Existence of Tangential Limits for α-Harmonic Functions on Half Spaces

Our aim in this paper is to prove the existence of tangential limits for Poisson integrals of the fractional order of functions in the L ^p Hölder space on half spaces.     

Harmonic functions associated to Dunkl operators

We prove some potential theoretical properties of harmonic functions associated to Dunkl operators. We solve the corresponding Dirichlet problem and establish the related Harnack principle and normality criteria.     

On boundary behavior of harmonic functions in Hölder domains

We analyze the boundary behavior of harmonic functions in a domain whose boundary is locally given by a graph of a Hölder continuous function. In particular we give a non-probabilistic proof of a Harnack-type principle, due to Bañuelos et al. and study some properties of the harmonic measure.     

Muckenhoupt weights and Lindelöf theorem for harmonic mappings

We extend the result of Lavrentiev which asserts that the harmonic measure and the arc-length measure are _A∞$A_{\infty }$ equivalent in a chord-arc Jordan domain. By using this result we extend the classical result of Lindelöf to the class of quasiconformal (q.c.) harmonic mappings by proving the following assertion. Assume that f is a quasiconformal harmonic mapping of the unit disk U onto a Jordan domain. Then the function A(z)=arg?(_∂φ(f(z))/z)$A(z)=\arg ?({\partial }_{\phi }(f(z))/z)$ where z=re^iφ$z=r{e}^{i\phi }$, is well-defined and smooth in U^?={z:0<|z|<1}${\mathbf{U}}^{?}=\{z:0<|z|<1\}$ and has a continuous extension to the boundary of the unit disk if and only if the image domain has C¹$C^1$ boundary.     

Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

We use the heat equation to establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric spaces. The metric spaces under consideration are those that are endowed with a doubling measure supporting a (1,2)-Poincaré inequality and in addition supporting a corresponding Sobolev-Poincaré-type inequality for the modification of the measure obtained via the heat kernel. Examples are given to illustrate the necessity of our assumptions on these spaces. We also provide an example to show that in the general setting the best possible regularity for the Cheeger-harmonic functions is Lipschitz continuity.     

Hölder estimates of p-harmonic extension operators

It is now a well-known fact that for 1<p<∞ the p-harmonic functions on domains in metric measure spaces equipped with a doubling measure supporting a (1,p)-Poincaré inequality are locally Hölder continuous. In this note we provide a characterization of domains in such metric spaces for which p-harmonic extensions of Hölder continuous boundary data are globally Hölder continuous. We also provide a link between this regularity property of the domain and the uniform p-fatness of the complement of the domain.     

The mixed problem for the Lamé system in a class of Lipschitz domains

We consider the mixed problem for the Lamé system

     

Application du prolongement analytique au problème inverse du potentiel

Sommaire Le but de cet article est établir quelques résultats nouveaux sur le problème inverse du potentiel newtonien. Nous démontrons deux théorèmes d'unicité: pour les polyédres convexes dansR ⁿ et pour les lemniscates dansR ². L'instrument principal est un lemme basé sur une idée de V. Kondrachkov rarement utilisé malgré sa puissance. Nous montrons son efficacité en liaison avec la méthode du prolongement analytique des potentiels.

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The goal of this paper is to establish some new results in the inverse Newtonian potential problem. We prove two uniqueness theorems: for convex polyhedra inR ⁿ and for lemniscates inR ². The main tool is a lemma based upon an idea of V. Kondrashkov which, though powerful, is rarely used. We show its efficiency applied together with the method of analytic continuation of potentials.

     

Potential theory of subordinate Brownian motions with Gaussian components

In this paper we study a subordinate Brownian motion with a Gaussian component and a rather general discontinuous part. The assumption on the subordinator is that its Laplace exponent is a complete Bernstein function with a Lévy density satisfying a certain growth condition near zero. The main result is a boundary Harnack principle with explicit boundary decay rate for non-negative harmonic functions of the process in C^1,1$C^{1,1}$ open sets. As a consequence of the boundary Harnack principle, we establish sharp two-sided estimates on the Green function of the subordinate Brownian motion in any bounded C^1,1$C^{1,1}$ open set D$D$ and identify the Martin boundary of D$D$ with respect to the subordinate Brownian motion with the Euclidean boundary.     

