Product integrals, young integrals and p-variation

  • R. M. Dudley
  • R. Norvaiša
Part of the Lecture Notes in Mathematics book series (LNM, volume 1703)


Let \(\mathbb{B}\)be a Banach algebra with identity \(\mathbb{I}\). Consider a function f defined on an interval [a, b] and with values in \(\mathbb{B}\). Let {a=x0<...<b=x n } be a partition of [a, b]. Then the product integral with respect to f over [a, b] is defined as the limit of the product from i=1 to n of \(\mathbb{I}\)+f(xi-1), if it exists, where the limit is taken under refinements of partitions. It is proved that the product integral with respect to f over [a, b] exists if fW p ([a,b];)\(\mathbb{B}\), 0<p<2, i.e., if f has bounded p-variation for some 0<p<2, as shown for f continuous by M. A. Freedman, Trans. Amer. Math. Soc. 279 (1983), 95–112. A necessary and sufficient condition for the existence of the product integral is given when \(\mathbb{B} = \mathbb{R}\). An operator \(\mathcal{P}_a\)from \(\mathcal{W}_p ([a,b];\mathbb{B})\)into itself is induced by an indefinite product integral. The main result says that \(\mathcal{P}_a\)is Fréchet differentiable. R. D. Gill and S. Johansen (Ann. Statist. 18, 1990, 1501–1555) had shown compact differentiability in the supremum norm, on sets uniformly bounded in 1-variation norm. The present paper shows that when restricted to rightor left-continuous elements of \(\mathcal{W}_p ([a,b];\mathbb{B})\), \(\mathcal{P}_a\)is analytic. To prove these results a generalized Stieltjes integral due to L. C. Young is developed, as are variants of it called left Young (LY) and right Young (RY) integrals, and the Duhamel formula is extended to (LY) and (RY) integrals. Also using Young integrals a logarithm operator \(\mathcal{L}_a\)is defined so that \(\mathcal{L}_a (f)\)exists for each \(f \in \mathcal{W}_p ([a,b];\mathbb{B})\), 0<p<2, such that the function xf(x)−1 is in \(\ell _\infty ([a,b];\mathbb{B})\). The operators \(\mathcal{P}_a\)and \(\mathcal{L}_a\)are shown to be inverses of each other. This allows one to determine a unique solution of a linear integral equation and to solve an evolution representation problem whenever the evolution has bounded p-variation for some 0<p<2.


Banach Algebra Product Integral Left Endpoint Left Limit Linear Integral Equation 
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  • R. M. Dudley
  • R. Norvaiša

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