Flux in axiomatic potential theory. II. Duality
Annales de l'Institut Fourier, Volume 19 (1969) no. 2, p. 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 H over W are given, such that the pair (W,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,H) does not admit a global potential (in particular, the case where W is compact). 2) Construction of a new fine resolution OHRLO of the sheaf H, in which 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 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,H) n’admet pas de potentiel global (cas particulier : W compact). 2) La construction d’une nouvelle résolution fine OHRLO de H, L étant un faisceau naturel de mesures sur W. 3) La construction d’une dualité naturelle entre Γ(W,H * ) et H K 1 (W,H) (K= supports compacts), faisant correspondre le flux à un élément positif distingué de H W * .

@article{AIF_1969__19_2_371_0,
     author = {Walsh, Bertram},
     title = {Flux in axiomatic potential theory. II. Duality},
     journal = {Annales de l'Institut Fourier},
     publisher = {Imprimerie Durand},
     address = {28 - Luisant},
     volume = {19},
     number = {2},
     year = {1969},
     pages = {371-417},
     doi = {10.5802/aif.331},
     zbl = {0181.11703},
     mrnumber = {42 \#2023},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1969__19_2_371_0}
}
Walsh, Bertram. Flux in axiomatic potential theory. II. Duality. Annales de l'Institut Fourier, Volume 19 (1969) no. 2, pp. 371-417. doi : 10.5802/aif.331. http://www.numdam.org/item/AIF_1969__19_2_371_0/

[1] H. Bauer, Harmonische Räume und ihre Potentialtheorie, Springer Lecture Notes in Mathematics 22 (1966). | Zbl 0142.38402

[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 32 #2603 | Zbl 0139.06604

[3] G. E. Bredon, Sheaf Theory, McGraw-Hill, (1967). | MR 36 #4552 | Zbl 0158.20505

[4] M. Brelot, Lectures on Potential Theory, Tata Institute, Bombay, 1960. | MR 22 #9749 | Zbl 0098.06903

[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 22 #8302 | Zbl 0084.10402

[7] R. C. Gunning, Lectures on Riemann Surfaces, Princeton Univ. Press, 1966. | MR 34 #7789 | Zbl 0175.36801

[8] R. C. Gunning and H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. | MR 31 #4927 | Zbl 0141.08601

[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 25 #3186 | Zbl 0101.08103

[10] P. A. Loeb, An axiomatic treatment of pairs of elliptic differential equations, Ann. Inst. Fourier (Grenoble) 16 (1966), 167-208. | Numdam | MR 37 #3039 | Zbl 0172.15101

[11] F.-Y. Maeda, Axiomatic treatment of full-superharmonic functions, J. Sci. Hiroshima Univ. Ser. A-1 30 (1966), 197-215. | Zbl 0168.09702

[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 29 #260 | Zbl 0116.30404

[13] B. Rodin and L. Sario, Principal Functions, van Nostrand, Princeton, 1968. | MR 37 #5378 | Zbl 0159.10701

[14] H. Schaefer, Topological Vector Spaces, Macmillan, New York, 1966. | MR 33 #1689 | Zbl 0141.30503

[15] H. Schaefer, Invariant ideals of positive operators in C(X), I, Illinois J. Math. 11 (1967), 703-715. | MR 36 #1996 | Zbl 0168.11801

[16] B. Walsh and P. A. Loeb, Nuclearity in axiomatic potential theory, Bull. Amer. Math. Soc. 72 (1966), 685-689. | MR 35 #407 | Zbl 0144.15503

[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 36 #6914 | Zbl 0165.14702

[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 37 #3040 | Zbl 0153.15501

[20] H.-G. Tillmann, Dualität in der Potentialtheorie, Port. Math. 13 (1954), 55-86. | MR 16,718b | Zbl 0056.33403

[21] B. Walsh, Flux in axiomatic potential theory. I : Cohomology, Inventiones Math. 8 (1969), 175-221. | MR 42 #532 | Zbl 0179.15203