Ordered Vector Sequence Spaces and Related Classes of Linear Operators.
Spaces of Continuous Functions Characterized by Kernel-Like Conditions.
Positive Approximate Identities and Lattice-Ordered Dual spaces.
Perturbation of harmonic structures and an index-zero theorem
In the framework of an axiomatic theory of sheaves of “harmonic” functions, a notion of perturbation of these sheaves is introduced which corresponds to the replacement of the operator by an operator , in the classical situation. The “harmonic” functions with which the paper is concerned are assumed to satisfy certain hypotheses (weaker than the axioms of Bauer); it is shown that the perturbed harmonic functions also satisfy these hypotheses. Moreover, the results obtained imply that the dimensions...
Flux in axiomatic potential theory. II. Duality
This is a continuation of an earlier paper [Inventiones Math., 8 (1969), 175-221]. It is assumed that a space and a sheaf over are given, such that the pair satisfies the Brelot axioms and also satisfies, locally, the additional hypotheses of the theory of adjoint sheaves. The following subjects are considered: 1) Extension of the adjoint-sheaf theory to the case where does not admit a global potential (in particular, the case where is compact). 2) Construction of a new fine resolution...
A maximal regular boundary for solutions of elliptic differential equations
Soit une classe harmonique de Brelot, définie sur . Il est donné un critère de régularité en termes de barrières, pour les points d’une frontière idéale. Soit un sous-treillis banachique de . Si est hyperbolique, la frontière idéale compactifiante déterminée par contient une “frontière harmonique” qui satisfait le critère de régularité et . Entre autres applications, on a la théorie des frontières de Wiener et Royden et des comparaisons de classes harmoniques.
The equivalence of Harnack's principle and Harnack's inequality in the axiomatic system of Brelot
Dans l’axiomatique des fonctions harmoniques de Brelot, où l’axiome 3 (de convergence) peut être appelé principe de Harnack, on démontre ici pour les fonctions harmoniques dans un domaine valant 1 en , la propriété d’égale continuité en qui peut se traduire par des “inégalités de Harnack”. Cela avait été établi par Mokobodzki grâce à l’hypothèse d’une base dénombrable d’ouverts, qui est évitée ici en utilisant le théorème d’Éberlein-Smulian.
Behavior of biharmonic functions on Wiener's and Royden's compactifications
Let be a smooth Riemannian manifold of finite volume, its Laplace (-Beltrami) operator. Canonical direct-sum decompositions of certain subspaces of the Wiener and Royden algebras of are found, and for biharmonic functions (those for which ) the decompositions are related to the values of the functions and their Laplacians on appropriate ideal boundaries.
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