Differentiability and integrability in \(n\) dimensions with respect to \(\alpha\)-regular intervals. (English) Zbl 0764.28005

Kurzweil, Jaroslav; Jarník, Jiří

Result. Math. 21, No. 1-2, 138-151 (1992).

The regularity of an interval (i.e., the ratio of its shortest and its longest edge) plays an important role in multidimensional differentiation and integration theory. The present paper is devoted to an analysis of its relevance in multidimensional generalized Riemann integrals and in differentiability theory for additive interval functions. If, for \(\alpha<1\), a concept of \(\alpha\)-regular differentiability of an additive interval function is defined in a natural way, it is proved in the paper that the \(\alpha\)-differentiability at one point implies the \(\beta\)-differentiability at this point for any \(0<\beta<\alpha\). The situation is quite different for \(\alpha\)-regular integrability based upon the Kurzweil-Henstock Riemann-type definition restricted of \(\alpha\)-regular intervals. In this case, it is shown that given \(\alpha\in]0,1[\), there exists a function \(f=f_ \alpha\) which is \(\alpha_ 1\)-regularly integrable for every \(\alpha_ 1\in]\alpha,1[\) and is not \(\alpha_ 2\)-regularly integrable for every \(\alpha_ 2\in]0,\alpha[\) on the interval \([-1,2]^ n\).

Reviewer: J.Mawhin (Louvain-La-Neuve)

MSC:

28A15	Abstract differentiation theory, differentiation of set functions
26B99	Functions of several variables
26A42	Integrals of Riemann, Stieltjes and Lebesgue type
26A39	Denjoy and Perron integrals, other special integrals

Keywords:

non absolute integration; generalized Perron integral; \(\alpha\)- differentiability; \(\alpha\)-regular integrability; regularity of an interval; multidimensional generalized Riemann integrals; additive interval functions

References:

[1]	Kurzweil J. and Jarník J: Equiintegrability and controlled convergence of Perron-type integrable functions. Real Analysis Exchange, (1991), in print. · Zbl 0754.26003
[2]	Mawhin J. and Pfeffer W.F.: Hake’s property of a multidimensional generalized Riemann integral. Czechoslovak Mathematical Journal 40 (115) (1990), 690-694 · Zbl 0746.26009