  This paper is devoted to the study of a generalization of Sobolev spaces for
small $L^{p}$ exponents, i.e. $0<p<1$. We consider spaces defined as abstract
completions of certain classes of smooth functions with respect to weighted
quasi-norms, simultaneously inspired by Carleman classes and classical Sobolev
spaces. If the class is restricted with a growth condition on the supremum
norms of the derivatives, we prove that there exists a condition which
guarantees that the resulting space can be embedded into
$C^{\infty}(\mathbb{R})$. This is sharp up to some regularity conditions on the
weight sequence. We also show that the growth condition is necessary, in the
sense that if we drop it entirely we can naturally embed $L^p$ into the
resulting completion.
