A General Perron Integral, II

Bullen, P. S.

Canadian journal of mathematics, Tome 19 (1967) no. 1, pp. 457-473

Résumé

This paper continues work begun in a previous paper of the same title (7), which will be called I; results from I will be referred to as Theorem 1.4, Axiom 1.1 etc. The notation used in the present paper will, except where noted, be that of I, to which reference should be made for further details.In § 2, certain ideas presented in I are modified to give a neater and more general theory and then some new results of this theory are added. The remaining two sections develop some of the examples mentioned in I, § 5.

Détail

DOI : 10.4153/CJM-1967-039-8

Bibliographie

[1] 1. Arsove, M. G., Functions representable as differences of subharmonic functions, Trans. Amer. Math. Soc., 75 (1953), 327–365. Google Scholar

[2] 2. Bauer, H., Axiomatische Behandlung des Dirichletschen Problems für elliptische und parabolische Differentialgleichungen, Math. Ann., 146 (1962), 1–59. Google Scholar

[3] 3. Bauer, H., Weiterführung einer axiomatischen Potentialtheorie ohne Kern (Existenz von Potentialen), Z. Wahrscheinlichkeitstheorie, 1 (1963), 197–229. Google Scholar

[4] 4. Bauer, H., Propriétés fines des fonctions surharmonique s dans une théorie axiomatique du potential, Colloq. Intern, du C.N.R.S. (Théorie du Potentiel), 1964. Google Scholar

[5] 5. Bonsall, F. F., On generalised subharmonic functions, Proc. Cambridge Philos. Soc., 46 (1950), 387–395. Google Scholar

[6] 6. Brelot, M. and Choquet, G., Espaces et lignes de Green, Ann. Inst. Fourier, Grenoble, 8 (1951), 199–263. Google Scholar

[7] 7. Bullen, P. S., A general Perron integral, Can. J. Math., 17 (1965), 17–30. Google Scholar

[8] 8. Burkill, J. C., The Cesàro-Perron scale of integration, Proc. London Math. Soc., 89, 2 (1935), 541–552. Google Scholar

[9] 9. Denjoy, A., Leçons sur le calcul des coefficients d'une série trigonométrique (Paris, 1941). Google Scholar

[10] 10. Doob, J., A probability approach to the heat equations, Trans. Amer. Math. Soc., 80 (1955), 216–280. Google Scholar

[11] 11. Dynkin, E. B., Markov processes (New York, 1965). Google Scholar

[12] 12. Gilbarg, D. and Serrin, J., On isolated singularities of second order elliptic differential equations, J. Analyse Math., 4 (1954-56), 309–340. Google Scholar

[13] 13. Hervé, R.-M., Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier, Grenoble, 12 (1962), 415–571. Google Scholar

[14] 14. Ito, S., On existence of Green function and positive superharmonic functions for linear elliptic operators of second order, J. Math. Soc. Japan, 16 (1964), 299–306. Google Scholar

[15] 15. James, R. D., A generalised integral, II, Can. J. Math., 2 (1950), 297–306. Google Scholar

[16] 16. James, R. D. and Gage, W. H., A generalised integral, Trans. Roy. Soc. Canada, 3rd Ser., Sec. III, 40 (1946), 25–35. Google Scholar

[17] 17. Meyer, P.-A., Brelot's axiomatic theory of the Dirichlet problem and Hunt's theory, Ann. Inst. Fourier, Grenoble, 18 (1963), 357–372. Google Scholar

[18] 18. Miranda, C., Equazione aile derivate parziali di tipo ellito (Berlin, 1955). Google Scholar

[19] 19. Rudin, W., Integral representation of continuous functions, Trans. Amer. Math. Soc., 70 (1951), 387–403. Google Scholar

[20] 20. Rudin, W., A theorem on subharmonic functions, Proc Amer. Math. Soc., 2 (1951), 209–212. Google Scholar

[21] 21. Rudin, W., Inversion of second order differential operators, Proc. Amer. Math. Soc., 8 (1952), 92–98. Google Scholar

[22] 22. Saks, S., On the operators of Blaschke and Privaloff for subharmonic functions, Mat. Sb., 9 (1941), 451–456. Google Scholar

[23] 23. Zygmund, A., Trigonometrical series, Vol. 1 (Cambridge, 1959). Google Scholar