Flux in axiomatic potential theory. II. Duality

Walsh, Bertram

Annales de l'Institut Fourier, Tome 19 (1969) no. 2, pp. 371-417

This is a continuation of an earlier paper [Inventiones Math., 8 (1969), 175-221]. It is assumed that a space $W$ and a sheaf $\mathbf{H}$ over $W$ are given, such that the pair $(W,\mathbf{H})$ 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 $(W,\mathbf{H})$ does not admit a global potential (in particular, the case where $W$ is compact). 2) Construction of a new fine resolution $O\to \mathbf{H}\to \mathbf{R}\to \mathbf{L}\to O$ of the sheaf $\mathbf{H}$, in which $\mathbf{L}$ 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 $H_W^{*}$.

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 $\mathbf{H}$ 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 $(W,\mathbf{H})$ n’admet pas de potentiel global (cas particulier : $W$ compact). 2) La construction d’une nouvelle résolution fine $O\to \mathbf{H}\to \mathbf{R}\to \mathbf{L}\to O$ de $\mathbf{H}$, $\mathbf{L}$ étant un faisceau naturel de mesures sur $W$. 3) La construction d’une dualité naturelle entre $\Gamma (W,{\mathbf{H}}^{*})$ et $H_K^1(W,\mathbf{H})$ ($K=$ supports compacts), faisant correspondre le flux à un élément positif distingué de ${\mathbf{H}}_W^{*}$.

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