In this wikipedia article, one can find the so-called "Gagliardo-Nirenberg inequality, which are of the form
$${\displaystyle \|D^{j}u\|_{L^{p}(\mathbb {R} ^{n})}\leq C\|D^{m}u\|_{L^{r}(\mathbb {R} ^{n})}^{\theta }\|u\|_{L^{q}(\mathbb {R} ^{n})}^{1-\theta }}$$
for coefficients $j,m,n,p,q,r,\theta$ satisfying certain relations. I am wondering:
What does the notation $D^{m}$ for $m\in\mathbb{N}$ mean in this context?
Of course, I am very well aware of the multiindex notation $D^{\alpha}=D^{\alpha_{1}}_{1}\dots D^{\alpha_{d}}_{n}$. However, in the case above, $j$ and $m$ are just natural numbers and not multiindices. So, I am wondering, what does $D^{j}$ and $D^{m}$ mean.
The expression $D^{m}$ might be a shortcut notation for $D^{\alpha}$ for any $\alpha$ with $\vert\alpha\vert=m$, or it might also be the supremum of all those norms. Any reference or explanation is appreciated.