Let $X_t$ be a continuous-time Gaussian process on time interval $[0,1]$ with $\mathbb{E}[X_t]=0$ and $$ \operatorname{cov}(X_t,X_s)=\frac{1}{2} \exp(-|t-s|) $$ Hence, $X_t$ is an O-U process. Would it be possible to prove that there exists a constant $C$ such that $$ \mathbb{E}[|X_t-X_s|^p] \leq C |t-s|^{1+\beta} $$
Key property: $$ \operatorname{var}(X_t-X_s)=1-\exp(-|t-s|) \leq |t-s| $$
Due to the $X_t$ is a Gaussian process, calculating the order-$p$ absolute moment of Gaussian variable, we have \begin{align*} \mathsf{E}[|X_t-X_s|^p]&=C_p(\mathsf{E}[|X_t-X_s|^2])^{p/2} \\ &=C_p(1-\exp(-|t-s|))^{p/2}\\ &\le C_p|t-s|^{p/2}, \qquad C_p=\dfrac{2^{p/2}\Gamma((p+1)/2)}{\sqrt{\pi}}. \end{align*} Hence taking $p>2$ and $\beta=\frac{p}{2}-1>0$, $\mathsf{E}[|X_t-X_s|^p] \le C_p|t-s|^{1+\beta}$ holds.