Potential kernels,probabilities of hitting a ball,harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal Lévy processes. We also prove a supremum estimate and a regularity result for functions harmonic with respect to a general isotropic unimodal Lévy process.In the second part we apply the recent results on the boundary Harnack inequality and Martin representation of harmonic functions for the class of isotropic unimodal Lévy processes. As a sample application, we provide sharp two-sided estimates of the Green function of a half-space.     

Estimates of operator moduli of continuity

In Aleksandrov and Peller (2010) [2] we obtained general estimates of the operator moduli of continuity of functions on the real line. In this paper we improve the estimates obtained in Aleksandrov and Peller (2010) [2] for certain special classes of functions. In particular, we improve estimates of Kato (1973) [18] and show that

     

Estimates for the orders of zeros of polynomials in some analytic functions

In the present paper, we consider estimates for the orders of zeros of polynomials in functions satisfying a system of algebraic differential equations and possessing a special D-property defined in the paper. The main result obtained in the paper consists of two theorems for the two cases in which these estimates are given. These estimates are improved versions of a similar estimate proved earlier in the case of algebraically independent functions and a single point. They are derived from a more general theorem concerning the estimates of absolute values of ideals in the ring of polynomials, and the proof of this theorem occupies the main part of the present paper. The proof is based on the theory of ideals in rings of polynomials. Such estimates may be used to prove the algebraic independence of the values of functions at algebraic points.     

Multilinear extrapolation and applications to the bilinear Hilbert transform

We present two extrapolation methods for multi-sublinear operators that allow us to derive estimates for general functions from the corresponding estimates on characteristic functions. Of these methods, the first is applicable to general multi-sublinear operators while the second requires working with the so-called (ε,δ)-atomic operators. Among the applications, we discuss some new endpoint estimates for the bilinear Hilbert transform.     

Analysis of jump processes with nondegenerate jumping kernels

We prove regularity estimates for functions which are harmonic with respect to certain jump processes. The aim of this article is to extend the method of Bass–Levin (2002) [3] and Bogdan–Sztonyk (2005) [6] to more general processes. Furthermore, we establish a new version of the Harnack inequality that implies regularity estimates for corresponding harmonic functions.     

A note on fast Fourier transforms for nonequispaced grids

In this paper, we are concerned with fast Fourier transforms for nonequispaced grids. We propose a general efficient method for the fast evaluation of trigonometric polynomials at nonequispaced nodes based on the approximation of the polynomials by special linear combinations of translates of suitable functions ϕ. We derive estimates for the approximation error. In particular, we improve the estimates given by Dutt and Rokhlin [7]. As a practical consequence, we obtain a criterion for the choice of the parameters involved in the fast transforms. This revised version was published online in August 2006 with corrections to the Cover Date.     

Generalized Solutions to Semilinear Elliptic PDE with Applications to the Lichnerowicz Equation

In this article we investigate the existence of a solution to a semi-linear, elliptic, partial differential equation with distributional coefficients and data. The problem we consider is a generalization of the Lichnerowicz equation that one encounters in studying the constraint equations in general relativity. Our method for solving this problem consists of solving a net of regularized, semi-linear problems with data obtained by smoothing the original, distributional coefficients. In order to solve these regularized problems, we develop a priori L ^∞-bounds and sub- and super-solutions to apply a fixed point argument. We then show that the net of solutions obtained through this process satisfies certain decay estimates by determining estimates for the sub- and super-solutions and utilizing classical, a priori elliptic estimates. The estimates for this net of solutions allow us to regard this collection of functions as a solution in a Colombeau-type algebra. We motivate this Colombeau algebra framework by first solving an ill-posed critical exponent problem. To solve this ill-posed problem, we use a collection of smooth, “approximating” problems and then use the resulting sequence of solutions and a compactness argument to obtain a solution to the original problem. This approach is modeled after the more general Colombeau framework that we develop, and it conveys the potential that solutions in these abstract spaces have for obtaining classical solutions to ill-posed non-linear problems with irregular data.     

