Flux in axiomatic potential theory. II. Duality

Walsh, Bertram

Annales de l'Institut Fourier, Tome 19 (1969), p. 371-417 / Harvested from Numdam

Résumé

Cet article est la suite d’une publication antérieure [Inventiones Math., 8 (1969), 175-221]. On développe, à partir d’un espace $W$ et d’un faisceau ${\bf H}$ défini là-dessus, satisfaisant aux axiomes de Brelot et, localement, aux hypothèses de la théorie des faisceaux adjoints, les sujets suivants : 1) l’extension de la théorie des faisceaux adjoints au cas où $(W,{\bf H})$ n’admet pas de potentiel global (cas particulier : $W$ compact). 2) La construction d’une nouvelle résolution fine $O\rightarrow {\bf H}\rightarrow {\bf R}\rightarrow {\bf L}\rightarrow O$ de ${\bf H}$, ${\bf L}$ étant un faisceau naturel de mesures sur $W$. 3) La construction d’une dualité naturelle entre $\Gamma (W,{\bf H}^*)$ et $H^1_K(W,{\bf H})$ ($K=$ supports compacts), faisant correspondre le flux à un élément positif distingué de ${\bf H}^*_W$.

Abstract

This is a continuation of an earlier paper [Inventiones Math., 8 (1969), 175-221]. It is assumed that a space $W$ and a sheaf ${\bf H}$ over $W$ are given, such that the pair $(W,{\bf H})$ 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 $(W,{\bf H})$ does not admit a global potential (in particular, the case where $W$ is compact). 2) Construction of a new fine resolution $O\rightarrow {\bf H}\rightarrow {\bf R}\rightarrow {\bf L}\rightarrow O$ of the sheaf ${\bf H}$, in which ${\bf L}$ is a (complete pre-)sheaf of measures on $W$. 3) Construction of a natural duality between the flux functional corresponds to a distinguished positive element of $H^*_W$.

Détail

Publié le : 1969-01-01

MR 42 #2023 | Zbl 0181.11703 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.331

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