_The following is a copy of my [earlier answer](https://mathoverflow.net/a/459919/3948) which resolves this problem for integer $t$_

Start with the [Gagliardo-Nirenberg-Sobolev interpolation inequality](https://en.wikipedia.org/wiki/Gagliardo%E2%80%93Nirenberg_interpolation_inequality), which in particular states that for compactly supported $C^k$ functions $g$ on $\mathbb{R}$, there exists a constant $C_k$ such that

$$ \|g\|_{\infty} \leq C_k\|g^{(k)}\|_{\infty}^\theta \|g\|_2^{1-\theta} $$

where $\theta = \frac{1}{1+2k}$. 

Now given $f$ in $\mathcal{H}(t)$, fix $\phi\in C^\infty_c((0,1))$, then we can apply GNS to $g = \phi f$. 

Observe that $g^{(k)}$ is an expression involving up to $k$ derivatives of $\phi$ and up to $k$ derivatives of $f$. The derivatives of $\phi$ are bounded by some constant $M_k$ once we fixed $\phi$. The derivatives of $f$ are all bounded by $1$. And so GNS implies, 
$$ \|g\|_\infty \leq C'_k \|g\|_2^{\frac{2k}{2k+1}} $$

If we choose $\phi$ such that $\phi$ takes values in $[0,1]$ and $\phi(1/2) = 1$, we have

$$ |f(1/2)| = |g(1/2)| \leq \|g\|_{\infty} \leq C'_k \|g\|_2^{\frac{2k}{2k+1}} \leq C'_k \|f\|_2^{\frac{2k}{2k+1}} $$

as conjectured. 

For non-integer orders, I would check standard references (Adams, or maybe Leoni) to see if there are GNS interpolation inequalities between $L^2$ and a Holder seminorm. (I don't have my copies near me right now.)


### Sharpness

The rate $2k/(2k+1)$ is sharp. 

Fix $\phi\in C^\infty_0((0,1))\cap \mathcal{H}(t)$, then for $\lambda > 1$, the function $\phi_\lambda(x) = \frac{1}{\lambda^{t}} \phi(\lambda (x-\frac12) + \frac12)$ is also in $C^\infty_0((-,1))\cap \mathcal{H}(t)$. And we have

$$ \phi_\lambda(1/2) = \lambda^{-t} \phi(1/2) $$

while

$$ \|\phi_\lambda\|_2 = \lambda^{-t-\frac12} \|\phi\|_2 $$