Chain rules and $p$-variation

R. Norvaiša¹

¹ Institute of Mathematics and Informatics Akademijos 4 Vilnius 2600, Lithuania

Studia Mathematica, Tome 149 (2002) no. 3, pp. 197-238

Résumé

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 $p1+\alpha $. If given $\alpha \in (0,1]$, the $p$-variation of $f$ is bounded for some $p2+\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.

Détail

DOI : 10.4064/sm149-3-1

Mots-clés : main result young stieltjes integral representation composition phi circ functions phi alpha phi has derivative satisfying lipschitz condition order alpha has bounded p variation alpha given alpha p variation bounded alpha phi has second derivative satisfying lipschitz condition order alpha similar result holds young stieltjes integral replaced its extension