I read the paper Vogt(2012) [doi:10.1214/12-AOS1043], and I'm stuck in the proof of Theorem 3.3.
My question is, if it holds that $|Y_{t}(u)-Y_{t}(v)|\le|u-v|U_t$ with randoms variables $U_t$ having the property that $\mathbb{E}[U_t^{\rho}]\le C$ for some $0<\rho<1$, then can one says that for some $0<q<\rho$, $$ \mathbb{E}\left[|Y_t(u)-Y_t(v)|I\left(U_t>\dfrac{C}{|u-v|^q}\right)\right] \le \tilde{C}|u-v|^r $$ for some $r>0$ and $\tilde{C}>0$ ?
In the paper, the author says that it is straightforward... but I think it is not that easy since only I know is about $\rho$-th moment, which is dominated by first moment. The condition of $0<q<\rho$ may be a clue, but I cannot find how to use it...