Uniform domains and quasiconformal mappings on the Heisenberg group

We prove that in the Heisenberg group the image of a uniform domain under a global quasiconformal homeomorphism is still a uniform domain. As a consequence, the class of NTA (non-tangentially accessible) domains in the Heisenberg group is also quasiconformally invariant. A large class of non-differentiable Lipschitz quasiconformal homeomorphisms is constructed. The images of smooth domains under these rough mappings give a class of non-smooth NTA domains in the Heisenberg group.     

A Fatou theorem for α-harmonic functions

We study functions which are harmonic in the upper half space with respect to (−Δ)^α/2, 0<α<2. We prove a Fatou theorem when the boundary function is L^p-Hölder continuous of order β and βp>1. We give examples to show this condition is sharp.     

Dynkin's formula and large coupling convergence

Let H?1 be a selfadjoint operator in H, let J be a linear and bounded operator from (D(H^1/2),∥H^1/2·∥) to H_aux and for β>0 let be the nonnegative selfadjoint operator in H satisfying

     

Dirichlet operators: A priori estimates and uniqueness problems II

We study some basic properties of the Dirichlet operator in weighted L^p$L^p$ spaces. We work under rather general assumptions on weights that destroy minimal local Sobolev regularity of the form domain.     

Asymptotic behaviour of equicoercive diffusion energies in dimension two

In this paper, we study the asymptotic behaviour of a given equicoercive sequence of diffusion energies F _n , , defined in L ²(Ω), for a bounded open subset Ω of . We prove that, contrary to the dimension three (or greater), the Γ-limit of any convergent subsequence of F _n is still a diffusion energy. We also provide an explicit representation formula of the Γ-limit when its domains contains the regular functions with compact support in Ω. This compactness result is based on the uniform convergence satisfied by some minimizers of the equicoercive sequence F _n , which is specific to the dimension two. The compactness result is applied to the period framework, when the energy density is a highly oscillating sequence of equicoercive matrix-valued functions. So, we give a definitive answer to the question of the asymptotic behaviour of periodic conduction problems under the only assumption of equicoerciveness for the two-dimensional conductivity.     

Problème inverse de la théorie du potentiel logarithmique pour les lemniscates

The paper is related to the question of uniqueness in the inverse logarithmic potential problem. This question is to find the conditions on which two domains D ₁ and D ₂ producing the same external potential must coincide. Assuming the general hypothesis of regularity and an additional condition of connectivity of (D₁D₂)_c, we prove a theorem of uniqueness in the case when one of the domains is a lemniscate. The main tool is one lemma for Cauchy's potential due to M. Sakai. We give a simple proof of its extension to Newtonian potential.

     

Weighted norm inequalities for operators of potential type and fractional maximal functions

In this paper, we study two-weight norm inequalities for operators of potential type in homogeneous spaces. We improve some of the results given in [6] and [8] by significantly weakening their hypotheses and by enlarging the class of operators to which they apply. We also show that corresponding results of Carleson type for upper half-spaces can be derived as corollaries of those for homogeneous spaces. As an application, we obtain some necessary and sufficient conditions for a large class of weighted norm inequalities for maximal functions under various assumptions on the measures or spaces involved.Research of the first author was supported in part by NSERC grant A5149.Research of the second author was supported in part by NSF grant DMS93-02991.     

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.     

Asymptotics of Green functions and Martin boundaries for elliptic operators with periodic coefficients

We give the asymptotics at infinity of a Green function for an elliptic equation with periodic coefficients on R^d. Basic ingredients in establishing the asymptotics are an integral representation of the Green function and the saddle point method. We also completely determine the Martin compactification of R^d with respect to an elliptic equation with periodic coefficients by using the exact asymptotics at infinity of the Green function.