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 of the sheaf , in which is a (complete pre-)sheaf of measures on . 3) Construction of a natural duality between the flux functional corresponds to a distinguished positive element of .
Cet article est la suite d’une publication antérieure [Inventiones Math., 8 (1969), 175-221]. On développe, à partir d’un espace et d’un faisceau 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 n’admet pas de potentiel global (cas particulier : compact). 2) La construction d’une nouvelle résolution fine de , étant un faisceau naturel de mesures sur . 3) La construction d’une dualité naturelle entre et ( supports compacts), faisant correspondre le flux à un élément positif distingué de .
@article{AIF_1969__19_2_371_0, author = {Walsh, Bertram}, title = {Flux in axiomatic potential theory. {II.} {Duality}}, journal = {Annales de l'Institut Fourier}, pages = {371--417}, publisher = {Institut Fourier}, address = {Grenoble}, volume = {19}, number = {2}, year = {1969}, doi = {10.5802/aif.331}, zbl = {0181.11703}, mrnumber = {42 #2023}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.331/} }
TY - JOUR AU - Walsh, Bertram TI - Flux in axiomatic potential theory. II. Duality JO - Annales de l'Institut Fourier PY - 1969 SP - 371 EP - 417 VL - 19 IS - 2 PB - Institut Fourier PP - Grenoble UR - https://aif.centre-mersenne.org/articles/10.5802/aif.331/ DO - 10.5802/aif.331 LA - en ID - AIF_1969__19_2_371_0 ER -
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. https://aif.centre-mersenne.org/articles/10.5802/aif.331/
[1] Harmonische Räume und ihre Potentialtheorie, Springer Lecture Notes in Mathematics 22 (1966). | Zbl
,[2] Axiomatic theory of harmonic functions : Nonnegative superharmonic functions, Ann. Inst. Fourier (Grenoble) 15 (1965), 283-312. | Numdam | MR | Zbl
, and ,[3] Sheaf Theory, McGraw-Hill, (1967). | MR | Zbl
,[4] Lectures on Potential Theory, Tata Institute, Bombay, 1960. | MR | Zbl
,[5] Lectures on Sheaf Theory, Tata Institute, Bombay, 1957.
,[6] Linear Operators I, Interscience, New York, 1958. | MR | Zbl
and ,[7] Lectures on Riemann Surfaces, Princeton Univ. Press, 1966. | MR | Zbl
,[8] Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. | MR | Zbl
and ,[9] Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962), 415-571. | Numdam | MR | Zbl
,[10] An axiomatic treatment of pairs of elliptic differential equations, Ann. Inst. Fourier (Grenoble) 16 (1966), 167-208. | Numdam | MR | Zbl
,[11] Axiomatic treatment of full-superharmonic functions, J. Sci. Hiroshima Univ. Ser. A-1 30 (1966), 197-215. | Zbl
,[12] Brelot's axiomatic theory of the Dirichlet problem and Hunt's theory, Ann. Inst. Fourier (Grenoble) 13 (1963), 357-372. | Numdam | MR | Zbl
,[13] Principal Functions, van Nostrand, Princeton, 1968. | MR | Zbl
and ,[14] Topological Vector Spaces, Macmillan, New York, 1966. | MR | Zbl
,[15] Invariant ideals of positive operators in C(X), I, Illinois J. Math. 11 (1967), 703-715. | MR | Zbl
,[16] Nuclearity in axiomatic potential theory, Bull. Amer. Math. Soc. 72 (1966), 685-689. | MR | Zbl
and ,[17] Intégration, Ch. V : Intégration des Mesures, Hermann et Cie, Paris, 1956.
,[18] Randintegrale und nukleare Funktionenräume, Ann. Inst. Fourier (Grenoble) 17 (1967), 225-271. | Numdam | MR | Zbl
,[19] 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] Dualität in der Potentialtheorie, Port. Math. 13 (1954), 55-86. | MR | Zbl
,[21] Flux in axiomatic potential theory. I : Cohomology, Inventiones Math. 8 (1969), 175-221. | MR | Zbl
,Cited by Sources: