This works for $s\ge 1$, by [Sobolev embedding for fractional orders:][1] if $u\in H^s=W^{s,2}$, then $u'\in L^p$ with $1/p=3/2-s$. Now your inequality follows by applying Hölder's inequality to $u(y)-u(x)=\int_x^y u'$.

  [1]: https://proofwiki.org/wiki/Fractional_Sobolev_Embedding_Theorem