Positivity 2: 19–45, 1998.
© 1998 Kluwer Academic Publishers. Printed in the Netherlands.
On Maps Of Bounded p -Variation With p>1
V. V. CHISTYAKOV
and O. E. GALKIN
Department of Mathematics, University of Nizhny Novgorod, 23 Gagarin Avenue, Nizhny Novgorod,
(Received: 25 April 1997; Accepted in revised form 3 November 1997)
Abstract. This paper addresses properties of maps of bounded p-variation (p>1) in the sense of
N. Wiener, which are deﬁned on a subset of the real line and take values in metric or normed spaces.
We prove the structural theorem for these maps and study their continuity properties. We obtain
the existence of a Hölder continuous path of minimal p-variation between two points and establish
the compactness theorem relative to the p-variation, which is an analog of the well-known Helly
selection principle in the theory of functions of bounded variation. We prove that the space of maps
of bounded p-variation with values in a Banach space is also a Banach space. We give an example
of a Hölder continuous of exponent 0 <γ <1 set-valued map with no continuous selection. In the
case p = 1 we show that a compact absolutely continuous set-valued map from the compact interval
into subsets of a Banach space admits an absolutely continuous selection.
Mathematics Subject Classiﬁcations (1991): Primary: 26A45, 26A16; Secondary: 49J45, 54C60,
Key words: maps of bounded p -variation, maps with values in metric spaces, Hölder continuous
maps, minimal paths, Helly’s selection principle, set-valued maps, selections
The purpose of this paper is to obtain properties of maps of bounded p-variation
in the classical sense of Norbert Wiener (cf. Wiener, 1924, and Young, 1937).
Consider a map f : E → X of bounded p-variation (see Sec. 2) deﬁned on the
nonempty subset E of the reals
with values in the metric or normed space X.
If p = 1, the properties of the variation in the sense of C. Jordan (cf. Schwartz,
1967) were recently studied by the ﬁrst author (Chistyakov, 1992 and 1997). Here
we concentrate mainly on the case where p>1. If X =
, p = 1andEis a
closed bounded interval [ a, b ] or an open interval ] a,b [, then, f : E →
function of bounded variation if and only if it is the difference of two bounded non-
decreasing functions (Jordan’s decomposition); see, e.g., Natanson (1965), Ch. 8.
However, this criterion is inapplicable if p>1, to say nothing of the case where
X is a metric or a normed vector space.
If p>1andXis a metric space, we show that f : E → X is a map of bounded
p-variation if and only if it is the composition of a bounded nondecreasing function
Partially supported by the Russian Fund for Fundamental Research, Grant No. 96-01-00278.
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