In Reed & Simon vol. II, an inequality called Sobolev's inequality is stated in Eq. (IX.19):
Let $0<\lambda<n$ and suppose that $f\in L^p(R^n)$, $h \in L^r(R^n)$ with $p^{-1} + r^{-1} + \lambda n^{-1} = 2$ and $1<p,r<\infty$. Then $$ \iint \frac{|f(x)||h(y)|}{|x-y|^\lambda} d^n x d^n y \leq C_{p,r,\lambda,n}\|f\|_p\|h\|_r.$$
The proof is by an extension of Young's inequality using weak Lebesgue spaces.
My question is: is this inequality related to what is usually called "the Sobolev inequality", bounding $L^q$ norms in terms of $W^{p,k}$ (semi)norms?
This is usually known as the Hardy-Littlewood-Sobolev inequality. By the way, there is no Sobolev space, in the statement! See also the classic paper by Elliot H. Lieb, Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities.
Most mathematicians call "Sobolev inequality" the main part of the Sobolev Embedding Theorem.