Derivative Estimates of Semigroups and Riesz Transforms on Vector Bundles

We use versions of Bismut type derivative formulas obtained by Driver and Thalmaier [9], to prove derivative estimates for various heat semigroups on Riemannian vector bundles. As an application, the weak (1,1) property for a class of Riesz transforms on a vector bundle is established. Some concrete examples of vector bundles (e.g., differential forms) are considered to illustrate the results.     

Polynomial Trace Estimates for Semigroups

For semigroups (e ^tA ) _t0 of operators on a Hilbert space, we give conditions guaranteeing trace estimates of the polynomial type 0$$ " align="middle" border="0"> , where denotes the trace class. As an application we present higher order analogues of results due to E.B. Davies, B. Simon and M. van den Berg of the type 0$$ " align="middle" border="0"> , for certain unbounded domains , e.g. spiny urchin domains.     

Generalized Logarithmic Derivative Estimates of Gol'dberg-Grinshtein Type

For f meromorphic in the complex plane and meromorphic in theunit disc, sharp upper bounds are obtained for and where k andj are integers satisfying k > j 0. The results generalizethe logarithmic derivative estimate due to Gol'dberg and Grinshteinto derivatives of higher order. 2000 Mathematics Subject Classification30D35.     

柯西型积分的高阶导数公式证明

本文给出一个较为简单的方法证明柯西型积分的高阶导数公式。     

A Priori Estimates for Quantum Dynamical Semigroups

We obtain a priori bounds for the solutions of master Markov quantum equations in the Heisenberg and Schrödinger pictures.     

Hessian型方程Neumann边值问题的梯度估计

通过构造辅助函数,利用基本对称函数的性质以及函数在极大值点的性质,得到Hessian型方程S_k(D~2u-A(x,u,Du))=B(x,u)的梯度内估计,构造不同的辅助函数,分近边、边界和内部3种情形讨论该方程Neumann边值问题,进而得到全局梯度估计.     

Gradient Estimates for Some Semi-Linear Hypoelliptic Equations

Given some smooth vector fields X ₁,X ₂,…,X _m on a compact manifold M, if they satisfy Hörmander’s condition, we establish global gradient estimates for the positive smooth solutions to the semi-linear hypoelliptic equations $$Lu+au\log u+bu=\partial_tu, \quad \mbox{on} \ M\times[0,\infty) $$ and $$Lu+au\log u+bu=0, \quad \mbox{on} \ M, $$ where a,b are constants, and $L=\sum_{i}X_{i}^{2}-X_{0}$ . We partially generalize the results of Cao and Yau (Math. Z. 211:485–504, 1992).     

Gradient Estimates for Solutions of Ultraparabolic Equations

We establish a priori estimates for solutions to ultraparabolic equations which play a crucial role in the solvability of the initial value problem. A class of these equations came from population dynamics, namely from a fish larvae model.      

Opial Type Inequalities for Semigroups

Various L_p form Opial type inequalities are given for semigroups with applications.     

Finite Derivation Type for Semigroups and Congruences

We consider a congruence ρ on a semigroup S as a subsemigroup of the direct product S × S. We prove that if ρ has finite derivation type (FDT), then so does S.     

Linear Differential Equations and Logarithmic Derivative Estimates

We prove two sharp inequalities for the growth of solutionsof certain linear differential equations in the unit disk. Forthe proofs of these inequalities, we use the method of successiveapproximations and sharp estimates for the logarithmic derivativesof finite order meromorphic functions in the unit disk. Thesetechniques can also be used to give an alternate proof of awell-known result in the plane. The sharp logarithmic derivativeestimates are a corollary of general estimates, and all theseestimates have independent interest. 2000 Mathematics SubjectClassification 34M10, 30D35.     

关于算子半群的一个一般概率表示公式

In this paper we present a general probabilistic representation theorem for semigroups of operators which essentially comprises almost all known representation theorems of this kind. Especially, so-called first and second main theorems turn out to be special cases of the same general formula. The general theorem also permits the calculation of rates of convergence involving the second modulus of continuity.     

Growth Estimates for Semigroups Generated by 2Sx2 Operator Matrices

In this paper we give an explicit formula for semigroups generated by 2×2 operator matrices. Using this formula we can derive an estimate for the growth bound of the associated matrix generator. Finally these results are applied to a complete second order Cauchy problem yielding an explicit formula and a growth estimate for the solution of this problem.     

Gradient Estimates for the Heat Semigroup on H-Type Groups

By utilizing the Poincaré inequality and representation formulae, it is shown that on the Heisenberg type group, ℍ(2n, m), there exists a constant C > 0 such that

Gradient Estimates in Fractional Dirichlet Problems

We obtain some fine gradient estimates near the boundary for solutions to fractional elliptic problems subject to exterior Dirichlet boundary conditions. Our results provide, in particular, the sign of the normal derivative of such solutions near the boundary of the underlying domain.

     

Logarithmic Gradient Estimates to Hessian-Type Equations

We study logarithmic gradient estimate to smooth admissible solutions of the Hessian-type equations on Sn. The equations contain a Monge-Ampere equation arising in designing a reflecting surface in geometric optics as a special case.     

Derivative Estimates of Iterated Spherical Average and Extension

In the paper, we study derivative estimates of the iterated spherical averages(A_t)~N(f).We obtain the optimal range of exponents(α, N, p) to ensure the Lpboundedness of P(?/(?_x))(A_1)~N(f)for 1 ≤ p ≤∞, where P is a homogeneous polynomial of degree α. The main theorem extends some known results. As an application, we obtain the smallest N such that(A_1)~N: L~p(R~n) → L_α~p(R~n).     

Non-uniform Gradient Estimates for SDEs with Local Monotonicity Conditions

By using the coupling method and the localization technique, we establish non-uniform gradient estimates for Markov semigroups of diffusions or stochastic differential equations driven by pure jump Le′vy noises, where the coefficients only satisfy local monotonicity conditions.     

f-Laplace非线性方程的梯度估计和Liouville定理

设(M,g,e~(-f)dv_g)是n维完备光滑的度量测度空间.考虑以下非线性椭圆方程△_f~u+hu~α=0,1α(n+m)/(n+m-2)(n+m≥4)和非线性抛物方程(△_f-?/?t)u+hu~α=0,α0正解的梯度估计.对于经典的Laplace情形,Li (Li J. Gradient estimates and harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds [J]. J Funct Anal,1991, 100:233-256.)证明了正解的梯度估计和Liouville定理.在本文中,对于上述的f-Laplace方程,作者将推导出相应的结果.     

Derivative Estimates for the Solution of Hyperbolic Affine Sphere Equation

Considering the hyperbolic affine sphere equation in a smooth strictly convex bounded domain with zero boundary values, the sharp derivative estimates of any order for its convex solution are obtained.