I'm looking at two different Sobolev inequalities on Wikipedia and I can't reconcile the differences between their corresponding Sobolev exponents. The first is: https://en.wikipedia.org/wiki/Sobolev_inequality#k_%3C_n/p
Let's take $n$ large, $k = 1$, and $p = 2$, so we are in the $k < n/p$ case. Then the Sobolev dimension on the left hand side satisfies $$ \frac{1}{q} = \frac{1}{p} - \frac{k}{n} $$ which implies $q = 2n/(n-2)$.
The second is the Gagliardo-Nirenberg interpolation formula (https://en.wikipedia.org/wiki/Gagliardo%E2%80%93Nirenberg_interpolation_inequality): $$ ||D^ju||_p \leq C||D^mu||_r^\theta||u||_q^{1-\theta} $$ for $j, m, n, q, r, \theta$ determining $p$. If I take the same parameters as before: $j = 0$, $m = 1$, $q = r = 2$ and $\theta = 1/2$, in which case $p = 2n/(n-1)$.
So the problem is that the Sobolev dimension appears different; I can use Young's inequality to get $$ ||D^mu||_r^\theta||u||_q^{1-\theta} = ||Du||_2^{1/2}||u||_2^{1/2} \leq \frac{1}{2}(||Du||_2 + ||u||_2) $$ which is equivalent to the Sobolev norm $||u||_W^{1,2}$. So I apparently have two different $L^p$ spaces on the left hand side and I can't see why. I also can't use rescaling to get the correct exponent since I have two terms on the right hand side and scaling $||Du||_2$ for instance small makes $||u||_2$ large. The only thing I've noticed is that the first Sobolev inequality is done on a bounded domain but there is a mention that it can be unbounded as well, and I don't know what I'm missing/why these differ.