 Continuity estimate for the Loewner driving function

Suppose the mapped increments $K_{s,t}$ have real-centred enclosing radii bounded by a nondecreasing modulus $\omega (t-s)\to 0$. Nested closures define the unique growing real point ${\xi }_s$. For $v\in K_{t,t+h}$, its preimage $z=g_{K_{s,t}}^{-1}(v)$ belongs to $K_{s,t+h}$. The uniform displacement estimate for a mapping-out function gives $|v-z|\le C\omega (t-s)$, while $|z-{\xi }_s|\le 2\omega (t-s+h)$. Letting $h\downarrow 0$ with $h\le t-s$ proves the displayed estimate. It proves uniform continuity of the Loewner transform on each bounded time interval without evaluating a conformal map on a rough boundary.