Calculus of directional subdifferentials and coderivatives in Banach spaces

In this work we study the directional versions of Mordukhovich normal cones to nonsmooth sets, coderivatives of set-valued mappings, and subdifferentials of extended-real-valued functions in the framework of general Banach spaces. We establish some characterizations and basic properties of these constructions, and then develop calculus including sum rules and chain rules involving smooth functions. As an application, we also explore the upper estimates of the directional Mordukhovich subdifferentials and singular subdifferentials of marginal functions.     

Approximations of Set-Valued Functions by Metric Linear Operators

In this paper we introduce new approximation operators for univariate set-valued functions with general compact images in Rⁿ. We adapt linear approximation methods for real-valued functions by replacing linear combinations of numbers with new metric linear combinations of finite sequences of compact sets, thus obtaining "metric analogues" of these operators for set-valued functions. The new metric linear combination extends the binary metric average of Artstein to several sets and admits any real coefficients. Approximation estimates for the metric analogue operators are derived. As examples we study metric Bernstein operators, metric Schoenberg operators, and metric polynomial interpolants.     

On the trace embedding and its applications to evolution equations

In this paper, we consider traces at initial times for functions with mixed time-space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity problems for fractional evolution equations and stochastic evolution equations, where uniform trace estimates on the half-line are shown.     

Extremal problems for quasiconformal maps of punctured plane domains

The main goal of this paper is to give an affirmative answer to the long-standing conjecture which asserts that the affine map is a uniquely extremal quasiconformal map in the Teichmüller space of the complex plane punctured at the integer lattice points. In addition we derive a corollary related to the geometry of the corresponding Teichmüller space. Besides that we consider the classical dual extremal problem which naturally arises in the tangent space of the Teichmüller space. In particular we prove the uniqueness of Hahn-Banach extension of the associated linear functional given on the Bergman space of the integer lattice domain. Several useful estimates related to the local and global properties of integrable meromorphic functions and the delta functional (see the definition below) are also obtained. These estimates are intended to study the behavior of integrable functions near singularities and they are valid in general settings.

     

Convolution Kernels of 2D Fourier Multipliers Based on Real Analytic Functions

In this paper, estimates are proven for convolution kernels associated to multipliers from a reasonably general class of compactly supported two-dimensional functions constructed out of real analytic functions. These estimates are both for overall decay rate and decay rate in specific directions. The estimates are sharp for a certain range of exponents appearing in the theorems.     

A Sum Operator with Applications to Self-Improving Properties of Poincaré Inequalities in Metric Spaces

We define a class of summation operators with applications to the self-improving nature of Poincaré–Sobolev estimates, in fairly general quasimetric spaces of homogeneous type. We show that these sum operators play the familiar role of integral operators of potential type (e.g., Riesz fractional integrals) in deriving Poincaré–Sobolev estimates in cases when representations of functions by such integral operators are not readily available. In particular, we derive norm estimates for sum operators and use these estimates to obtain improved Poincaré–Sobolev results.     

The multisublinear maximal type operators in Banach function lattices

The main aim of this paper is to study a general multisublinear operators generated by quasi-concave functions between weighted Banach function lattices. These operators, in particular, generalize the Hardy–Littlewood and fractional maximal functions playing an important role in harmonic analysis. We prove that under some general geometrical assumptions on Banach function lattices two-weight weak type and also strong type estimates for these operators are true. To derive the main results of this paper we characterize the strong type estimate for a variant of multilinear averaging operators. As special cases we provide boundedness results for fractional maximal operators in concrete function spaces.     

Error Estimates and Reproduction of Polynomials for Biorthogonal Local Trigonometric Bases

A general approach for biorthogonal local trigonometric bases in the two-overlapping setting was given by Chui and Shi. In this paper, we give error estimates for the approximation with such basis functions. In particular, it is shown that for a partition of the real axis into small intervals one obtains better approximation order if polynomials are reproduced locally. Furthermore, smooth trigonometric bases are constructed, which reproduce constants resp. linear functions by only one resp. a small number of basis functions for each interval.     

