= Davie discrete sewing lemma
{c}
{title2=$a_{s,t}\leq C\omega(s,t)^\theta,\quad\theta>1$}
= Davie lemma
{c}
{synonym}
For a finite time grid, suppose a nonnegative error is zero on neighboring points and obeys $a_{s,t}\leq a_{s,u}+a_{u,t}+L\omega(s,t)^{1/p}a_{s,u}+A\omega(s,t)^\theta$. For a <control function for rough paths> and $\theta>1$, sufficiently small control intervals have the displayed bound with constants independent of the grid. Splitting near half the control size makes the higher-order errors contract, and handles a crossing single grid interval by its zero error. Applying this to local Euler composition errors yields compactness of approximate pathlevel solutions.
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