This works for $s\ge 1$, by Sobolev embedding for fractional orders: 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 $u(y)-u(x)=\int_x^y u'$.
Christian Remling
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