Minimal coefficients in Hölder conditions in the Wiener space

For almost every $x$ in the Wiener space ${C_w}$, the Hölder condition $|x(t’) - x(t'')| \leqq h|t’ - t''{|^\alpha }$ holds for some $h > 0$ when $\alpha \in (0,\tfrac {1} {2})$. Let ${\phi _\alpha }[x]$ be the infimum of all $h > 0$ for fixed $x$ and $\alpha$. In the present paper we prove that every positive power of ${\phi _\alpha }[x]$ is Wiener integrable over ${C_w}$ and give an estimate for the Wiener integral.