Approximations and solution estimates in optimization

Approximation is central to many optimization problems and the supporting theory provides insight as well as foundation for algorithms. In this paper, we lay out a broad framework for quantifying approximations by viewing finite- and infinite-dimensional constrained minimization problems as instances of extended real-valued lower semicontinuous functions defined on a general metric space. Since the Attouch-Wets distance between such functions quantifies epi-convergence, we are able to obtain estimates of optimal solutions and optimal values through bounds of that distance. In particular, we show that near-optimal and near-feasible solutions are effectively Lipschitz continuous with modulus one in this distance. Under additional assumptions on the underlying metric space, we construct approximating functions involving only a finite number of parameters that still are close to an arbitrary extended real-valued lower semicontinuous functions.     

Dimension-Independent Estimates for Heat Operators and Harmonic Functions

We establish dimension-independent estimates related to heat operators e ^tL on manifolds. We first develop a very general contractivity result for Markov kernels which can be applied to diffusion semigroups. Second, we develop estimates on the norm behavior of harmonic and non-negative subharmonic functions. We apply these results to two examples of interest: when L is the Laplace–Beltrami operator on a Riemannian manifold with Ricci curvature bounded from below, and when L is an invariant subelliptic operator of Hörmander type on a Lie group. In the former example, we also obtain pointwise bounds on harmonic and subharmonic functions, while in the latter example, we obtain pointwise bounds on harmonic functions when a generalized curvature-dimension inequality is satisfied.     

New trends towards lower energy estimates and optimality for nonlinearly damped vibrating systems

We present an approach based on comparison principles for energy and interpolation properties to derive lower energy estimates for nonlinearly either locally damped or boundary damped vibrating systems. We show how the dissipation relation provides strong information on the asymptotic behavior of the energy of solutions. The geometrical situations are either one-dimensional, or radial two-dimensional or three-dimensional for annulus domains. We also consider the case of general domains, but in this case, for solutions with bounded velocities in time and space. In all these cases, the nonlinear damping function is assumed to have arbitrary (strictly sublinear) growth at the origin. We give results for strong solutions and stronger lower estimates for smoother solutions. The results are presented in two forms, either on the side of energy comparison principles, or through time-pointwise lower estimates. Under additional geometric assumptions, we give the resulting lower and upper estimates for four representative examples of damping functions. We further give a “weak” lower estimate (in the sense of a certain lim supt→∞) and an upper estimate of the velocity for smoother solutions in case of general damping functions and for radial, as well as multi-dimensional domains. We also discuss these estimates in the framework of optimality, which is not proved here, and indicate open problems raised by these results.     

利用$(s,t)$-微分算子和拟从属定义的双单叶函数新子类的系数相关问题研究

本文中,我们引进了由$(s,t)$-微分算子和拟从属定义的一类广义双单叶函数类. 对该新函数类及其子类,利用Faber多项式展开式我们得到了前两项系数$|a_2|$, $|a_3|$和一般项系数$|a_n|~(n\geq 4)$的估计,然后也解决了Fekete-Szeg\"{o}问题.     

Lower order eigenvalues of Dirichlet Laplacian

In this paper, we investigate an eigenvalue problem for the Dirichlet Laplacian on a domain in an n-dimensional compact Riemannian manifold. First we give a general inequality for eigenvalues. As one of its applications, we study eigenvalues of the Laplacian on a domain in an n-dimensional complex projective space, on a compact complex submanifold in complex projective space and on the unit sphere. By making use of the orthogonalization of Gram–Schmidt (QR-factorization theorem), we construct trial functions. By means of these trial functions, estimates for lower order eigenvalues are obtained. Qing-Ming Cheng research was partially supported by a Grant-in-Aid for Scientific Research from JSPS. Hejun Sun and Hongcang Yang research were partially supported by NSF of China.