= Solution
\b[With the standard Sobolev norm, the printed constant one is not valid for arbitrary $a$.] The constant function $u=1$ has supremum norm one and $H^1$ norm $\sqrt{2a}$, which is smaller when $a<1/2$.
For the continuous <one-dimensional Sobolev representative>, let $\bar u=(2a)^{-1}\int_Iu$. The <Cauchy-Schwarz inequality> gives $|\bar u|\le(2a)^{-1/2}\|u\|_2$. Using the preceding <Hölder seminorm> estimate,
$$
|u(x)-\bar u|\le\frac1{2a}\int_I|u(x)-u(y)|\,dy\le\sqrt{2a}\|u'\|_2.
$$
Consequently the corrected <interval Sobolev supremum estimate> is
$$
\boxed{\|u\|_\infty\le(2a)^{-1/2}\|u\|_2+\sqrt{2a}\|u'\|_2\le\sqrt{(2a)^{-1}+2a}\,\|u\|_{H^1(I)}.}
$$
One may first prove this for smooth approximations and pass on the full-measure convergence set. Averaging differences also handles complex-valued functions without assuming that a function takes its complex average at some point. At the fixed interval used below the constant is absolute, which suffices for the intended interior estimate.
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