Let $I=(a, b)$, $a<y<b$ an arbitrary point and $J\subseteq I$ an open interval. Then the norms $\left\Vert \cdot \right\Vert_{\sharp}$, $\left\Vert \cdot \right\Vert_{\flat}$ on the Sobolev space $W^{1, p}(I)$ defined by
$\left\Vert u \right\Vert_\sharp:=\left\Vert u' \right\Vert_p+\left|\int_{J}udx\right|$ and $\left\Vert u \right\Vert_\flat:=\left\Vert u' \right\Vert_p+\left|u\left(y\right)\right|$
are equivalent to the standard norm $\left\Vert u \right\Vert_{1, p}$.
Under the assumption $a=0$, $b=1$ I was able to show that $\left\Vert u \right\Vert_{\sharp}\leq \left\Vert u \right\Vert_{1, p}$. For everything else I have no idea so far.