariate density and conditional density. We consider the case of double kernel estimators anal trigonometric series estimators in detail.     

Sharp constants for a class of multilinear integral operators and some applications

We investigate a class of multilinear integral operators with the nonnegative kernels, and prove that the norms of the operators can be obtained by integral of the product of the kernel function and finitely many basic functions. Using the integral, we can easily calculate the sharp constants for the multilinear Hilbert inequality, the generalized Hardy-Littlewood-Sobolev inequality and the multilinear Hardy operator.     

ON A HYPER HILBERT TRANSFORM＊＊＊＊

The authors define the directional hyper Hilbert transform and give ita mixed norm estimate. The similar conclusions for the directional fractional integral of one dimension are also obtained in this paper. As an application of the above results, the authors give the Lp-boundedness for a class of the hyper singular integrals and the fractional integrals with variable kernel. Moreover, as another application of the above results, the authors prove the dimension free estimate for the hyper Riesz transform. This is an extension of the related result obtained by Stein.     

Equivariant heat invariants of the laplacian and nonmininmal operators on differential forms

In this paper,we compute the first two equivariant heat kernel coeffcients of the Bochner Laplacian on differential forms.The first two equivariant heat kernel coeffcients of the Bochner Laplacian with torsion are also given.We also study the equivariant heat kernel coeffcients of nonminimal operators on differential forms and get the equivariant Gilkey-Branson-Fulling formula.     

Drift perturbation of subordinate Brownian motions with Gaussian component

Let d ≥ 1 and Z be a subordinate Brownian motion on R~d with infinitesimal generator ? + ψ(?),where ψ is the Laplace exponent of a one-dimensional non-decreasing L′evy process(called subordinator). We establish the existence and uniqueness of fundamental solution(also called heat kernel) pb(t, x, y) for non-local operator L~b= ? + ψ(?) + b ?, where Rb is an Rd-valued function in Kato class K_(d,1). We show that p~b(t, x, y)is jointly continuous and derive its sharp two-sided estimates. The kernel pb(t, x, y) determines a conservative Feller process X. We further show that the law of X is the unique solution of the martingale problem for(L~b, C_c~∞(R~d)) and X is a weak solution of Xt = X0+ Zt + integral from n=0 to t(b(Xs)ds, t ≥ 0).Moreover, we prove that the above stochastic differential equation has a unique weak solution.     

SHARP DISTORTION THEOREMS FOR A CLASS OF BIHOLOMORPHIC MAPPINGS IN SEVERAL COMPLEX VARIABLES

In this paper,we first establish the sharp growth theorem and the distortion theorem of the Frechet derivative for biholomorphic mappings defined on the unit ball of complex Banach spaces and the unit polydisk in Cⁿ with some restricted conditions.We next give the distortion theorem of the Jacobi determinant for biholomorphic mappings defined on the unit ball of Cⁿ with an arbitrary norm and the unit polydisk in Cⁿ under certain restricted assumptions.Finally we ...     

LITTLEWOOD-PALEY THEOREM FOR SCHR DINGER OPERATORS

Let H be a Schrodinger operator on Rn. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.     

Global gradient estimate on graph and its applications

Continuing our previous work(ar Xiv:1509.07981v1),we derive another global gradient estimate for positive functions,particularly for positive solutions to the heat equation on finite or locally finite graphs.In general,the gradient estimate in the present paper is independent of our previous one.As applications,it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs.These global gradient estimates can be compared with the Li–Yau inequality on graphs contributed by Bauer et al.[J.Differential Geom.,99,359–409(2015)].In many topics,such as eigenvalue estimate and heat kernel estimate(not including the Liouville type theorems),replacing the Li–Yau inequality by the global gradient estimate,we can get similar results.     

本刊英文版Vol.27(2011),No.10论文摘要(英文)

<正>On the Boundedness of Rough Oscillatory Singular Integrals on Triebel-Lizorkin Spaces Hussain AL-QASSEM Leslie CHENG Yi Biao PAN Abstract We obtain appropriate sharp bounds on Triebel-Lizorkin spaces for rough oscillatory integrals with polynomial phase.By using these bounds and using an extrapolation argument we obtain some new and previously known results for oscillatory integrals under very weak size conditions on the kernel functions.     

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.     

Sharp Distortion Theorems for a Subclass of Biholomorphic Mappings Which Have a Parametric Representation in Several Complex Variables

In this paper, the sharp distortion theorems of the Fr′echet-derivative type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball of complex Banach spaces are established, and the corresponding results of the above generalized mappings on the unit polydisk in C~n are also given. Meanwhile, the sharp distortion theorems of the Jacobi determinant type for a subclass of biholomorphic mappings which have a parametric representation on the unit ball with an arbitrary norm in C~n are obtained, and the corresponding results of the above generalized mappings on the unit polydisk in C~n are got as well. Thus, some known results in prior literatures are generalized.     

Heat Kernel on Analytic Subvariety

In this paper, the author extends Peter Li and Tian Gang's results on the heat kernel from projective varieties to analytic varieties. The author gets an upper bound of the heat kernel on analytic varieties and proves several properties. Moreover, the results are extended to vector bundles. The author also gets an upper bound of the heat operators of some Schr¨ondinger type operators on vector bundles. As a corollary, an upper bound of the trace of the heat operators is obtained.