Representations of hyperharmonic cones

Eriksson, Sirkka-Liisa

doi:10.1090/S0002-9947-1988-0920157-2

Representations of hyperharmonic cones
HTML articles powered by AMS MathViewer

by Sirkka-Liisa Eriksson PDF

Trans. Amer. Math. Soc. 305 (1988), 247-262 Request permission

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

Maynard Arsove and Heinz Leutwiler, Algebraic potential theory, Mem. Amer. Math. Soc. 23 (1980), no. 226, v+130. MR 550855, DOI 10.1090/memo/0226
Heinz Bauer, Harmonische Räume und ihre Potentialtheorie, Lecture Notes in Mathematics, No. 22, Springer-Verlag, Berlin-New York, 1966 (German). Ausarbeitung einer im Sommersemester 1965 an der Universität 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 Höllein. MR 613980
Nicu Boboc and Aurel Cornea, Cônes convexes ordonnés. $H$-cônes et biadjoints des $H$-cônes, C. R. Acad. Sci. Paris Sér. A-B 270 (1970), A1679–A1682 (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, Die Grundlehren der mathematischen Wissenschaften, Band 158, Springer-Verlag, New York-Heidelberg, 1972. With a preface by H. Bauer. MR 0419799
Aurel Cornea and Sirkka-Liisa Eriksson, Order continuity of the greatest lower bound of two functionals, Analysis 7 (1987), no. 2, 173–184. MR 885123, DOI 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. Ştiinţ. Univ. Al. I. Cuza Iaşi Secţ. I a Mat. 31 (1985), no. 2, 109–116. MR 858048
G. Mokobodzki, Cônes de potentiels et noyaux subordonnés, Potential Theory (C.I.M.E., I Ciclo, Stresa, 1969) Edizioni Cremonese, Rome, 1970, pp. 207–248 (French). MR 0274791

Structure des cônes de potentials

Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039

Cônes de fonctions et potentiels