Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Representations of hyperharmonic cones


Author: Sirkka-Liisa Eriksson
Journal: Trans. Amer. Math. Soc. 305 (1988), 247-262
MSC: Primary 31D05
DOI: https://doi.org/10.1090/S0002-9947-1988-0920157-2
MathSciNet review: 920157
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Hyperharmonic cones are ordered convex cones possessing order properties similar to those of hyperharmonic functions on harmonic spaces. The dual of a hyperharmonic cone is defined to be the set of extended real-valued additive and left order-continuous mappings $(\not \equiv \infty )$. The second dual gives a representation of certain hyperharmonic cones in which suprema of upward directed families are pointwise suprema, although infima of pairs of functions are not generally pointwise infima. We obtain necessary and sufficient conditions for the existence of a representation of a hyperharmonic cone in which suprema of upward directed families are pointwise suprema and infima of pairs of functions are pointwise infima.


References [Enhancements On Off] (What's this?)

  • Maynard Arsove and Heinz Leutwiler, Algebraic potential theory, Mem. Amer. Math. Soc. 23 (1980), no. 226, v+130. MR 550855, DOI https://doi.org/10.1090/memo/0226
  • Heinz Bauer, Harmonische R盲ume und ihre Potentialtheorie, Lecture Notes in Mathematics, No. 22, Springer-Verlag, Berlin-New York, 1966 (German). Ausarbeitung einer im Sommersemester 1965 an der Universit盲t Hamburg gehaltenen Vorlesung. MR 0210916
  • Nicu Boboc, Gheorghe Bucur, and Aurel Cornea, Order and convexity in potential theory: $H$-cones, Lecture Notes in Mathematics, vol. 853, Springer, Berlin, 1981. In collaboration with Herbert H枚llein. MR 613980
  • Nicu Boboc and Aurel Cornea, C么nes convexes ordonn茅s. $H$-c么nes et biadjoints des $H$-c么nes, C. R. Acad. Sci. Paris S茅r. A-B 270 (1970), A1679鈥揂1682 (French). MR 273048
  • Marcel Brelot, On topologies and boundaries in potential theory, Lecture Notes in Mathematics, Vol. 175, Springer-Verlag, Berlin-New York, 1971. Enlarged edition of a course of lectures delivered in 1966. MR 0281940
  • Corneliu Constantinescu and Aurel Cornea, Potential theory on harmonic spaces, Springer-Verlag, New York-Heidelberg, 1972. With a preface by H. Bauer; Die Grundlehren der mathematischen Wissenschaften, Band 158. MR 0419799
  • Aurel Cornea and Sirkka-Liisa Eriksson, Order continuity of the greatest lower bound of two functionals, Analysis 7 (1987), no. 2, 173鈥184. MR 885123, DOI https://doi.org/10.1524/anly.1987.7.2.173
  • Sirkka-Liisa Eriksson, Hyperharmonic cones and hyperharmonic morphisms, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes 49 (1984), 75. MR 746342
  • Sirkka-Liisa Eriksson, Hyperharmonic cones and cones of hyperharmonics, An. 艦tiin牛. Univ. Al. I. Cuza Ia艧i Sec牛. I a Mat. 31 (1985), no. 2, 109鈥116. MR 858048
  • G. Mokobodzki, C么nes de potentiels et noyaux subordonn茅s, Potential Theory (C.I.M.E., I Ciclo, Stresa, 1969) Edizioni Cremonese, Rome, 1970, pp. 207鈥248 (French). MR 0274791
  • ---, Structure des c么nes de potentials, S茅minaire Bourbaki 377, 1970. pp. 239-252.
  • Helmut H. Schaefer, Banach lattices and positive operators, Springer-Verlag, New York-Heidelberg, 1974. Die Grundlehren der mathematischen Wissenschaften, Band 215. MR 0423039
  • D. Sibony, C么nes de fonctions et potentiels, Cours de 3 茅me Cycle 脿 la Facult茅 des Sciences de Paris (1967-68) et 脿 l鈥橴niversit茅 McGill de Montr茅al (茅t茅 1968).

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 31D05

Retrieve articles in all journals with MSC: 31D05


Additional Information

Article copyright: © Copyright 1988 American Mathematical Society