Hyperbolic Harmonic Functions: Weak Approach with Applications in Function Spaces

Harmonic functions with respect to the Poincare metric on the unit ball are called hyperbolic harmonic functions. We establish the weak formulation of hyperbolic harmonic functions and use it in the study of hyperbolic harmonic function spaces. In particular, we give the Carleson measure characterization for the whole spectrum of spaces, whose analytic counterparts include among else Bloch spaces, Bergman-spaces, Besov-spaces, and Q_p-spaces. The second author was supported by the Finnish Cultural Foundation.     

双曲空间中Laplace-Beltrami方程的一个带位移的边值问题

讨论了双曲空间中Laplace-Beltrami方程的一个带位移的边值问题.首先将双曲空间中的Laplace-Beltrami方程的解转化为Clifford分析中的超正则函数.然后给出了超正则函数的Plemelj公式并讨论了相关奇异积分算子的性质,最后利用积分方程的方法和压缩不动点原理证明了Laplace-Beltrami方程的一个带位移的边值问题的解的存在性和唯一性.     

On the existence and uniqueness of solutions to an asymptotic equation of a variational wave equation

We prove the global existence and uniqueness of admissible weak solutions to an asymptotic equation of a nonlinear hyperbolic variational wave equation with nonnegative L ²(ℝ) initial data. The work of Ping Zhang is supported by the Chinese postdoctor’s foundation, and that of Yuxi Zheng is supported in part by NSF DMS-9703711 and the Alfred P. Sloan Research Fellows award.     

On $$$$-Hypermonogenic Functions and Their Mean Value Properties

We study a hyperbolic version of holomorphic functions to higher dimensions. In this frame work, a generalization of holomorphic functions are called \(k\)-hypermonogenic functions. These functions are depending on several real variables and their values are in a Clifford algebra. They are defined in terms of hyperbolic Dirac operators. They are connected to harmonic functions with respect to the Riemannian metric \(ds^{2}_k=x_{n}^{2k/\left( 1-n\right) }\left( dx_{0}^{2}+\cdots +dx_{n}^{2}\right) \) in the same way as the usual harmonic function to holomorphic functions. We present the mean value property for \(k\)-hypermonogenic functions and related results. Earlier the mean value properties has been proved for hypermonogenic functions. The key tools are the invariance properties of the hyperbolic metric.     

Shadowing and Inverse Shadowing for C^1 Endomorphisms

In this paper, we consider the shadowing and the inverse shadowing properties for C^1 endomorphisms. We show that near a hyperbolic set a C^1 endomorphism has the shadowing property, and a hyperbolic endomorphism has the inverse shadowing property with respect to a class of continuous methods. Moreover, each of these shadowing properties is also ＂uniform＂ with respect to C^1 perturbation.     

Proto-derivative formulas for basic subgradient mappings in mathematical programming

Subgradient mappings associated with various convex and nonconvex functions are a vehicle for stating optimality conditions, and their proto-differentiability plays a role therefore in the sensitivity analysis of solutions to problems of optimization. Examples of special interest are the subgradients of the max of finitely manyC ² functions, and the subgradients of the indicator of a set defined by finitely manyC ² constraints satisfying a basic constraint qualification. In both cases the function has a property called full amenability, so the general theory of existence and calculus of proto-derivatives of subgradient mappings associated with fully amenable functions is applicable. This paper works out the details for such examples. A formula of Auslender and Cominetti in the case of a max function is improved in particular.This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under grant OGP41983 for the first author and by the National Science Foundation under grant DMS-9200303 for the second author.     

Semyanistyi’s Integrals and Radon Transforms on Matrix Spaces

We introduce a new analytic family of intertwining operators which include the Radon transform over matrix planes and its inverse. These operators generalize integral transformations introduced by Semyanistyi (Dokl. Akad. Nauk SSSR 134:536–539, [1960]) in his research related to the hyperplane Radon transform in ℝ ⁿ . We obtain an extended version of Fuglede’s formula, connecting generalized Semyanistyi’s integrals, Radon transforms and Riesz potentials on the space of real rectangular matrices. This result combined with the matrix analog of the Hilbert transform leads to variety of new explicit inversion formulas for the Radon transform of functions of matrix argument. The authors were supported in part by the Edmund Landau Center for Research in Mathematical Analysis and Related Areas, sponsored by the Minerva Foundation (Germany). The first author was also supported by Abraham and Sarah Gelbart Research Institute for Mathematical Sciences. The second author was also supported by the NSF grants EPS-0346411 (Louisiana Board of Regents) and DMS-0556157).     

Local Well-posedness of Kinetic Chemotaxis Models

We present a general functional analytic setting in which the Cauchy problem for mild solutions of kinetic chemotaxis models is well-posed, locally in time, in general physical dimensions. The models consist of a hyperbolic transport equation that is non-linearly and non-locally coupled to a reaction-diffusion system through kernel operators. Three examples are elaborated throughout the paper in which the latter system is (1) a single linear equation, (2) a FitzHugh-Nagumo system and (3) a piecewise linear approximation thereof. Finally we present a limit argument to obtain results on solutions in L ¹ ∩ L ^∞. Sander C. Hille: This work is supported by PIONIER grant 600-61410 of the Netherlands Organisation for Scientific Research, NWO.     

Rough Isometry and <Emphasis Type="Italic">p</Emphasis>-Harmonic Boundaries of Complete Riemannian Manifolds

In this paper, we describe the behavior of bounded energy finite solutions for certain nonlinear elliptic operators on a complete Riemannian manifold in terms of its p-harmonic boundary. We also prove that if two complete Riemannian manifolds are roughly isometric to each other, then their p-harmonic boundaries are homeomorphic to each other. In the case, there is a one to one correspondence between the sets of bounded energy finite solutions on such manifolds. In particular, in the case of the Laplacian, it becomes a linear isomorphism between the spaces of bounded harmonic functions with finite Dirichlet integral on the manifolds. This work was supported by grant No. R06-2002-012-01001-0(2002) from the Basic Research Program of the Korea Science & Engineering Foundation.     

ExponentialL 2-convergence andL 2-spectral gap for Markov processes

This paper deals with the exponentialL ²-convergence for jump processes. We introduce some reduction methods and improve some previous results. Then we prove that for birth-death processes, exponentialL ²-convergence coincides indeed with exponential ergodicity which is widely studied in the Markov chain theory. Research supported in part by the National Natural Science Foundation of China and Fok Ying-Tung Educational Foundation     

Ordering of sets, hyperbolic metrics, and harmonic measures

We establish a series of inequalities which relate solutions to certain partial differential equations defined on a given system of open sets with similar solutions defined on the ordered system of sets. As a corollary, we prove a comparison theorem for the hyperbolic metric that allows us to interpret this metric as a Choquet capacity. Using a similar comparison theorem for harmonic measures, we give a solution to S. Segawa's problem on the set having the minimal harmonic measure among all compact sets that lie on the diameter of the unit disk and have a given linear measure. Bibliography: 26 titles.Dedicated to the 90th anniversary of G. M. Goluzin's birthTranslated fromZapiski Nauchnykh Seminarov POMI, Vol. 237, 1997, pp. 129–147.This research was partially supported by the Russian Foundation for Basic Research, grant 97-01-00259.     

Reflection of transversal progressing waves for semilinear strictly hyperbolic systems

We discuss the reflection of weak singularities of solutions to semilinear hyperbolic systems by using the method of energy estimates in the space of conormal distributions, and consequently obtain the general results on the reflection of weak singularities of the solution to a high order semilinear strictly hyperbolic equation. Research partially supported by the National Natural Science Foundation of China     

Hardy–Littlewood Inequalities for Fractional Derivatives of Invariant Harmonic Functions

We establish Hardy–Littlewood inequalities for fractional derivatives of M?bius invariant harmonic functions over the unit ball of \mathbb Rⁿ{\mathbb R^n} in mixed-norm spaces. In doing so we introduce a new criteria for the boundedness of operators in mixed-norm L ^p -spaces in terms of hyperbolic geometry of the real unit ball.     

Solutions for the prescribing mean curvature equation

By variational methods, for a kind of Yamabe problem whose scalar curvature vanishes in the unit ball BN and on the boundary S^N-1 the mean curvature is prescribed, we construct multi-peak solutions whose maxima are located on the boundary as the parameter tends to 0^＋ under certain assumptions. We also obtain the asymptotic behaviors of the solutions.     

Exponential dichotomies for linear differential equations with delays: Periodic and autonomous equations

Summary We consider a homogeneous linear differential equation with delays u ^. +Mu=0,where u takes values in a Banach space and M is a memory with bounded recall under natural Carathéodory conditions, i.e., a linear mapping from the continuous to the locally Bochner-integrable functions such that Mu on [a, b] depends only on u on [a–l, b] for a suitable recall bound l, and M satisfies a reasonable boundedness condition. The presence of the conditional exponential stability behaviour called an exponential dichotomy—which is known to be related to the existence of bounded solutions of the inhomogeneous equation for right-hand sides in a suitable function space — is examined when the equation is periodic; it is shown to be equivalent to a splitting of the spectrum of the transition operators corresponding to a period. More detailed results are obtained when the equation is autonomous.Dedicated toJosé Luis Massera This work was supported in part by National Science Foundation Grant MCS 8200894.     

Landweber iterative methods for angle-limited image reconstruction

We introduce a general iterative scheme for angle-limited image reconstruction based on Landwebet＇s method. We derive a representation formula for this scheme and consequently establish its convergence conditions. Our results suggest certain relaxation strategies for an accelerated convergence for angle-limited image reconstruction in L^2-norm comparing with alternative projection methods. The convolution-backprojection algorithm is given for this iterative process.     

Triangle subgroups of Kleinian groups

We exhibit an interesting new phenomenon concerning certain triangle subgroups Δ of Kleinian groups Γ. Namely the hyperbolic plane Π stabilized by Δ has a precisely invariant tubular neighbourhood. Thus the corresponding 2-orbifoldF ²=∏/Γ _∏ is always embedded in the hyperbolic 3-orbifoldM ³=ℍ³/Γ. We deduce that any two such triangle groups can algebraically intersect only in a finite cyclic subgroup. We give sharp estimates for the radius of these tubular neighbourhoods and present applications concerning the estimation of co-volumes of Kleinian groups containing these triangle subgroups. for J. A. Kalman on the occasion of his 65th birthday Research supported in part by grants from the Australian Research Council, the New Zealand Foundation for Research Science and Technology and the U.K. Scientific and Engineering Research Council.     

Algebras on the unit disk and Toeplitz operators on the Bergman space

This paper studies algebras of functions on the unit disk generated byH ^∞(D) and bounded harmonic functions. Using these algebras, we characterize compact semicommutators and commutators of Toeplitz operators with harmonic symbols on the Bergman space. Supported in part by the National Science Foundation and the University Research Council of Vanderbilt University.     

Boundary structure of hyperbolic 3-manifolds admitting toroidal fillings at large distance

We show that if a hyperbolic 3-manifold M has two toroidal Dehn fillings with distance at least 3, then ∂M consists of at most three tori. As a result, we can obtain an optimal estimate for the number of exceptional slopes on hyperbolic 3-manifolds with boundary a union of at least 4 tori. S. Lee was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund) (KRF-2007-314-C00024). M. Teragaito was supported by Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (C), 19540089.