Previous |  Up |  Next


Henstock integral; Stieltjes integral; Young integral; $\phi $-variation
In 1938, L. C. Young proved that the Moore-Pollard-Stieltjes integral $\int _a^bf\mathrm{d}g$ exists if $f\in \mathop {{\mathrm BV}}_\phi [a,b]$, $g\in \mathop {{\mathrm BV}}_\psi [a,b]$ and $\sum _{n=1}^\infty \phi ^{-1}({1}/{n})\psi ^{-1} ({1}/{n})<\infty $. In this note we use the Henstock-Kurzweil approach to handle the above integral defined by Young.
[1] Boonpogkrong Varayu, Tuan Seng Chew: On integrals with integrators in $\mathop {{\mathrm BV}}_p$. Real Anal. Exch. 30 (2004/2005), 193–200. MR 2127525
[2] R. M. Dudley, R. Norvaisa: Differentiability of Six Operators on Nonsmooth Functions and $p$-Variation. Springer, Berlin, 1999. MR 1705318
[3] I. J. L. Garces, Peng-Yee Lee, Dongsheng Zhao: Moore-Smith limits and the Henstock integral. Real Anal. Exchange 24 (1998/1999), 447–455. MR 1691764
[4] P. Y. Lee, R. Výborný: The Integral. An Easy Approach after Kurzweil and Henstock. Cambridge University Press, 2000. MR 1756319
[5] R. Lesniewicz, W. Orlicz: On generalized variations (II). Stud. Math. 45 (1973), 71–109. MR 0346509
[6] J. Musielak, W. Orlicz: On generalized variations (I). Stud. Math. 18 (1959), 11–41. MR 0104771
[7] E. R. Love, L. C. Young: On fractional integration by parts. Proc. London Math. Soc., II. Ser., 44 (1938), 1–35. MR 1575481
[8] E. R. Love: Integration by parts and other theorems for $R^3S$-integrals. Real Anal. Exch. 24 (1998/1999), 315–336. MR 1691754
[9] R. Norvaisa: Quadratic Variation, p-Variation and Integration with Applications to Strock Price Modelling. Preprint, 2003.
[10] Š. Schwabik: A note on integration by parts for abstract Perron-Stieltjes integrals. Math. Bohem. 126 (2001), 613–629. MR 1970264 | Zbl 0980.26005
[11] L. C. Young: An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67 (1936), 251–282. MR 1555421 | Zbl 0016.10404
[12] L. C. Young: General inequalities for Stieltjes integrals and the convergence of Fourier series. Math. Ann. 115 (1938), 581–612. MR 1513204
Partner of
EuDML logo