Flux in axiomatic potential theory. II. Duality

Walsh, Bertram

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

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 ${\mathbf{H}}_W^{*}$.

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 $H_W^{*}$.

[1] H. Bauer, Harmonische Räume und ihre Potentialtheorie, Springer Lecture Notes in Mathematics 22 (1966). | Zbl

[2] N. Boboc, C. Constantinescu and A. Cornea, Axiomatic theory of harmonic functions : Nonnegative superharmonic functions, Ann. Inst. Fourier (Grenoble) 15 (1965), 283-312. | Numdam | MR | Zbl

[3] G. E. Bredon, Sheaf Theory, McGraw-Hill, (1967). | MR | Zbl

[4] M. Brelot, Lectures on Potential Theory, Tata Institute, Bombay, 1960. | MR | Zbl

[5] C. H. Dowker, Lectures on Sheaf Theory, Tata Institute, Bombay, 1957.

[6] N. Dunford and J. T. Schwartz, Linear Operators I, Interscience, New York, 1958. | MR | Zbl

[7] R. C. Gunning, Lectures on Riemann Surfaces, Princeton Univ. Press, 1966. | MR | Zbl

[8] R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. | MR | Zbl

[9] R.-M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962), 415-571. | Numdam | MR | Zbl

[10] P. A. Loeb, An axiomatic treatment of pairs of elliptic differential equations, Ann. Inst. Fourier (Grenoble) 16 (1966), 167-208. | Numdam | MR | Zbl

[11] F.-Y. Maeda, Axiomatic treatment of full-superharmonic functions, J. Sci. Hiroshima Univ. Ser. A-1 30 (1966), 197-215. | Zbl

[12] P. A. Meyer, Brelot's axiomatic theory of the Dirichlet problem and Hunt's theory, Ann. Inst. Fourier (Grenoble) 13 (1963), 357-372. | Numdam | MR | Zbl

[13] B. Rodin and L. Sario, Principal Functions, van Nostrand, Princeton, 1968. | MR | Zbl

[14] H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966. | MR | Zbl

[15] H. Schaefer, Invariant ideals of positive operators in C(X), I, Illinois J. Math. 11 (1967), 703-715. | MR | Zbl

[16] B. Walsh and P. A. Loeb, Nuclearity in axiomatic potential theory, Bull. Amer. Math. Soc. 72 (1966), 685-689. | MR | Zbl

[17] N. Bourbaki, Intégration, Ch. V : Intégration des Mesures, Hermann et Cie, Paris, 1956.

[18] D. Hinrichsen, Randintegrale und nukleare Funktionenräume, Ann. Inst. Fourier (Grenoble) 17 (1967), 225-271. | Numdam | MR | Zbl

[19] A. De La Pradelle, Approximation et caractère de quasi-analyticité dans la théorie axiomatique des fonctions harmoniques, Ann. Inst. Fourier (Grenoble) 17 (1967), 383-399. | Numdam | MR | Zbl

[20] H.-G. Tillmann, Dualität in der Potentialtheorie, Port. Math. 13 (1954), 55-86. | MR | Zbl

[21] B. Walsh, Flux in axiomatic potential theory. I : Cohomology, Inventiones Math. 8 (1969), 175-221. | MR | Zbl