= Interval Sobolev supremum estimate
{title2=$\|u\|_\infty\le C(a)\|u\|_{H^1(-a,a)}$}
On $I=(-a,a)$, a <first-order Sobolev space> function has a continuous representative and obeys
$$
\|u\|_\infty\le(2a)^{-1/2}\|u\|_2+\sqrt{2a}\|u'\|_2\le\sqrt{(2a)^{-1}+2a}\,\|u\|_{H^1(I)}.
$$
Bound the average by the <Cauchy-Schwarz inequality> and average the identity $u(x)-u(y)=\int_y^xu'$. This also gives $[u]_{C^{0,1/2}}\le\|u'\|_2$. A unit constant with the standard <Sobolev norm> cannot be asserted on arbitrary interval lengths: $u=1$ already disproves it when $2a<1$.
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