Chain rules and $p$-variation
Volume 149 / 2002
Studia Mathematica 149 (2002), 197-238
MSC: Primary 26A42; Secondary 26A45, 60H05.
DOI: 10.4064/sm149-3-1
Abstract
The main result is a Young–Stieltjes integral representation of the composition $\phi \circ f$ of two functions $f$ and $\phi $ such that for some $\alpha \in (0,1]$, $\phi $ has a derivative satisfying a Lipschitz condition of order $\alpha $, and $f$ has bounded $p$-variation for some $p<1+\alpha $. If given $\alpha \in (0,1]$, the $p$-variation of $f$ is bounded for some $p<2+\alpha $, and $\phi $ has a second derivative satisfying a Lipschitz condition of order $\alpha $, then a similar result holds with the Young–Stieltjes integral replaced by its extension.
